1/384
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a178 a59 a62 a196 a229 a198
a152a117a140a198a212a110a88
a28 a147
2/384
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trianglerightsld
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a86a135
a163a227a62a94a131a112a138a94a27a178a59a110a216
a51a247a247a42a42a186a221a221a27a27a154a218a238a229a198a110a216 a197,a226a102mnegationslash= 0
a178a59a124a216 (a62a94a124,a218a229a124)
a67a147a212a110a27a86a103,a78a88a218a144a144a123a123 a124,a218a152,a14a67
3/384
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trianglerightsld
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4/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a19a225a134a214
a103a63a249a194(pdfa130a170)a218a19a89(pdfa130a170)a249a76a27a220a169a169a152a152a51a31a228a198a44
〈a62a62a196a196a229a198〉a167 a72a97a245a167 a112a31a19a152a209a135a22a167 a49a19a135
〈a62a62a196a196a229a198〉a167 a220a220a76a76a146a167a235a254a205a167 a152a117a140a198a209a135a22
〈a178a59a59a62a62a62a196a196a229a198〉a254a101a254a167[a123]J.D a35a142a214a205a167a193a23a254a200
a60a172a19a152a209a135a22
〈Classical Electrodynamics〉a167 Third Edition
John David Jackson,U.C.Berkeleya167John Wiley & Sons,Inc.
〈a62a62a196a196a229a198a123a178a19a167〉,a124a35a114a205a167 a16a174a140a198a209a135a22
〈a62a62a196a196a229a198〉a167 a217a253a205a167 a72a174a140a198a209a135a22
〈a62a62a196a196a229a198〉a167 a52a250a178a205a167 a201a199a140a198a209a135a22
〈a178a59a124a216〉a167 a220a233a59a205a167 a137a198a209a135a22
5/384
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a19a198a140a106
a234a198a79a23,a165a254a169a219;a124a216;a173a130a139a73a182δa188a234;Helmholtza189a110
a62a94a121a121a150a150a27a202a72a53a198,a62a94a131a112a138a94a27a13; a62a94a131a112a138a94a27a124a134a253
a152a165a27a196a29a162a8a189a198; a253a152a165a62a94a131a112a138a94a27a124a144a167; a240
a142a100a137a144a167a124a27a200a169a47a170a134a135a169a47a170; a48a159a165a62a94a131a112a138
a94a27a124a144a167; a62a94a124a27a62a138a39a88; a62a94a131a112a138a94a85a254a196a254a27
a61a122a134a197a240
a183a62a124a218a173a240a62a54a27a27a62a62a94a124,a183a62a179a57a217a135a169a144a167; a141a152a53a189a110
a57a65a65a94a94; a46a202a46a100a144a167,a169a108a67a254a123; a186a148a123;a130a21a188a234; a183a62
a124a27a85a254; a240a189a189a62a62a54a27a27a62a62a124; a173a240a62a54a27a94a124,a165a165a179a179; a94a124a175
a75a27a152a132a41a123; a245a52a208a109
a62a94a197a27a68a194,a110a142a253a14a48a159a165a27a197a196a144a167a57a178a161a62a94a197a41; a189a21
a197a196a144a167a57a178a161a197a41; a62a94a197a51a46a161a254a27a135a19a218a242a19; a20a8
a110; a62a94a197a27a27a189a189a149a68a194; a62a94a197a27a65a219a219a49a49a198a52a129
a207a165a127a193 a156
6/384
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a19a198a140a106
a62a94a197a27a203a19,a62a94a124a27a165a179a218a73a179; a237a180a179; a203a19a62a94a124; a62a94
a197a27a251a19
a100a194a131a233a216,a131a233a216a27a162a8a196a58a182 a131a233a216a196a29a6a110a167a226a212a91a67
a134; a131a233a216a27a158a152a110a216; a131a233a216a110a216a27a14a67a47a170; a131a233a216a229
a198; a131a233a216a62a62a196a196a229a198
a145a62a226a102a218a62a94a124a27a131a112a138a94,a36a196a145a62a226a102a27a62a94a124; a112a132a36a196
a145a62a226a102a27a203a19; a203a19a170a204a169a219; a131a212a133a197a203a19; a145a62a226a102
a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94
a40a229a138,a62a62a196a196a229a198a51a121a147a212a110a198a165a27a27a47a47a160
a207a34a127a193 a156
7/384
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a138a146a134a127a193
a207a19a181a28a94a163 a141a250a191a51a110a137a1622412,a62a12362772794
wsz04@mails.tsinghua.edu.cn
a122a177a2a152a103a138a146a167a80a164a49; a138a146a8510%a80a92a111a164a49
a207a165a127a193a83a78a134a207a34a127a193a83a78a216a173a69
a207a165a127a193a109a242a186a207a34a127a193a52a242a186a136a8545%a80a92a111a164a49
a178a158a137a166a207a76e-maila189a62a123a233a88a21a158a109
wangq@mail.tsinghua.edu.cn
62772791(O); 62783615(H); a110a137a1623408
a83a75a145a216a252a213a254a167a250a64a27a74a75a140a145a254a249
8/384
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a239a198a134a167a119
a55a76a135a107a158a109a218a176a229a27a221a92
a145a167a129a208a135a253a83a182 a152a189a135a107a69a83; a135a103a67a193a88a22a237a19
a183a65a245a120a78a101a27a252a204a182 a116a152a250a170a170a180a180a212a110a39a88a132a180a180a234a234a198a39a88
a39a53a86a103a218a144a144a123a123; a227a229a108a234a198a165a214a212a110,a108a212a110a165a21a234a198!
a216a135a249a75a137a216a209a53; a138a146a103a67a137; a29a177a138a146a216a135a246a20a101a177
a213a225a103a127a167 a125a193a88a74a175a75a167 a207a166a216a211a27a110a41a218a252a204a27a144a144a123a123
a8a92a117a121a102a216a195a189a139a216a254a63a221a158a167a152a189a135a134a80a147a233a88 a156
a107a175a75a53a134a80a147a3a3a3a3a167a159a111a209a140a177a33 a90a25a216a135a31a31a20a20a127a193a131a0 !
9/384
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a202a207a212a110a145a68a199a134a92a130a10
a56a66a123
a70a34a51a100a145a254a92a130a85a198a20
a252a204a123
10/384
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a234a198a79a23
11/384
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a165a254a169a219,a165a254a27a27a189a189a194
a110a145a152a109a27a110a135a213a225a144a149a167a110a135a112a131a82a134a27a252a160a165a254
vectore1 vectore2 vectore3 a189 vectorex vectorey vectorez a189 vectori vectorj vectork
a63a191a191a152a152a135a110a145a152a109a27a165a254 vectorAa94a110a135a196a165a254a208a109a181
vectorA = A1vectore1 + A2vectore2 + A3vectore3 =
3summationdisplay
i=1
Aivectorei
a165a254a27a28a181
|vectorA|=
radicalbig
A21 + A22 + A33 =
radicaltpradicalvertex
radicalvertexradicalbt 3summationdisplay
i=1
AiAi
a252a160a165a254a181
|vectore1|=|vectore2|=|vectore3|= 1 a189|vectorei|= 1
12/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
a92a92a126a126a123a181
vectorA±vectorB = (A1vectore1 + A2vectore2 + A3vectore3)±(B1vectore1 + B2vectore2 + B3vectore3)
= (A1±B1)vectore1 + (A2±B2)vectore2 + (A3±B3)vectore3
=
3summationdisplay
i=1
(Ai±Bi)vectorei
a58a166a181 (a58a200a167a73a200a167a83a200)
a196a165a27a27a58a58a166a181
vectore1·vectore1 =vectore2·vectore2 =vectore3·vectore3 = 1
vectore1·vectore2 =vectore2·vectore1 =vectore1·vectore3 =vectore3·vectore1 =vectore2·vectore3 =vectore3·vectore2 = 0
a189
vectorei·vectorej = δij≡
braceleftbigg 1 i = j
0 inegationslash= j
13/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
vectorA·vectorB = (A1vectore1 + A2vectore2 + A3vectore3)·(B1vectore1 + B2vectore2 + B3vectore3)
= A1B1vectore1·vectore1 + A1B2vectore1·vectore2 + A1B3vectore1·vectore3
+A2B1vectore2·vectore1 + A2B2vectore2·vectore2 + A2B3vectore2·vectore3
+A3B1vectore3·vectore1 + A3B2vectore3·vectore2 + A3B3vectore3·vectore3
= A1B1 + A2B2 + A3B3 =
3summationdisplay
i=1
AiBi = vectorB·vectorA
δija228a107a88a101a53a159a181
δij = δji
3summationdisplay
j=1
Ajδij =
3summationdisplay
j=1
Ajδji = Ai
3summationdisplay
i=1
δii = 3
a124a94δija27a53a159a167
vectorA·vectorB =
3summationdisplay
i=1
Aivectorei·
3summationdisplay
j=1
Bjvectorej =
3summationdisplay
i,j=1
AiBjvectorei·vectorej =
3summationdisplay
i,j=1
AiBjδij =
3summationdisplay
i=1
AiBi
14/384
triangleleftsldtriangleleftsld
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
a58a166a27a191a191a194a194a181
a253a233a127a221 (a28),vectorA·vectorA =
3summationdisplay
i=1
AiAi =|vectorA|2
a252a160a165a254a181
vectorA
|vectorA| a28a1431,a169a254a143a144a149a14
a89a14a181
vectorA·vectorB
|vectorA||vectorB|≡cosθ
vectorA·vectorB = 0vectorA⊥vectorB
a8a8
a8a8
a8a8
a8a8a8a42
a45θ vectorB
vectorA
a144a149a14a181 cosθi≡ Ai|vectorA| a169a254Aia180a28a27a221a75
15/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
a3a3a166a166a181 (a3a200a167a165a200a167a9a200)
a196a165a27a3a3a166a166a181
vectore1×vectore1 =vectore2×vectore2 =vectore3×vectore3 = 0
vectore1×vectore2 =?vectore2×vectore1 =vectore3
vectore2×vectore3 =?vectore3×vectore2 =vectore1
vectore3×vectore1 =?vectore1×vectore3 =vectore2
a189
vectorei×vectorej =
3summationdisplay
k=1
epsilon1ijkvectorek
epsilon1ijk≡
1 1,2,3a27a243a152a134 a126a181 epsilon1123,epsilon1231,epsilon1312
1 1,2,3a27a219a152a134 a126a181 epsilon1132,epsilon1213,epsilon1321
0 a217a167 a126a181 epsilon1111,epsilon1223,epsilon1313
16/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
vectorA×vectorB = (A1vectore1 + A2vectore2 + A3vectore3)×(B1vectore1 + B2vectore2 + B3vectore3)
= A1B1vectore1×vectore1 + A1B2vectore1×vectore2 + A1B3vectore1×vectore3
+A2B1vectore2×vectore1 + A2B2vectore2×vectore2 + A2B3vectore2×vectore3
+A3B1vectore3×vectore1 + A3B2vectore3×vectore2 + A3B3vectore3×vectore3
= (A1B2?A2B1)vectore3 + (A3B1?A1B3)vectore2 + (A2B3?A3B2)vectore1
=?vectorB×vectorA =
3summationdisplay
i,j,k=1
epsilon1ijkAiBjvectorek
epsilon1ijka228a107a88a101a53a159a181
epsilon1ijka2a134a63a252a141a73a152a103a67a210 epsilon1ijk =?epsilon1jik =?epsilon1kji =?epsilon1ikj
3summationdisplay
k=1
epsilon1ijkepsilon1lmk = δilδjm?δimδjl
17/384
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trianglerightsld
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
3summationdisplay
j,k=1
epsilon1ijkepsilon1ljk =
3summationdisplay
j=1
(δilδjj?δijδjl) = 3δil?δil = 2δil
3summationdisplay
i,j,k=1
epsilon1ijkepsilon1ijk =
3summationdisplay
i,j=1
(δiiδjj?δijδji) = 9?3 = 6
vectorA×vectorBa27a79a142a76a167a140a123a21a143a181
vectorA×vectorB =
3summationdisplay
i=1
Aivectorei×
3summationdisplay
j=1
Bjvectorej =
3summationdisplay
i,j=1
AiBjvectorei×vectorej=
3summationdisplay
i,j,k=1
AiBjepsilon1ijkvectorek
vectorA×vectorA = 0
(vectorA×vectorB)·vectorC = (
3summationdisplay
i,j,k=1
epsilon1ijkAiBjvectorek)·
3summationdisplay
n=1
Cnvectoren =
3summationdisplay
i,j,k,n=1
epsilon1ijkAiBjCnvectorek·vectoren
=
3summationdisplay
i,j,k,n=1
epsilon1ijkAiBjCnδkn =
3summationdisplay
i,j,k=1
epsilon1ijkAiBjCk a49a15a170;a204a130
18/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
a3a3a166a166a27a191a191a194a194a181
a253a233a127a221(a28),|vectorA×vectorB|=|vectorA||vectorB|sinθ
a144a149a181 vectorA×vectorB⊥vectorA,vectorB vectorAbardblvectorBvectorA×vectorB = 0
a24a24a24a24
a24a24a58
a80a80a80
a80a80a80
a80a80a80a113
a54vectorA×vectorB
θ
vectorA
vectorB
|vectorA×vectorB|2 = (vectorA×vectorB)·(vectorA×vectorB) = (vectorA·vectorA)(vectorB·vectorB)?(vectorA·vectorB)(vectorA·vectorB)
= |vectorA|2|vectorB|2?(|vectorA||vectorB|cosθ)2 =|vectorA|2|vectorB|2(1?cos2θ)
= |vectorA|2|vectorB|2 sin2θ
(vectorA×vectorB)·vectorA = 0 (vectorA×vectorB)·vectorB = 0
19/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
(vectorA×vectorB)·(vectorC×vectorD)
= [(A1B2?A2B1)vectore3 + (A3B1?A1B3)vectore2 + (A2B3?A3B2)vectore1]
·[(C1D2?C2D1)vectore3 + (C3D1?C1D3)vectore2 + (C2D3?C3D2)vectore1]
= (A1B2?A2B1)(C1D2?C2D1) + (A3B1?A1B3)(C3D1?C1D3)
+(A2B3?A3B2)(C2D3?C3D2)
= A1C1B2D2 + A2C2B1D1 + A3C3B1D1 + A1C1B3D3
+A2C2B3D3 + A3C3B2D2?A1D1B2C2?A2D2B1C1
A3D3B1C1?A1D1B3C3?A2D2B3C3?A3D3B2C2
= (A1C1 + A2C2 + A3C3)(B1D1 + B2D2 + B3D3)
(A1D1 + A2D2 + A3D3)(B1C1 + B2C2 + B3C3)
= (vectorA·vectorC)(vectorB·vectorD)?(vectorA·vectorD)(vectorB·vectorC)
(vectorA×vectorB)·(vectorA×vectorB) = (vectorA·vectorA)(vectorB·vectorB)?(vectorA·vectorB)(vectorB·vectorA)
a52a165a254a181vectorA a152a109a135a19→?vectorA a182a165a254a181vectorA a152a109a135a19→ vectorA
a52a165a254×a52a165a254=a182a165a254
20/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
a152a135a14a160a117a139a73a6a58a58a27a27a110a14a47a167a217a108a6a58a209a117a27a252a94a62a8a164a165a254
vectorr =
3summationdisplay
i=1
xivectorei cosθi = xir r≡|vectorr|=
radicalbig
x21 + x22 + x23
vectorrprime =
3summationdisplay
i=1
xprimeivectorei cosθprimei = x
prime
i
rprime r
prime≡|vectorrprime|=
radicalBig
xprime12 + xprime22 + xprime32
a49a110a94a62a8a164a27a165a254
vectorR =vectorr?vectorrprime =
3summationdisplay
i=1
(xi?xprimei)vectorei a1a1
a1
a1a1a21
a16a16a16
a16a16a16
a16a16a16a49
a80a80a80
a80a80a80a105vectorr
vectorr prime
vectorR
cosφi =
vectorR·vectorei
R =
xi?xprimei
R R =
radicaltpradicalvertex
radicalvertexradicalbt 3summationdisplay
i=1
(xi?xprimei)(xi?xprimei)
21/384
triangleleftsldtriangleleftsld
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trianglerightsld
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
a110a135a83a14a169a79a80a143α(vectorr,vectorrprime);β(vectorrprime,vectorR);γ(vectorr,vectorR)
a1
a1
a1
a1a1a21
a16a16a16
a16a16a16
a16a16a16a49
a80a80a80
a80a80a80a105vectorr
vectorr prime
vectorR
α
γ β
sinα = |vectorr×vectorr
prime|
rrprime
sin(pi?β) = |
vectorR×vectorrprime|
Rrprime =
|vectorr×vectorrprime|
Rrprime sinγ =
|vectorR×vectorr|
Rr =
|vectorr×vectorrprime|
Rr
sinα
R =
sinβ
r =
sinγ
rprime =
|vectorr×vectorrprime|
rrprimeR
R2 = (vectorr?vectorrprime)·(vectorr?vectorrprime) =vectorr·vectorr +vectorrprime·vectorrprime?2vectorr·vectorrprime= r2 + rprime2?2rrprimecosα
r2 = (vectorR +vectorrprime)·(vectorR +vectorrprime) = vectorR·vectorR+vectorrprime·vectorrprime+2vectorR·vectorrprime= R2+rprime2?2rrprimecosβ
rprime2 = (vectorr?vectorR)·(vectorr?vectorR) =vectorr·vectorr + vectorR·vectorR?2vectorr·vectorR= r2 + R2?2rRcosγ
22/384
triangleleftsldtriangleleftsld
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a165a254a169a219,a220a254a237a50
a110a145a152a109a165a27a19a30a220a254,arrowrighttophalfarrowrighttophalfT≡
3summationdisplay
i,j=1
Tijvectoreivectorej
arrowrighttophalfarrowrighttophalfI ≡ 3summationdisplay
i,j=1
δijvectoreivectorej =
3summationdisplay
i=1
vectoreivectorei
arrowrighttophalfarrowrighttophalfI ·vectorA = vectorA·arrowrighttophalfarrowrighttophalfI = vectorA
na30a220a254:
3summationdisplay
i1,i2,...,in=1
Ti1,i2,...,invectorei1vectorei2···vectorein
23/384
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a220a254a237a50arrowrighttophalfarrowrighttophalfS ±arrowrighttophalfarrowrighttophalfT =
3summationdisplay
i,j=1
Sijvectoreivectorej±
3summationdisplay
i,j=1
Tijvectoreivectorej =
3summationdisplay
i,j=1
(Sij±Tij)vectoreivectorej
arrowrighttophalfarrowrighttophalfT ·vectorA = ( 3summationdisplay
i,j=1
Tijvectoreivectorej)·(
3summationdisplay
k=1
Akvectorek) =
3summationdisplay
i,j,k=1
TijAkvectorei(vectorej·vectorek) =
3summationdisplay
i,j=1
TijAjvectorei
vectorA·arrowrighttophalfarrowrighttophalfT = (
3summationdisplay
k=1
Akvectorek)·(
3summationdisplay
i,j=1
Tijvectoreivectorej) =
3summationdisplay
i,j,k=1
TijAk(vectorek·vectorei)vectorej =
3summationdisplay
i,j=1
AiTijvectorej
arrowrighttophalfarrowrighttophalfT ×vectorA = ( 3summationdisplay
i,j=1
Tijvectoreivectorej)×(
3summationdisplay
k=1
Akvectorek) =
3summationdisplay
i,j,k=1
TijAkvectorei(vectorej×vectorek)
=
3summationdisplay
i,j,k,l=1
TijAkepsilon1jklvectoreivectorel
vectorA×arrowrighttophalfarrowrighttophalfT = (
3summationdisplay
k=1
Akvectorek)×(
3summationdisplay
i,j=1
Tijvectoreivectorej) =
3summationdisplay
i,j,k=1
TijAk(vectorek×vectorei)vectorej
=
3summationdisplay
i,j,k,l=1
AkTijepsilon1kilvectorelvectorej
24/384
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a124a216,a124a27a27a189a189a194
a73a254a124,a152a109a139a73 x,y,z a27a73a254a188a234a169 φ(vectorr) = φ(x,y,z)
a165a254a124,a152a109a139a73 x,y,za27a165a254a188a234a169 vectorA(vectorr) =
3summationdisplay
i=1
Ai(x,y,z)vectorei
a19a30a220a254a124,a152a109a139a73 x,y,za27a19a30a220a254a188a234a169
arrowrighttophalfarrowrighttophalfT (vectorr) = 3summationdisplay
i=1
Tij(x,y,z)vectoreivectorej
......
a124a207a126a61a157a54a152a109a139a73a167a143a143a157a157a54a158a109a139a73a34
a178a126a94a229a130a163a227a152a109a27a165a254a124a167 a229a130a27a144a149a134a165a254a124a27a144a149a131
a211a167a229a130a27a151a221a20a39a117a165a254a124a28a27a140a2a169
25/384
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26/384
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a124a216,a124a27a135a169a53a159
a135a169a142a206a27a27a189a189a194:
a110a145a152a109a165a140a177a169a79a233a110a135a139a73a63a49a135a251
1≡x?2≡y?3≡z?ivectorej =vectorej?i
≡vectore1x +vectore2y +vectore3z =
3summationdisplay
i=1
vectorei?i
2≡?·?=
3summationdisplay
i=1
i?i =x2 +y2 +z2?×?= 0
a144a107a233a134a14a139a73a88a167a135a169a142a206a226a134a196a165a233a180a34
a107a86a173a14a218a167a61a107a135a169a142a206a27a53a159(?i),a113a180a180a165a165a254(vectorei)a34
a185?a27a144a167a252a62a167a76a166?a27a135a169a142a206a218a165a254a53a159a211a158a26a177
a247a118a226a85a178a239a156
27/384
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a124a216,a124a27a135a169a53a159
a73a254a124φa27a70a221a181?φ =
3summationdisplay
i=1
vectorei?iφ=vectore1?φ?x +vectore2?φ?y +vectore3?φ?z
a165a254a124vectorAa27a209a221a181
·vectorA = (
3summationdisplay
i=1
vectorei?i)·(
3summationdisplay
j=1
Ajvectorej) =
3summationdisplay
i,j=1
vectorei·vectorej?iAj =
3summationdisplay
i=1
iAi
=?A1?x +?A2?y +?A3?z
a165a254a124vectorAa27a94a221a181
×vectorA = (
3summationdisplay
i=1
vectorei?i)×(
3summationdisplay
j=1
Ajvectorej) =
3summationdisplay
i,j=1
vectorei×vectorej?iAj =
3summationdisplay
i,j,k=1
epsilon1ijk(?iAj)vectorek
=vectore3(?A2?xA1?y ) +vectore2(?A1?zA3?x ) +vectore1(?A3?yA2?z )
28/384
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a124a216,a124a27a135a169a53a159
a135a169a142a206a233a252a135a124a27a138a94
·(vectorf×vectorg)
=
3summationdisplay
i,j,k,l=1
ivectorei·(fjgkepsilon1jklvectorel) =
3summationdisplay
i,j,k,l=1
vectorei·vectorelepsilon1jkl?i(fjgk)
=
3summationdisplay
i,j,k,l=1
δilepsilon1jkl(gk?ifj + fj?igk) =
3summationdisplay
i,j,k=1
epsilon1jki(gk?ifj + fj?igk)
=
3summationdisplay
i,j,k=1
epsilon1ijkgk?ifj?epsilon1jikfj?igk
=vectorg·(?×vectorf)?vectorf·(?×vectorg)
29/384
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a124a216,a124a27a135a169a53a159
×(vectorf×vectorg)
=
3summationdisplay
i,j,k,l=1
ivectorei×(fjgkepsilon1jklvectorel) =
3summationdisplay
i,j,k,l=1
vectorei×vectorelepsilon1jkl?i(fjgk)
=
3summationdisplay
i,j,k,l,m=1
epsilon1ilmvectoremepsilon1jkl(gk?ifj + fj?igk)
=
3summationdisplay
i,j,k,m=1
vectorem(?δijδmk +δikδmj)(gk?ifj + fj?igk)
=
3summationdisplay
i,j,k,m=1
vectorem(?δijδmkgk?ifj?δijδmkfj?igk +δikδmjgk?ifj +δikδmjfj?igk)
=?
3summationdisplay
i,k=1
(vectorekgk?ifi + fi?igkvectorek) +
3summationdisplay
i,j=1
(gi?ifjvectorej +vectorejfj?igi)
=?vectorg?·vectorf?vectorf·?vectorg +vectorg·?vectorf +vectorf?·vectorg
30/384
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a124a216,a124a27a135a169a53a159
a135a169a142a206a233a69a220a124a27a138a94
φ(u) =
3summationdisplay
i=1
vectorei?iφ(u) =
3summationdisplay
i=1
vectorei?φ?u?iu= φprime(u)?u
·vectorf(u) =
3summationdisplay
i=1
ifi(u) =
3summationdisplay
i=1
fi
u?iu=
vectorfprime(u)·?u
×vectorf(u) =
3summationdisplay
i,j,k=1
epsilon1ijkvectorek?ifj(u) =
3summationdisplay
i,j,k=1
epsilon1ijkvectorek?fj?u?iu=?u×vectorfprime(u)
a135a169a142a206a233vectorRa27a138a94
vectorr =
3summationdisplay
i=1
xivectorei?=
3summationdisplay
i=1
vectorei?i?i =x
i
vectorrprime =
3summationdisplay
i=1
xprimeivectorei?prime =
3summationdisplay
i=1
vectorei?primei?primei =xprime
i
31/384
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a124a216,a124a27a135a169a53a159vectorR =vectorr?vectorrprime = 3summationdisplay
i=1
(xi?xprimei)vectorei
ixj = δij?primeixprimej = δij?ixprimej =?primeixj = 0
·vectorR =
3summationdisplay
i=1
i(xi?xprimei)= 3 =prime·vectorR
×vectorR =
3summationdisplay
i,j,k=1
epsilon1ijk?i(xj?xprimej)vectorek =
3summationdisplay
i,j,k=1
epsilon1ijkδijvectorek= 0 =prime×vectorR
R =
3summationdisplay
i=1
vectorei?iR = 12R
3summationdisplay
i,j=1
vectorei?i[(xj?xprimej)(xj?xprimej)]
= 1R
3summationdisplay
i,j=1
vectorei(xj?xprimej)?i(xj?xprimej) = 1R
3summationdisplay
i,j=1
vectorei(xj?xprimej)δij
= 1R
3summationdisplay
i=1
vectorei(xi?xprimei)=
vectorR
R =
primeR
32/384
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a124a216,a124a27a200a169a53a159
a70a221a221a27a27a130a200a169:
integraldisplay B
A
dvectorl·?φ =
3summationdisplay
i,j=1
integraldisplay B
A
vectoreidxi·?jφvectorej =
3summationdisplay
i,j=1
integraldisplay B
A
dxi?jφvectorei·vectorej
=
3summationdisplay
i,j=1
integraldisplay B
A
dxi?jφδij =
3summationdisplay
i=1
integraldisplay B
A
dxi?iφ =
integraldisplay B
A
dφ
= φB?φA
a45a27 a54
a63
a0a0a18a64a64a73
a64a64a82a0a0a9
a115?φ
a27a191a191a194a194a181
vectorl·?φ = φB?φA a189 |?φ|cosθ = φB?φA|?vectorl| a144a149a19a234====?φ|?vectorl|
φa27a140a2a143φa247a136a144a149a27a19a234a165a129a140a27a19a234a138a167a144a149a247a228a107 a129a140
a19a234a27a144a149a34a63a191a144a149a27a144a149a19a234a61a180?φ a51a100a144a149a254a27a221a75a34
φa27a144a149a180a31a138a161a27a123a130a144a149a169
33/384
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a124a216,a124a27a200a169a53a159a209a221a221a27a27a78a200a169:
a78a200 τ a165a27a78a200a169
integraldisplay
τ
dτ?·vectorf =
contintegraldisplay
τ
dvectorS·vectorf a78a200 τ a76a161a27a161a200a169
a192za144a149a166a121,τa76a161a169a143a254a101a252a76a161:SB～zB(x,y)a218SA～zA(x,y)
integraldisplay
τ
dτzf3(x,y,z)
=
integraldisplay
dxdy
integraldisplay zB(x,y)
zA(x,y)
dzzf3(x,y,z)
=
integraldisplay
dxdy[f3(x,y,zB(x,y))?f3(x,y,zA(x,y))]
=
integraldisplay
SB
dS3f3(x,y,zB(x,y))?
integraldisplay
SA
dS3f3(x,y,zA(x,y))
=
contintegraldisplay
τ
dS3 f3(x,y,z) a250a170a144a183a94a94a117a117a78a200a165a195a154a201a201a27a27a156a185
34/384
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a124a216,a124a27a200a169a53a159
integraldisplay
τ
dτ?=
contintegraldisplay
τ
dvectorS
→
integraldisplay
τ
dτ?φ =
contintegraldisplay
τ
dvectorSφ
integraldisplay
τ
dτ?×vectorf =
contintegraldisplay
τ
dvectorS×vectorf
·vectorfa27a191a191a194a194a181
contintegraldisplay
τ
dvectorS·vectorf a189a194a143a181vectorfa108τa54a209a27a207a254
τ?·vectorf =
contintegraldisplay
τ
dvectorS·vectorf
·vectorf = lim
τ→0
contintegraltext
τ d
vectorS·vectorf
τ = a252a160a78a200a108a124a58a54a209a27a207a254
·vectorf = 0a195a192a207a254a9a196a167a195a13a182?·vectorf negationslash= 0a107a192a207a254a9a196a167a107a13!
a45a27 a54
a63
a0a0a18a64a64a73
a64a64a82a0a0a9
a115 a27a0a9a63a64a64a82a45
a0a18a54a64a64a73
a115 a252a135a31a254a20a75a58a58a62a62a214a13a27a27a62a62a124a130a28a28a91a91
35/384
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a124a216,a124a27a200a169a53a159
a94a221a221a27a27a161a200a169,a173a161 S a165a27a161a200a169
integraldisplay
S
dvectorS ·?×vectorf =
contintegraldisplay
S
dvectorl·vectorf a173a161 S a62a46a27a173a130a200a169
xa144a149,a173a161a144a167:z = g(x,y),a123a130:vectorn = n1vectore1 + n2vectore2 + n3vectore3
dz?gprimexdx?gprimeydy = 0 dvectorr = dxvectore1 + dyvectore2 + dzvectore3
vectorn·dvectorr = 0? vectorn∝?gprimexvectorex?gprimeyvectorey +vectorez
vectorn = c(?gprimexvectorex?gprimeyvectorey +vectorez) vectorn·vectorn=1====?c2(1 + (gprimex)2 + (gprimey)2) = 1
nx =? g
prime
xradicalBig
1 + gprimex2 + gprimey2
ny =? g
prime
yradicalBig
1 + gprimex2 + gprimey2
nz = 1radicalBig
1 + gprimex2 + gprimey2
36/384
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a124a216,a124a27a200a169a53a159a94a221a221a27a27a161a200a169:
integraldisplay
S
dvectorS ·?×vectorf =
contintegraldisplay
S
dvectorl·vectorf
xa144a149,a173a161a144a167:z = g(x,y),a123a130:vectorn = n1vectore1 + n2vectore2 + n3vectore3
nx,y =? g
prime
x,yradicalBig
1 + gprimex2 + gprimey2; nz = 1radicalBig
1 + gprimex2 + gprimey2
dS2
dS3 =
n2dS
n3dS =?g
prime
y
a173a161a62a46a51 xy a178a161a221a75a171a141a173a130a62a46a169a143:y = yB(x),y = yA(x)integraldisplay
S
(dS2?3f1?dS3?2f1) =?
integraldisplay
S
dS3 (gprimey?3f1 +?2f1)
=?
integraldisplay
S
dS3 ddyf1(x,y,g(x,y))
=?
integraldisplay
dx
integraldisplay yB(x)
yA(x)
dy ddyf1(x,y,g(x,y))
=?
integraldisplay
dx [f1(x,yB(x),g(x,yB(x)))?f1(x,yA(x),g(x,yA(x)))]
=
contintegraldisplay
S
dx f1(x,y,g(x,y)) a173a161a9a123a130a85a109a195a39a88a53a189xya178a161a27a173a130a27a20a149a143a95a158a2
a250a170a144a183a94a94a117a117a173a161a165a195a154a201a201a27a27a156a185!
37/384
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a124a216,a124a27a200a169a53a159
integraldisplay
S
dvectorS ·?×vectorf =
integraldisplay
S
dvectorS×?·vectorf =
contintegraldisplay
S
dvectorl·vectorf?
integraldisplay
S
dvectorS×?=
contintegraldisplay
S
dvectorl
integraldisplay
S
dvectorS×?φ =
contintegraldisplay
S
dvectorlφ
integraldisplay
S
(dvectorS×?)×vectorf =
contintegraldisplay
S
dvectorl×vectorf
×vectorfa27a191a191a194a194a181
vectorS ·?×vectorf =
contintegraldisplay
S
dvectorl·vectorf ≡vectorfa55Sa130a55a27a130a254
|?×vectorf|cosθ = lim
S→0
contintegraltext
Sd
vectorl·vectorf
S
×vectorf a140a2a143a124a58a63a129a140a140a252a252a160a161a200a200a130a130a254a138a167a144a149a247
a228a107a129a140a130a254a27a173a161a144a149a169a63a191a144a149a173a161a27a130a254
a180?×vectorfa51a173a161a144a149a254a221a75a34
×vectorf = 0a195a192a130a254,a195a94;?×vectorf negationslash= 0a192a130a254,a107a94a34
a98a54a105a54a54a14a13a15a12a54
a22a21
a23a20
a30a29
a31a28
38/384
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a173a130a139a73,a206a139a73
a139a73a3a181 r =
radicalbig
x2 + y2,θ = arctan yx,z a196a165a181 vectorer,vectoreθ,vectorez
a38a37
a39a36
a101a114a0
a0
a0a0a18a54
a64
a64
a64a64a73 a45
vectorervectoreyvectoreθ
vectorexθθ
vectorez
vectorer·vectorex = cosθ vectorer·vectorey = sinθ vectorer·vectorez = 0
vectoreθ·vectorex =?sinθ vectoreθ·vectorey = cosθ vectoreθ·vectorez = 0
vectorez·vectorex = 0 vectorez·vectorey = 0 vectorez·vectorez = 1
vectorer = cosθvectorex + sinθvectorey vectoreθ =?sinθvectorex + cosθvectorey
vectorex = cosθvectorer?sinθvectoreθ vectorey = sinθvectorer + cosθvectoreθ
vectorer
r =0
vectorer
θ =vectoreθ
vectorer
z =
vectoreθ
r =0
vectoreθ
θ =?vectorer
vectoreθ
z =
vectorez
r =
vectorez
θ =
vectorez
z =0
a247a196a165a144a149a27a195a161a2a130a3a181 dl1 = dr,dl2 = rdθ,dl3 = dz
a135a169a142a206a181?=vectorerr+vectoreθ1rθ+vectorezz a165a254a124a181vectorf = frvectorer+fθvectoreθ+fzvectorez
a73a254a124a27a70a221a181?φ =vectorer?φ?r+vectoreθ1r?φ?θ+vectorez?φ?z
a165a254a124a27a209a221a181?·vectorf = [vectorerr+vectoreθ1rθ+vectorezz]·(frvectorer+fθvectoreθ+fzvectorez)
39/384
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a173a130a139a73,a206a139a73
a139a73a3a181 r =
radicalbig
x2 + y2,θ = arctan yx,z a196a165a181 vectorer,vectoreθ,vectorez
vectorer = cosθvectorex + sinθvectorey vectoreθ =?sinθvectorex + cosθvectorey
vectorer
r =0
vectorer
θ =vectoreθ
vectorer
z =
vectoreθ
r =0
vectoreθ
θ =?vectorer
vectoreθ
z =
vectorez
r =
vectorez
θ =
vectorez
z =0
a135a169a142a206a181?=vectorerr+vectoreθ1rθ+vectorezz a165a254a124a181vectorf = frvectorer+fθvectoreθ+fzvectorez
a73a254a124a27a70a221a181?φ =vectorer?φ?r+vectoreθ1r?φ?θ+vectorez?φ?z
a165a254a124a27a209a221a181?·vectorf = [vectorerr+vectoreθ1rθ+vectorezz]·(frvectorer+fθvectoreθ+fzvectorez)
=?fr?r +1r?fθ?θ +?fz?z +vectoreθ1r·(fr?vectorer?θ +fθ?vectoreθ?θ )
=?fr?r +1r?fθ?θ +?fz?z +frr
vectorf →vector2 = vector?·vector?=?2
r2+
1
r
2
θ2+
2
z2+
1
r
r
40/384
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a173a130a139a73,a206a139a73
vectorer
r =0
vectorer
θ =vectoreθ
vectorer
z =
vectoreθ
r =0
vectoreθ
θ =?vectorer
vectoreθ
z =
vectorez
r =
vectorez
θ =
vectorez
z =0
a135a169a142a206a181?=vectorerr+vectoreθ1rθ+vectorezz?2=?
2
r2+
1
r
2
θ2+
2
z2+
1
r
r
a73a254a124a27a70a221a181?φ =vectorer?φ?r+vectoreθ1r?φ?θ+vectorez?φ?z
a165a254a124a27a209a221a181?·vectorf =?fr?r +1r?fθ?θ +?fz?z +frr
a165a254a124a27a94a221a181?×vectorf = [vectorerr+vectoreθ1rθ+vectorezz]×(frvectorer+fθvectoreθ+fzvectorez)
=vectorez?fθ?r?vectoreθ?fz?r?vectorez?frr?θ+vectorer?fzr?θ+vectoreθ?fr?z?vectorer?fθ?z
+frrvectoreθ×?vectorer?θ +fθrvectoreθ×?vectoreθ?θ
=vectorez[?fθ?rfrr?θ+fθr]+vectoreθ[?fr?zfz?r]+vectorer[?fzr?θfθ?z]
41/384
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a173a130a139a73,a165a139a73
a0
a0a0a9
a45
a54
a0
a0a0a18
a64
a64a64a82
z
θ
φ
vectorr
x
y
vectorρ
a38a37
a39a36
a0
a0
a0a0a18a54
a64
a64
a64a64a73 a45
vectorervectorez?vectoreθ
vectoreρθ
θ
a38a37
a39a36
a101a114a0
a0
a0a0a18a54
a64
a64
a64a64a73 a45
vectoreρvectoreyvectoreφ
vectorexφφ
vectorez
a139a73a3a181 r=
radicalbig
x2+y2+z2,θ=arctan(
radicalbig
x2+y2/z),φ=arctan(y/x)
a196a165a181 vectorer,vectoreθ,vectoreφ x,ya178a161a196a165a181 vectoreρ
vectoreρ= cosφvectorex+sinφvectorey vectoreφ=?sinφvectorex+cosφvectorey
vectorex= cosφvectoreρ?sinφvectoreφ vectorey= sinφvectoreρ+cosφvectoreφ
vectorer·vectoreρ = sinθ vectorer·vectorez = cosθ vectorer·vectoreφ = 0
vectorer·vectorex = cosφvectorer·vectoreρ = cosφsinθ vectorer·vectorey = sinφvectorer·vectoreρ = sinφsinθ
vectoreθ·vectoreρ = cosθ vectoreθ·vectorez =?sinθ vectoreθ·vectoreφ = 0
vectoreθ·vectorex = cosφvectoreθ·vectoreρ =cosφcosθ vectoreθ·vectorey = sinφvectoreθ·vectoreρ =sinφcosθ
vectoreφ·vectorex =?sinφ vectoreφ·vectorey = cosφ vectoreφ·vectorez = 0?
vectorer
vectoreθ
vectoreφ
=
cosφsinθ sinφsinθ cosθ
cosφcosθ sinφcosθ?sinθ
sinφ cosφ 0
vectorex
vectorey
vectorez
vectorex
vectorey
vectorez
=
cosφsinθ cosφcosθ?sinφ
sinφsinθ sinφcosθ cosφ
cosθ?sinθ 0
vectorer
vectoreθ
vectoreφ
42/384
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a173a130a139a73,a165a139a73
a38a37
a39a36
a0
a0
a0a0a18a54
a64
a64
a64a64a73 a45
vectorervectorez?vectoreθ
vectoreρθ
θ
a38a37
a39a36
a101a114a0
a0
a0a0a18a54
a64
a64
a64a64a73 a45
vectoreρvectoreyvectoreφ
vectorexφφ
vectorez
vectorer
vectoreθ
vectoreφ
=
cosφsinθ sinφsinθ cosθ
cosφcosθ sinφcosθ?sinθ
sinφ cosφ 0
vectorex
vectorey
vectorez
vectorex
vectorey
vectorez
=
cosφsinθ cosφcosθ?sinφ
sinφsinθ sinφcosθ cosφ
cosθ?sinθ 0
vectorer
vectoreθ
vectoreφ
vectorer
r =
vectoreθ
r =
vectoreφ
r = 0
vectorer
θ =vectoreθ
vectoreθ
θ =?vectorer
vectoreφ
θ = 0
vectorer
φ = sinθvectoreφ
vectoreθ
φ = cosθvectoreφ
vectoreφ
φ =?sinθvectorer?cosθvectoreθ
a247a196a165a144a149a27a195a161a2a130a3a181 dl1 = dr,dl2 = rdθ,dl3 = rsinθdφ
a135a169a142a206a181?=vectorerr+vectoreθ1rθ+vectoreφ 1rsinθφ a165a254a124a181vectorf=frvectorer+fθvectoreθ+fφvectoreφ
a73a254a124a27a70a221a181?φ =vectorer?φ?r+vectoreθ1r?φ?θ+vectoreφ 1rsinθ?φ?φ
43/384
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a173a130a139a73,a165a139a73
vectorer
r =
vectoreθ
r =
vectoreφ
r = 0
vectorer
θ =vectoreθ
vectoreθ
θ =?vectorer
vectoreφ
θ = 0
vectorer
φ = sinθvectoreφ
vectoreθ
φ = cosθvectoreφ
vectoreφ
φ =?sinθvectorer?cosθvectoreθ
a135a169a142a206a181?=vectorerr+vectoreθ1rθ+vectoreφ 1rsinθφ a165a254a124a181vectorf=frvectorer+fθvectoreθ+fφvectoreφ
a73a254a124a27a70a221a181?φ =vectorer?φ?r+vectoreθ1r?φ?θ+vectoreφ 1rsinθ?φ?φ
a165a254a124a27a209a221a181?·vectorf=[vectorerr+vectoreθ1rθ+vectoreφ 1rsinθφ]·(frvectorer+fθvectoreθ+fφvectoreφ)
=?fr?r +1r?fθ?θ + 1rsinθ?fφ?φ+vectoreθ1r·(fr?vectorer?θ +fθ?vectoreθ?θ )+vectoreφ 1rsinθ·(fr?vectorer?φ +fθ?vectoreθ?φ)
=?fr?r +1r?fθ?θ + 1rsinθ?fφ?φ+2frr +fθ cosθrsinθ
vectorf →vector2=vector?·vector?=?2
r2+
1
r2
2
θ2+
1
r2 sin2θ
2
φ2+
2
r
r+
cosθ
r2 sinθ
θ
44/384
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a173a130a139a73,a165a139a73
vectorer
r =
vectoreθ
r =
vectoreφ
r = 0
vectorer
θ =vectoreθ
vectoreθ
θ =?vectorer
vectoreφ
θ = 0
vectorer
φ = sinθvectoreφ
vectoreθ
φ = cosθvectoreφ
vectoreφ
φ =?sinθvectorer?cosθvectoreθ
a73a254a124a27a70a221a181?φ =vectorer?φ?r+vectoreθ1r?φ?θ+vectoreφ 1rsinθ?φ?φ
a165a254a124a27a209a221a181?·vectorf=?fr?r +1r?fθ?θ + 1rsinθ?fφ?φ+2frr +fθ cosθrsinθ
2=?
2
r2+
1
r2
2
θ2+
1
r2 sin2θ
2
φ2+
2
r
r+
cosθ
r2 sinθ
θ
a165a254a124a27a94a221a181?×vectorf=[vectorerr+vectoreθ1rθ+vectoreφ 1rsinθφ]×(frvectorer+fθvectoreθ+fφvectoreφ)
=vectoreφ?fθ?r?vectoreθ?fφ?r?vectoreφ?frr?θ+vectorer?fφr?θ+vectoreθ?frrsinθ?φ?vectorer?fθrsinθ?φ
+1rvectoreθ×[fr?vectorer?θ +fθ?vectoreθ?θ ]+ 1rsinθvectoreφ×[fr?vectorer?φ +fθ?vectoreθ?φ +fφ?vectoreφ?φ]
45/384
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a173a130a139a73,a165a139a73
vectorer
r =
vectoreθ
r =
vectoreφ
r = 0
vectorer
θ =vectoreθ
vectoreθ
θ =?vectorer
vectoreφ
θ = 0
vectorer
φ = sinθvectoreφ
vectoreθ
φ = cosθvectoreφ
vectoreφ
φ =?sinθvectorer?cosθvectoreθ
a165a254a124a27a94a221a181?×vectorf=[vectorerr+vectoreθ1rθ+vectoreφ 1rsinθφ]×(frvectorer+fθvectoreθ+fφvectoreφ)
=vectoreφ?fθ?r?vectoreθ?fφ?r?vectoreφ?frr?θ+vectorer?fφr?θ+vectoreθ?frrsinθ?φ?vectorer?fθrsinθ?φ
+1rvectoreθ×[fr?vectorer?θ +fθ?vectoreθ?θ ]+ 1rsinθvectoreφ×[fr?vectorer?φ +fθ?vectoreθ?φ +fφ?vectoreφ?φ]
=vectoreφ?fθ?r?vectoreθ?fφ?r?vectoreφ?frr?θ+vectorer?fφr?θ+vectoreθ?frrsinθ?φ?vectorer?fθrsinθ?φ
+1rvectoreθ×[frvectoreθ?fθvectorer]? fφrsinθvectoreφ×[sinθvectorer + cosθvectoreθ]
=vectorer[?fφr?θfθrsinθ?φ+fφcosθrsinθ]+vectoreθ[?frrsinθ?φfφ?r?fφr]+vectoreφ[?fθ?rfrr?θ+fθr]
46/384
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δ a188a234 a152a145a152a109a189a194a134a53a159
δ(x?x0) =
braceleftbigg 0 xnegationslash= x
0
∞ x = x0
integraldisplay
a157a41x0
dx δ(x?x0) = 1
δ(x?x0)≥0 δ(x?x0) = δ(x0?x) f(x)δ(x?x0) = f(x0)δ(x?x0)
δ(f(x)) =
summationdisplay
i
1
|fprime(xi)|δ(x?xi) xia180f(x) = 0a27a27a49a49 i a135a135a138a138
δepsilon1(x?xprime)≡1pi epsilon1(x?xprime)2 +epsilon12 = Re1pi ix?xprime + iepsilon1 epsilon1→0
+
→
braceleftbigg 0 xnegationslash= xprime
∞ x = xprimeintegraldisplay
∞
∞
dx 1pi epsilon1(x?xprime)2 +epsilon12 = 1pi
integraldisplay ∞
∞
dt 1t2 + 1 = 1
δ(x?xprime) = lim
epsilon1→0+
δepsilon1(x?xprime) = Re
integraldisplay ∞
0
dk
pi e
ik(x?xprime+iepsilon1)
= 12pi[
integraldisplay ∞
0
dk eik(x?xprime)?|k|epsilon1?
integraldisplay?∞
0
d(?k) ei(?k)(x?xprime)?|?k|epsilon1] =
integraldisplay ∞
∞
dk
2pie
ik(x?xprime)?epsilon1|k|
47/384
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δ a188a234 a110a145a152a109a189a194a134a53a159
δ(vectorr?vectorrprime) = δ(x?xprime)δ(y?yprime)δ(z?zprime)
δ(vectorr?vectorrprime)≥0 δ(vectorr?vectorrprime) = δ(vectorrprime?vectorr)
f(vectorr)δ(vectorr?vectorrprime) = f(vectorrprime)δ(vectorr?vectorrprime) δ(a(vectorr?vectorrprime)) = 1|a|δ(vectorr?vectorrprime)
a41a219a76a136,δ(vectorr?vectorrprime) =?14pi?2 1|vectorr?vectorrprime|
48/384
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a124a216,Helmholtza189a110
a63a152a171a141a165a27a165a254a124a100a217a209a221a218a94a221a57a51a171a141a254a27a62a46a94a135a17a28a251a189!
H.V.Helmholtz
U(vectorr)≡
integraldisplay
dτprime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
vectorW(vectorr)≡
integraldisplay
dτprime?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
U(vectorr) =?
integraldisplay
dτprime[? 1|vectorr?vectorrprime|]?
prime·vectorF(vectorrprime)
4pi =
integraldisplay
dτprime[?prime 1|vectorr?vectorrprime|]?
prime·vectorF(vectorrprime)
4pi
=
integraldisplay
dτprime{?prime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|?
prime[?prime·vectorF(vectorrprime)]
4pi|vectorr?vectorrprime| }
×vectorW(vectorr) =
integraldisplay
dτprime? 1|vectorr?vectorrprime|×?
prime×vectorF(vectorrprime)
4pi =?
integraldisplay
dτprime?prime 1|vectorr?vectorrprime|×?
prime×vectorF(vectorrprime)
4pi
=
integraldisplay
dτprime{prime×?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime| +
prime×[?prime×vectorF(vectorrprime)]
4pi|vectorr?vectorrprime| }
=
integraldisplay
dτprime{prime×?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime|+
prime[?prime·vectorF(vectorrprime)]prime2vectorF(vectorrprime)
4pi|vectorr?vectorrprime| }
49/384
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a124a216,Helmholtza189a110
a63a152a171a141a165a27a165a254a124a100a217a209a221a218a94a221a57a51a171a141a254a27a62a46a94a135a17a28a251a189!
U(vectorr) =
integraldisplay
dτprime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
vectorW(vectorr) =
integraldisplay
dτprime?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
U(vectorr) =
integraldisplay
dτprime{?prime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|?
prime[?prime·vectorF(vectorrprime)]
4pi|vectorr?vectorrprime| }
×vectorW(vectorr) =
integraldisplay
dτprime{prime×?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime| +
prime[?prime·vectorF(vectorrprime)]prime2vectorF(vectorrprime)
4pi|vectorr?vectorrprime| }
integraldisplay
dτprime?
prime2vectorF(vectorrprime)
4pi|vectorr?vectorrprime| =?
integraldisplay
dτprime[?prime·?
primevectorF(vectorrprime)
4pi|vectorr?vectorrprime|
prime 1
4pi|vectorr?vectorrprime|·?
primevectorF(vectorrprime)]
=?
integraldisplay
dτprime{?prime·?
primevectorF(vectorrprime)
4pi|vectorr?vectorrprime|
prime[vectorF(vectorrprime)·?prime 1
4pi|vectorr?vectorrprime|] +
vectorF(vectorrprime)?prime2 1
4pi|vectorr?vectorrprime|}
=?
integraldisplay
dτprime{?prime·?
primevectorF(vectorrprime)
4pi|vectorr?vectorrprime|
prime[vectorF(vectorrprime)·?prime 1
4pi|vectorr?vectorrprime|]}+
vectorF(vectorr)
50/384
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a124a216,Helmholtza189a110
a63a152a171a141a165a27a165a254a124a100a217a209a221a218a94a221a57a51a171a141a254a27a62a46a94a135a17a28a251a189!
U(vectorr) =
integraldisplay
dτprime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
vectorW(vectorr) =
integraldisplay
dτprime?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
U(vectorr)?
integraldisplay
dτprime?prime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime| =?
integraldisplay
dτprime?
prime[?prime·vectorF(vectorrprime)]
4pi|vectorr?vectorrprime|
×vectorW(vectorr) +
integraldisplay
dτprime?prime×?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime| =
integraldisplay
dτprime?
prime[?prime·vectorF(vectorrprime)]prime2vectorF(vectorrprime)
4pi|vectorr?vectorrprime|integraldisplay
dτprime{?prime·?
primevectorF(vectorrprime)
4pi|vectorr?vectorrprime|
prime[vectorF(vectorrprime)·?prime 1
4pi|vectorr?vectorrprime|]}=
integraldisplay
dτprime?
prime2vectorF(vectorrprime)
4pi|vectorr?vectorrprime|+
vectorF(vectorr)
vectorF(vectorr) =U(vectorr)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
a112a124
+?×vectorW(vectorr)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
a238a124
+
integraldisplay
dτprime{prime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
+?prime×?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime| +?
prime·?
primevectorF(vectorrprime)
4pi|vectorr?vectorrprime|
prime[vectorF(vectorrprime)·?prime 1
4pi|vectorr?vectorrprime|]}
a233a195a46a152a109,a101vectorF(vectorrprime) r
prime→∞
→0,a75a62a46a145a143a34!
51/384
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a124a216,Helmholtza189a110
U(vectorr) =
integraldisplay
dτprime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
vectorW(vectorr) =
integraldisplay
dτprime?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
vectorF(vectorr) =U(vectorr)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
a112a124
+?×vectorW(vectorr)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
a238a124
+
integraldisplay
dτprime{prime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
+?prime×?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime| +?
prime·?
primevectorF(vectorrprime)
4pi|vectorr?vectorrprime|
prime[vectorF(vectorrprime)·?prime 1
4pi|vectorr?vectorrprime|]}
a165a254a124a100a217a209a221a218a94a221a57a62a46a94a135a17a28a251a189!
a209a221a180a23a41a41a165a165a254a124a229a130a117a209a189a224a241a27a47a13a48a48a34a34
a94a221a180a23a41a41a165a165a254a124a229a130a94a61a27a47a13a48a48a34a34
a40a189a165a254a124a73a239a225a163a227a249a252a171a47a13a48a27a212a110a94a135a156
52/384
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a62a94a121a150a27a202a72a53a198
53/384
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a62a94a131a112a138a94a27a13,a62a214,a62a214a151a221
a62a214a180a62a131a112a138a94a131a13,a212a159a27a62a131a112a138a94a114a221a134a217a16a145a27a62a214a164
a20a39,a62a214a140a20a140a75,a143a140a143a34(a61a216a145a62),a62a214a27a229a13? a31a222a41a62
a58a58a62a62a214,a195a78a200,a145a62a254,a180a196a29a62a254a252a160a27a18a234a21
(1.6021917×10?19a165a212)a34a169a225a53a51a247a247a42a42a212a110a165a140a3a209a34
a62a214a151a221,a62a214a169a217a51a152a109a78a200a165.
ρ = lim
τ→0
q
τ = a124a58a63a63a252a252a160a78a200a27a27a62a62a254
σ = lim
S→0
q
S = a124a58a63a63a252a252a160a161a200a27a27a62a62a254
η = lim
l→0
q
l = a124a58a63a63a252a252a160a127a221a221a27a27a27a62a62a254
a58a58a62a62a214a27a27a62a62a214a151a221,a160a117vectorr0a63a62a254a143q a27a58a62a214a51a152a109a27a62
a214a151a221a169a217a143ρ(vectorr,vectorr0)
integraldisplay
τ
dτ ρ(vectorr,vectorr0) = q →ρ(vectorr,vectorr0) =
braceleftbigg 0 vectorrnegationslash=vectorr
0
∞ vectorr =vectorr0 = qδ(vectorr?vectorr0)
54/384
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a62a94a131a112a138a94a27a13,a62a54,a62a54a151a221
a36a196a196a62a62a214a47a164a62a54,a107a144a149a218a62a254a140a2a34
a62a54a151a221a165a254vectorja27a140a2a143a252a160a158a109a54a76a82a134a117a62a54a144a149a252a160a161a200a27
a62a254a167a144a149a143a62a54a144a149a34
a233a63a191a152a135a216a152a189a82a134a117a62a54a144a149a27a161a3?vectorS,a252a160a158a109a207a76a27a62
a254a143j?S⊥ =vectorj·?vectorS,a233a107a129a140a2a161a3,a252a160a158a109a54a76a27a62a254a23a62a54
a114a221 a16a16a16
a16a16a16a45
a45a45 a45a45
a45?S⊥
vectorj J =
integraltext
Sd
vectorS·vectorj
a177a132a221vectorva36a196a196a27a27a27a62a62a214a228a107a62a54a151a221 vectorj = ρvectorv
a62a54a151a221a94a227a120a209a143a62a54a130,a167a144a149a143vectorja144a149,
a151a221a143vectorja140a2a34a51a62a54a130a227a254a247vectorja144a149a138a152a2a62a54a43:
Jdl = j?Sdvectorl =vectorjdτ = ρvectorvdτ a245a171a132a221====
summationdisplay
i
ρivectorvidτ =
summationdisplay
i
qivectorvi
ρ? =?ρ+→ρ = ρ? +ρ+ = 0 vectorj = ρ+vectorv+ +ρ?vectorv? = ρ+(vectorv+?vectorv?)
55/384
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a62a94a131a112a138a94a27a13,a62a54,a62a54a151a221; a94a94a252a252a52
a62a54a169a217a51a76a161a254a94a161a62a54a151a221vectori,a144a149a143a62a54a144a149,a140a2a143 a252a160a158a109a54a76a82a134a117a62
a54a144a149a252a160a127a221a221a27a27a27a62a62a254a34a233a63a152a130a3?vectorl,a252a160a158a109a207a76a27a27a62a62a254vectori·?vectorla34
a233a177a132a221vectorvia36a196a196a27a27a161a62a214 σi,vectori =
summationdisplay
i
σivectorvia161a62a54a151a221a134a78a62a54a151a221a107a39,vectorj = vectoriδ,
Jdl = i?ldvectorl =vectoridS =
summationdisplay
i
σivectorvidS =
summationdisplay
i
qivectorvi
a94a94a252a252a52:
a94a131a112a138a94a23a41a13a27a64a163a123a164a181
a94a99?a94a214(a94a94a252a252a52) Qm; a94a214a151a221ρm; a94a54a151a221vectorjm
a94a214a131a109a132a108a97a113a117a165a213a189a198a27a178a144a144a135a135a39a199
a62a54a143a140a177a23a41a94a138a94
a83a23a74a209a169a102a62a54a103a94a98a96?a94a131a112a138a94a94a100a100a62a54a36a196a196a62a62a214a23a41
a132a107a118a107a94a94a252a252a52?
56/384
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57/384
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triangleleftsld
trianglerightsld
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a62a94a131a112a138a94a27a124a134a253a152a165a27a196a29a162a8a189a198,a62a214a197a240; a85a92a6a110
a62a214a197a240a189a198:
a62a214a216a172a103a196a23a41a189a7a171,a216a154a45a254a217a167a62a214
contintegraldisplay
τ
dvectorS·vectorj =?ddt
integraldisplay
τ
dτρ
a233a152a135a63a191a78a200τ,a101a51a44a152a227a158a109a83a217a62a254a126a8,a75a126a8a27a62a254a152
a189a180 a108a108a100a100a78a200a27a76a161Sa54a209(a189a54a63a31a254a27a75a62a214)a34
a85a92a6a110:
a62a94a131a112a138a94a229a228a107a140a83a92a53a159
vectorf = summationdisplay
i
vectorfi
58/384
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a62a94a131a112a138a94a27a124a134a253a152a165a27a196a29a162a8a189a198,a165a213a189a198
a165a213a189a198:
a1
a1
a1
a1a1a21
a16a16a16
a16a16a16
a16a16a16a49
a80a80a80
a80a80a80a105
a115
a115
vectorrprime
q1
vectorr primeprime
vectorR q
2
a233a183a142a58a58a62a62a214,a211a210a131a189,a201a210a131a225,a131a112a138a94a229a247a62a214
a235a130a144a149,a140a2a20a39a117a62a214 a27a27a62a62a254,a135a39a117a62a214a131a109a229a108a178a144
vectorf21 = k1q1q2
R3
vectorR k1 = 1
4piepsilon10 epsilon10 = 8.854×10
12a165a2122/a218a238·a1462
vectorf21a180q2a233q1a27a138a94a229,q1a233q2a27a138a94a229a143vectorf12 =?vectorf21,vectorRa180a108q2a141a149q1a27a165a254a34
a178a144a144a135a135a39a1863→3+epsilon1 epsilon1<3×10?16?a144a149?a233a161a53a182a52a165a254?vectorR = 0?
a233a121a162a162a173a173a46a127a51a27a252a172a145a62a78,τ2a233τ1a27a138a94a229vectorF21
vectorF21 =
integraldisplay
dvectorf21 =
integraldisplay
τ1
integraldisplay
τ2
k1ρ1(vectorrprime)dτprimeρ2(vectorrprimeprime)dτprimeprime
R3
vectorR vectorR =vectorrprime?vectorrprimeprime
vectorf21 =
integraldisplay
τ1
dτprime
integraldisplay
τ2
dτprimeprimek1q1δ(vectorr
prime?vectorr1)q2δ(vectorrprimeprime?vectorr2)
|vectorrprime?vectorrprimeprime|3 (vectorr
prime?vectorrprimeprime) = k1q1q2
|vectorr1?vectorr2|3(vectorr1?vectorr2)
59/384
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a165a213a189a198a134a25a107a218a229a189a198a27a39a22
a229 a189a198 a214 a8a65
a218a229 vectorf = Gm1m2R3 vectorR a159a254 a131a112a225a218
a62a229 vectorf = k1q1q2R3 vectorR a62a214 a211a53a131a189a163a189a229a164a164a182a182a201a53a131a225a163a218a229a164
a218a229a166a212a159a207a143a131a112a225a218a224a56a20a152a229a167a212a159a245a27a47a144a172a22a53a22a245a156 a218a238a238a27a27a42a58=====?
a137a187a187a165a165a23a41a40a8a27a212a110a196a58
a211a24a27a145a62a212a159a62a229a69a164a212a159a109a27a131a112a252a189a167a126a102a211a171a212a159a27a118a224a8a65!
a252a159a102a109a27a27a62a62a229a39a217a218a229a1141036a21!
a131a2291a49a99a27a159a102a109a27a27a62a62a229a134a131a2291a102a146a27a159a102a109a27a218a229a131a31!
60/384
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a62a94a131a112a138a94a27a124a134a253a152a165a27a196a29a162a8a189a198,a62a54a3a27a131a112a138a94
a62a54a3a27a131a112a138a94a181
a240a189a189a62a62a54a3J2dvectorl2a233J1dvectorl1a27a138a94a229a143
a80a80a80
a80a80a80a113
a0a0a18
a54J2d
vectorl2 vector
R J1dvectorl1
d2vectorF21 = k2J1dvectorl1×(J2d
vectorl2×vectorR
R3 ) =
k2J1J2
R3 [(
vectorR·dvectorl1)dvectorl2?(dvectorl1·dvectorl2)vectorR]
a62a54a3J1dvectorl1a233 J2dvectorl2a27a138a94a229a143
d2vectorF12 = k2J1J2R3 [?(vectorR·dvectorl2)dvectorl1 + (dvectorl1·dvectorl2)vectorR] a52a165a254
a217a165,a83a23a129a64a27a243a138a255a254a27a216a180a62a54a3a167a13a180a62a54a23a131a109a27a131a112a138a94a229a156
vectorR =vectorr1?vectorr2 k2 = μ0
4pi μ0 = 4pi×10
7a250a54·a146/a165a2122
a252a62a54a3a131a109a27a138a94a229a134a135a138a94a229a216a152a152a24a24,vectorF1 + vectorF2negationslash= 0
(a126:a252a36a196a196a58a58a58a62a62a214vectorv2bardblvectorR⊥vectorv1)→ dvectorP1dt + dvectorP2dt negationslash= 0a189 vectorP1 + vectorP2negationslash= 0,
a196a254a197a240a216a50a164a225a34
61/384
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a62a94a131a112a138a94a27a124a134a253a152a165a27a196a29a162a8a189a198,a124
a124a181
a252a135a62a54a3a27a131a112a138a94a196a254a197a240a216a164a225a27a121a150a233a171a252a135a62a54a3a8a164
a27 a78a88a191a191a216a216a180a152a135a181a52a78a88,a132a107a217a167a16a145a196a254a27a212a159a162a51a34
a124a27a86a103a129a64a180a123a46a49a74a209a27a181a124a180a219a141a27a182a145a251a117a18a135a152a109a34a13
a23a41a124a167 a124a135a138a94a94a117a117a13a34a124a180a68a52a13a131a109a131a112a138a94a27a49a78a34
vectorF21 =
integraldisplay
τ1
dτprime [ ρ1(vectorrprime)
integraldisplay
τ2
dτprimeprimek1ρ2(vectorr
primeprime)
R3
vectorR + vectorj1(vectorrprime)×
integraldisplay
τ2
dτprimeprimek2
vectorj2(vectorrprimeprime)×vectorR
R3 ]
=
integraldisplay
τ1
dτprime [ ρ1(vectorrprime)vectorE1(vectorrprime) + vectorj1(vectorrprime)×vectorB1(vectorrprime) ] vectorR =vectorrprime?vectorrprimeprime
vectorE1(vectorrprime) = k1
integraldisplay
dτprimeprimeρ2(vectorr
primeprime)
R3
vectorR vectorB1(vectorrprime) = k2
integraldisplay
dτprimeprime
vectorj2(vectorrprimeprime)
R3
vectorR
a165a212a189a198a218a83a23a39a105a189a198a165a13a201a229a27a250a170a169a41a143 a13a23a41a124a250a170a218a13a13a51a51a124a165a201a229a250a170a27a40a220a156
a62a94a229a131a109a27a27a83a83a92a6a110a61a61a122a143a124a27a27a83a83a92a6a110,?a124a144a167a180a130a53a27 !
a52a165a254 a171a169a52a33a182a165a254a27a126a102 a182a165a254 Biot-Savarta189a198!
62/384
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a62a94a131a112a138a94a27a124a134a253a152a165a27a196a29a162a8a189a198,a124
a39a117a226a226a213a213a91a229a181
vectorF =
integraldisplay
dτ[ρ(vectorr)vectorE(vectorr) +vectorj(vectorr)×vectorB(vectorr)] =
integraldisplay
dτρ(vectorr)vectorE(vectorr) +
integraldisplay
Jdvectorl×vectorB(vectorr)
a233a177a132a221vectorva36a196,a62a254a143qa27a27a58a58a58a62a62a214,Jdvectorl = qvectorv,ρdτ = q,
vectorf = q(vectorE +vectorv×vectorB) a226a226a212a212a91a250a170
a62a94a124a140a169a143a201a229a13a23a41a27a27a62a62a94a124vectorEs,vectorBsa218a9a62a94a124vectorEe,vectorBe
vectorE = vectorEs + vectorEe vectorB = vectorBs + vectorBe
a103a62a94a124a6a29a29a216a216a209a121a51a250a170a165a167a2a127a196a13a144a85a97a201a111a124a167a195a123a171a169a124a27a209a28a156
a13a164a201a27a103a103a138a138a94a229a27a63a110a191a191a216a216a152a189a103a84! a58a13a33a67a122a62a94a124....
vectorFs≡
integraldisplay
dτ[ρ(vectorr)vectorEs(vectorr) +vectorj(vectorr)×vectorBs(vectorr)] =
integraldisplay
τ
integraldisplay
τ
dτprimedτ”[k1ρ(vectorrprime)ρ(vectorr”) +k2vectorj(vectorrprime)×vectorj(vectorr”)×] (
vectorrprime?vectorr”)
|vectorrprime?vectorr”|3= 0
a252a252a226a226a102a2a134 a52a220a163a180
63/384
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a62a94a131a112a138a94a27a124a134a253a152a165a27a196a29a162a8a189a198,a123a46a49a49a62a62a94a97a65a189a198
a62a124a100a62a214a23a41,a94a124a100a62a54a61a36a196a27a62a214a23a41,a207a13a62a124a134a94a124a27
a39a88a152a189a134a36a196a61a158a109a27a67a122a107a39.
a152a138a52a220a19a130a164a157a140a27a94a207a27a85a67,a242a51a100a52a220a19a130a254a23a41a97a65a62
a196a179,a217a234a138 a31a117a252a160a158a109a83a94a207a27a67a122a199,a100a100a23a41a27a97a65a62a196a179
a164a251a189a189a27a27a97a65a62a54a144a149 a123a142a19a130a83a94a207a27a67a122.
ε =?dΦdt ε =
contintegraldisplay
S
dvectorl·vectorE Φ =
integraldisplay
S
dvectorS·vectorB
vectorSa144a149a134vectorla144a149a164a109a195a218a94a39a88a34 a62a94a97a65a28a28a91a91
64/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a200a169a47a170
a62a214a197a240:
contintegraldisplay
τ
dvectorS·vectorj =?
integraldisplay
τ
dτ?ρ?t a165a212a189a198:
vectorE = k1
integraldisplay
dτprimeρ(vectorr
prime)
R3
vectorR =?k1
integraldisplay
dτprimeρ(vectorrprime)?1R =bracketleftbig
integraldisplay
dτprimeρ(vectorr
prime)
R
bracketrightbig
φ(vectorr) = k1
integraldisplay
dτprimeρ(vectorr
prime)
R +a126a234
vectorE(vectorr) =φ(vectorr)
contintegraldisplay
S
dvectorl·vectorE =
integraldisplay
S
dvectorS·(?×vectorE) =?
integraldisplay
S
dvectorS·[?×(?φ)]= 0
contintegraldisplay
τ
dvectorS·vectorE =
integraldisplay
τ
dτ?·vectorE =?
integraldisplay
τ
dτ?·(?φ) =?
integraldisplay
τ
dτ?2
integraldisplay
dτprimek1ρ(vectorr
prime)
R
=?
integraldisplay
τ
dτ
integraldisplay
dτprimek1ρ(vectorrprime)?2 1R =?
integraldisplay
τ
dτ
integraldisplay
dτprimek1ρ(vectorrprime)(?4pi)δ(vectorr?vectorrprime)
= 1epsilon1
0
integraldisplay
τ
dτρ(vectorr)= 1epsilon1
0
Qa83
65/384
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a233a112a100a189a110a27a95a149a110a41contintegraldisplay
τ
dvectorS·vectorE = 1epsilon1
0
Qa83
a98a189a152a109a165a27a164a107a62a214a209a169a217a117a139a73a6a58a78a67a27a152a135a2a171a141a165,
a51a177a139a73a6a58a143a11a37a140a187a143ra27a165a161a254
a101a62a214a154a165a233a161a169a217,a211a165a161a254a216a211a58a58a27a27a207a254a140a85a112a216a131a211
E = a207a254/?S
a178a254a96a53:
E =a152a135a27a189a165a161a254a252a160a161a200a254a27a178a254a207a254= a111a207a2544pir2
a216a211a140a187a27a165a161a254a27a111a207a254= Q/epsilon10a209a131a211,a134a140a187ra195a39
arrowtripleright a178a144a144a135a135a39a199,E = Q/(4piepsilon10r2)
66/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a200a169a47a170
a39a105a189a198,vectorB =
integraldisplay k
2J(vectorrprime)dvectorlprime×vectorR
R3 =?
integraldisplay
k2J(vectorrprime)dvectorlprime×?1R
=?×
integraldisplay k
2J(vectorrprime)dvectorlprime
R
vectorA(vectorr) = k2
integraldisplay
dvectorlJ(vectorr
prime)
R +?χ =
integraldisplay
dτprime
vectorj(vectorrprime)
R +?χ
vectorB(vectorr) =?×vectorA(vectorr)
contintegraldisplay
τ
dvectorS·vectorB =
integraldisplay
τ
dτ?·vectorB =
integraldisplay
τ
dτ?·(?×vectorA)= 0
67/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a200a169a47a170contintegraldisplay
S
dvectorl·vectorB =
integraldisplay
S
dvectorS·(?×vectorB) =?
integraldisplay
S
dvectorS·?×[
integraldisplay
dvectorlprime×k2J(vectorrprime)?1R]
=?
integraldisplay
S
dvectorS·
integraldisplay
∞
dτprimek2?×[vectorj(vectorrprime)×?1R]
= k2
integraldisplay
S
dvectorS·
integraldisplay
∞
dτprime[vectorj(vectorrprime)·vectorj(vectorrprime)?2]1R
integraldisplay
∞
dτprimevectorj(vectorrprime)·1R =?
integraldisplay
∞
dτprimevectorj(vectorrprime)·?prime?1R
=
integraldisplay
∞
dτprimebracketleftbig?prime·[?vectorj(vectorrprime)?1R] + [?prime·vectorj(vectorrprime)]?1Rbracketrightbig
=?
integraldisplay
∞
dvectorSprime·vectorj(vectorrprime)?1R +
integraldisplay
∞
dτprime[?prime·vectorj(vectorrprime)]?1R
= 0
0 =
contintegraldisplay
τ
dvectorS·vectorj =
integraldisplay
τ
dτ?·vectorj→?·vectorj = 0
68/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a200a169a47a170
contintegraldisplay
S
dvectorl·vectorB = k2
integraldisplay
S
dvectorS·
integraldisplay
dτprime[vectorj(vectorrprime)·vectorj(vectorrprime)?2]1R
=?k2
integraldisplay
S
dvectorS·
integraldisplay
dτprimevectorj(vectorrprime)?2 1R
= 4pik2
integraldisplay
S
dvectorS·
integraldisplay
dτprimevectorj(vectorrprime)δ(vectorr?vectorrprime)
= 4pik2
integraldisplay
S
dvectorS·vectorj(vectorr)= μ0J
a123a46a49a49a62a62a94a97a65a189a198:
contintegraldisplay
S
dvectorl·vectorE =?
integraldisplay
S
dvectorS·?
vectorB
t
a165a212a189a198
contintegraldisplay
S
dvectorl·vectorE = 0a27a237a50!
69/384
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a233a83a23a130a180a189a110a27a95a149a110a41contintegraldisplay
S
dvectorl·vectorB = μ0J
a98a189a152a109a165a27a164a107a62a54a209a247a169a217a117za182a78a67a27a152a135a2a171a141a165,
a51a177a182a143a11a37a140a187a143ra27a9a130a180a254
a101a62a54a154a206a233a161a169a217,a211a130a180a254a216a211a58a27a94a97a65a114a221a140a85a112
a216a131a211
a178a254a96a53:
B =a152a135a27a189a130a180a254a252a160a127a221a254a27a178a254a130a254= a130a2542pir
a216a211a140a187a27a130a180a254a27a130a254= μ0Ja209a131a211,a134a140a187ra195a39
arrowtripleright a134a130a62a54a94a124a27a229a108a135a39a199,B = μ0J/(2pir)
70/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a135a169a47a170
contintegraldisplay
τ
dvectorS·vectorj =?
integraldisplay
τ
dτ?ρ?t →
integraldisplay
τ
dτ(?·vectorj +?ρ?t) = 0 →?ρ?t +?·vectorj = 0
contintegraldisplay
τ
dvectorS·vectorE = 1epsilon1
0
integraldisplay
τ
dτρ →
integraldisplay
τ
dτ(?·vectorE? 1epsilon1
0
ρ) = 0 →?·vectorE = 1epsilon1
0
ρ
contintegraldisplay
S
dvectorl·vectorB = μ0
integraldisplay
S
dvectorS·vectorj→
integraldisplay
S
dvectorS·(?×vectorB?μ0vectorj) = 0→?×vectorB = μ0vectorj
contintegraldisplay
τ
dvectorS·vectorB = 0 →
integraldisplay
τ
dτ?·vectorB = 0 →?·vectorB = 0
contintegraldisplay
τ
dvectorl·vectorE =?
integraldisplay
τ
dvectorS·?
vectorB
t →
integraldisplay
τ
dvectorS·(?×vectorE +?
vectorB
t ) = 0
→?×vectorE =
vectorB
t
vectorF =
integraldisplay
τ
dτ[ρ(vectorr)vectorE(vectorr) +vectorj×vectorB(vectorr)] →
integraldisplay
τ
dτ(vectorf?ρvectorE?vectorj×vectorB) = 0
→ vectorf = ρvectorE +vectorj×vectorB
71/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a135a169a47a170
ρ
t +?·
vectorj = 0?·vectorE = 1
epsilon10ρ?×
vectorB = μ0vectorj
ρ
t = epsilon10
t?·
vectorE = epsilon10?·?vectorE
t
·vectorj = 1μ
0
·(?×vectorB) = 1μ
0
(?×?)·vectorB = 0
×vectorB →?×vectorB = μ0vectorj +μ0epsilon10?
vectorE
t ←a160a163a62a54a145(a62a41a94)
72/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a135a169a47a170
·vectorE = 1epsilon1
0
ρ
3summationdisplay
i=1
iEi = 1epsilon1
0
ρ a62a214a180a62a124a117a209a224a241a27a13
×vectorE =
vectorB
t
3summationdisplay
i,j=1
epsilon1ijk?iEj =Bk?t a94a124a27a158a109a67a122a199a180a62a124a94a61a27a13
·vectorB = 0
3summationdisplay
i=1
iBi = 0 a94a124a216a117a209a218a224a241
×vectorB = μ0vectorj +μ0epsilon10?
vectorE
t
3summationdisplay
i,j=1
epsilon1ijk?iBj = μ0jk +μ0epsilon10?Ek?t
a62a54a151a221a218a62a124a27a158a109a67a122a199a180a94a124a94a61a27a13
vectorf = ρvectorE +vectorj×vectorB fk = ρEk +
3summationdisplay
i,j=1
epsilon1ijkjiBj
73/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a135a169a47a170
a108 Helmholtz a189a110a119a240a142a100a137a144a167a124:
vectorF(vectorr) =U(vectorr)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
a112a124
+?×vectorW(vectorr)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
a238a124
+
integraldisplay
dτprime{prime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
+?prime×?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime| +?
prime·?
primevectorF(vectorrprime)
4pi|vectorr?vectorrprime|
prime[vectorF(vectorrprime)·?prime 1
4pi|vectorr?vectorrprime|]}
U(vectorr) =
integraldisplay
dτprime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
vectorW(vectorr) =
integraldisplay
dτprime?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
a195a46a152a109a181 vectorF(vectorr) = 14pi
integraldisplay
dτprime?
prime·vectorF(vectorrprime)
|vectorr?vectorrprime|3 (vectorr?vectorr
prime) + 1
4pi
integraldisplay
dτprime[?
prime×vectorF(vectorrprime)]
|vectorr?vectorrprime|3 ×(vectorr?vectorr
prime)
vectorE(vectorr) = 1
4pi
integraldisplay
dτprime
1
epsilon10ρ(vectorr
prime)
|vectorr?vectorrprime|3(vectorr?vectorr
prime) vectorB(vectorr) = 1
4pi
integraldisplay
dτprimeμ0
vectorj(vectorrprime)
|vectorr?vectorrprime|3×(vectorr?vectorr
prime)
·vectorE = 1epsilon1
0
ρ?×vectorE =
vectorB
t?·
vectorB = 0?×vectorB = μ0vectorj +μ0epsilon10?vectorE
t
74/384
triangleleftsldtriangleleftsld
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triangleleftsld
trianglerightsld
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a135a169a47a170
a92a92a94a94a252a252a52a151a221ρma218a94a94a252a252a52a54a151a221vectorjma27a0a122:
·vectorE = 1epsilon1
0
ρ
×vectorE =?vectorjm
vectorB
t
·vectorB = ρm
×vectorB = μ0vectorj +μ0epsilon10?
vectorE
t
a62a94a233a243a181
ρ→
radicalbiggepsilon1
0
μ0ρm
vectorj→
radicalbiggepsilon1
0
μ0
vectorjm vectorE→ 1√
epsilon10μ0
vectorB
ρm→?
radicalbiggμ
0
epsilon10ρ
vectorjm→?
radicalbiggμ
0
epsilon10
vectorj vectorB→?√epsilon10μ0vectorE
75/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a62a94a233a243
·vectorE = 1epsilon1
0
ρ?×vectorE =?vectorjm
vectorB
t
·vectorB = ρm?×vectorB = μ0vectorj +μ0epsilon10?
vectorE
t
·(vectorE + i√epsilon1
0μ0
vectorB) = 1√
epsilon10(
1√
epsilon10ρ+
i√
μ0ρm)
×(vectorE + i√epsilon1
0μ0
vectorB) =?vectorjm + i
radicalbiggμ
0
epsilon10
vectorj + i√μ0epsilon10?
t(
vectorE + i√
epsilon10μ0
vectorB)
a62a94a233a243a181 θa134a158a152a139a73a195a39
vectorEprime + i√
epsilon10μ0
vectorBprime = eiθ(vectorE + i√
epsilon10μ0
vectorB)
1√
epsilon10ρ
prime + i√
μ0ρ
prime
m = e
iθ( 1√
epsilon10ρ+
i√
μ0ρm)
vectorjprimem + i
radicalbiggμ
0
epsilon10
vectorjprime = eiθ(?vectorjm + i
radicalbiggμ
0
epsilon10
vectorj)
76/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a212a159a27a27a62a62a94a53a159
a121a162a173a46a191a216a180a253a152,a180a100a212a159a8a164a27a34a247a42a137a140a27a212a159a209a100a6
a102a124a164,a6a102a180a152a135a177a145a20a62a27a6a102a216a143a165a37,a177a140a55a88a145a75a62a27a62
a102a31a94a61a27a88a218a34a229a124a27a138a94a27a180a62a94a131a112a138a94,a61a6a102a134a62a102,a62
a102a134a62a102a131a109a27a62a94a131a112a138a94a229,a167a251a189a10a18a135a6a102a27a49a143a34a122a135
a6a102a209a180a152a135a2a27a62a94a88a218,a175a245a27a2a62a94a88a218a92a200a229a53 a8a164a140a27
a62a94a88a218a34a23a41a136a171a136a24a69a44a27a69a220a62a94a8a65,a47a164a195a119a245a231a27a121
a162a162a173a173a46a34
a6a75a254a122a135a6a102a27a62a94a131a112a138a94a49a143a79a142a152a217a10,a247a42a212a159a27a62
a94a49a143a210a127a23a10a34a207a143a212a78a165a6a102a27a234a56a150a8a1801023a254a63,1023a254a63
a27 a83a92a51a56a99a27a79a142a85a229a89a178a254a180a195a123a162a121a27,a191a133a122a135a6a102a88a218
a27a141a91a49a143a143a180a195a123a127a23a27a34a207a100,a212a159a27a62a94a49a143 a216a140a85a207a76a114
a217a165a27a122a135a6a102a27a62a94a49a143a99a91a239a196a152a217a50a63a216,a13a144a85a56a209a228a107
a1a53a27 a65a58,a191a124a94a249a10a65a58a63a216a34a252a135a6a102a27a176a91a49a143a51a247a42a254
a207a126a180a118 a107a8a74a27,a144a107a64a10a140a220a169a6a102a209a228a107a27a53a159a226a172a107a247a42
a8a65a34
77/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a212a159a27a27a62a62a94a53a159
a108a62a214a14a221a119a181 a106+
vectorEnegationslash=0
→ a106+
a45a45
a45
vectorE
a233a154a52a53a27a6a102a181a122a135a6a102a88a218a209a180a152a135a165a37a143a20a62a214(a6a102a162)a177a140a143a75a62a214a27
a88a218a34a195a9a46a75a143a158,a75a62a214 a27a169a217a180a233a161a27,a18a135a6a102a180a62a165a53,a212a78a18a78a143a210a180
a62a165a53a27a34a101a127a51a9a62a124,a108a229a198a254a119a167a20a62a214a165a37a134a107a8a75a62a214a165a37a117a41a160a108,
a47a164a152a62a243a52a102vectorp = qvectorl,a212a159a159a165a165a27a2a62a243a52a102a247a9a62a124a252a15,a52a122a47a164a247a247a42a42a62a124a34
a233a52a53a27a6a102a181 a122a135a6a102a88a218a29a28a210a180a67a113a27a62a243a52a102a34 a108a218a79a254a119a167a62a243
a52a102a51a195a9a124a158a100a57a222a225a145a197a252a15a167a9a62a124a27a209a121a166a243a52a102a107a247a9a124a144a149a252a15a27
a170a179W =?vectorp·vectorE,a166a26a209a207a252a15a27a27a62a62a243a52a102a109a169a211a149a252a15a34
a189a194a52a122a114a221:
vectorP = lim
τ→0
summationdisplay
τ
vectorp
τ = npvectorp np,a252a160a78a200a27a27a62a62a243a52a229
vectorP = vectorP(vectorE,vectorB) a212a159a27a27a62a62a94a53a159a144a167
78/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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Close
a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a212a159a27a27a62a62a94a53a159
a108a62a54a14a221a119:
a122a135a6a102a88a218a165a55a6a102a216a94a61a27a62a102a47a164a152a2a62a54a23,a8a164a152a2a94a243a52a102a34a195a9
a46a75a143a158,a2a94a243a52a102a144a149a44a207,a233a9a46a27a75a143a131a112a45a158,a216a23a41a247a42a8a65a34a8a107a9
a124a158,a108a229a198a14a221a119a167a2a94a243a52a102a201a229a221 vectorL = vectorm× vectorB = dvectorJdt,vectorJ bardbl vectorm a47a164Larmora63
a196a250a218a63a196a167a13a23a41a31a8a27a135a94a124a144a149a61a196a31a100a135a94a124a94a83a252a15a27a94a221a163a95a94a164,a94
a122a47a164a247a42a94a124a34 a108a218a79a14a221a119a167a94a243a52a102a51a9a94a124a107a247a9a124a144a149a252a15a27a170a179
W=?vectorm·vectorB,a166a26a100a57a222a225a209a207a252a15a27a94a243a52a102a247a94a124a211a149a252a15a163a94a94a164a207a126a140a117a95a94a34
a189a194a94a122a114a221:
vectorM = lim
τ→0
summationdisplay
τ
vectorm
τ = nmvectorm nm,a252a160a78a200a27a94a243a52a229
vectorM = vectorM(vectorE,vectorB) a212a159a27a27a62a62a94a53a159a144a167
79/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a52a122a62a214a134a94a122a62a54
a52a122a62a214a134a94a122a62a54(a154a19a78a156a185):
a62a94a138a94a144a163a79a62a214,a62a54,a233a62a165a53a27a212a78a180a223a178a27a34a62a94a124a233
a212a159a27a138a94a162a83a254 a180a62a94a124a233a212a159a165a16a145a51a6a102a216,a62a102a254a27a62a214
a57a217a36a196a47a164a27a62a54a27a138a94a34a249a10 a62a214a218a62a54a78a88a51a6a102a254,a233a27
a21a212a159a6a102a113a26a27a189a51a172a130a254,a207a100a167a130a180a26a229a80 a52a27a34a23a41a52a122
a8a65a27a62a214a161a143a52a122a62a214,a217a62a214a151a221a80a143ρprime,a23a41a94a122a8a65a27a62
a54a161a143a94a122a62a54,a217a62a54a151a221a80a143vectorjprimea34a233a65a27a103a100a62a214a27a62a214a151a221a80
a143ρf,a68a19a19a62a62a54a27a27a62a62a54a151a221a80a143vectorjca34
a100a100a117a117ρprimea180a100a100a117a117a52a122a69a164a27,a175a75a122a143a152a230a100a52a122a114a221vectorPa163a227a27 a2a243
a52a102a23a41a27ρprimea180a245a8?
a100a117vectorjprimea180a100a117a94a122a69a164a27,a175a75a122a143a152a230a100 a52a122a114a221vectorPa163a227a27a2a62
a243a52a102a218a94a122a114a221vectorMa163a227a27a2a94a243a52a102 a23a41a27vectorjprimea180a245a8?
80/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a52a122a62a214a134a94a122a62a54
φ(vectorr) =
integraldisplay
dτprimek1ρ
prime(vectorrprime)
R +
integraldisplay
dSprimek1σ
prime(vectorrprime)
R +a126a234
vectorR =vectorr?vectorrprime
φ(vectorr) =
integraldisplay k
1vectorR·vectorP(vectorrprime)dτprime
R3 +a126a234 =
integraldisplay
dτprimek1(?prime1R)·vectorP +a126a234
=
integraldisplay
dτprimek1[?prime·(
vectorP
R)?
prime·vectorP
R ] +a126a234
=
integraldisplay
dvectorSprime·k1
vectorP
R +
integraldisplay
dτprimek1(
prime·vectorP)
R +a126a234
=
integraldisplay
dτprimek1(
prime·vectorP)
R +
integraldisplay
dSprimek1vectorn
prime·vectorP
R +a126a234
ρprime(vectorr) =·vectorP(vectorr) σprime(vectorr) = vectorn·vectorP(vectorr)
a254a33a52a122a158a195a78a229a80a62a214; a52a122a51a212a159a76a161a23a41a41a229a229a80a161a62a214a34
81/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a52a122a62a214a134a94a122a62a54
vectorjprime =vectorj1 +vectorj2?ρprime
t +?·
vectorj1 = 0 ρprime =·vectorP vectorj1 =?vectorP
t
vectorA(vectorr) =
integraldisplay
dτprimek2
vectorj2(vectorrprime)
R +
integraldisplay
dSprimek2
vectoriprime(vectorrprime)
R +a126a234
vectorR =vectorr?vectorrprime
vectorA(vectorr) =
integraldisplay k
2 vectorM(vectorrprime)dτprime×vectorR
R3 +a126a234 =
integraldisplay
dτprimek2 vectorM×?prime1R +a126a234
=?
integraldisplay
dτprimek2(?prime1R)× vectorM =?k2
integraldisplay
dτprime[?prime×(
vectorM
R)?
1
R(?
prime× vectorM)]
=
integraldisplay
dτprimek2(?
prime× vectorM)
R +
integraldisplay
dSprimek2
vectorMprime×vectornprime
R +a126a234
vectorj2(vectorr) =?× vectorM(vectorr) vectoriprime(vectorr) = vectorM(vectorr)×vectorn vectorjprime(vectorr) =?vectorP(vectorr)
t +?×
vectorM(vectorr)
a254a33a94a122a191a133a52a122a216a145a158a109a67a122a158a195a78a94a122a62a54; a94a122a51a212a159a76a161
a23a41a94a122a161a62a54a34
82/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a212a159a159a165a165a27a27a62a62a94a124a144a167
·vectorE = 1epsilon1
0
(ρf +ρprime)?×vectorE =
vectorB
t
·vectorB = 0?×vectorB = μ0(vectorjc +vectorjprime) +μ0epsilon10?
vectorE
t
·vectorE = 1epsilon1
0
(ρf·vectorP) →?·(epsilon10vectorE + vectorP) = ρf
×vectorB = μ0(vectorjc +?× vectorM +?
vectorP
t ) +μ0epsilon10
vectorE
t
→?×( 1μ
0
vectorB? vectorM) =vectorjc +?
t(epsilon10
vectorE + vectorP)
a62a160a163a165a254,vectorD = epsilon10vectorE + vectorP a94a124a114a221a165a254,vectorH = 1μ
0
vectorB? vectorM
·vectorD = ρf?×vectorH =vectorjc +?
vectorD
t
83/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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trianglerightsld
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a212a159a159a165a165a27a27a62a62a94a53a159a144a167
a62a48a159,a162a8a137a209
vectorP = χeepsilon10vectorE χea181a52a122a199 a195a254a106a181 P a134 epsilon1
0E a211a254a106· vectorE = ρ/epsilon10
vectorD = epsilon10vectorE + vectorP = epsilon10(1 +χe)vectorE = epsilon10epsilon1rvectorE= epsilon1vectorE
epsilon1r = 1 +χe a181a131a233a48a62a126a234 epsilon1 = epsilon10epsilon1r a181a253a233a48a62a126a234
a130a53a48a159,χea134vectorEa195a39a34 a20a75a62a214a165a37a160a108a229a108a20a39a117a62a124vectorEa34
a154a130a53a48a159a181
Di =
3summationdisplay
j=1
εijEj +
3summationdisplay
j,k=1
εijkEjEk +
3summationdisplay
j,k,l=1
εijklEjEkEl +···
a102a124a67a113a156
84/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a212a159a159a165a165a27a27a62a62a94a53a159a144a167
a19a78,a238a238a48a48a189a198
J =?φR R = λlS
R,a62a123 λ,a62a123a199 γ = 1λ a62a62a19a19a199
j = JS =?φRS = Sλl?φS = Eλ = γE
vectorj = γvectorE vectorE,a218a229a62a54a27a27a62a62a124
85/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a212a159a159a165a165a27a27a62a62a94a53a159a144a167
a94a48a159a181
a94a94,a95a94a48a159:
vectorM = χmvectorH χma94a122a199
vectorB = μ0(vectorH + vectorM) = μ0(1 +χm)vectorH = μ0μrvectorH= μvectorH
μr = 1 +χm a131a233a94a19a199 μ = μ0μr a253a233a94a19a199
μr >1,a94a94a48a159a182 μr <1,a95a94a48a159
a99a94a225a14:
a51Ha2a27a171a141,Ba134Ha180a130a53a39a88,μra67a113a143a126a234,a2a233a140a34
a8Ha140a20a152a189a167a221,μr～1,a136a20a4a218a71a21a34
a77a94a225a14a181
μra134a94a122a123a164a107a39a34 a8Ha108a195a20a107,
a113a108a107a20a195a158,a209a121a144a123a94a221a34
a91a94a225a14a181
a94a221a174a28a220a136a20a4a218,χm = 0,vectorM = vectorM0,vectorB = μ0vectorH +μ0 vectorM0
a206a71a94a99a94a124a27a28a28a91a91 a47a165a94a124a27a28a28a91a91 a102a124a67a113a156
86/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a62a94a124a27a62a138a39a88,a123a149a169a254
·vectorD = ρf?
contintegraldisplay
dvectorS·vectorD =
integraldisplay
dτρf
a152a207a76a46a161a27a161a200a143?Sa27a2a11a206a78τ
vectorna143a108a48a1591a141a149a48a1592a27a46a161a123a149a252a160a165a254?vectorS = vectorn?S
σf =
integraltext dτρ
f
S =
1
S
contintegraldisplay
τ
dvectorS·vectorD = 1?S[vectorD1·(vectorS) + vectorD2·?vectorS]
= vectorD1·(?vectorn) + vectorD2·vectorn =?D1n + D2n
vectorn·(vectorD2?vectorD1) = σf
contintegraldisplay
dvectorS·vectorB = 0 vectorD→vectorB; ρf →0,σf →0
vectorn·(vectorB2?vectorB1) = 0
a46a161a254a161a62a214a69a164vectorDa51a46a161a252a224a107a30a23,a46a161a254vectorBa123a149a169a254a254a235a235a89a34
87/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
Back
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a62a94a124a27a62a138a39a88,a131a149a169a254
×vectorH =vectorjc +?
vectorD
t?
contintegraldisplay
dvectorl·vectorH =
integraldisplay
dvectorS·(vectorjc +?
vectorD
t )
a152a207a76a46a161a27a163a180,a51a48a1591a2182a252a62a27a247a131a149a144a149a27
a62a127a143?l,vectorna143a108a48a1591a141a149a48a1592a27a46a161a123a130a252a160
a165a254,vectorta143a163a180a157a140a76a161a27a123a149(a46a161a27a131a149)a252a160a165a254
vectort·vectoric?l =vectort·[vectoric?l +?vectorD?t?lδ]vextendsinglevextendsingle
δ→0 =
integraldisplay
dvectorS·(vectorjc +?
vectorD
t ) =
contintegraldisplay
dvectorl·vectorH
=?l[vectorH1·(?vectort×vectorn) + vectorH2·(vectort×vectorn)] =?lvectort·[vectorn×(vectorH2?vectorH1)]
vectorta51a46a161a254a180a63a191;vectoric,vectorn×(vectorH2?vectorH1)a195a82a134a46a161a27a169a254
vectorn×(vectorH2?vectorH1) =vectoriccontintegraldisplay
dvectorl·vectorE =?
integraldisplay
dvectorS·?
vectorB
t
vectorH→vectorE vectorD→?vectorB vectorjc→0 vectoric→0
vectorn×(vectorE2?vectorE1) = 0
a46a161a254vectorEa131a149a169a254a254a235a235a89,a46a161a254a161a62a54 a69a164vectorHa51a46a161a252a224a107a30a23a34
88/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
Back
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a85a196a254a27a61a122a134a197a240,a124a218a62a214a88a218a85a196a254a61a122a218a197a240a189a198a27a152a132a47a170
a124a27a85a254a151a221w,a252a160a78a200a27a124a164a145a27a85a254,a180a158a152a139a73a27
a188a234 w = w(vectorr,t)
a124a27a196a254a151a221vectorg,a252a160a78a200a27a124a164a145a27a196a254,a180a158a152a139a73a27a188
a234 vectorg =vectorg(vectorr,t)
a124a27a85a54a151a221vectorS,a163a227a85a254a27a68a194,a140a2a31a117a252a160a158a109a82a134a54
a76a252a160a238 a31a161a27a85a254,a144a149a141a149a85a254a68a194a27a144a149,a180a158a152a139
a73a27a188a234 vectorS = vectorS(vectorr,t)
a124a27a196a254a54a151a221
arrowrighttophalfarrowrighttophalfJ,
a167a163a227a196a254a27a68a194,?vectorS· arrowrighttophalfarrowrighttophalfJ =?vectorpa189
a194a143a252a160a158a109a207a76a161a3?vectorSa54a209a27a196a254,a180a158a152a139a73a27a188a234
arrowrighttophalfarrowrighttophalfJ =arrowrighttophalfarrowrighttophalfJ (vectorr,t)
89/384
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a85a196a254a27a61a122a134a197a240,a124a218a62a214a88a218a85a196a254a61a122a218a197a240a189a198a27a152a132a47a170
a85a254a61a122a134a197a240a189a198,a252a160a158a109a207a76a76a161a54a92Va27a85a254a31a117
a252a160a158a109a124a233 Va83a62a214a164a137a27a245(a61a245a199)a134Va83a252a160a158a109
a62a94a124a85a254a27a79a92a131a218a34
contintegraldisplay
V
dvectorσ·vectorS =
integraldisplay
V
dVvectorf·vectorv + ddt
integraldisplay
V
dVw
vectorf,a124a233a233a62a62a214a138a94a229a151a221 vectorv,a62a214a36a196a132a221
·vectorS =vectorf·vectorv +?w?t
a196a254a61a122a134a197a240a189a198,a252a160a158a109a207a76a76a161a54a92Va27a196a254a31
a117V a83a62a214a164a201a27a229a134Va83a252a160a158a109a62a94a124a196a254a27a79a92a131
a218a34
contintegraldisplay
V
dvectorσ· arrowrighttophalfarrowrighttophalfJ =
integraldisplay
V
dVvectorf + ddt
integraldisplay
V
dVvectorg
· arrowrighttophalfarrowrighttophalfJ =vectorf +?vectorg?t
90/384
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a85a196a254a27a61a122a134a197a240,a124a218a62a214a88a218a85a196a254a61a122a218a197a240a189a198a27a152a132a47a170
a212a110a110a254a254 a151a221 a54a151a221 a197a240a189a198 a197a240a240a214a214
a62a214 ρ vectorj?·vectorj +?ρ?t = 0 Q = integraltext dVρ
a85a254 w vectorS?·vectorS +?w?t +vectorf·vectorv = 0 Ea85a254 = integraltext dVw
a196a254 vectorg arrowrighttophalfarrowrighttophalfJ?· arrowrighttophalfarrowrighttophalfJ +?vectorg?t +vectorf = 0 vectorPa196a254 = integraltext dVvectorg
91/384
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a62a94a131a112a138a94a85a254a196a254a27a61a122a134a197a240,a85a254a151a221a134a85a54a151a221
vectorf·vectorv = (ρfvectorE +ρfvectorv×vectorB)·vectorv = ρfvectorv·vectorE =vectorjc·vectorE = vectorE·(?×vectorHvectorD
t )
= vectorE·(?×vectorH)?vectorE·?
vectorD
t =·(
vectorE×vectorH) + vectorH·(?×vectorE)?vectorE·?vectorD
t
=·(vectorE×vectorH)?vectorE·?
vectorD
t?
vectorH·?vectorB
t =·
vectorSw
t
vectorS = vectorE×vectorH?w
t =
vectorE·?vectorD
t +
vectorH·?vectorB
t
w
t = epsilon1
vectorE·?vectorE
t +
1
μ
vectorB·?vectorB
t =
t
1
2(epsilon1E
2 + 1
μB
2) =?
t
1
2(
vectorE·vectorD + vectorH·vectorB)
w = 12(vectorE·vectorD + vectorH·vectorB) a124a228a107a85a254a96a178a124a180a212a159a27a152a171a47a21a156
92/384
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a62a94a131a112a138a94a85a254a196a254a27a61a122a134a197a240,a196a254a151a221a134a196a254a54a151a221
vectorf = ρfvectorE +vectorjc×vectorB = (?·epsilon10vectorE)vectorE + (?× 1
μ0
vectorBepsilon10vectorE
t )×
vectorB
= epsilon10(?·vectorE)vectorE + 1μ
0
(?×vectorB)×vectorB?epsilon10?
vectorE
t ×
vectorB
= epsilon10(?·vectorE)vectorE + 1μ
0
[(?·vectorB)vectorB + (?×vectorB)×vectorB]
epsilon10[?
vectorE
t ×
vectorB + vectorE×?vectorB
t +
vectorE×(?×vectorE)]
= epsilon10[(?·vectorE)vectorE + (?×vectorE)×vectorE] + 1μ
0
[(?·vectorB)vectorB + (?×vectorB)×vectorB]
epsilon10t(vectorE×vectorB)
= epsilon10[(?·vectorE)vectorE + (vectorE·?)vectorE?(?vectorE)·vectorE]
+ 1μ
0
[(?·vectorB)vectorB + (vectorB·?)vectorB?(?vectorB)·vectorB]?epsilon10t(vectorE×vectorB)
93/384
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a62a94a131a112a138a94a85a254a196a254a27a61a122a134a197a240,a196a254a151a221a134a196a254a54a151a221
vectorf = epsilon10[(?·vectorE)vectorE + (vectorE·?)vectorE?(?vectorE)·vectorE]
+ 1μ
0
[(?·vectorB)vectorB + (vectorB·?)vectorB?(?vectorB)·vectorB]?epsilon10t(vectorE×vectorB)
= epsilon10[?·(vectorEvectorE)?12?E2] + 1μ
0
[?·(vectorBvectorB)?12?B2]?epsilon10t(vectorE×vectorB)
= epsilon10?·(vectorEvectorE?12 arrowrighttophalfarrowrighttophalfI E2) + 1μ
0
·(vectorBvectorB?12 arrowrighttophalfarrowrighttophalfI B2)?epsilon10t(vectorE×vectorB)
=· arrowrighttophalfarrowrighttophalfJvectorg?t
vectorg = epsilon10vectorE×vectorB = epsilon10μ0vectorS
arrowrighttophalfarrowrighttophalfJ =?epsilon1
0vectorEvectorE?
1
μ0
vectorBvectorB + 1
2
arrowrighttophalfarrowrighttophalfI (epsilon1
0E2 +
1
μ0B
2)
a252a145a62a78a131a109a27a218a238a238a49a49a110a189a198a152a132a216a164a225a156 vectorf21 = dP1dt vectorf12 = dP2dt
94/384
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a62a94a131a112a138a94a85a254a196a254a27a61a122a134a197a240,a62a94a85a254a68a209
a152a152a152a152a152a227a54a107a62a54a27a27a19a19a130,a165a165a182a182a143za182,a252a224a31a161a169a79a143Aa218B,a253a253a76a76a161C,A,Ba254a62a124a247za182a144a149a134a62a54a144a149a149a131a131a211,a85a54
a151a221a195za144a149a169a254,a85a254a216a85a108Aa190Ba161a54a92a189a54a209a19a78.a233a253a161C,
vectorE = Etvectornz + Envectornr epsilon10En = σ Eta143a19a78a83a27za144a149a62a124
vectorH = J
2piavectornφ vectornφ,a253a253a76a76a161a82a134a117za144a149a27a131a165a254
vectorS = vectorE×vectorH = J
2pia(?Etvectornr + Envectornz) =
J
2pia(?Etvectornr +
σ
epsilon10vectornz)
a247za144a149a27a85a54a195a123a63a209a19a78,a247vectornra144a149a85a54a134Eta134Ja144a149a152a151
(JEt > 0)a111a180a54a92a19a78a27,a85a254a108a19a130a253a161a54a92a19a78a83,a216a180a247a62
a54a144a149a94a19a130a54a92.a252a160a158a109a108a253a161C(a140a187a143a,a127a143l)a54a92a27a85a254
a143:
Sr2pial =?vectorS·vectornr2pial = JEt2pia2pial = JEtl = JVAB = J2RAB
VAB,RAB,a19a130a252a224a224a27a27a27a62a62a179a11,a62a123
95/384
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a62a94a131a112a138a94a85a254a196a254a27a61a122a134a197a240,a62a94a85a254a68a209
a252a160a158a109a108a253a161C(a140a187a143a,a127a143l)a54a92a27a85a254|vectorS|2pial = J2RABa20
a208a31a117a252a160a158a109a19a130a254a164a158a209a27a57a85a34a88a74a164a63a216a27a180a62a179,A,Ba161
a254a211a24a195a85a254a54a209a92,a85a254a144a85a100a253a161a54a209a92,a100a117a51a62a179a83
a220Ea134Ja135a149,vectorSr =? J2piaEtvectornra238a143a20,a61a85a254a240a180a108a62a179a253a161a54a209a34
a152a135a248a62a163a180,a85a254a216a180a247a19a130a83a220a68a52a137a75a49,a13a180a108a62a179a76a161a54
a20a152a109,a50a108a152a109(a204a135a180a19a130a76a161a78a67)a54a92a19a130a218a75a49.
a64a143a85a254a180a108a68a209a19a130a83a62a102a36a196a196a85a247a19a130a68a53a27a42a58a180a134a216
a27,(a196a75,a233a2a54a62,a62a54a145a158a85a67a144a149,a85a254a75a152a158a54a53,a152a158a54a22)
a175a162a254,a101a108vectorj = nevectorva15a79,a18j =1a83/a206a1462=1a165a212/a166·a206a1462,e = 1.6×10?19a165a212,n =
1023/a102a1463 = 1020/a206a1463,a26|vectorv|= j/ne = 1/16a206a146/a166a34
a88a100a2a27a27a62a62a102a178a254a164a163a132a221(a2a167a152a166a168a174a170a76a101020/3/16≈3×105a135a6a102)
a216a140a85a248a248a137a137a137a75a75a49 a254a158a209a27a85a254a34
96/384
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a62a94a131a112a138a94a85a254a196a254a27a61a122a134a197a240,a220a254a254a229a229
contintegraldisplay
V
dvectorσ· arrowrighttophalfarrowrighttophalfJ =
integraldisplay
V
dVvectorf + ddt
integraldisplay
V
dVvectorg
vectorF≡
integraldisplay
dVvectorf = d
vectorPa197a22
dt
vectorPa62a94≡
integraldisplay
dVvectorg → ddt[vectorPa197a22+vectorPa62a94] =
contintegraldisplay
dvectorS·(?arrowrighttophalfarrowrighttophalfJ )
arrowrighttophalfarrowrighttophalfJa180a252a160a76a161a164a201a27a229.a51a63a152a76a161a78a67,a101a107a62a94a124,a75a249a135a76a161
a210a135a201a229.
a152a76a161,a123a149a144a149a252a160a165a254vectorn,a62a124a134vectorna27a89a14θE,a62a124a221a75a164a51a27
a131a149a144a149a252a160a165a254vectoreE,a94a124a134vectorna27a89a14θB,a94a124a221a75a164a51a27a131a149a144a149
a252a160a165a254vectoreB,a100a76a161a252a160a76a161a164a201a27a229a143
vectorfa76a161 =?vectorn· arrowrighttophalfarrowrighttophalfJ = epsilon10(vectorn·vectorEvectorE?1
2vectornE
2) + 1
μ0(vectorn·
vectorBvectorB?1
2vectornB
2)
a64
a64
a64
a64
a64a64a73
a0
a0a18a54
a12
a12a12
vectorn
θE
vectorEvectorfEvectoreE
= (EcosθEvectorn + EsinθEvectoreE)epsilon10EcosθE?12epsilon10vectornE2
+(BcosθBvectorn + BsinθBvectoreB)Bμ
0
cosθB? 12μ
0
B2 =vectorfE +vectorfB
vectorfE = 1
2epsilon10E
2(vectorncos2θE +vectoreE sin2θE) vectorfB = 1
2μ0B
2(vectorncos2θB +vectoreB sin2θB)
97/384
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trianglerightsld
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a183a62a124a218a173a240a62a54a27a27a62a62a94a124
98/384
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a183a62a179a57a217a135a169a169a144a144a167
·vectorD = ρf?×vectorE =
vectorB
t
·vectorB = 0?×vectorH =vectorjc +?
vectorD
t
a183a21a62a94a124a94a135:
vectorE
t = 0
vectorB
t = 0
a183a21a62a94a124a247a118:
braceleftbigg?·vectorD = ρ
f
×vectorE = 0
braceleftbigg?·vectorB = 0
×vectorH =vectorjc
a183a62a124a134a183a94a124a112a216a39a233,a140a177a169a79a63a49a63a216!
99/384
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a183a62a179a57a217a135a169a169a144a144a167,a62a179a57a217a247a118a27a135a169a169a144a144a167
braceleftbigg?·vectorD = ρ
f
×vectorE = 0
braceleftbigg vectorD = epsilon1vectorE
vectorf = ρvectorE a62a94a135:
braceleftbiggvectorn·(vectorD
2?vectorD1) = σf
vectorn×(vectorE2?vectorE1) = 0 vectorn,1→2
contintegraldisplay
dvectorl·vectorE =
integraldisplay
dvectorS·(?×vectorE) = 0 →
integraldisplay 2
1
dvectorl·vectorEa134a180a187a195a39
a189a194a62a179a188a234,φ2?φ1 =?
integraldisplay 2
1
dvectorl·vectorE vectorE =φ
a195a94a124a152a189a140a177a76a143a73a254a124a27a70a221 !
φ|∞ = 0 → φ1 =
integraldisplay ∞
1
dvectorl·vectorE =
integraldisplay ∞
1
dvectorl·vectorfa252a160a62a214
φa27a191a191a194a194,a252a160a20a62a214a108a124a58a163a20a195a161a15 (a189a114a252a160a75a62a214a108a195a161a15
a163a20a124a58)a62a124a229a164a137a27a245a34
a62a124a180a62a179a27a129a140a152a109a67a122a199a27a75a138,a144a149a82a134a117a31a179a161(a62a229a130a134
a31a179a161a82a134,a144a149a100a112a62a179a141a149a36a36a62a62a179)a34
100/384
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a183a62a179a57a217a135a169a169a144a144a167,a62a179a57a217a247a118a27a135a169a169a144a144a167
·vectorD = ρf →?(epsilon1?φ) =?ρf
vectorD2n?vectorD1n = σf → epsilon12?φ2
n?epsilon11
φ1
n =?σf
φ2?φ1 =?
integraldisplay 2
1
dvectorl·E → φ2?φ1 =?1epsilon1vectorn·vectorτf
epsilon1a180a152a109a139a73a27a188a234?φ?n = vectorn·?φ φ(vectorr) =
integraldisplay
dτprime ρ(vectorr
prime)
4piepsilon10|vectorr?vectorrprime|+a126a234
a1081a202a180a187a192a143a20a208a207a76a46a161,a144a149a247a46a161a123a149vectorn,dvectorl =vectorndl
++
++
++
+
1 2σf
vectorn
dl
a45
a46a161a27a243a52a0a151a221,τf = σfdl E = σfepsilon1
0
= τfepsilon1
0dl
vectorE = vectorτf
epsilon10dl
vectorE·dvectorl = vectorτf
epsilon10dl·vectorndl =
1
epsilon10vectorτf·vectorn
101/384
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a183a62a179a57a217a135a169a169a144a144a167,a154a52a138a189a110
a254a33a48a159a159a165a165a195a62a214a169a217a27a171a141a83a27a63a219a152a58a58a27a27a183a62a179
a209a216a140a85a18a52a140a189a52a2a138 a156
a98a23a100a189a110a216a164a225a34
a75a51a254a33a48a159a165a195a62a214a169a217a27a171a141a83a127a51a152a58aa27a183a62a179φaa18a52
a140a163a2a164a34
a207a13a127a51a157a140a217a27a118a10a2a181a52a173a161sa167a51a217a83a195a62a214a167a62a46a254a27
a122a152a58a27a62a179a27a9a123a149a135a251a240a143a75a163a20a164a167a191a133a216a85a28a143a34a167 a177
a2a121a108a62a46a112a149a9a62a179a111a180a126a2a163a79a92a164a34
a249a134sa83a62a214a143a34a131a112a103a241a1810 = Q =
contintegraldisplay
dvectors·vectorD =?epsilon1
contintegraldisplay
ds?φ?n > (<)0
a183a62a179a27a52a138a138a144a144a140a85a160a117a62a46a189a107a62a214a63 a156
102/384
triangleleftsldtriangleleftsld
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a83a75
2.14 a120a209a188a234dδ(x)dx a27a227,a96a178ρ =?(vectorp·?)δ(vectorr) a180a152a135a160a117a6a58a27
a243a52a102a27a27a62a62a214a151a221.
dδ(x)
dx = lim?x→0
δ(x +?x)?δ(x)
x a45
a54
x =x
x = 0
integraldisplay
dV ρ =?
integraldisplay
dV pi?iδ(vectorr) = 0
integraldisplay
dVxiρ =?
integraldisplay
dVxipj?jδ(vectorr) =?
integraldisplay
dVbraceleftbig?j[xipjδ(vectorr)]?piδ(vectorr)bracerightbig = pi
integraldisplay
dV xixjρ =?
integraldisplay
dV xixjpk?kδ(vectorr)
=?
integraldisplay
dV braceleftbig?k[xixjpkδ(vectorr)]?(pixj + pjxi)δ(vectorr)bracerightbig = 0
integraldisplay
dV xi1xi2···xinρ = 0 n≥2
103/384
triangleleftsldtriangleleftsld
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trianglerightsld
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a83a75
1.5 a121a178a62a243a52a221a27a67a122a199dvectorpdt = integraltext dVvectorj(vectorr,t)
dvectorp
dt =
integraldisplay
dV vectorrtρ(vectorr,t) =?
integraldisplay
dV vectorr?·vectorj(vectorr,t)
=?vectorei
integraldisplay
dV xi?kjk(vectorr,t) =vectorei
integraldisplay
dV braceleftbigk[xijk(vectorr,t)] + ji(vectorr,t)bracerightbig
=
integraldisplay
dVvectorj
=
integraldisplay
dV braceleftbig·(vectorjvectorr) +vectorj·?vectorrbracerightbig
negationslash=
integraldisplay
dV braceleftbig(vectorj·vectorr) + (?vectorr)·vectorjbracerightbig
negationslash=
integraldisplay
dV [vectorrtρ(vectorr,t) +vectorvρ(vectorr,t)]
=·vectorjvectorr=0 vectorj
104/384
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trianglerightsld
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a83a75
1.10 a121a178a252a135a62a54a130a23a27a131a112a138a94a140a2a131a31a167a144a149a149a131a131a135a34
a240a189a189a62a62a54a3J2dvectorl2a233J1dvectorl1a27a138a94a229a143 a80a80a80a80
a80a80a105
a0a0a18
a54J1d
vectorl1 vector
R J2dvectorl2
vectorF21 = k2
contintegraldisplay
2
contintegraldisplay
1
J1dvectorl1×(J2d
vectorl2×vectorR
R3 )
vectorR =vectorr1?vectorr2
=
contintegraldisplay
2
contintegraldisplay
1
J1dvectorl1×J2dvectorl2×
vectorR
R3
=?
contintegraldisplay
2
contintegraldisplay
1
J2dvectorl2×J1dvectorl1×
vectorR
R3
=?k2
contintegraldisplay
2
contintegraldisplay
1
J2dvectorl2×(J1d
vectorl1×vectorR
R3 ) =?
vectorF12
vectorA×(vectorB×vectorC)negationslash=?vectorB×(vectorA×vectorC)
105/384
triangleleftsldtriangleleftsld
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a83a75
1.10 a121a178a252a135a62a54a130a23a27a131a112a138a94a140a2a131a31a167a144a149a149a131a131a135a34
a240a189a189a62a62a54a3J2dvectorl2a233J1dvectorl1a27a138a94a229a143 a80a80a80a80
a80a80a105
a0a0a18
a54J1d
vectorl1 vector
R J2dvectorl2
vectorF21 = k2
contintegraldisplay
2
contintegraldisplay
1
J1dvectorl1×(J2d
vectorl2×vectorR
R3 )
vectorR =vectorr1?vectorr2
=
contintegraldisplay
2
contintegraldisplay
1
k2J1J2
R3 [(
vectorR·dvectorl1)dvectorl2?(dvectorl1·dvectorl2)vectorR]
a62a54a3J1dvectorl1a233 J2dvectorl2a27a138a94a229a143:1?2; vectorRvectorR
contintegraldisplay
2
contintegraldisplay
1
k2J1J2
R3 (
vectorR·dvectorl1)dvectorl2 =
contintegraldisplay
2prime
contintegraldisplay
1prime
k2J1J2
R3 (
vectorR·dvectorl1prime)dvectorl2prime
1prime=2,2prime=1======
contintegraldisplay
1
contintegraldisplay
2
k2J1J2
R3 (
vectorR·dvectorl2)dvectorl1 =?
contintegraldisplay
1
contintegraldisplay
2
k2J1J2
R3 (?
vectorR·dvectorl2)dvectorl1
106/384
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a83a751.10 a121a178a252a135a62a54a130a23a27a131a112a138a94a140a2a131a31a167a144a149a149a131a131a135a34
a240a189a189a62a62a54a3J2dvectorl2a233J1dvectorl1a27a138a94a229a143 a80a80a80a80
a80a80a105
a0a0a18
a54J1d
vectorl1 vector
R J2dvectorl2
vectorF21 = k2
contintegraldisplay
2
contintegraldisplay
1
J1dvectorl1×(J2d
vectorl2×vectorR
R3 )
vectorR =vectorr1?vectorr2
=
contintegraldisplay
2
contintegraldisplay
1
k2J1J2
R3 [(
vectorR·dvectorl1)dvectorl2?(dvectorl1·dvectorl2)vectorR]
a62a54a3J1dvectorl1a233 J2dvectorl2a27a138a94a229a143:1?2; vectorRvectorR
contintegraldisplay
2
contintegraldisplay
1
k2J1J2
R3 (
vectorR·dvectorl1)dvectorl2 =k2
contintegraldisplay
1
J2dvectorl2
contintegraldisplay
1
J1dvectorl1·?1 1R
=?k2
integraldisplay
dV2 vectorj2
integraldisplay
dV1(?1 1R)·vectorj1
=?k2
integraldisplay
dV2 vectorj2
integraldisplay
dV1braceleftbig[?1·
vectorj1
R]?
1
R?1·
vectorj1bracerightbig = 0
a52a220a163a180
107/384
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a83a75
a49a152a217a49a110a33a214a191a138a146(b),a166a121a94a243a52a102(a62a54a23a161a200S → 0,a62
a54J→∞,JSa27a189)a23a41a27a165a254a179a143,
vectorA = μ0
4pi
vectorm×vectorr
r3 +a126a234
vectorS = 1
2
contintegraldisplay
vectora×dvectorl vectorm = JvectorS
(1)vector
A = k2
contintegraldisplay
dvectorl J|vectorr?vectora| = k2
integraldisplay
dvectorS×?a J|vectorr?vectora| = k2
integraldisplay
dvectorS×Jvectorrr3
vectorA = k2
contintegraldisplay
dvectorl J|vectorr?vectora| = k2
contintegraldisplay
dvectorl J√r2?2vectorr·vectora = k2
contintegraldisplay
dvectorl Jrradicalbig1?2vectorr·vectora/r
= k2
contintegraldisplay
dvectorlJr[1 +vectorr·vectorar ] = k2Jr2
contintegraldisplay
dvectorlvectorr·vectora = k2J2r2
contintegraldisplay
[dvectorlvectorr·vectora?vectoradvectorl·vectorr]
= k2J2r2
contintegraldisplay
(vectora×dvectorl)×vectorr = k2Jr2 vectorS×vectorr = μ04pi vectorm×vectorrr3
contintegraldisplay
J[dvectorlvectorr·vectora +vectora dvectorl·vectorr] =
integraldisplay
dτa[vectorj(vectora) vectorr·vectora +vectoravectorj(vectora)·vectorr] =
integraldisplay
dτa?a·[vectorj(vectora)vectoravectorr·vectora]
= 0
108/384
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a141a152a53a189a110a57a65a65a94a94,a183a62a175a75a141a152a53a189a110
a152a135a171a141τ,a62a46S0,S1,S2,···,S0a180a129a9a157a140a161,S1,S2,···a180a83
a220a152a201a157a140a161a34
a38a37
a39a36
a105
a102
S0
S2S1τa83a48a159a180a169a171a235a89a27(epsilon1
i(vectorr)a180a235a89a188a234,a133epsilon1i >0),
τa83ρf,σfa169a217a174a127,a133a101a15a94a135a131a152a164a225:
(1),φvextendsinglevextendsingleS
i
a174a127
(2),?φ?nvextendsinglevextendsingleS
i
=?EnvextendsinglevextendsingleS
i
a174a127
a101S0a51a195a161a15,a75a230a94a195a161a15a103a44a62a94a135a181
φvextendsinglevextendsingler→∞→1r a189 Evextendsinglevextendsingler→∞→ 1r2
a75,a100a183a62a124a135a169a169a144a144a167a57a62a94a135a251a189a189a27a27τa165a27a27a62a62a124a141a152a34
109/384
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a141a152a53a189a110a57a65a65a94a94,a183a62a175a75a141a152a53a189a110
a38a37
a39a36S
02S01
21
Sprime?·[epsilon11?φ
prime
1] =?ρf1?·[epsilon12?φ
prime
2] =?ρf2
·[epsilon11?φprimeprime1] =?ρf1?·[epsilon12?φprimeprime2] =?ρf2
φprime1vextendsinglevextendsingleSprime = φprime2vextendsinglevextendsingleSprime (epsilon11?φ
prime
1
n?epsilon12
φprime2
n )
vextendsinglevextendsingle
Sprime = σf
φprimeprime1vextendsinglevextendsingleSprime = φprimeprime2vextendsinglevextendsingleSprime (epsilon11?φ
primeprime
1
n?epsilon12
φprimeprime2
n )
vextendsinglevextendsingle
Sprime = σf
φprime1vextendsinglevextendsingleS
01
= φprimeprime1vextendsinglevextendsingleS
01
= a174a127 a189nφprime1vextendsinglevextendsingleS
01
=nφprimeprime1vextendsinglevextendsingleS
01
= a174a127
φprime2vextendsinglevextendsingleS
02
= φprimeprime2vextendsinglevextendsingleS
02
= a174a127 a189nφprime2
vextendsinglevextendsingle
S02 =
nφ
primeprime
2
vextendsinglevextendsingle
S02 = a174a127
·[epsilon11?(φprime1?φprimeprime1)] = 0 (φprime1?φprimeprime1)vextendsinglevextendsingleSprime = (φprime2?φprimeprime2)vextendsinglevextendsingleSprime
·[epsilon12?(φprime2?φprimeprime2)] = 0 epsilon11n(φprime1?φprimeprime1)vextendsinglevextendsingleSprime = epsilon12n(φprime2?φprimeprime2)vextendsinglevextendsingleSprime
(φprime1?φprimeprime1)vextendsinglevextendsingleS
01
= 0 a189n(φprime1?φprimeprime1)
vextendsinglevextendsingle
S01 = 0
(φprime2?φprimeprime2)vextendsinglevextendsingleS
02
= 0 a189n(φprime2?φprimeprime2)vextendsinglevextendsingleS
02
= 0
110/384
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a141a152a53a189a110a57a65a65a94a94,a183a62a175a75a141a152a53a189a110integraldisplay
τ1
dτepsilon11|vectorEprime1?vectorEprimeprime1|2 +
integraldisplay
τ2
dτepsilon12|vectorEprime2?vectorEprimeprime2|2=? 0 → vectorEprime1 = vectorEprimeprime1 vectorEprime2 = vectorEprimeprime2
=
integraldisplay
τ1
dτepsilon11|?(φprime1?φprimeprime1)|2 +
integraldisplay
τ2
dτepsilon12|?(φprime2?φprimeprime2)|2
=
integraldisplay
τ1
dτepsilon11?(φprime1?φprimeprime1)·?(φprime1?φprimeprime1) +
integraldisplay
τ2
dτepsilon12?(φprime2?φprimeprime2)·?(φprime2?φprimeprime2)
=
integraldisplay
τ1
dτ{?·[epsilon11(φprime1?φprimeprime1)?(φprime1?φprimeprime1)]?(φprime1?φprimeprime1)?·[epsilon11?(φprime1?φprimeprime1)]}
+
integraldisplay
τ2
dτ{?·[epsilon12(φprime2?φprimeprime2)?(φprime2?φprimeprime2)]?(φprime2?φprimeprime2)?·[epsilon12?(φprime2?φprimeprime2)]}
=
contintegraldisplay
τ1
dvectorS1·[epsilon11(φprime1?φprimeprime1)?(φprime1?φprimeprime1)] +
contintegraldisplay
τ2
dvectorS2·[epsilon12(φprime2?φprimeprime2)?(φprime2?φprimeprime2)]
=
integraldisplay
S01
dvectorS1·[epsilon11(φprime1?φprimeprime1)?(φprime1?φprimeprime1)] +
integraldisplay
S02
dvectorS2·[epsilon12(φprime2?φprimeprime2)?(φprime2?φprimeprime2)]
+
integraldisplay
Sprime
dvectorS·[epsilon11(φprime1?φprimeprime1)?(φprime1?φprimeprime1)]?
integraldisplay
Sprime
dvectorS·[epsilon12(φprime2?φprimeprime2)?(φprime2?φprimeprime2)]
111/384
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a141a152a53a189a110a57a65a65a94a94,a107a19a78a127a51a158a27a141a152a53a189a110
a19a78a83a220a107a233a245a140a177a103a100a185a196a27a62a102.a242a19a78a152a92a174a127a27a9a62a124
a165,a19a78a254a62a214a201a62a124a229a138a94a135a173a35a169a217,a134a20a167a23a41a27a97a65a62a124a218
a9a62a124a140a2a152a24,a144a149a131a135,a166a19a78a83a111 a62a124a143a34a167a103a100a62a102a66a216a50
a189a149a36a196a136a20a178a239a71a21,a19a78a57a217a51a9a62a124a101a136a20a178a239a27a28a28a91a91 a249a158
a19a78a180a31a179a78 φ1?φ2 =
integraldisplay 2
1
dvectorl·vectorE a19a78a76a161a180a31a179a161
a83a220a220a216a216a145a62,a62a214a169a217a51a76a161 vectorEa83 = 0→ρf =?·vectorD =?·(epsilon1vectorE) = 0
a19a78a9a76a161a181 vectorEt = 0 vectorEn = σfepsilon1vectorn
vectorn×(vectorE2?vectorE1) = 0 vectorn·(epsilon12vectorE2?epsilon11vectorE1) = σf a38a37
a39a36
a106 a101
S02S01
21
Sprime
a134a254a33a211a24a27a171a141,a2S1,S2,...a165a107a152a10a180a19a78,a75a101τa169a171a235a89,
epsilon1i >0,τa83ρf,σfa174a127,a133a233a233a19a19a78a62a46,a101a15a94a135a131a152 a164a225:
(1).a19a78a254a62a179a174a127 (2).a19a78a254a111a62a214a174a127
a75a41a141a152 a34
112/384
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a141a152a53a189a110a57a65a65a94a94,a107a19a78a127a51a158a27a141a152a53a189a110
a192a144a107a152a135a9a62a46S0a218a152a135a83a62a46S1,a83a62a46a157a140a27a180a19a78
a38a37
a39a36
a109
S0
S1?·[epsilon1?φprime] =?ρf
contintegraldisplay
S1
dSepsilon1?φ
prime
n =?Q a189 φ
primevextendsinglevextendsingle
S1 = a126a234
·[epsilon1?φprimeprime] =?ρf
contintegraldisplay
S1
dSepsilon1?φ
primeprime
n =?Q a189 φ
primeprimevextendsinglevextendsingle
S1 = a126a234
a217a165S0a254a27a62a94a135a211a99a161a152a152a24a24.
·[epsilon1?(φprime?φprimeprime)] = 0
contintegraldisplay
S1
dSepsilon1n(φprime?φprimeprime) = 0 a189 (φprime?φprimeprime)vextendsinglevextendsingleS
1
= 0
integraldisplay
dτepsilon1|vectorEprime?vectorEprimeprime|2 =
integraldisplay
dτepsilon1?(φprime?φprimeprime)·?(φprime?φprimeprime)
=
integraldisplay
dτ{?[epsilon1(φprime?φprimeprime)?(φprime?φprimeprime)]?(φprime?φprimeprime)?·[epsilon1?(φprime?φprimeprime)]}
=
contintegraldisplay
S0
dvectorS·epsilon1(φprime?φprimeprime)?(φprime?φprimeprime) +
contintegraldisplay
S1
dvectorS·epsilon1(φprime?φprimeprime)?(φprime?φprimeprime)
= epsilon1(φprime?φprimeprime)vextendsinglevextendsingleS
1
contintegraldisplay
S1
dvectorS·epsilon1?(φprime?φprimeprime) = 0 → vectorEprime = vectorEprimeprimea41a141a152!
113/384
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a141a152a53a189a110a57a65a65a94a94,a107a19a78a127a51a158a27a141a152a53a189a110
a126,a152a135a152a37a19a78a110,a19a78a254a140a177a145a143a140a177a216a145a62a214
a101a110a83a195a62a214:
a19a78a83a76a161a254a62a207a254a131a218a1430,a110a83?2φ = 0
→a110a83φ =a126a234,vectorE = 0,σf = epsilon1En,a110a83a76a161a195a62a214,a62a214a144a85a169
a217a51a9a76a161(a101a19a78a254a164a145a27a111a62a254Qa189a9a62a124vectorEa85a67,a110a83vectorE a69a145
a1770,a83a76a161a69a145a177a195a62a214,a2φa27a253a233a234a138a85a67)a100a121a121a150a150a23a183a62a182a45a34
a101a110a83a107a62a214,a110a83a111a62a254a143q:
a19a78a9a76a161a62a207a254a131a218a1431epsilon1(Q + q),qa143a19a78a110a254a27a111a62a214.a100
a252a135a94a135a216a145a110a83a62a214a169a217a85a67a13a85a67,a207a13 a110a9a62a124a216a145a110
a83a62a214a169a217a85a67a13a85a67(a2a110a83a62a124a135a67),a65a79a47,a8a9a76a161a180a165
a158,a110a9a62a124a61a180a152a135a100a160a117a165a165a37a27a62a214 q + Qa23a41a27a62a124a34
114/384
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115/384
triangleleftsldtriangleleftsld
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triangleleftsld
trianglerightsld
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a141a152a53a189a110a57a65a65a94a94,a107a19a78a127a51a158a27a141a152a53a189a110
a17a225a19a78a23a41a27a27a62a62a179a169a217a20a39a117a167a167a164a164a145a27a111a62a254a34
a23a19a78τ,a164a145a111a62a254Q,a19a78a9a62a179a169a217φ(vectorr).
·(epsilon1?φ) = 0
contintegraldisplay
τ
dvectorS·epsilon1?φ = Q a195a161a15a103a44a62a94a135
a121,a8Qa134a143λQa0,a62a179a169a217a217a67a67a143λφ(vectorr).
a144a167a62a94a135a247a118a141a152a53a189a110,a144a73a121a8Qa134a143λQa0,λφ(vectorr)a69a180a41.
·(epsilon1?λφ) = 0
contintegraldisplay
τ
dvectorS·epsilon1λ?φ = λQ a195a161a15a103a44a62a94a135
λφa180a41.
116/384
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a141a152a53a189a110a57a65a65a94a94,a107a19a78a127a51a158a27a141a152a53a189a110
a19a78a124a164a27a78a88,a49ia135a19a78a254a27a111a62a214qi,a19a78a9a140a177a107a48a159,a2
a195a103a100a62a214,a166a121a152a109a63a191a191a152a152a58vectorra63a62a179a134qia27a39a88a180a130a53a224a103a27.
φ(vectorr,q1,q2,...,qn) =
nsummationdisplay
i=1
pi(vectorr)qi
pi(vectorr)a134qia195a39.
a23a49ia135a19a78a254a27a27a62a62a179a143φi,a75φi =
nsummationdisplay
j=1
pijqj,a135a41a26qi =
nsummationdisplay
j=1
cijφj,
pij:a62a179a88a234,a180a19a78ja145a145a252a252a160a62a254,a217a167a19a78a122a135a19a78a254a111a62a214a1430a158,
a49ia135a19a78a27a27a62a62a179a138.cij:a62a78a88a234,a189a194 cj(vectorr) =
nsummationdisplay
i=1
pi(vectorr)cij a75,
φ(vectorr) =
nsummationdisplay
i=1
pi(vectorr)qi =
nsummationdisplay
i,j=1
pi(vectorr)cijφj =
nsummationdisplay
j=1
cj(vectorr)φj
cj(vectorr)a180a19a78ja143a252a160a62a179,a217a167a19a78a1430a62a179a179a158a158,a152a109a27a27a62a62a179a169a217.
117/384
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a141a152a53a189a110a57a65a65a94a94,a107a19a78a127a51a158a27a141a152a53a189a110
a233a254a33a130a53a48a159,a62a179a169a217a247a118?2φ =?ρf/epsilon1 =?ρf/(epsilon10epsilon1r)
a233a39a253a152a165a27a27a62a62a179a169a217a144a167?2φ =?ρ/epsilon10
ρ = ρf +ρprime = ρfepsilon1
r
a61 ρprime = (1epsilon1
r
1)ρf
a51a254a33a130a53a48a159a159a165a165,a52a122a62a214a209a121a51a103a100a62a214a63,ρprime = (1epsilon1
r
1)ρf,a51a195
a103a100a62a214a63a143a195a52a122a62a214,a52a122a166a103a100a62a214a63a27a111a62a214a254a67a143a6a53
a271epsilon1
r
(a182a45a8a65),a101a180a195a161a140a254a33a130a53a48a159,a62a179a169a217a247a118
2φ =?ρfepsilon1 =? ρfepsilon1
0epsilon1r
φvextendsinglevextendsingler→∞→ 1r
a233a39a253a152a165a27a27a62a62a179a169a217a144a167
2φf =?ρfepsilon1
0
φfvextendsinglevextendsingler→∞→1r
φ = φfepsilon1
r
vectorE = vectorEf
epsilon1r a180a41
118/384
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a46a202a46a100a144a167,a169a108a67a254a123
a254a33a48a159,a51a152a109a171a141τa165,a118a107a103a100a62a214a158a27a27a62a62a179a169a217a247a118a144a167:
2φ = 0
a107a103a100a62a214a156a47a101,a62a179a169a217a53a103a252a220a169,a78a62a214ρ = ρf/epsilon1r a27a0
a122; τa76a161a161a161a161a62a214a57τa171a141a9a27a27a62a62a214a27a0a122 (a80a143φprime).
a78a62a214a169a217ρf/epsilon1ra152a132a180a174a127a27,a167a233a233a62a62a179a169a217a27a0a122a143,14piepsilon1
0
integraltext dτprime
τ
ρf
epsilon1rR,
a242a217a169a209,φ = 14piepsilon1 integraltext dτprimeρfR +φprime,φprimea51τa165a247a118a46a202a46a100a144a167.
2φprime =?2[φ? 14piepsilon1
integraldisplay
dτprimeρfR] =?ρfepsilon1? 14piepsilon1
integraldisplay
dτprimeρf?2 1R= 0
a161a62a214a233a233a62a62a179a169a217a27a0a122a247a118a224a103a46a202a46a100a144a167.
a252a171a156a185,a175a75a209a122a143a166a41a224a103a46a202a46a100a144a167?2φ = 0
119/384
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a46a202a46a100a144a167,a169a108a67a254a123,a165a139a73
a0
a0a0a9
a45
a54
a0
a0a0a18
a64
a64a64a82
z
θ
ψ
vectorr
x
ya165a139a73a27a110a135a139a73a73a67a67a254a143r,θ,ψ,
a169a108a67a234a123a137a209a224a103a46a202a46a100a144a167a207a41a27a152a132a47a170a143:
φ(r,θ,ψ) =
summationdisplay
n,m
(an,mrn + bn,mrn+1)Pmn (cosθ)cosmψ
+
summationdisplay
n,m
(cn,mrn + dn,mrn+1)Pmn (cosθ)sinmψ
a101a164a63a216a27a175a75a39a117za182a233a161,a75φa134ψa195a39:
φ(r,θ) =
∞summationdisplay
n=0
(anrn + bnrn+1)Pn(cosθ)
Pmn (cosθ),Pn(cosθ)a143a53a220a86a52a25a188a234a218a86a52a25a188a234.
a88a234a100a62a94a135a251a189.
120/384
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a46a202a46a100a144a167,a169a108a67a254a123,a165a139a73
1
|vectorr?vectorrprime| = (r
2 + rprime2?2rrprimecosθ)?12 =
∞summationdisplay
n=0
rn<
rn+1> Pn(cosθ)
integraldisplay 1
1
dxPm(x)Pn(x) = 2δmn2n + 1
∞summationdisplay
n=0
Pn(x)Pn(xprime)2n + 12 = δ(x?xprime)
Pn(x) = 12nn!( ddx)n(x2?1)n
P0(x) = 1 P1(x) = x P2(x) = 12(3x2?1)
121/384
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a46a202a46a100a144a167,a169a108a67a254a123,a206a139a73
a0
a0a0a9
a45
a54
a64
a64a64a82
z
θ
vectorrx
ya139a73a27a110a135a139a73a73a67a67a254a143r,θ,z,a169a108a67a234a123a137a209
a224a103a46a202a46a100a144a167a134za195a39a207a41a27a152a132a47a170a143:
φ(r,θ) = A0 lnr + B0θ+ C0 +
∞summationdisplay
n=1
(Anrn + Bnr?n)cos(nθ+αn)
ln|vectorr?vectorrprime|= ln(r2 + rprime2?2rrprimecosθ)12 =?
∞summationdisplay
n=1
parenleftbigr<
r>
parenrightbign cosnθ+ lnr
>
integraldisplay 2pi
0
dθcosnθcosmθ = piδmn
∞summationdisplay
n=0
cosnθcosnθprime = piδ(θ?θprime)
122/384
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a46a202a46a100a144a167,a169a108a67a254a123,a165a139a73a126
a26a25
a27a24a45a45
a45
epsilon1
vectorE0?2φ
1 = 0 φ1
vextendsinglevextendsingle
r=a = φ2
vextendsinglevextendsingle
r=a φ1
vextendsinglevextendsingle
r=0a107a129
2φ2 = 0 epsilon1r?φ1?r vextendsinglevextendsingler=a =?φ2?r vextendsinglevextendsingler=a φ2vextendsinglevextendsingler→∞→?E0rP1(cosθ) + C
φ1 =
∞summationdisplay
n=0
AnrnPn φ2 =
∞summationdisplay
n=0
Bn
rn+1Pn + B
primeP0 + BprimeprimerP1
A0 = B0a + Bprime; epsilon1rA1 =?2B1a3 + Bprimeprime; A1a = B1a2 + Bprimeprimea;?B0a = 0
Anan = Bnan+1; epsilon1rnAnan?1 =?(n + 1) Bnan+2 n>1; Bprimeprime =?E0; Bprime = C
A0 = C; A1 =?3E0epsilon1
r + 2; B1 = epsilon1r?1epsilon1
r + 2
E0a3; An = Bn = 0 n>0
φ1 = C?3
vectorE0·vectorr
epsilon1r + 2 φ2 = C +
epsilon1r?1
epsilon1r + 2a
3vectorE0·vectorr
r3?
vectorE0·vectorr
a8epsilon1r = 1a158,φ1 = φ2,a8epsilon1r→∞a158,a48a159a67a164a19a78.
123/384
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a46a202a46a100a144a167,a169a108a67a254a123,a206a139a73a126
φ = φ0 +φ1
φ0=?2k1ηln|vectorr?vectorr0|+C=2k1η
∞summationdisplay
n=1
1
n
parenleftbigr<
r>
parenrightbigncosnθ?2k
1ηlnr>+C
2φ1 = 0 →φ1(r,θ) = A0 lnr + B0 +
∞summationdisplay
n=1
Bnr?n cos(nθ+αn)
φ(r,θ) = φ(r,?θ)→αn = 0; φ(a,θ) = a126a234;?a
integraldisplay 2pi
0
dθ?φ?rvextendsinglevextendsingler=a= η
prime
epsilon10
C+2k1η
∞summationdisplay
n=1
1
n(
a
r0)
ncosnθ?2k1ηlnr0+A0lna+B0+
∞summationdisplay
n=1
Bna?ncosnθ=a126a234
integraldisplay 2pi
0
dθ[2k1η
∞summationdisplay
n=1
( ar
0
)n?1 1r
0
cosnθ+ A0a +
∞summationdisplay
n=1
nBna?n?1 cosnθ] =?η
prime
aepsilon10
2k1η( ar
0
)n1n + Bna?n = 0; Bn =?2k1η( ar
0
)na
n
n ; 2pi
A0
a =?
ηprime
aepsilon10
124/384
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a46a202a46a100a144a167,a169a108a67a254a123,a206a139a73a126
φ(r,θ) = φ0(r,θ) +φ1(r,θ)
φ0(r,θ) = 2k1η
∞summationdisplay
n=1
1
n
parenleftbigr<
r>
parenrightbigcosnθ+ lnr
> + C
=?2k1ηln|vectorr?vectorr0|+ C
φ1(r,θ) =?2k1ηprimelnr?2k1η
∞summationdisplay
n=1
( a
2
r0r)
n1
n cosnθ
= 2k1ηln|a
2
r20vectorr0?vectorr|?2k1(η
prime +η)lnr
a11a206a19a130a254a27a62a214a233a206a9a62a179a27a0a122a140a94a160a117a186a150a58a151a221?η,a218
a160a117a206a37,a151a221ηprime +ηa27a134a206a206a182a182a178a49a27a130a62a214a23a41a27a179a147a79.
a38a37
a39a36
a114vectorr0
η
a114 a114
a2
r20 vectorr0
η
η + ηprime
125/384
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a46a202a46a100a144a167,a169a108a67a254a123,a165a139a73a126
za182a254,a6a58a58a20a20a9a130a44a152a58a58a27a27a165a254vectorc,c =√a2 + b2
a167a20za182a254a124a58a229a108:R =|zvectorez?vectorc| 1R=
∞summationdisplay
n=0
rn<
rn+1> Pn(cosγ)
γ,a9a130a233za182a27a220a14 r< = min(z,c) r> = max(z,c)
φ(z,0) =
contintegraldisplay
a130
dlk1
q
2pia
R =
k1q
R = k1q
∞summationdisplay
n=0
rn<
rn+1> Pn(cosγ)
a152a132a154a9a130a63,?2φ = 0 a6a58a58a62a62a179a107a129; a195a161a15a103a44a62a94a135
φ(r,θ) =
∞summationdisplay
n=0
Bn
rn+1Pn(cosθ) r>c
∞summationdisplay
n=0
AnrnPn(cosθ) r<c
za182a254a62a179a140a189a209a88a234,
Bn = k1qcnPn(cosγ)
An = k1q 1cn+1Pn(cosγ)
b = 0? cosγ = 0
φ(r,θ) = k1q
∞summationdisplay
n=0
rn<
rn+1> Pn(cosγ)Pn(cosθ) P2n+1(0) = 0,P2n(0) = (?1)
n
2 1·3·5···(2n?1)2·4·6···2n
126/384
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a46a202a46a100a144a167,a169a108a67a254a123,a165a139a73a126
a152a132a154a9a130a63,φ(r,θ) =
contintegraldisplay
a130
dlk1
q
2pia
R a134a26a200a169a79a142
vectorr = rcosθvectorez + rsinθcosφvectorex + rsinθsinφvectorey
vectora = acosφ0vectorex + asinφ0vectorey dl = adφ0
R2 = r2 + a2?2vectorr·vectora = r2 + a2?2rasinθcos(φ0?φ)
φ(r,θ) = k1qa2pia
integraldisplay φ+2pi
φ
dφ0 1radicalbigr2 + a2?2rasinθcos(φ
0?φ)
φ0?φ = 2α+pi cos(φ0?φ) =?cos(2α) = 2sin2α?1 dφ0 = 2dα
φ(r,θ) = k1qpi
integraldisplay pi/2
pi/2
dα 1radicalbigr
2 + a2?2rasinθ(2sin2α?1)
φ(r,θ) = 2k1qpi√r2 + a2 + 2rasinθ
integraldisplay pi/2
0
dα 1radicalBig
1? 4rasinθr2+a2+2rasinθ sin2α
= 2k1qpi√r2+a2+2rasinθK(
radicalbigg 4rasinθ
r2+a2+2rasinθ) a49a152a97a17a28a253a11a200a169
127/384
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a186a148a123,a196a29a103a142
a253a152a165,?2φ =?ρepsilon1
0
a83a92a152a135a100a160a117vectorr0a63a62a254a143qa27a27a58a58a58a62a62a214a62a179φprime = φ+ q4piepsilon1
0|vectorr?vectorr0|
2φprime =?2φ? qepsilon1
0
δ(vectorr?vectorr0)=?ρepsilon1
0
qepsilon1
0
δ(vectorr?vectorr0)
a58a58a62a62a214a160a117a166a41a171,φprimea134φa247a118a216a211a27a144a167
a58a58a62a62a214a216a51a166a41a171,φprimea134φa247a118a131a211a27a144a167
a154a166a41a171a92a58a58a62a62a214a216a75a143a124a144a167 !
a51a154a166a41a171a92a62a214a53a78a33a41a27a49a143,a166a131a216a61a247a118a124a144a167,a132
a247a118a62a94a135
a51a154a41a171a92a27a62a214a23a150a62a214a34a152a132a216a180a162a83a127a51a27a62a214,a2
a167a51a166a41a171a23a41a27 a62a179a134a162a83a51a62a46a254a218a154a166a41a171a127a51a27
a62a214a51a166a41a171a23a41a27a27a62a62a179a131a211!
a186a150a123a180a124a94a186a150a62a214a27a62a179a53a8a69a224a103a46a202a46a100a144a167a41
a207a41,a44a0a50a124a94a62a94a135a189a209a207a41a165a27a63a191a126a234
128/384
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a186a148a123,a1261,2,3
a1261,a9a76a161a143a165a27a27a19a19a78a138,a110a83a62a254Q,a138a254a62a254 q,
φvextendsinglevextendsinglea165a161a31a179,
contintegraldisplay
a165a76a161
dvectorS·?φ =?1epsilon1
0
(q + Q)
2φ = 0 → φ = q + Q4piepsilon1
0r
a150a62a214a160a117a165a37a27a27a62a62a254a143q + Q !
a1262,a195a161a140a26a47a47a19a19a78a134a9a107a152a58a58a62a62a214q,a114x0
q
a114
q
x0
a150a62a214a160a117a19a78a134a83a186a150a233a161a58,a62a254a143?q !a85a216a85a216a26a47a186
a1263,a140a187a143r0a27a26a47a47a19a19a78a165a9a229a165a37da63a107a152a58a58a62a62a214q,
φ(vectorr) = 14piepsilon1
0
bracketleftbig q
|vectorr?vectord|?
qr0
d
|vectorr? r20d2vectord|
bracketrightbig
a150a62a214a51qa134a165a37a233a130a254,a229a165a37r20da63,a62a254a143qprime =?qr0d !
a38a37
a39a36
a114
vectord
qqprime
a114
r20
d2
vectord
129/384
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a186a148a123,a1263
a51a165a161a254 a181
φ(vectorr)vextendsinglevextendsingler=r
0
= 14piepsilon1
0
bracketleftbig q
|vectorr?vectord|?
qr0
d
|vectorr? r20d2vectord|
bracketrightbig
r=r0
= 14piepsilon1
0
bracketleftbig qradicalbig
r20 + d2?2r0dcosθ?
qr0
dradicalBig
r20 + r40d2?2r0r20d cosθ
bracketrightbig
= 14piepsilon1
0
bracketleftbig qradicalbig
r20 + d2?2r0dcosθ?
qradicalbig
d2 + r20?2r0dcosθ
bracketrightbig
= 0
a216a26a47a78a111a141a186
a137a189a189a19a19a78a111a62a214Q?a51a165a37a92a148a62a214Q+qr0d
a137a189a189a19a19a78a62a179φ0?a51a165a37a92a148a62a2144piepsilon10r0φ0 +qr0d
130/384
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a186a148a123,a1264,5
a1264,a114q
φ0
a114?q
φ1
φ1 + φ0
a114?q
φ2
φ0 + φ2
a114q
φ3
φ3 + φ2
a114q
φ4
φ1 + φ4
a114?q
φ5
φ5 + φ4
a114?q
φ6
φ3 + φ6··· ···
φ = φ0 +φ1 +φ2 +φ3 +φ4 +φ5 +φ6 +···
a1265,?2φ1 =? q0epsilon1
r1epsilon10
δ(x?x0)δ(y)δ(z)?2φ2 = 0
φ1vextendsinglevextendsinglex=0 = φ2vextendsinglevextendsinglex=0 epsilon1r1?φ1?xvextendsinglevextendsinglex=0 = epsilon1r2?φ2?xvextendsinglevextendsinglex=0
a114 a114x0?x0
epsilon11epsilon12
q2 q0epsilon1r1,q3
φ1 = k1q0epsilon1
r1r1
+ k1q2r
2
φ2 = k1q0epsilon1
r1r1
+ k1q3r
3
vectorr2 =vectorr + x0vectorex vectorr3 =vectorr1 =vectorr?x0vectorex q2 = q3 = epsilon1r1?epsilon1r2epsilon1
r1(epsilon1r1 +epsilon1r2)
q0
131/384
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a186a148a123,a152a132a156a185
φ = φa83 +φa9 +φa76a161= φa83 +φa150
φa150 = φa9 +φa76a161
a51a107a48a159a27a171a141,a103a100a62a214a177a140a135a78a88a52a122a62a214,a166a111a62a254a67
a143a6a53a62a254a27 1/epsilon1ra21a34
a186a150a62a214a27a23a152a73a135a179a178a8,a152a135a216a49,a210a23a245a135,a134a20a85a247
a118a62a94a135a143a142a34
132/384
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a130a21a188a234,a152a132a156a185
a137a189Va83a62a214a169a217ρa218
Va62a46Sa254a136a58a58a27a27a27a62a62a179φS
a189a62a124a123a149a169a254?φ?n
vextendsinglevextendsingle
S
a189a195a46a152a109a158a27a195a161a15a103a44a62a94a135
a166Va83a136a58a58a27a27a27a62a62a179a138?
2φ(vectorr) =?1epsilon1
0
ρ(vectorr)
a62a94a135
a233a65a27a27a58a58a58a62a62a214a62a138a175a75=======?
2G(vectorr,vectorrprime) =?1epsilon1
0
δ(vectorr?vectorrprime)
a62a94a135
a242a154a219a103a62a214a169a217a27a27a62a62a179a166a41a122a143a58a58a62a62a214a27a27a62a62a179a166a41a175a75
a58a58a62a62a214a27a27a62a62a179a140a94a186a148a123a31a144a144a123a123a166a41
133/384
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a130a21a188a234,a152a132a156a185
φ(vectorr)
=
integraldisplay
V
dτprimeφ(vectorrprime)δ(vectorrprime?vectorr) =?epsilon10
integraldisplay
V
dτprimeφ(vectorrprime)?prime2G(vectorrprime,vectorr)
=?epsilon10
integraldisplay
V
dτprimebraceleftbig?prime·[φ(vectorrprime)?primeG(vectorrprime,vectorr)]primeφ(vectorrprime)·?primeG(vectorrprime,vectorr)bracerightbig
= epsilon10
integraldisplay
V
dτprimebraceleftbig?prime·bracketleftbig?φ(vectorrprime)?primeG(vectorrprime,vectorr) + [?primeφ(vectorrprime)]G(vectorrprime,vectorr)bracketrightbig
prime2φ(vectorrprime)G(vectorrprime,vectorr)bracerightbig
= epsilon10
integraldisplay
V
dτprimebraceleftbig?prime·bracketleftbig?φ(vectorrprime)?primeG(vectorrprime,vectorr) + [?primeφ(vectorrprime)]G(vectorrprime,vectorr)bracketrightbig
+1epsilon1
0
ρ(vectorrprime)G(vectorrprime,vectorr)bracerightbig
=
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr) +epsilon10
contintegraldisplay
S
dvectorSprime·[G(vectorrprime,vectorr)?primeφ(vectorrprime)?φ(vectorrprime)?primeG(vectorrprime,vectorr)]
=
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr) +epsilon10
contintegraldisplay
S
dSprime[G(vectorrprime,vectorr)?φ(vectorr
prime)
nprime?φ(vectorr
prime)?G(vectorr
prime,vectorr)
nprime ]
134/384
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a130a21a188a234,a62a138a175a75
φ(vectorr) =
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr) +epsilon10
contintegraldisplay
S
dSprime[G(vectorrprime,vectorr)?φ(vectorr
prime)
nprime?φ(vectorr
prime)?G(vectorr
prime,vectorr)
nprime ]
a49a152a97a62a138a175a75,φvextendsinglevextendsinglevectorrprime?Sa174a127,a18,G(vectorrprime,vectorr)vextendsinglevextendsinglevectorrprime?S = 0
φ(vectorr) =
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr)?epsilon10
contintegraldisplay
S
dSprimeφ(vectorrprime)?G(vectorr
prime,vectorr)
nprime
a49a19a97a62a138a175a75,?φ?nprimevextendsinglevextendsingleSa174a127,a18,?G(vectorr
prime,vectorr)
nprime
vextendsinglevextendsingle
vectorrprime?S = c 1 =?epsilon10cSV
φ(vectorr) =
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr) +epsilon10
contintegraldisplay
S
dSprimeG(vectorrprime,vectorr)?φ(vectorr
prime)
nprime
a49a152,a19a97a183a218a62a138a175a75,S = S1 + S2,φvextendsinglevextendsingleS
1
,?φ?nprimevextendsinglevextendsingleS
2
a174a127,
G(vectorrprime,vectorr)vextendsinglevextendsinglevectorrprime?S
1
= 0,?G(vectorr
prime,vectorr)
nprime
vextendsinglevextendsingle
vectorrprime?S2 = c
a195a46a152a109(Va143a143a195a195a161a140),φvextendsinglevextendsingle∞a247a118a195a161a15a103a44a62a94a135
G(vectorrprime,vectorr)vextendsinglevextendsinglevectorrprime?∞a247a118a195a161a15a103a44a62a94a135 φ(vectorr) =
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr)
135/384
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a130a21a188a234,a49a19a97a62a138a175a75
φ(vectorr) =
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr) +epsilon10
contintegraldisplay
S
dSprime[G(vectorrprime,vectorr)?φ(vectorr
prime)
nprime?φ(vectorr
prime)?G(vectorr
prime,vectorr)
nprime ]
a49a19a97a62a138a175a75:?φ?nprimevextendsinglevextendsingleSa174a127,a18,?G(vectorr
prime,vectorr)
nprime
vextendsinglevextendsingle
vectorrprime?S = 0
1 =
integraldisplay
V
dτprimeδ(vectorrprime?vectorr) =?epsilon10
integraldisplay
V
dτprime?prime2G(vectorrprime,vectorr)
=?epsilon10
integraldisplay
V
dτprime?prime·[?primeG(vectorrprime,vectorr)] =?epsilon10
integraldisplay
V
dvectorSprime·[?primeG(vectorrprime,vectorr)]
=?epsilon10
integraldisplay
V
dSprimenprimeG(vectorrprime,vectorr)
G(vectorrprime,vectorr)
nprime
vextendsinglevextendsingle
vectorrprime?S = c 1 =?epsilon10cSV
φ(vectorr) =
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr) +epsilon10
contintegraldisplay
S
dSprimeG(vectorrprime,vectorr)?φ(vectorr
prime)
nprime
136/384
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a130a21a188a234,a126a195a46a152a109a27a130a21a188a234:
G(vectorr,vectorrprime) = 14piepsilon1
0
1
|vectorr?vectorrprime| φ(vectorr) =
1
4piepsilon10
integraldisplay
∞
dτprime ρ(vectorr
prime)
|vectorr?vectorrprime|
a254a140a152a109(z > 0)a49a152a97a62a138a175a75a27a130a21a188a234:
G(vectorr,vectorrprime) = 14piepsilon1
0
parenleftbig 1
|vectorr?vectorrprime|?
1
|vectorr +vectorrprime|
parenrightbig
φ(vectorr) = 14piepsilon1
0
integraldisplay
∞
dτprimeρ(vectorrprime)parenleftbig 1|vectorr?vectorrprime|? 1|vectorr +vectorrprime|parenrightbig
14pi
integraldisplay
z=0
dSprimeφ(vectorrprime)zprimeparenleftbig 1|vectorr?vectorrprime|? 1|vectorr +vectorrprime|parenrightbig
a165a9a152a109(r>r0)a49a152a97a62a138a175a75a27a130a21a188a234:
G(vectorr,vectorrprime) = 14piepsilon1
0
parenleftbig 1
|vectorr?vectorrprime|?
r0
rprime
|vectorr? r20rprime2vectorrprime|
parenrightbig
φ(vectorr) = 14piepsilon1
0
integraldisplay
∞
dτprimeρ(vectorrprime)parenleftbig 1|vectorr?vectorrprime|?
r0
r
|vectorrprime?r20r2vectorr|
parenrightbig
14pi
integraldisplay
r0
dSprimeφ(vectorrprime)rprimeparenleftbig 1|vectorr?vectorrprime|?
r0
r
|vectorrprime?r20r2vectorr|
parenrightbig
137/384
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a183a62a124a85a254
W = 12
integraldisplay
∞
dτvectorE·vectorD = 12
integraldisplay
∞
dτ(φ)·vectorD
= 12
integraldisplay
∞
dτ[·(φvectorD) +φ?·vectorD]
=?12
contintegraldisplay
∞
dvectorS·φvectorD + 12
integraldisplay
∞
dτρfφ
a233a183a62a124,φvextendsinglevextendsingler→∞～1r,Dvextendsinglevextendsingler→∞～ 1r2 →
contintegraldisplay
∞
dvectorS·φvectorD→0
W = 12
integraldisplay
dτvectorE·vectorD = 12
integraldisplay
dτρfφ = 18piepsilon1
0
integraldisplay
dτdτprimeρf(vectorr)ρ(vectorr
prime)
R
a233a253a152a165a27a27a62a62a214a169a217,ρ(vectorr) = ρf(vectorr)
W = 18piepsilon1
0
integraldisplay
dτdτprimeρf(vectorr)ρf(vectorr
prime)
R
138/384
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a183a62a124a85a254,a131a112a138a94a85
W = 18piepsilon1
0
integraldisplay
dτdτprimeρf(vectorr)ρf(vectorr
prime)
R
a253a152a165a107a252a124a62a214a169a217a124a164a27a78a88,ρ = ρ1 +ρ2
W = W1 + W2 + Wa131a112
W = 18piepsilon1
0
integraldisplay
dτdτprime1R[ρ1(vectorr) +ρ2(vectorr)][ρ1(vectorrprime) +ρ2(vectorr)prime]
= 18piepsilon1
0
integraldisplay
dτdτprime1R[ρ1(vectorr)ρ1(vectorrprime) +ρ2(vectorr)ρ2(vectorrprime) +ρ1(vectorr)ρ2(vectorrprime) +ρ2(vectorr)ρ1(vectorrprime)]
Wa131a112 = 14piepsilon1
0
integraldisplay
dτdτprime1Rρ1(vectorr)ρ2(vectorrprime) =
integraldisplay
dτρ1φ2 =
integraldisplay
dτρ2φ1
a62a214ρa51a9a124φa165a27a85a254,Wa131a112 =
integraldisplay
dτρφ
a58a58a62a62a214a51a9a124a165a27a85a254,qφ a183a62a160a85a180a62a124a85a27a141a150a76a227a156
139/384
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a183a62a124a85a254,a131a112a138a94a85
a70a126a41a185a165a107 m = m1 + m2 Frank.Wilczek,a218a238a238a49a49a34a189a198
a85 E = mc2 a110a41 W = W1 + W2 + Wa131a112
a218a238a238a49a49a34a189a198a216a50a164a225a156 Wa131a112a178a126< 0
a22a20a135a42 Wa131a112 a229a27a138a94a22a140a156
a131a112a138a94a51a135a42a43a141a229a216a37a138a94
a51a212a159a216a211a27a0a103a107a216a211a27a198a137a59a128a239a196Wa131a112a156
140/384
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a183a62a124a85a254,a41a48a154a189a110
a62a214a51a216a26a47a47a27a27a27a19a19a78a254a27a183a21a169a217a240a166a183a62a85a18a52a2a138 !
a127a196a19a78a254a27a62a214a27a67a122(a253a14a48a159a165a27a62a214a195a123a67,a122a135a19a78a254
a27a111a62a254a143a195a123a67)a19a151a27a27a62a62a124a67a122δvectorE
δW = 12
integraldisplay
dτ epsilon1(vectorE +δvectorE)·(vectorE +δvectorE)?12
integraldisplay
dτ epsilon1E2
=
integraldisplay
dτ epsilon1[vectorE·δvectorE + 12δvectorE·δvectorE] a217a165:
integraldisplay
dτ epsilon1vectorE·δvectorE =?
integraldisplay
dτ?φ·δvectorD =?
integraldisplay
dτ[?·(φδvectorD)?φ?·δvectorD]
=?
contintegraldisplay
dvectorS·φδvectorD +
integraldisplay
dτ φδ?·vectorD =
integraldisplay
dτ φδρ =
summationdisplay
a19a78i
φiδqi = 0
δW = 12
integraldisplay
dτ epsilon1δvectorE·δvectorE≥0
141/384
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a183a62a124a85a254,a41a48a154a189a110
a62a214a51a216a26a47a47a27a27a27a19a19a78a254a27a183a21a169a217a240a166a183a62a85a18a52a2a138 !
a127a196a19a78a254a27a62a214a27a67a122(a253a14a48a159a165a27a62a214a195a123a67,a122a135a19a78a254
a27a111a62a254a143a195a123a67)a19a151a27a27a62a62a124a67a122δvectorE
δW = 12
integraldisplay
dτ epsilon1(vectorE+δvectorE)·(vectorE+δvectorE)?12
integraldisplay
dτ epsilon1E2 = 12
integraldisplay
dτ epsilon1δvectorE·δvectorE≥0
a135a131a101a62a214a169a217a166a183a62a85a136a20a52a2.a167a131a8a117a88a101a21a229a67a169a175a75:
δqi =
integraldisplay
a19a78i
dτδρ = 0
summationdisplay
a19a78i
integraldisplay
a19a78i
dτ φδρ = 0
a218a92a46a188a166a102φi:
summationdisplay
a19a78i
integraldisplay
a19a78i
dτ (φ?φi)δρ=0 φ(vectorr)=φi a19a78a143a31a179a78!
142/384
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a240a189a189a62a62a54a27a27a62a62a124
·vectorD = ρf?×vectorE = 0 vectorD = epsilon1vectorE vectorjc = γvectorE
a240a189a189a62a62a54a88a218,?
vectorjc
t = 0 →
vectorE
t = 0
a173a21a88a218,a108a62a124a85a61a122a53a27a62a102a27a196a85,a207a76a62a102a134a172a130a45a69,a242
a196a85a68a52a137a172a130,a47a164a57a85a181
a0a15a189a198,a57a245a199a151a221 W = Q?φ?t?τ = jc?t?SδlE?t?S?l =vectorjc·vectorE = j
2
c
γ
a73a9a46a74a248a85a254a156
a218a92a154a183a62a229a57a131a65a27a27a31a31a8a62a124vectorEprime = vectorFa154/q + +
a63a85a27a238a238a48a48a189a198a181vectorjc = γ(vectorE + vectorEprime)
a62a62a196a196a179a181 ε =
contintegraldisplay
dvectorl·vectorFa154/q =
contintegraldisplay
dvectorl·(vectorE + vectorEprime)
=
contintegraldisplay
dvectorl·vectorjc/γ =
contintegraldisplay
dlJλS = JR
a9a53a85a254a207a76a154a183a62a229a27a62a196a179a216a228a61a122a164a62a124a85a167a129a0a158a209a180a27
a62a123a254a61a67a143a57a85a156
143/384
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a240a189a189a62a62a54a27a27a62a62a124,a124a144a167a124
vectorj,vectorEprimea216a145a158a109a85a67→vectorEa216a145a158a109a85a67.
·vectorD = ρf?×vectorE = 0
ρf
t +?·
vectorjc = 0 vectorjc = γ(vectorE + vectorEprime)
ρf
t = 0 →?·
vectorjc = 0 →?·(γvectorE) =·(γvectorEprime)
×vectorE = 0?·(epsilon1vectorE) = ρf
γa233a65epsilon1,·(γvectorEprime)a233a65a65a117a117ρf.
144/384
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a240a189a189a62a62a54a27a27a62a62a124,a1261
a62a52A,B(γ =∞)a152a117a19a62a120a159γ1a165,γ1a9 a157a140a44a152a171a19a62a120a159γ2:
×vectorE1 = 0?×vectorE2 = 0 γ1E1n = γ2E2na62a54a235a89
·(γ1vectorE1) = 0?·(γ1vectorE2) = 0 E1t = E2t
a154a62a52a63,vectorEprime = 0,γ1,γ2a229epsilon1a27a138a94,a124a94vectorE =φ
2φ1 = 0?2φ2 = 0 γ1?φ1?n = γ2?φ2?n φ1 = φ2
a92a254a62a52A,Ba27a53a159a57a195a161a15a27a103a44a62a94a135,a62a52a180a110a142a19a78a181
γ =∞ vectorjc = γvectorE(vectorEprime = 0) → vectorE→0 a62a52a143a31a179a78!
a62a94a135a181
braceleftbiggφ
A?φB = V
QA =?QB
braceleftbiggφ
B = 0
φA = V
a92a254a195a161a15a103a44a62a94a135φvextendsinglevextendsingler→∞→ 1r,a144a167a124a27a41a180a141a152a27.
ρf =?·vectorD =?·(epsilon1vectorE) σf = D2n?D1n σ = epsilon10(E2n?E1n)
a101a120a159a254a33a27,γ,epsilon1a143a126a234,a75ρf = 0.
145/384
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a240a189a189a62a62a54a27a27a62a62a124,a1262
a140a187a143aa27a140a165a47a62a52a238a51a62a19a199a143γa27a47a101,a62a52a178a161a134 a47a161a178
a224,a108a62a52a83a54a209a27a27a62a62a54a114a221I,a166a47a101a27a27a62a62a54a169a217?
a47a101a154a62a52a63a27a27a62a62a179φa247a118?2φ = 0
a51a47a161a62a54a144a85a247a131a149a54a196,a107a62a94a135En = 0
a51a62a52a27a165a76a161I = integraltext dvectorS·γvectorE
a192a193a38a41φ = cr,a167a247a118a51a47a161a27a62a94a135En = 0
I =
integraldisplay
a140a165
dvectorS·cγvectorrr3 =
integraldisplay
a140a165
dS cγ 1a2 =?2picγ
→ c = I2piγ → φ = I2piγr
a62a52a229a152a135a145a152a189a189a62a62a214a27a27a19a19a78a27a138a94.
146/384
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a173a240a62a54a27a94a124,a165a165a179a179
braceleftbigg?×vectorH =vectorj
c
·vectorB = 0
braceleftbigg vectorB = μvectorH
vectorf =vectorjc×vectorB a62a94a135,
braceleftbiggvectorn·(vectorB
2?vectorB1) = 0
vectorn×(vectorH2?vectorH1) =vectoric
a195a209a27a165a254a124a152a189a140a177a76a143a44a135a165a254a124a27a94a221 a156
a189a194a181vectorA(vectorr)≡
integraldisplay 1
0
dλ vectorB(λvectorr)×(λvectorr)
×(vectorf×vectorg) =?vectorg?·vectorf?vectorf·?vectorg +vectorg·?vectorf +vectorf?·vectorg
×vectorA(vectorr) =
integraldisplay 1
0
dλ λ?×[vectorB(λvectorr)×vectorr]
=
integraldisplay 1
0
dλ λ[?vectorr?·vectorB(λvectorr)?vectorB(λvectorr)·?vectorr +vectorr·?vectorB(λvectorr) + vectorB(λvectorr)?·vectorr]
=
integraldisplay 1
0
dλ λ[?vectorB(λvectorr)+λ ddλvectorB(λvectorr)+3vectorB(λvectorr)]=
integraldisplay 1
0
dλ λ[2vectorB(λvectorr)+λ ddλvectorB(λvectorr)]
=
integraldisplay 1
0
dλ ddλ[λ2vectorB(λvectorr)] = [λ2vectorB(λvectorr)]vextendsinglevextendsingleλ=1λ=0 = vectorB(vectorr)
147/384
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a173a240a62a54a27a94a124,a165a165a179a179
a189a194 vectorB =?×vectorAa40a189a189a27a27vectorAa216a141a152.
vectorB =?×vectorA =?×(vectorA +?χ) vectorAprime = vectorA +?χ ←a53a137a67a134
a53a137a67a134a166vectorAa144a85a79a40a20a131a11a152a135a63a191a70a221a188a234.
a143a166vectorA a40a189,a76a78a92a152a135a53a137a27a189a94a135a22a75a53a137a103a100a221:
a165a212a53a137a27a189a94a135,?·vectorA = 0
a126a181 a233a126a234a124vectorB,a192vectorA·vectorr = 0,a75vectorA = 12vectorB×vectorr
×vectorA = 12(?·vectorr)vectorB?12(vectorB·?)vectorr = 32vectorB?12vectorB·arrowrighttophalfarrowrighttophalfI = vectorB
a53a137a27a189a94a135a233a245,a212a110a134a53a137a27a189a94a135a27a192a74a195a39.
vectorB =?×vectorAa137a209?·vectorB =?·(?×vectorA) = 0,a191a155a155a195a195a94a94a252a252a52.
148/384
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a173a240a62a54a27a94a124,a165a165a179a179
a165a212a53a137a181vectorAvextendsinglevextendsinglea165a212a53a137 =
integraldisplay
dvectorlprimeμ0J4piR =
integraldisplay
dτprime μ0
vectorj
4piR
a51a165a212a53a137a101a27a165a254a179a247a118a181
·vectorAvextendsinglevextendsinglea165a212a53a137 = μ04pi
integraldisplay
dτprimevectorj·?1R =?μ04pi
integraldisplay
dτprimevectorj·?prime1R
=?μ04pi
integraldisplay
dτprime?prime·
vectorj
R =?
μ0
4pi
integraldisplay
∞
dvectorSprime·
vectorj
R = 0
a165a165a179a179a57a217a247a118a27a144a167,(a254a33a48a159)
2vectorA(?·vectorA) =×(?×vectorA) =×vectorB =?μ?×vectorH=?μvectorjc
a233a165a212a53a137,?2vectorA =?μvectorjc.
149/384
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a173a240a62a54a27a94a124,a165a165a179a179,a173a240a62a54a94a124a27a85a254
W = 12
integraldisplay
∞
dτvectorB·vectorH = 12
integraldisplay
∞
dτ(?×vectorA)·vectorH
= 12
integraldisplay
∞
dτ[?·(vectorA×vectorH) + vectorA·(?×vectorH)]
= 12
integraldisplay
∞
dvectorS·(vectorA×vectorH) + 12
integraldisplay
∞
dτvectorA·vectorjc
a233a173a240a62a54a27a94a124,vectorAvextendsinglevextendsingler→∞～1r,vectorHvextendsinglevextendsingler→∞～ 1r2
→
integraldisplay
∞
dvectorS·(vectorA×vectorH)→ 0
W = 12
integraldisplay
dτvectorB·vectorH = 12
integraldisplay
dτvectorA·vectorjc = μ08pi
integraldisplay
dτdτprime
vectorjc(vectorr)·vectorj(vectorrprime)
R
a233a253a152a165a27a27a62a62a54a169a217:vectorj(vectorr) =vectorjc(vectorr) W = μ08pi
integraldisplay
dτdτprime
vectorjc(vectorr)·vectorjc(vectorrprime)
R
150/384
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a173a240a62a54a27a94a124,a165a165a179a179,a173a240a62a54a94a124a27a85a254
a233a253a152a165a27a27a62a62a54a169a217:vectorj(vectorr) =vectorjc(vectorr) W = μ08pi
integraldisplay
dτdτprime
vectorj(vectorr)·vectorj(vectorrprime)
R
a252a124a62a54a169a217a124a164a27a78a88:vectorj =vectorj1 +vectorj2,W = W1 + W2 + Wa131a112
W = μ08pi
integraldisplay
dτdτprime1R[vectorj1(vectorr) +vectorj2(vectorr)]·[vectorj1(vectorrprime) +vectorj2(vectorr)prime]
= μ08pi
integraldisplay
dτdτprime1R[vectorj1(vectorr)·vectorj1(vectorrprime) +vectorj2(vectorr)·vectorj2(vectorrprime) +vectorj1(vectorr)·vectorj2(vectorrprime)
+vectorj2(vectorr)·vectorj1(vectorrprime)]
Wa131a112 = μ04pi
integraldisplay
dτdτprime1Rvectorj1(vectorr)·vectorj2(vectorr)prime =
integraldisplay
dτvectorA1·vectorj2 =
integraldisplay
dτvectorA2·vectorj1
a62a54vectorja51a9a124vectorAa165a27a85a254,Wa131a112 =
integraldisplay
dτvectorj·vectorA
151/384
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a94a124a175a75a27a152a132a41a123,a195a68a19a19a62a62a54a195a144a94
a152a135a131a233a94a19a199μra27a140a187a143aa27a254a33a165,a152a51a254a33a94a124a114a221vectorH0 a27
a9a94a124a165,a166a166a165a165a83a9a27a94a124
a26a25
a27a24a45a45
a45
μ
vectorH0
braceleftbigg?×vectorH
1 = 0
·vectorB1 = 0
braceleftbigg?×vectorH
2 = 0
·vectorB2 = 0braceleftbigg
H1tvextendsinglevextendsingler=a = H2tvextendsinglevextendsingler=a
B1nvextendsinglevextendsingler=a = B2nvextendsinglevextendsingler=a
braceleftbigg vectorB
2 = μ0vectorH2vector
B1 = μvectorH1
vectorH2vextendsinglevextendsingle
r→∞→
vectorH0
a131a211a140a2a27a27a62a62a48a159a165a152a51a254a33a62a124vectorE0a165:
a26a25
a27a24a45a45
a45
epsilon1
vectorE0
braceleftbigg?×vectorE
1 = 0
·vectorD1 = 0
braceleftbigg?×vectorE
2 = 0
·vectorD2 = 0braceleftbigg
E1tvextendsinglevextendsingler=a = E2tvextendsinglevextendsingler=a
D1nvextendsinglevextendsingler=a = D2nvextendsinglevextendsingler=a
braceleftbigg vectorD
2 = epsilon10vectorE2vector
D1 = epsilon1vectorE1
vectorE2vextendsinglevextendsingle
r→∞→
vectorE0
vectorH?vectorE vectorB?vectorD μ?epsilon1
152/384
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a46a202a46a100a144a167,a169a108a67a254a123,a165a139a73a126
2φ1 = 0 φ1vextendsinglevextendsingler=a = φ2vextendsinglevextendsingler=a φ1vextendsinglevextendsingler=0a107a129
2φ2 = 0 epsilon1r?φ1?r vextendsinglevextendsingler=a =?φ2?r vextendsinglevextendsingler=a φ2vextendsinglevextendsingler→∞→?E0rP1(cosθ) + C
a26a25
a27a24a45a45
a45
epsilon1
vectorE0
φ1 =
∞summationdisplay
n=0
AnrnPn φ2 =
∞summationdisplay
n=0
Bn
rn+1Pn + B
primeP0 + BprimeprimerP1
A0 = B0a + Bprime; epsilon1rA1 =?2B1a3 + Bprimeprime; A1a = B1a2 + Bprimeprimea;?B0a = 0
Anan = Bnan+1; epsilon1rnAnan?1 =?(n + 1) Bnan+2 n>1; Bprimeprime =?E0; Bprime = C
A0 = C; A1 =?3E0epsilon1
r + 2; B1 = epsilon1r?1epsilon1
r + 2
E0a3; An = Bn = 0 n>0
φ1 = C?3
vectorE0·vectorr
epsilon1r + 2 φ2 = C +
epsilon1r?1
epsilon1r + 2a
3vectorE0·vectorr
r3?
vectorE0·vectorr
a8epsilon1r = 1a158,φ1 = φ2,a8epsilon1r→∞a158,a48a159a67a164a19a78.
153/384
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a94a124a175a75a27a152a132a41a123,a195a68a19a19a62a62a54a195a144a94
vectorH1 =φm1 vectorH2 =φm2
2φm1 = 0 φm1vextendsinglevextendsingler=a = φm2vextendsinglevextendsingler=a φm1vextendsinglevextendsingler=0a107a129
2φm2 = 0 μr?φm1?r vextendsinglevextendsingler=a =?φm2?r vextendsinglevextendsingler=a φm2vextendsinglevextendsingler→∞?H0rP1(cosθ) + C
φm1 =? 3H0μ
r + 2
rP1(cosθ)
φm2 =?H0rP1(cosθ) + μr?1μ
r + 2
a3H0r2P1(cosθ)
vectorH1 = 3
μr + 2
vectorH0 vectorH2 = vectorH0 + 1
4pi[?
vectorm
r3 +
3vectorr(vectorm·vectorr)
r5 ]
vectorB1 = 3μr
μr + 2
vectorB0 vectorB2 = μ0vectorH2 vectorm = 4pia3μr?1
μr + 2
vectorH0
a195a68a19a19a62a62a54?×vectorH = 0→vectorH =φm,vectorjc =?×(φm) = 0.
154/384
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a94a124a175a75a27a152a132a41a123,a195a68a19a19a62a62a54a107a144a94
a152a172a144a94a225a14,a48a159a83
braceleftbigg?×vectorH
1 = 0
·vectorB1 = 0,a48a159a9
braceleftbigg?×vectorH
2 = 0
·vectorB2 = 0
a62a46a254
braceleftbigg H
1t=H2t
B1n=B2n a62a94a53a159a144a167
braceleftbigg vectorB
2 = μ0vectorH2vector
B1 = μ1vectorH1+μ0 vectorM0 a94
vectorHa76a136
braceleftbigg?·(μvectorH
1) =·(μ0 vectorM0)
×vectorH1 = 0
braceleftbigg?·(μ
0vectorH2) = 0
×vectorH2 = 0
braceleftbiggμ
0H2n?μH1n = μ0M0nvector
H2t = vectorH1t
braceleftbigg?·(epsilon1vectorE
1) = ρf
×vectorE1 = 0
braceleftbigg?·(epsilon1
0vectorE2) = 0
×vectorE2 = 0
braceleftbiggepsilon1
0E2n?epsilon1E1n = σfvector
E2t = vectorE1t
μ～epsilon1 μ0～epsilon10 μr～epsilon1r vectorH～vectorE·(μ0 vectorM0)～ρf μ0M0n～σf
155/384
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a94a124a175a75a27a152a132a41a123,a195a68a19a19a62a62a54a107a144a94
μ～epsilon1 vectorH～vectorE·(μ0 vectorM0)～ρf μ0M0n～σf
a91a200a94a99,μ = μ0
a233a65epsilon1 = epsilon10,a195a48a159,a207a13a195a52a122a62a214,a62a124a28a220a100ρf,σfa23a41
vectorE =
integraldisplay
dτprime ρf
vectorR
4piepsilon10R3 +
integraldisplay
dSprime σf
vectorR
4piepsilon10R3
vectorH =?
integraldisplay
dτprime?
prime·(μ0 vectorM0)vectorR
4piμ0R3 +
integraldisplay
dSprimeμ0M0n
vectorR
4piμ0R3
a101a154a91a200a94a99,a75a65a92a92a52a122a62a214(a94a214)a27a0a122.
156/384
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a94a124a175a75a27a152a132a41a123,a107a68a19a19a62a62a54
braceleftbigg?×vectorH =vectorj
c
·vectorB = 0
vectorB = μvectorH
braceleftbiggvectorn×(vectorH
2?vectorH1) =vectoric
vectorn·(vectorB2?vectorB1) = 0
a23a123a242a100a175a75a122a143a143a195a195a68a19a19a62a62a54a27a175a75:
vectorB0 =
integraldisplay
dτprimeμ0
vectorjc×vectorR
4piR3 +
integraldisplay
dSprimeμ0
vectoric×vectorR
4piR3
·vectorB0 = 0?×vectorB0 = μ0vectorjc
vectorn×(vectorB02?vectorB01) = μ0vectoric vectorn·(vectorB02?vectorB01) = 0
157/384
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a94a124a175a75a27a152a132a41a123,a107a68a19a19a62a62a54
·vectorB0=0?×vectorB0=μ0vectorjc vectorn×(vectorB02?vectorB01)=μ0vectoric vectorn·(vectorB02?vectorB01)=0
vectorH = 1
μ0
vectorB0 + vectorHprime vectorB = μvectorH = μ
μ0
vectorB0 +μvectorHprime
×vectorHprime =?×(vectorH? 1μ
0
vectorB0) =vectorjc? 1
μ0?×
vectorB0 =vectorjc?vectorjc = 0
·(μvectorHprime) =?·(vectorB? μμ
0
vectorB0) =?·vectorBμ
μ0 ·
vectorB0? μ
μ0?·
vectorB0 =μ·vectorB0
μ0
vectorn×(vectorHprime2?vectorHprime1) = vectorn×bracketleftbig(?1μ
0
vectorB02 + vectorH2)?(?1
μ0
vectorB01 + vectorH1)bracketrightbig
=?1μ
0
vectorn×(vectorB02?vectorB01) +vectorn×(vectorH2?vectorH1) =vectoric?vectoric = 0
vectorn·(μ2vectorHprime2?μ1vectorHprime1) = vectorn·bracketleftbigvectorB2?μ2μ
0
vectorB02?vectorB1 + μ1
μ0
vectorB01bracketrightbig = μ1?μ2
μ0 vectorn·
vectorB0
vectorHprime～vectorE μ～epsilon1?(?μ)vectorB0
μ0 ～ρf
μ1?μ2
μ0 vectorn·
vectorB0～σf
a233a245a183a94a175a75a134a183a62a175a75a107a233a65a39a88,a131a65a27a212a110a121a121a150a150a143a107a233a65.
158/384
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a94a124a175a75a27a152a132a41a123,a94a186a148a123braceleftbigg
×vectorB1 = μ1vectorjc
·vectorB1 = 0
braceleftbigg?×vectorB
2 = 0
·vectorB2 = 0
braceleftBigg
1
μ1
vectorB1tvextendsinglevextendsingle
x=0 =
1
μ2
vectorB2tvextendsinglevextendsingle
x=0
B1nvextendsinglevextendsinglex=0 = B2nvextendsinglevextendsinglex=0
a94a122a62a54a152a220a169a78a88a51vectorjca254,a166a167a67a143μrvectorjc,a44a152a220a169a51a48a159 a2a46a161
a254vectoriprime,a23μrvectorjca23a41a27a94a97a65a114a221vectorB0,vectoriprimea511,2a48a159a165a23a41a27 a94a97a65a114
a221a169a79a143vectorB10,vectorB20,vectorB1 = vectorB0 + vectorB10 vectorB2 = vectorB0 + vectorB20
a186a150a123a242vectoriprimea23a41a27a94a97a65a114a221a94a160a117a154a166a41a171a27a186a150a62a54vectorjprime,vectorjprimeprimea23a41
a27a94a97a65a114a221a147a76.
a0a18a64a64a73 a65
a65a75
a8a8
a8a8a8a42 x0?x0
vectorR2 vectorR1 μ1μ2
vectorj” μ1rvectorj,vectorjprimea100a100a117a117vectoriprimea51a46a161a252a62a23a41a27a124a233a161,a23a41a249a135a124
a27a186a150a62a54a27a160a152a144a149a143a233a161.vectorjprimea134vectorjc a211a160a152,
vectorjprimeprimea160a117a48a1592a165a27vectorjca27a186a150a233a161a160a152.
vectorjprimen =?vectorjprimeprimen vectorjprimet =vectorjprimeprimet
a23vectorjca20a124a58a58a27a27a165a254vectorR1,vectorjprimeprimea20a124a58a58a27a27a165a254vectorR2,
vectorB0 = μ0μ1rvectorjc×vectorR1
4piR31 dτ
prime vectorB10 = μ0vectorj
primeprime×vectorR2
4piR32 dτ
prime vectorB20 = μ0vectorj
prime×vectorR1
4piR31 dτ
prime
159/384
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a94a124a175a75a27a152a132a41a123,a94a186a148a123a51a48a159a2a2a46a46a161a254,vector
R1 =?x0vectore1 + yvectore2 + zvectore3 vectorR2 = x0vectore1 + yvectore2 + zvectore3 R1 = R2≡R
a0a18a64a64a73 a64
a64a64a73
a0
a0a0a18 x0?x0
vectorR2 vectorR1 μ1μ2
vectorj” μ1rvectorj,vectorjprimevectorB1 = μ0
4pidτ
prime(vectorj
primeprime×vectorR2
R32 +
μ1rvectorjc×vectorR1
R31 )
= μ04piR3dτprime[(vectorjprimeprime?μ1rvectorjc)×(x0vectore1) + (vectorjprimeprime +μ1rvectorjc)×(yvectore2 + zvectore3)]
= μ04piR3dτprime[(vectorjprimeprimet?μ1rvectorjct)×(x0vectore1) + (vectorjprimeprimet +vectorjprimeprimen +μ1rvectorjct +μ1rvectorjcn)
×(yvectore2 + zvectore3)]
vectorB2 = μ0
4pidτ
prime(vectorj
prime×vectorR1
R31 +
μ1rvectorjc×vectorR1
R31 )
= μ04piR3dτprime[(vectorjprime +μ1rvectorjc)×(?x0vectore1) + (vectorjprime +μ1rvectorjc)×(yvectore2 + zvectore3)]
= μ04piR3dτprime[(vectorjprimet +μ1rvectorjct)×(?x0vectore1) + (vectorjprimet +vectorjprimen +μ1rvectorjct +μ1rvectorjcn)
×(yvectore2 + zvectore3)]
= μ04piR3dτprime[(vectorjprimeprimet +μ1rvectorjct)×(?x0vectore1) + (vectorjprimeprimet?vectorjprimeprimen +μ1rvectorjct +μ1rvectorjcn)
×(yvectore2 + zvectore3)]
160/384
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a94a124a175a75a27a152a132a41a123,a94a186a148a123
(vectorjprimeprimet +μ1rvectorjt)×(yvectore2 + zvectore3) = (vectorjprimeprimet +μ1rvectorjt)×(yvectore2 + zvectore3)
1
μ1[(
vectorjprimeprimet?μ1rvectorjt)×(x0vectore1) + (vectorjprimeprimen +μ1rvectorjn)×(yvectore2 + zvectore3)]
= 1μ
2
[(vectorjprimeprimet +μ1rvectorjt)×(?x0vectore1) + (?vectorjprimeprimen +μ1rvectorjn)×(yvectore2 + zvectore3)]
1
μ1(
vectorjprimeprimet?μ1rvectorjct) =?1
μ2(
vectorjprimeprimet +μ1rvectorjct) 1
μ1(
vectorjprimeprimen +μ1rvectorjcn) = 1
μ2(?
vectorjprimeprimen +μ1rvectorjcn)
vectorjprimet =vectorjprimeprimet = μ2?μ1
μ2 +μ1μ1r
vectorjct vectorjprimen =?vectorjprimeprimen = μ2?μ1
μ2 +μ1μ1r
vectorjcn
Jprime = Jprimeprime = μ2?μ1μ
2 +μ1
μ1rJc
161/384
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a94a124a175a75a27a152a132a41a123,a126a181a195a161a127a134a19a130
a206a139a73a181 vectorB(r,θ,z) = B(r)vectoreθ
μ0J =
contintegraldisplay
dvectorl·vectorB = 2pirB(r)
vectorB(r,θ,z) = μ0J
2pirvectoreθ
a134a19a130a94a124a27a28a28a91a91
162/384
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a94a124a175a75a27a152a132a41a123,a126a181a130a47a19a130
a206a139a73,vectorA(r,θ,z) =
contintegraldisplay
a130
dvectorl μ0J4piR a134a26a200a169a79a142
vectorr = zvectorez + rvectorer vectora = acosθprimevectorer + asinθprimevectoreθ
R2 = z2+ r2+ a2?2vectorr·vectora = z2+ r2+ a2?2racosθprime
dvectorl=adθprimevectoreθprime=adθprime(vectoreθprime·vectoreθvectoreθ+vectoreθprime·vectorervectorer)=adθprime(cosθprimevectoreθ?sinθprimevectorer)
vectorA(r,θ,z) = aμ0J
4pi
integraldisplay 2pi
0
dθprime cosθ
primevectoreθ?sinθprimevectorer
√z2 + r2 + a2?2racosθprime
θprime = 2α+pi cosθprime =?cos(2α) = 2sin2α?1 dθprime = 2dα
vectorA(r,θ,z) = aμ0Jvectoreθ
4pi
integraldisplay pi/2
pi/2
dα 2sin
2α?1
radicalbigz
2 + r2 + a2?2ra(2sin2α?1)
= 2μ0Javectoreθpi√z2 + r2 + a2 + 2ra
integraldisplay pi/2
0
dα sin
2α?1/2
radicalBig
1? 4raz2+r2+a2+2ra sin2α
= 2μ0Javectoreθpi√z2+r2+a2+2ra
integraldisplay pi/2
0
dαsin
2α?1/2
√1?k
2sin2α k
2≡ 4ra
z2+r2+a2+2ra
163/384
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a94a124a175a75a27a152a132a41a123,a126a181a130a47a19a130
a206a139a73,vectorA(r,θ,z) =
contintegraldisplay
a130
dvectorl μ0J4piR a134a26a200a169a79a142
vectorr = zvectorez + rvectorer vectora = acosθprimevectorer + asinθprimevectoreθ
R2 = z2+ r2+ a2?2vectorr·vectora = z2+ r2+ a2?2racosθprime
dvectorl = adθprimevectoreθprime = adθprime(cosθprimevectoreθ?sinθprimevectorer) k2≡ 4raz2+r2+a2+2ra
vectorA(r,θ,z) = 2μ0Jvectoreθ
pi√z2 + r2 + a2 + 2ra
integraldisplay pi/2
0
dα sin
2α?1/2
√1?k
2 sin2α
sin2α?1/2 =?1k2(1?k2 sin2α) + 1k2?12
vectorA(r,θ,z)= 2μ0Javectoreθ
pi√z2+r2+a2+2ra
integraldisplay pi/2
0
dα[ 1/k
2?1/2
√1?k
2sin2α?
1
k2
radicalbig
1?k2sin2α]
= 2μ0Javectoreθpi√z2 + r2 + a2 + 2ra[( 1k2?12)K(k)? 1k2E(k)]
= μ0Jvectoreθ
√z2 + r2 + a2 + 2ra
2rpi [(1?
k2
2 )K(k)?E(k)]
164/384
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a94a124a175a75a27a152a132a41a123,a126a181a130a47a19a130
a206a139a73,vectorA(r,θ,z) =
contintegraldisplay
a130
dvectorl μ0J4piR = Aθ(r,z)vectoreθ
Aθ(r,z) = μ0Jvectoreθ
√z2+r2+a2+2ra
2rpi [(1?
k2
2 )K(k)?E(k)]
k2≡ 4raz2+r2+a2+2ra?k?z =?zk
3
4ar
k
r =
k
2r
a2+z2?r2
(a+r)2+z2
vectorB(r,θ,z) =?×vectorA(r,θ,z) =Aθ(r,z)
z vectorer + [
Aθ(r,z)
r +
Aθ(r,z)
r ]vectorez
K(k) =
integraldisplay pi/2
0
dα 1√1?k2 sin2α E(k) =
integraldisplay pi/2
0
dα
radicalbig
1?k2 sin2α
K(k)
k =
E(k)
k(1?k2)?
K(k)
k
E(k)
k =
E(k)?K(k)
k
vectorB(r,θ,z) = μ0J
2piradicalbig(a + r)2 + z2{
vectorer
r [
a2 + r2 + z2
(a?r)2 + z2E(k)?K(k)]
+vectorez[a
2?r2?z2
(a?r)2 + z2E(k) + K(k)]} a130a47a19a130a94a124a27a28a28a91a91
165/384
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a245a52a208a109
a152a236a62a214a51a51a15a15a63a27a27a62a62a198a198a8a8a65
≡a152a135a58a58a62a62a214+a152a135a62a243a52a102+a152a135a62a111a52a102+···
a51a51a15a15a63a27a27a62a62a198a198a8a8a65
a152a236a62a54a51a51a15a15a63a27a94a198a198a8a8a65
≡a152a135a94a243a52a102+a152a135a94a111a52a102+···a51a51a15a15a63a27a94a198a198a8a8a65
166/384
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a245a52a208a109g(xprime) a51xprime = 0a63a208a109:
g(xprime) = g(0) +
∞summationdisplay
n=1
1
n!
ng(xprime)
xprimen
vextendsinglevextendsingle
xprime=0 x
primen
f(vectorrprime) a51vectorrprime = 0a63a208a109:
f(vectorrprime) = f(0) +
∞summationdisplay
n=1
3summationdisplay
i1i2...in=1
1
n!
bracketleftbig?
xprimei1
xprimei2···
xprimeinf(vectorr
prime)bracketrightbig
vectorrprime=0x
prime
i1x
prime
i2···x
prime
in
= f(0) +
∞summationdisplay
n=1
1
n!
bracketleftbig[vectorrprime·?”]n f(vectorr”)bracketrightbig
vectorr”=0
1
|vectorr?vectorrprime| =
1
r +
∞summationdisplay
n=1
3summationdisplay
i1i2...in=1
1
n!
bracketleftbig?
xprimei1
xprimei2···
xprimein
1
|vectorr?vectorrprime|
bracketrightbig
vectorrprime=0x
prime
i1x
prime
i2···x
prime
in
= 1r +
∞summationdisplay
n=1
3summationdisplay
i1i2...in=1
(?1)n
n!
bracketleftbig?
xi1
xi2···
xin
1
|vectorr?vectorrprime|
bracketrightbig
vectorrprime=0x
prime
i1x
prime
i2···x
prime
in
= 1r +
∞summationdisplay
n=1
3summationdisplay
i1i2...in=1
(?1)n
n!
bracketleftbig?
xi1
xi2···
xin
1
r
bracketrightbigxprime
i1x
prime
i2···x
prime
in
167/384
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a245a52a208a109,a62a179a27a27a245a245a52a208a109,a62a62a245a245a52a221
φ(vectorr) = 14piepsilon1
0
integraldisplay
dτprime ρ(vectorr
prime)
|vectorr?vectorrprime|
= 14piepsilon1
0
braceleftbig1
r
integraldisplay
dτprimeρ(vectorrprime) +
∞summationdisplay
n=1
3summationdisplay
i1i2...in=1
(?1)n
n!
bracketleftbig?
xi1
xi2···
xin
1
r
bracketrightbig
×
integraldisplay
dτprimexprimei1xprimei2···xprimeinρ(vectorrprime)bracerightbig =
∞summationdisplay
n=0
φn(vectorr)
φ0(vectorr) = 14piepsilon1
0r
integraldisplay
dτprimeρ(vectorrprime)
n>0,
φn(vectorr) = 14piepsilon1
0
3summationdisplay
i1i2...in=1
(?1)n
n!
bracketleftbig?
xi1
xi2···
xin
1
r
bracketrightbigq
i1i2...in
qi1i2...in =
integraldisplay
dτprimexprimei1xprimei2···xprimeinρ(vectorrprime) a79a142a165(a253a165)a47a169a217a62a214a27qi1i2...in,a96a178a217a112a30a145a233a233a62a62a179a195a0a122!
vectorE = vectorE0 + vectorE1 +··· vectorEn =φn φn+1/φn～rprime/r
168/384
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a245a52a208a109,a62a179a27a27a245a245a52a208a109,a62a62a245a245a52a221
a62a214:
φ0(vectorr) = 14piepsilon1
0
Q
r
Q =
integraldisplay
dτprimeρ(vectorrprime)
vectorE0(vectorr) = Qvectorr
4piepsilon10r3
a78a88a26a160a164a152a135a62a254a143Qa27a27a58a58a58a62a62a214,a78a88a88a130a130a221a221a27a27a8a65a26a209a22a10.
169/384
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a245a52a208a109,a62a179a27a27a245a245a52a208a109,a62a62a245a245a52a221
a62a243a52a102:
φ1(vectorr) =? 14piepsilon1
0
3summationdisplay
i1=1
bracketleftbig?
xi1
1
r
bracketrightbigq
i1 =
1
4piepsilon10
3summationdisplay
i1=1
xi1
r3 qi1 =
vectorr·vectorp
4piepsilon10r3
vectorp =
3summationdisplay
i=1
qivectorei =
integraldisplay
dτprimevectorrprimeρ(vectorrprime) vectorE1 = 14piepsilon1
0
[?vectorpr3 + 3vectorr(vectorp·vectorr)r5 ]
a78a88a26a160a164a152a135a62a243a52a221a143vectorPa27a62a243a52a102,a152a135a100a252a135a62a254±q,a131
a229vectorl(a108a75a141a149a20),a133l→0,q→∞,a2a2a177qla143a107a129a27a58a62a214a8
a164a27a78a88,a167a140a140a119a119a164a180a134a160a27a20a75a62a214a78a88a34
a1
a1
a1
a1a1a21
a16a16a16
a16a16a16
a16a16a16a49
a80a80a80
a80a80a80a113a109
a109
vectorr2
vectorr1
vectorl
+
a233a100a78a88,a111a62a254Q = q?q = 0,a129a36a30a27a210a180a243a52a221a27a0a122,
vectorp =
integraldisplay
dτprimevectorrprimeρ = q
integraldisplay
dτprimevectorrprime[δ(vectorrprime?vectorrq)?δ(vectorrprime?vectorr?q)] = q(vectorrq?vectorr?q) = qvectorl
a112a30a145a20a39a117 ql(lr)n→0(n>1)a8l→0 a158a170a1170.
170/384
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a245a52a208a109,a62a179a27a27a245a245a52a208a109,a62a62a245a245a52a221a62a111a52a102:
φ2(vectorr) =? 14piepsilon1
0
3summationdisplay
i1,i2=1
1
2
bracketleftbig?
xi1
xi2
1
r
bracketrightbigq
i1i2 =
1
8piepsilon10
3summationdisplay
i1,i2=1
[3xi1xi2r5?δi1i2r3 ]qi1i2
= 18piepsilon1
0r5
3summationdisplay
i1,i2=1
integraldisplay
dτprimeρ(vectorrprime)[3xi1xprimei1xprimei2xi2?rprime2r2]
= 18piepsilon1
0r5
3summationdisplay
i1,i2=1
xi1xi2
integraldisplay
dτprimeρ(vectorrprime)[3xprimei1xprimei2?rprime2δi1i2] =
3summationdisplay
i1,i2=1
xi1Qi1i2xi2
8piepsilon10r5
= vectorr·
arrowrighttophalfarrowrighttophalfQ ·vectorr
8piepsilon10r5 Qij =
integraldisplay
dτprime[3xprimeixprimej?rprime2δij]ρ(vectorrprime)
summationdisplay
i
Qii = 0
arrowrighttophalfarrowrighttophalfQ= 3summationdisplay
i,j=1
Qijvectoreivectorej =
integraldisplay
dτprime[3vectorrprimevectorrprime?rprime2 arrowrighttophalfarrowrighttophalfI ]ρ(vectorrprime)
vectorE2 = 1
8piepsilon10[
(arrowrighttophalfarrowrighttophalfQ ·vectorr +vectorr·arrowrighttophalfarrowrighttophalfQ)
r5?
5vectorr(vectorr·arrowrighttophalfarrowrighttophalfQ ·vectorr)
r7 ]
171/384
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a245a52a208a109,a62a179a27a27a245a245a52a208a109,a62a62a245a245a52a221
a78a88a26a160a164a152a135a62a111a52a221a143
arrowrighttophalfarrowrighttophalfQ
a27 a62a111a52a102.
a152a135a100a111a135a152a152a51a152a135a178a49a111a62a47a111a135a186a14A B C D(a62a127vectorl1 Aa141a149D,vectorl2 Aa141a149B)a254
a27a169a79a143q,?q,q,?q,a133l1→0,l2→0,q→∞,a2a2a177ql1l2a143a107a129a27a58a62a214a8a164a27a78
a88,a167a140a140a119a119a164a180a134a160a27a20a135a243a52a102a88a218a34
a45a1
a1a1a21
a1
a1a1a21
a45
vectorl1
vectorl2
A D
B C
a233a100a78a88,a111a62a254Q = q?q + q?q = 0,a111a62a243a52a221
vectorp = qvectorrA?qvectorrB+ qvectorrC?qvectorrD =?qvectorl2+ qvectorl2 = 0,a129a36a30a180a62a111a52a221a0a122,
arrowrighttophalfarrowrighttophalfQ= q(3vectorr
AvectorrA?r2A
arrowrighttophalfarrowrighttophalfI?3vectorr
BvectorrB + r2B
arrowrighttophalfarrowrighttophalfI +3vectorr
CvectorrC?r2C
arrowrighttophalfarrowrighttophalfI
3vectorrDvectorrD + r2D arrowrighttophalfarrowrighttophalfI )
= q[3vectorrAvectorrA?r2A arrowrighttophalfarrowrighttophalfI?3(vectorrA +vectorl2)(vectorrA +vectorl2) + (vectorrA +vectorl2)·(vectorrA +vectorl2) arrowrighttophalfarrowrighttophalfI
+3(vectorrA +vectorl1 +vectorl2)(vectorrA +vectorl1 +vectorl2)?(vectorrA +vectorl1 +vectorl2)·(vectorrA +vectorl1 +vectorl2) arrowrighttophalfarrowrighttophalfI
3(vectorrA +vectorl1)(vectorrA +vectorl1) + (vectorrA +vectorl1)·(vectorrA +vectorl1) arrowrighttophalfarrowrighttophalfI ]
= q(3vectorl1vectorl2 + 3vectorl2vectorl1?2vectorl1·vectorl2 arrowrighttophalfarrowrighttophalfI )
a112a30a145a20a39a117 ql2(lr)n→0(n>1)a8l→0 a158a170a1170.
172/384
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φ(vectorr) = φ(0) +
∞summationdisplay
n=1
3summationdisplay
i1i2...in=1
1
n!
bracketleftbig?
xi1
xi2···
xinφ(vectorr)
bracketrightbig
vectorr=0xi1xi2···xin
= φ(0)?
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
xi1xi2···xinbracketleftbigx
i2
xi3···
xinEi1(vectorr)
bracketrightbig
vectorr=0
vectorE(vectorr) = vectorE(0) +
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
xi1xi2···xinbracketleftbigx
i1
xi2···
xin
vectorE(vectorr)bracketrightbig
vectorr=0
W =
integraldisplay
dτρ(vectorr)φ(vectorr) ρ,a102a53a169a217
=
integraldisplay
dτρ(vectorr)braceleftbigφ(0)?
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
xi1xi2···xinbracketleftbigx
i2
···x
in
Ei1(vectorr)bracketrightbigvectorr=0bracerightbig
= Qφ(0)?
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
qi1i2...inbracketleftbigx
i2
xi3···
xinEi1(vectorr)
bracketrightbig
vectorr=0 =
∞summationdisplay
n=0
Wn
W0 = Qφ(0); Wn =?1n!
3summationdisplay
i1i2...in=1
qi1i2...inbracketleftbigx
i2
xi3···
xinEi1(vectorr)
bracketrightbig
vectorr=0
173/384
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a245a52a208a109,a145a62a78a51a9a124a165a27a85a254,a201a229a218a229a221
W =
∞summationdisplay
n=0
Wn
W0 = Qφ(0)
Wn =?1n!
3summationdisplay
i1i2...in=1
qi1i2...inbracketleftbigx
i2
xi3···
xinEi1(vectorr)
bracketrightbig
vectorr=0
W1 =?
3summationdisplay
i=1
qiEi(0) =?vectorp·vectorE(0)
W2 =?12
3summationdisplay
i,j=1
qijx
j
Ei =?16
3summationdisplay
i,j=1
(3qij?δij
3summationdisplay
k=1
qkk)x
j
Ei
=?16
3summationdisplay
i,j=1
Qijx
j
Ei?·vectorEvextendsinglevextendsinglevectorr=0 = 0
174/384
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a245a52a208a109,a145a62a78a51a9a124a165a27a85a254,a201a229a218a229a221vectorF =
integraldisplay
dτρ(vectorr)vectorE(vectorr)
=
integraldisplay
dτρ(vectorr)braceleftbigvectorE(0) +
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
xi1xi2···xinbracketleftbigx
i1
···x
in
vectorE(vectorr)bracketrightbig
vectorr=0
bracerightbig
= QvectorE(0) +
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
qi1i2...inbracketleftbigx
i1
xi2···
xin
vectorE(vectorr)bracketrightbig
vectorr=0 =
∞summationdisplay
n=0
vectorFn
vectorF0 = QvectorE(0) vectorFn = 1
n!
3summationdisplay
i1i2...in=1
qi1i2...inbracketleftbigx
i1
xi2···
xin
vectorE(vectorr)bracketrightbig
vectorr=0
vectorF1 =
3summationdisplay
i=1
qix
i
vectorE(vectorr)vextendsinglevextendsingle
vectorr=0 = vectorp·?
vectorE(vectorr)vextendsinglevextendsingle
vectorr=0
vectorF2 =?1
2
3summationdisplay
i,j=1
qij[x
i
xj
vectorE(vectorr)]vextendsinglevextendsingle
vectorr=0 =
1
6
3summationdisplay
i,j=1
(3qij?δij
3summationdisplay
k=1
qkk)?
2
xi?xj
vectorEvextendsinglevextendsingle
vectorr=0
= 16
3summationdisplay
i,j=1
Qij[?
2
xi?xj
vectorE(vectorr)]vextendsinglevextendsingle
vectorr=0?
2vectorEvextendsinglevextendsingle
vectorr=0 =[?
2φ]vextendsinglevextendsingle
vectorr=0 = 0
175/384
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a245a52a208a109,a145a62a78a51a9a124a165a27a85a254,a201a229a218a229a221vectorL = integraldisplay dτρ(vectorr)vectorr×vectorE(vectorr)
=
integraldisplay
dτρ(vectorr)vectorr×
∞summationdisplay
n=0
1
n!
3summationdisplay
i1i2...in=1
xi1···xinbracketleftbigx
i1
···x
in
vectorE(vectorr)bracketrightbig
vectorr=0
=
3summationdisplay
i,j,k=1
epsilon1ijk
integraldisplay
dτρ(vectorr)xivectorek
∞summationdisplay
n=0
1
n!
3summationdisplay
i1i2...in=1
xi1···xinbracketleftbigx
i1
···x
in
Ej(vectorr)bracketrightbigvectorr=0
=
∞summationdisplay
n=1
1
n!
3summationdisplay
i,j,k,i1i2...in=1
epsilon1ijkqii1...invectorekbracketleftbigx
i1
···x
in
Ej(vectorr)bracketrightbigvectorr=0 =
∞summationdisplay
n=0
vectorLn
vectorLn = 1
n!
3summationdisplay
i,j,k,i1i2...in=1
epsilon1ijkqii1...invectorekbracketleftbigx
i1
···x
in
Ej(vectorr)bracketrightbigvectorr=0
vectorL0 = summationdisplay
i,j,k
epsilon1ijkqivectorekEj(0) = vectorp×vectorE(0)
summationdisplay
ijk
epsilon1ijkvectorekx
i
Ej =?×vectorE = 0
vectorL1 = summationdisplay
i,j,k,i1
epsilon1ijkqii1vectorekx
i1
Ejvextendsinglevextendsinglevectorr=0 =
3summationdisplay
i,j,k,i1=1
epsilon1ijk[qii1?δii13
3summationdisplay
k=1
qkk]vectorek?Ej?x
i1
vextendsinglevextendsingle
vectorr=0
176/384
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a245a52a208a109,a165a254a179a27a27a245a245a52a208a109,a94a245a52a221
vectorA(vectorr) = μ0
4pi
integraldisplay
dτprime
vectorj(vectorrprime)
|vectorr?vectorrprime|
= μ04pi
∞summationdisplay
n=0
3summationdisplay
i1i2...in=1
(?1)n
n!
bracketleftbig?
xi1
xi2···
xin
1
r
bracketrightbigintegraldisplay dτprimexprime
i1x
prime
i2···x
prime
invectorj(vectorr
prime)
=
∞summationdisplay
n=0
vectorAn(vectorr)
vectorAn(vectorr) = μ0
4pi
3summationdisplay
i1i2...in=1
(?1)n
n!
bracketleftbig?
xi1
xi2···
xin
1
r
bracketrightbigvectorJ
i1i2...in
vectorJi
1i2...in =
integraldisplay
dτprimexprimei1xprimei2···xprimein vectorj(vectorrprime)
vectorB = vectorB0 + vectorB1 +··· vectorBn =?×vectorAn
177/384
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a245a52a208a109,a165a254a179a27a27a245a245a52a208a109,a94a245a52a221
a94a214:
vectorA0(vectorr) = μ0
4pi
vectorJ
r
vectorJ =
integraldisplay
dτprimevectorj(vectorrprime) =
integraldisplay
dτprime?prime·[vectorj(vectorrprime)vectorrprime] =
integraldisplay
∞
dvectorSprime·[vectorj(vectorrprime)vectorrprime] = 0
vectorA0(vectorr) = 0 vectorB0(vectorr) = 0
a78a88a88a130a130a221a221a27a27a8a65a26a209a22,a26a160a164a152a135a58a94a214,a100a100a117a117a195a94a214,a94a124a1430.
178/384
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a245a52a208a109,a165a254a179a27a27a245a245a52a208a109,a94a245a52a221a94a243a52a221:
vectorA1(vectorr) =?μ0
4pi
3summationdisplay
i1=1
bracketleftbig?
xi1
1
r
bracketrightbigvectorJ
i1 =
μ0
4pi
3summationdisplay
i1=1
xi1
r3
vectorJi
1 =
μ0
4pivectorm×
vectorr
r3
vectorJi =
3summationdisplay
k=1
vectorek
integraldisplay
dτprimexprimeijk(vectorrprime) = 12
3summationdisplay
k=1
vectorek
integraldisplay
dτprime[(xprimeijk?xprimekji) + (xprimeijk + xprimekji)]
= 12
3summationdisplay
k=1
vectorek
integraldisplay
dτprime(xprimeijk?xprimekji)
integraldisplay
dτprime(xprimeijk + xprimekji) =
3summationdisplay
l=1
integraldisplay
dτprimexprime
l
(jlxprimeixprimek) =
integraldisplay
∞
dvectorSprime·(vectorjxprimeixprimek) = 0
3summationdisplay
i=1
xivectorJi = 12
3summationdisplay
i,k=1
integraldisplay
dτprime(xixprimeijk?xprimekjixi)vectorek = 12
integraldisplay
dτprime(vectorr·vectorrprimevectorj?vectorrprimevectorr·vectorj)
= 12
integraldisplay
dτprime(vectorrprime×vectorj)×vectorr = vectorm×vectorr vectorm = 12
integraldisplay
dτprimevectorrprime×vectorj
179/384
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a245a52a208a109,a165a254a179a27a27a245a245a52a208a109,a94a245a52a221a94a243a52a221:
vectorA1(vectorr) = μ0
4pivectorm×
vectorr
r3
vectorB1 = μ0
4pi[?
vectorm
r3 +
3vectorr(vectorm·vectorr)
r5 ] vectorm =
1
2
integraldisplay
dτprimevectorrprime×vectorj
a78a88a26a160a164a152a135a94a243a52a221a143vectorma27a94a243a52a102,a152a135a100 a62a54Ja254a33a169a217
a51a161vectorSa254,a133S → 0,J →∞,a2a2a177JSa143a107a129a27a62a54a23a8a164a27a78
a88,a233a100a78a88,a129a36a30a27a210a180a94a243a52a221a27a0a122,
vectorm = 12
integraldisplay
dτprimevectorrprime×vectorj =?12
integraldisplay
dτprimevectorj×vectorrprime =?12J
contintegraldisplay
dvectorlprime×vectorrprime
=?12J
integraldisplay
(dvectorSprime×?prime)×vectorrprime =?12J
3summationdisplay
i,j=1
integraldisplay
(dvectorSprime×vectorei)×vectorej?primeixprimej
=?12J
3summationdisplay
i=1
integraldisplay
(dvectorSprime×vectorei)×vectorei =?12J
3summationdisplay
i=1
integraldisplay
[(dvectorSprime·vectorei)vectorei?dvectorSprime(vectorei·vectorei)]
= J
integraldisplay
dvectorSprime = JvectorS
a112a30a145a20a39a117Jlprimen(n > 3,lprime2～S)～JSn/2 = JSSn?22 →0a167a180a100a94
a111a52a221,a94a108a52a221...a31a31a31a31a0a122a27a8a65.
180/384
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a62a94a197a27a68a194
181/384
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a110a142a253a14a48a159a159a165a165a27a197a196a144a167a57a178a161a62a94a197a41,a197a196a144a167
vectorB = μvectorH vectorD = epsilon1vectorE
a110a142a253a14a48a159a159a165a165γ = 0,a127a196a103a100a62a214a214a218a218a68a19a19a62a62a54a143a34a27a156a185,
×vectorE=
vectorB
t
1
μ?×
vectorB+?1
μ×
vectorB=epsilon1?vectorE
t epsilon1?·
vectorE=epsilon1·vectorE?·vectorB=0
×vectorB?(?lnμ)×vectorB = μepsilon1?
vectorE
t?·
vectorE =?(?lnepsilon1)·vectorE
2vectorE =[?·vectorE + (?lnepsilon1)·vectorE]+?2vectorE =×(?×vectorE)[(?lnepsilon1)·vectorE]
=?×?
vectorB
t[(?lnepsilon1)·
vectorE] =μepsilon1?2
t2
vectorE+(?lnμ)×?vectorB
t[(?lnepsilon1)·
vectorE]
=μepsilon1?
2
t2
vectorE?(?lnμ)×(?×vectorE)[(?lnepsilon1)·vectorE]
=μepsilon1?
2
t2
vectorE+(?lnμ)·?vectorE?(?vectorE)·(?lnμ)[(?lnepsilon1)·vectorE]
2vectorE?1v2?
2vectorE
t2 =(?lnμ)·?
vectorE?(?vectorE)·(?lnμ)[(?lnepsilon1)·vectorE] v2= 1
μepsilon1
182/384
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a110a142a253a14a48a159a159a165a165a27a197a196a144a167a57a178a161a62a94a197a41,a197a196a144a167
vectorB = μvectorH vectorD = epsilon1vectorE
a110a142a253a14a48a159a159a165a165γ = 0,a127a196a103a100a62a214a214a218a218a68a19a19a62a62a54a143a34a27a156a185,
×vectorE=
vectorB
t?×
vectorB?(?lnμ)×vectorB=μepsilon1?vectorE
t?·
vectorE=?(?lnepsilon1)·vectorE?·vectorB=0
2vectorE?1v2?
2vectorE
t2 =(?lnμ)·?
vectorE?(?vectorE)·(?lnμ)[(?lnepsilon1)·vectorE] v2= 1
μepsilon1
2vectorB=(?·vectorB)+?2vectorB=×(?×vectorB)=×[(μepsilon1?
vectorE
t )+?(lnμ)×
vectorB]
= μepsilon1?
2
t2
vectorB(μepsilon1)×?vectorE
t×[?(lnμ)×
vectorB]
= μepsilon1?
2
t2
vectorBln(μepsilon1)×[?×vectorB?(?lnμ)×vectorB]×[?(lnμ)×vectorB]
2vectorB? 1v2?
2vectorB
t2 =ln(μepsilon1)×[?×
vectorB?(?lnμ)×vectorB]×[?(lnμ)×vectorB]
183/384
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a110a142a253a14a48a159a159a165a165a27a197a196a144a167a57a178a161a62a94a197a41,a253a152a165a197a196a144a167
(?2? 1c2?
2
t2)
vectorB = 0 (?2? 1
c2
2
t2)
vectorE = 0
a108a254a102a27a14a221a119a181
iplanckover2pi1vectorp iplanckover2pi1t?E
21planckover2pi12vectorp·vectorp?1c2?2?t2? 1c2planckover2pi12E2
(?vectorp·vectorp + E2/c2)vectorB = 0 (?vectorp·vectorp + E2/c2)vectorE = 0
E2/c2?vectorp·vectorp = m2c2? m2a49a102 = 0
a88a74a74a49a49a102a107a154a34a183a142a142a159a159a254m2a49a102:
(?vectorp·vectorp + E2/c2?m2a49a102c2)vectorB = 0 (?vectorp·vectorp + E2/c2?m2a49a102c2)vectorE = 0
(?2? 1c2?
2
t2?
m2a49a102c2
planckover2pi12 )
vectorB = 0 (?2? 1
c2
2
t2?
m2a49a102c2
planckover2pi12 )
vectorE = 0
184/384
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a110a142a253a14a48a159a159a165a165a27a197a196a144a167a57a178a161a62a94a197a41,a178a161a197a41
a254a33a48a159a181 vectorE(vectorr,t) = vectorE(t?vectorn·vectorrv ) vectorB(vectorr,t) = vectorB(t?vectorn·vectorrv ) vectorn·vectorn=1
vectorE(t?vectorn·vectorrv ) =?(t?vectorn·vectorrv )?
vectorE
t =?
vectorn
v
vectorE
t
·?vectorE =?vectornv·
vectorE
t =
vectorn
v·?(t?
vectorn·vectorr
v )
2vectorE
t2 =
1
v2
2vectorE
t2
a64
a64
a64a64
a64
a64
a64a64
a0
a0
a0
a0a18
a2
a2a2a14
a8a8
a8a42
a0a18
t1
t2vectorn
vectorr2
vectorr1
vectorr2?vectorr1
a233a137a189a158a143,a31a138a161a143a134vectorna82a134a27a178a161,vectorr⊥·vectorn = 0 vectorn·vectorr =
vectorn·(vectorr +vectorr⊥)
t?vectorn·vectorrv = t?vectorn·(vectorr +vectorr⊥)v
a31a138a161a177a132a221vectorva247vectorna144a149a36a196,a252a135a158a143t1 < t2,a31a138a161
a100vectorr1a163a20vectorr2,
t1?vectorn·vectorr1v = t2?vectorn·vectorr2v → v = vectorn·parenleftbigvectorr2?vectorr1t
2?t1
parenrightbig
185/384
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a110a142a253a14a48a159a159a165a165a27a197a196a144a167a57a178a161a62a94a197a41,a178a161a197a41
×vectorE =?(t?vectorr·vectornv )×?
vectorE
t =?
vectorn
v×
vectorE
t =?
vectorB
t
×vectorB =?(t?vectorr·vectornv )×?
vectorB
t =?
vectorn
v×
vectorB
t = μ?×
vectorH = μ?vectorD
t
= μepsilon1?
vectorE
t =
1
v2
vectorE
t
t(v
vectorB?vectorn×vectorE) = 0 → vvectorB = vectorn×vectorE +a158a109a195a39a145
t(vectorn×v
vectorB + vectorE) = 0 → vectorE =?vectorn×vvectorB +a158a109a195a39a145
=?vectorv×vectorB +a158a109a195a39a145
186/384
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a110a142a253a14a48a159a159a165a165a27a197a196a144a167a57a178a161a62a94a197a41,a178a161a197a41
vvectorB = vectorn×vectorE +a158a109a195a39a145 vectorE =?vectorv×vectorB +a158a109a195a39a145
0 =?·vectorE =?(t?vectorr·vectornv )·?
vectorE
t =?
t
parenleftbigvectorn
v·
vectorEparenrightbig → vectorn·vectorE = a158a109a195a39a145
0 =?·vectorB =?(t?vectorr·vectornv )·?
vectorB
t =?
t
parenleftbigvectorn
v·
vectorBparenrightbig → vectorn·vectorB = a158a109a195a39a145
a209a22a158a109a195a39a145(a183a21a62a94a124),vectorE,vectorB,vectorn a209a108a109a195a39a88! vB = E
We = 12epsilon1E2 = 12epsilon1v2B2 = 12μB2 = Wm
vectorS = vectorE×vectorH = vectorE×vectorB
μ =?(vectorv×
vectorB)×vectorB
μ = vectorv
B2
μ = vectorv(We + Wm)
187/384
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a189a21a197a196a144a167a57a178a161a197a41,a19a78
a19a78a134a253a14a48a159a27a11a79a51a117a233a233a19a19a78a127a51a238a238a48a48a189a198vectorj = γvectorE,γ = 0a75
a163a20a110a142a253a14a48a159.
ρ
t =·
vectorj =·(γvectorE) =?γ
epsilon1?·
vectorD =?γ
epsilon1ρ
ρ(t) = ρ(0)e?γepsilon1t
a51a19a78a165,a62a214a151a221a180a233a175a80a126a27!
(a51a183a62a158,a19a78a83a220a220a216a216a145a62,a62a214a144a169a217a51a76a161a254).
a63a216a88a101a252a171a156a185:
γ～0 ρ(0) = 0
γgreatermuchepsilon1ω ω,a62a94a197a27a14a170a199
a67a113a47a18 ρ(t) = 0
188/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197
vectorD(ω) = epsilon1(ω)vectorE(ω) vectorB(ω) = μ(ω)vectorH(ω) vectorj(ω) = γ(ω)vectorE(ω)
a189a21a62a94a197,vectorE(vectorr,t) = vectorE(vectorr)e?iωt vectorB(vectorr,t) = vectorB(vectorr)e?iωt?
×vectorE = iωvectorB
×vectorH = vectorjc?iωvectorD
·vectorD = 0
·vectorB = 0
a147a92a212a159a62a94a144a167
×vectorE = iωvectorB
×vectorB = μ(γ?iωepsilon1)vectorE
·vectorE = 0
·vectorB = 0
2vectorE =?2vectorE(?·vectorE) =×(?×vectorE) =?iω?×vectorB
=?iωμ(γ?iωepsilon1)vectorE
2vectorB =?2vectorB(?·vectorB) =×(?×vectorB) =?μ(γ?iωepsilon1)?×vectorE
=?iωμ(γ?iωepsilon1)vectorB
(?2 + k2)vectorE = 0
·vectorE = 0
vectorB =?i
ω?×
vectorE
vectorE,vectorBa131a112a216a213a225
←→
k2≡ω2μepsilon1+ iωμγ
(?2 + k2)vectorB = 0
·vectorB = 0
vectorE = 1
μ(γ?iωepsilon1)?×
vectorB
189/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
vectorE(vectorr) = vectorE0eivectork·vectorr vectorE(vectorr,t) = vectorE0ei(vectork·vectorr?ωt) vectork·vectork≡k2 = ω2μepsilon1+ iωμγ
vectorB(vectorr) = vectorB0eivectork·vectorr vectorB(vectorr,t) = vectorB0ei(vectork·vectorr?ωt)
vectorka181 a197a165 vectork = vectorkR + ivectorkI.
vectorE0,vectorB0,a69a8a204 E0n =bardblE0nbardbleiδEn B0n =bardblB0nbardbleiδBn.
vectorE(vectorr,t) =
3summationdisplay
n=1
vectorenbardblE0nbardble?vectorkI·vectorrei(vectorkR·vectorr?ωt+δEn)
vectorB(vectorr,t) =
3summationdisplay
n=1
vectorenbardblB0nbardble?vectorkI·vectorrei(vectorkR·vectorr?ωt+δBn)
bardblE0nbardble?vectorkI·vectorr a180a62a124a114a221a51na144a149a169a254a27a8a204.
bardblB0nbardble?vectorkI·vectorr a180a94a97a65a114a221a51na144a149a169a254a27a8a204.
vectorkR·vectorr?ωt +δEn a180a62a124a114a221a51na144a149a169a254a27a160a131.
vectorkR·vectorr?ωt +δBn a180a94a97a65a114a221a51na144a149a169a254a27a160a131.
190/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
vectorE(vectorr,t) =
3summationdisplay
n=1
vectorenbardblE0nbardble?vectorkI·vectorrei(vectorkR·vectorr?ωt+δEn)
a51a137a189a158a143t,a31a131a161a134vectorkRa82a134.
vectorkR·vectorr = vectorkR·(vectorr +vectorr⊥) vectorr⊥·vectorkR = 0
a31a131a161a177a132a199vφ = ω/kRa247vectorkRa144a149a36a196.
ωt1+vectorkR·vectorr1 =?ωt2+vectorkR·vectorr2 → 1 =
vectorkR
ω ·
(vectorr2?vectorr1)
t2?t1 =
vectorkR
ω ·vectorvφ
a8vectorkI = 0a158,a31a131a161a210a180a31a138a161.
a8vectorkInegationslash= 0a158,a31a138a171a141a100 vectorkR·vectorr =a126a234a218 vectorkI·vectorr =a126a234a233a220a166
a41a251a189,a152a132a180a152a94a130,a247vectorkIa144a149a167a8a204a80a126a129a175a167a135vectorkIa144
a149a167a8a204a79a114a129a175,
191/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
a197a165:
vectorkRa27a144a149a135a78a27a180a160a131a27a68a194a144a149,vectorkIa27a144a149a135a78a27a180a8
a204a27a129a140a80a126a144a149.
k2 = ω2μepsilon1+ iωμγ = vectork·vectork = k2R?k2I + 2ivectorkR·vectorkI
k2R?k2I = μepsilon1ω2 vectorkR·vectorkI = 12ωμγ
vectorkR·vectorkI >0? a247a160a131a68a194a144a149a197a111a180a80a126a27a156
kIa147a76a62a94a197a8a204a27a80a126.a143a135a78a10a212a159a233a62a94a197a27a225a194.
a233a253a14a48a159,γ = 0,kI = 0,a233a253a14a48a159,a62a94a197a195a225a194,a62a94
a85a254a216a209a209.
a197a204a252a150a6a1381/ea27 a68a194a229a108a143a66a223a29a221δ = 1k
I cosθ
.
kIa134ωa107a39,a135a78a249a171a212a159 a233a44a10a170a199a27a27a62a62a94a197a225a194a27a114a102.
192/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
a8a204,a69a8a204vectorE0,vectorB0a247a118 vectork·vectorE0 = 0 vectorB0 = vectorkω×vectorE0
kI = 0,vectork,vectorE0,vectorB0 a164a109a195a39a88,B0 = kωE0,vectorBa134vectorEa211a160a131.
kInegationslash= 0,a109a195a39a88a216a164a225,vectorBa134vectorEa216a211a160a131.
vectorE0 = vectorE0R + ivectorE0I vectorB0 = vectorB0R + ivectorB0I
vectorkR·vectorE0R?vectorkI·vectorE0I = 0 vectorkR·vectorE0I +vectorkI·vectorE0R = 0
vectorB0R = vectorkR
ω ×
vectorE0R?vectorkI
ω ×
vectorE0I vectorB0I = vectorkR
ω ×
vectorE0I + vectorkI
ω ×
vectorE0R
vectorE0R =
3summationdisplay
n=1
bardblE0nbardblvectoren cosδEn vectorE0I =
3summationdisplay
n=1
bardblE0nbardblvectoren sinδEn
vectorB0R =
3summationdisplay
n=1
bardblB0nbardblvectoren cosδBn vectorB0I =
3summationdisplay
n=1
bardblB0nbardblvectoren sinδBn
193/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
a160a8,vectorka143a162,za182a143vectorka144a149(a62a94a197a247za182a182a20a20a144a149a68a194)
E0z = 0
vectorE(vectorr,t) =bardblE0xbardblei(kz?ωt+δEx)vectore1 +bardblE0ybardblei(kz?ωt+δEy)vectore2
a51a44a152za18a189a189a27a27xya178a161a254,a62a124a114a221a165a254a145a158a109a216a228a67a122
δEx?δEy = 0,pi a130a160a8
bardblE0xbardbl=bardblE0ybardbl
δEx?δEy = pi2 a134a94
δEx?δEy = 3pi2 a109a94
δEx?δEya216a27a189 a103a44a49
a152a132,a253a11
194/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
a85a254a151a221,a85a54a151a221:
We = 12epsilon1vectorER·vectorER Wm = 12μvectorBR·vectorBR vectorS = 1μvectorER×vectorBR
vectorER·vectorER =
3summationdisplay
n=1
ERnERn =
3summationdisplay
n=1
bardblE0nbardbl2e?2vectorkI·vectorr cos2(vectorkR·vectorr?ωt +δEn)
vectorBR·vectorBR =
3summationdisplay
n=1
BRnBRn =
3summationdisplay
n=1
bardblB0nbardbl2e?2vectorkI·vectorr cos2(vectorkR·vectorr?ωt +δBn)
vectorER×vectorBR =
3summationdisplay
n,m,p=1
epsilon1nmpERnBRmvectorep
=
3summationdisplay
n,m,p=1
bardblE0nbardblbardblB0mbardble?2vectorkI·vectorrvectorep cos(vectorkR·vectorr?ωt +δEn)
×cos(vectorkR·vectorr?ωt +δBn)
195/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
a85a254a151a221,a85a54a151a221:
cos2(vectorkR·vectorr?ωt +δEn) = 12[1 + cos2(vectorkR·vectorr?ωt +δEn)] = 12
cos2(vectorkR·vectorr?ωt +δBn) = 12[1 + cos2(vectorkR·vectorr?ωt +δBn)] = 12
cos(vectorkR·vectorr?ωt +δEn)cos(vectorkR·vectorr?ωt +δBn)
= 12[cos(2vectorkR·vectorr?2ωt +δEn +δBn) + cos(δEn?δBn)] = 12 cos(δEn?δBn)
vectorER·vectorER = 1
2
3summationdisplay
n=1
bardblE20nbardble?2vectorkI·vectorr = 12Re(vectorE·vectorE?)
vectorBR·vectorBR = 1
2
3summationdisplay
n=1
bardblB20nbardble?2vectorkI·vectorr = 12Re(vectorB·vectorB?)
vectorER×vectorBR = 1
2
3summationdisplay
n,m,p=1
epsilon1nmpbardblE0nbardblbardblB0mbardblvectorepe?2vectorkI·vectorr cos(δEn?δBn)
= 12Re(vectorE×vectorB?)
196/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
a85a254a151a221,a85a54a151a221:
We = 12epsilon1vectorER·vectorER = 14epsilon1Re(vectorE·vectorE?)
Wm = 12μvectorBR·vectorBR = 14μRe(vectorB·vectorB?) = 14μRebracketleftbig(
vectork
ω×
vectorE)·(vectork?
ω ×
vectorE?)bracketrightbig
= 14μω2Rebracketleftbig(vectork·vectork?)(vectorE·vectorE?)?(vectork·vectorE?)(vectork?·vectorE)bracketrightbig
a233a253a14a78,vectork = vectork?,vectork·vectorE = 0→ We = Wm a233a233a19a19a78,Wenegationslash= Wm
vectorS = vectorER×vectorBR
μ =
1
2μRe(
vectorE×vectorB?)
= 12μRe(vectorE?×vectorB) = 12μRe[vectorE?×(
vectork
ω×
vectorE)]
= 12μωRe[vectork(vectorE·vectorE?)?vectorE(vectork·vectorE?)]
197/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a62a46a94a135
a92a19a197,vectorA1 = vectorA10ei(vectork1·vectorr?ωt) a223a19a197,vectorA2 = vectorA20ei(vectork2·vectorr?ω2t)
a135a19a197,vectorA3 = vectorA30ei(vectork3·vectorr?ω3t) vectorA?(vectorE,vectorH)
a0
a0a0a18
a64
a64a64a73
a45zθ3
θ1
epsilon11,μ1,γ1 = 0 epsilon12,μ2,γ2
vectork1 = k1(sinθ1vectore1 + cosθ1vectore3) vectork3 = k3xvectore1+ k3yvectore2+ k3zvectore3 vectork2 = vectork2R+ivectork2I
k1 = ωv =√μ1epsilon11ω k3 = ω3v =√μ1epsilon11ω3
braceleftbigg k2
2R?k
2
2I = μ2epsilon12ω
2
2vector
k2R·vectork2I = 12ω2μ2γ2
a174a127,vectork1,ω,vectorA10,μ1,epsilon11,γ1,μ2,epsilon12,γ2
a73a166,vectork2,vectork3,ω2,ω3,vectorA20,vectorA30
a62a94a135,vectorn×(vectorE2?vectorE1) = 0,vectorn×(vectorH2?vectorH1) =vectoric
vectorn·(vectorD2?vectorD1) = σf,vectorn·(vectorB2?vectorB1) = 0
vectoric = 0,σf = 0 vectorn×(vectorA1 + vectorA3)vextendsinglevextendsingle
z=0 = vectorn×
vectorA2vextendsinglevextendsingle
z=0
vectorn×[vectorA10ei(k1xx+k1yy?ωt) + vectorA30ei(k3xx+k3yy?ω3t)] = vectorn×vectorA20ei(k2xx+k2yy?ω2t)?
ω = ω2 = ω3
k1x = k2x = k3x = k1 sinθ1 =√μ1epsilon11 sinθ1
k1y = k2y = k3y = 0
vectorn×(vectorA10 + vectorA30) = vectorn×vectorA20
198/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a197a165
a0
a0a0a18
a64
a64a64a73
a45zθ3
θ1
epsilon11,μ1,γ1 = 0 epsilon12,μ2,γ2
a135a19a197:
ω3 = ω,k3 = k1
vectork3 = k3 sinθ3vectore1?k3 cosθ3vectore3
k3x = k1x → k3 sinθ3 = k1 sinθ1 → θ3 = θ1 a135a19a189a198
a223a19a197:
ω2 = ω,k2Rx =√μ1epsilon11ωsinθ1,k2Ix = 0,k2Ry = k2Iy = 0
k22R?k22I = μ2epsilon12ω22,vectork2R·vectork2I = 12ω2μ2γ2
k22Rz?k22Iz = k22R?k22Rx?k22Ry?k22I + k22Ix + k22Iy
= μ2epsilon12ω2?μ1epsilon11ω2 sin2θ1
k2Rzk2Iz = vectork2R·vectork2I?k2Rxk2Ix?k2Ryk2Iy = 12ωμ2γ2
199/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a197a165
a223a19a197:
ω2 = ω,k2Rx =√μ1epsilon11ωsinθ1,k2Ix = 0,k2Ry = k2Iy = 0
k22Rz?k22Iz = μ2epsilon12ω2?μ1epsilon11ω2 sin2θ1 k2Rzk2Iz = 12ωμ2γ2
k2Rz
= 1√2
radicalbigg
(μ2epsilon12?μ1epsilon11 sin2θ1)ω2 +
radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1)2ω4 +μ22γ22ω2
k2Iz
= ωμ2γ2√
2
radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1)ω2 +radicalbig(μ2epsilon12?μ1epsilon11 sin2θ1)2ω4 +μ22γ22ω2
200/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a197a165～a253a14a223a19a48a159
a0
a0a0a18
a64
a64a64a73
a45zθ3
θ1
epsilon11,μ1,γ1 = 0 epsilon12,μ2,γ2 = 0
γ2 = 0 a242a19a199,n≡√μrepsilon1r = c√μepsilon1
k2 =√μ2epsilon12ω = n2c ω k2k
1
= n2n
1
k2Rz = ω√2
radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1) +bardblμ2epsilon12?μ1epsilon11 sin2θ1bardbl
k2Iz = ωμ2γ2√2radicalbig(μ
2epsilon12?μ1epsilon11 sin2θ1) +bardblμ2epsilon12?μ1epsilon11 sin2θ1bardbl
μ2epsilon12≥μ1epsilon11→ n2≥n1,a49a213→a49a151
k2Rz = ω
radicalBig
μ2epsilon12?μ1epsilon11 sin2θ1 k2Iz = 0 a195a80a126
vectork2 = k2 sinθ2vectore1 + k2 cosθ2vectore3 = ω√μ2epsilon12(sinθ2vectore1 + cosθ2vectore3)
→ ω√μ2epsilon12 sinθ2 = ω√μ1epsilon11 sinθ1→n2 sinθ2 = n1 sinθ1 a242a19a189a198
sinθ2 = n1n
2
sinθ1≤1
201/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a197a165～a253a14a223a19a48a159
a0
a0a0a18
a64
a64a64a73
a45zθ3
θ1
epsilon11,μ1,γ1 = 0 epsilon12,μ2,γ2 = 0
γ2 = 0 a242a19a199,n≡√μrepsilon1r = c√μepsilon1
k2 =√μ2epsilon12ω = n2c ω k2k
1
= n2n
1
k2Rz = ω√2
radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1) +bardblμ2epsilon12?μ1epsilon11 sin2θ1bardbl
k2Iz = ωμ2γ2√2radicalbig(μ
2epsilon12?μ1epsilon11 sin2θ1) +bardblμ2epsilon12?μ1epsilon11 sin2θ1bardbl
μ2epsilon12 <μ1epsilon11 sinθ1≤
radicalbiggμ
2epsilon12
μ1epsilon11 =
n2
n1,a49a151→a49a213
k2Rz = ω
radicalBig
μ2epsilon12?μ1epsilon11 sin2θ1 k2Iz = 0 a195a80a126
vectork2 = k2 sinθ2vectore1 + k2 cosθ2vectore3 = ω√μ2epsilon12(sinθ2vectore1 + cosθ2vectore3)
→ ω√μ2epsilon12 sinθ2 = ω√μ1epsilon11 sinθ1→n2 sinθ2 = n1 sinθ1 a242a19a189a198
sinθ2 = n1n
2
sinθ1≤1
202/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a197a165～a253a14a223a19a48a159
a0
a0a0a18
a64
a64a64a73
a45zθ3
θ1
epsilon11,μ1,γ1 = 0 epsilon12,μ2
γ2 = 0
μ2epsilon12 <μ1epsilon11 sinθ1 >
radicalbiggμ
2epsilon12
μ1epsilon11 =
n2
n1,a49a151→a49a213
k2Rz = ω√2
radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1) +bardblμ2epsilon12?μ1epsilon11 sin2θ1bardbl= 0
√2k
2Iz =
ωμ2γ2radicalbig
(μ2epsilon12?μ1epsilon11 sin2θ1) +bardblμ2epsilon12?μ1epsilon11 sin2θ1bardbl =
0
0
= ωμ2γ2radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1)ω2 +radicalbig(μ2epsilon12?μ1epsilon11 sin2θ1)2ω4 +μ22γ22ω2
= ωμ2γ2radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1)ω2+(μ1epsilon11 sin2θ1?μ2epsilon12)ω2[1+ μ22γ22ω22(μ
1epsilon11 sin2 θ1?μ2epsilon12)2ω4
]
vextendsinglevextendsingle
γ2→0
=√2ω
radicalBig
μ1epsilon11 sin2θ1?μ2epsilon12 vectork2I = ω
radicalBig
μ1epsilon11 sin2θ1?μ2epsilon12vectore3
vectork2R =√μ1epsilon11ωsinθ1vectore1 v2φ = v2φ,x = ω
kR =
1√
μ1epsilon11 sinθ1 =
v1
sinθ1
vectorA2=vectorA20e?ωz√μ1epsilon11sinθ1?μ2epsilon12ei(√μ1epsilon11ωsinθ1x?ωt) a28a135a19a27a28a28a91a91a252a171 a162a126a252a171
203/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a197a165～a19a19a62a62a223a19a48a159
a0
a0a0a18
a64
a64a64a73
a45zθ3
θ1
epsilon11,μ1,γ1 = 0 epsilon12,μ2,γ2
γnegationslash= 0 vectorA2 = vectorA20e?k2Izzei(k2Rxx+k2Rzz?ωt)
k2Rx =√μ1epsilon11ωsinθ1,k2Ix = 0,k2Ry = k2Iy = 0
k2Rz
= 1√2
radicalbigg
(μ2epsilon12?μ1epsilon11 sin2θ1)ω2 +
radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1)2ω4 +μ22γ22ω2
k2Iz
= ωμ2γ2√
2
radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1)ω2 +radicalbig(μ2epsilon12?μ1epsilon11 sin2θ1)2ω4 +μ22γ22ω2
a223a19a197a31a131a161a247vectork2R = k2Rxvectore1 + k2Rzvectore3a144a149a68a194,a170a199ω,a132a199v =
ω
k2R =
ω√
k22Rx+k22RZ,a223a19a197a107a80a126,a144a149a51za144a149.
204/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a8a204～a92a19a197a62a124a160a117a92a19a161
a0
a0a0a18
a64
a64a64a73
a45
a54
z
x
y
θ
vectorE10 θ
epsilon11,μ1,γ1 = 0 epsilon12,μ2,γ2
a102a114
a102a114a64a64a73
braceleftbiggvectork
3·vectorE30 = 0vector
k2·vectorE20 = 0 →
braceleftbigg k
3xE30x + k3zE30z = 0
k2xE20x + k2zE20z = 0
→
braceleftbigg E
30z =?k3xk3zE30x
E20z =?k2xk
2z
E20x
braceleftBigg vector
B30 = vectork3ω ×vectorE30
vectorB20 = vectork2
ω ×
vectorE20
vectorE10 = E10(vectore1 cosθ1?vectore3 sinθ1) vectorB10 = vectork1
ω ×
vectorE10 = k1
ωE10vectorey
vectorE10t + vectorE30t = vectorE20t vectorH10t + vectorH30t = vectorH20t
braceleftbigg E
30y = E20y?
1
μ1B30x =
1
μ2B20x?
braceleftbigg E
10 cosθ1 + E30x = E20x?
1
μ1
k1
ω E10 +
1
μ1B30y =
1
μ2B20y?
205/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a8a204～a92a19a197a62a124a160a117a92a19a161
braceleftbigg E
30y = E20y?
1
μ1B30x =
1
μ2B20x?
braceleftbigg E
10 cosθ1 + E30x = E20x?
1
μ1
k1
ω E10 +
1
μ1B30y =
1
μ2B20y?
→ 1μ
1
(
vectork3
ω ×
vectorE30)x = 1
μ2(
vectork2
ω ×
vectorE20)x
→ 1μ
1
(k3yE30z?k3zE30y) = 1μ
2
(k2yE20z?k2zE20y)
→ 1μ
1
k3zE30y = 1μ
2
k2zE20y?→E30y = E20y = 0?→B30x = B20x = 0
vectore2bardbl=k2zk
2
vectore1?k2xk
2
vectore3⊥vectork2 vectore3bardbl=k3zk
3
vectore1?k3xk
3
vectore3=?cosθ1vectore1?sinθ1vectore3⊥vectork3
vectorE20 = E20xvectore1 + E20zvectore3 = E20xvectore1?k2x
k2zvectore3E20x =vectore2bardbl
k2
k2zE20x
vectorE30=E30xvectore1+E30zvectore3=E30xvectore1?k3x
k3zvectore3E30x=vectore3bardbl
k3
k3zE30x=?
vectore3bardbl
cosθ1E30x
206/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a8a204～a92a19a197a62a124a160a117a92a19a161
braceleftbigg E
30y = E20y?
1
μ1B30x =
1
μ2B20x?
braceleftbigg E
10 cosθ1 + E30x = E20x?
1
μ1
k1
ω E10 +
1
μ1B30y =
1
μ2B20y?
→ 1μ
2
(
vectork2
ω ×
vectorE20)y? 1
μ1(
vectork3
ω ×
vectorE30)y = 1
μ1
k1
ωE10
→ 1μ
2
(k2zE20x?k2xE20z)? 1μ
1
(k3zE30x?k3xE30z) = k1μ
1
E10
→ 1μ
2
(k2z + k
2
2x
k2z)E20x?
1
μ1(k3z +
k23x
k3z)E30x =
1
μ1k1E10
→ 1μ
2
k22
k2zE20x?
1
μ1
k23
k3zE30x =
1
μ1k1E10
+?→ μ1μ
2
k22
k1k2zE20x +
1
cosθ1E30x = E10 E20x?E30x = E10 cosθ1
→E20x = 2cosθ11 + μ
1k22 cosθ1
μ2k1k2z
E10 E30x = 1?
μ1k22 cosθ1
μ2k1k2z
1 + μ1k22 cosθ1μ
2k1k2z
cosθ1E10
207/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a8a204～a92a19a197a62a124a160a117a92a19a161
→E20x = 2cosθ11 + μ
1k22 cosθ1
μ2k1k2z
E10 E30x = 1?
μ1k22 cosθ1
μ2k1k2z
1 + μ1k22 cosθ1μ
2k1k2z
cosθ1E10
vectorE20bardbl = 2cosθ1
k2z
k2 +
k2μ1
k1μ2 cosθ1
E10bardblvectore2bardbl vectorE30bardbl =?
k2z
k2 +
k2μ1
k1μ2 cosθ1
k2z
k2 +
k2μ1
k1μ2 cosθ1
E10bardblvectore3bardbl
γ2 = 0,μ2 = μ1→k2z = k2 cosθ2; k2x = k2 sinθ2
→vectore2bardbl = cosθ2vectore1?sinθ2vectore3; k2/k1 = n2/n1 = sinθ1/sinθ2
vectorE20bardbl = 2cosθ1
cosθ2 + sinθ1sinθ
2
cosθ1E10bardblvectore2bardbl =
2cosθ1 sinθ2
1
2 sin2θ2 +
1
2 sin2θ1
E10bardblvectore2bardbl
= 2cosθ1 sinθ2sin(θ
1 +θ2)cos(θ1?θ2)
E10bardblvectore2bardbl
vectorE30bardbl =?sinθ2 cosθ2 + sinθ1 cosθ1
sinθ2 cosθ2 + sinθ1 cosθ1 E10bardblvectore3bardbl =
sin2θ2 + sin2θ1
sin2θ2 + sin2θ1 E10bardblvectore3bardbl
= tan(θ1?θ2)tan(θ
1 +θ2)
E10bardblvectore3bardbl
208/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a8a204～a92a19a197a62a124a160a117a92a19a161
a0
a0a0a18
a64
a64a64a73
a17a17
a17a17a17a51
a45
a54
z
x
y
θ1 θ2
vectorE10
vectore3bardbl
vectore2bardbl
θ1
epsilon11,μ1=μ2,γ1= 0 epsilon12,γ2= 0
a102a114
a102a114
a102a114
a64a64a73
a0a9
a102a114a74a74a93γ
2 = 0,μ2 = μ1 k2/k1 = n2/n1 = sinθ1/sinθ2
vectore2bardbl = cosθ2vectore1?sinθ2vectore3 vectore3bardbl =?cosθ1vectore1?sinθ1vectore3
vectorE30bardbl = tan(θ1?θ2)
tan(θ1 +θ2)E10bardblvectore3bardbl = E10bardblvectore3bardbl×
n2?n1
n2+n1 θ1→0
1 θ1→ pi2
n1 = n2 sinθ2 ≤ n2
a49a213→a49a151(n2 >n1)? vectorE30bardbl
a178a49a117 θ1 → 0
a135a178a49a117 θ1 → pi2
vectore3bardbl? a140a197a155a148
vectorE20bardbl = 2cosθ1 sinθ2
sin(θ1 +θ2)cos(θ1?θ2)E10bardblvectore2bardbl = E10bardblvectore2bardbl×
n1
n2+n1 θ1→0
0 θ1→ pi2
a49a213→a49a151 (n2 >n1)? vectorE20bardbla178a49a117vectore2bardbl? a195a140a197a155a148
209/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a8a204～a92a19a197a62a124a82a134a92a19a161
a0
a0a0a18
a64
a64a64a73
a45
a54
z
x
y
θ
vectorE10
vectorB10
θ
epsilon11,μ1,γ1 = 0 epsilon12,μ2,γ2
a102a114
a102a114
a64a64a82
braceleftbiggvectork
3·vectorE30 = 0vector
k2·vectorE20 = 0 →
braceleftbigg k
3xE30x + k3zE30z = 0
k2xE20x + k2zE20z = 0
→
braceleftbigg E
30z =?k3xk3zE30x
E20z =?k2xk
2z
E20x
braceleftBigg vector
B30 = vectork3ω ×vectorE30
vectorB20 = vectork2
ω ×
vectorE20
vectorE10 = E10vectore2 vectorB10 = vectork1
ω ×
vectorE10 = k1
ωE10(?vectore1 cosθ1 +vectore3 sinθ1)
vectorE10t + vectorE30t = vectorE20t vectorH10t + vectorH30t = vectorH20t
braceleftbigg E
30x = E20x?
1
μ1B30y =
1
μ2B20y?
braceleftbigg E
10y + E30y = E20y?
1
μ1(B10x + B30x) =
1
μ2B20x?
210/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a8a204～a92a19a197a62a124a82a134a92a19a161
vectorE20⊥ = 2
1 + μ1k2zμ
2k1 cosθ1
E10⊥vectore2 vectorE30⊥ = 1?
μ1k2z
μ2k1 cosθ1
1 + μ1k2zμ
2k1 cosθ1
E10⊥vectore2
γ2 = 0,μ2 = μ1→k2z = k2 cosθ2 = k1 sinθ1 cosθ2/sinθ2
vectorE30⊥ = 1?
sinθ1 cosθ2
sinθ2 cosθ1
1 + sinθ1 cosθ2sinθ
2 cosθ1
E10⊥vectore2 = E10⊥vectore2×
n1?n2
n1+n2 θ1→0
1 θ1→ pi2
a49a213→a49a151? n2 >n1? vectorE30bardbl a135a178a49a117vectore2 a140a197a155a148
vectorE20⊥ = 2
1 + sinθ1 cosθ2sinθ
2 cosθ1
E10⊥vectore2 = E10⊥vectore2×
2n2
n1+n2 θ1→0
0 θ1→ pi2
a49a213→a49a151? n2 >n1? vectorE20bardbl a178a49a117vectore2 a195a140a197a155a148
211/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a197a27a8a204～a152a132a156a185
vectork1 = k1(sinθ1vectore1 + cosθ1vectore3) vectork3 = k3(sinθ1vectore1?cosθ1vectore3) k1 = k3
vectorE10 = vectorE10bardbl + vectorE10⊥ = E10bardbl(vectore1 cosθ1?vectore3 sinθ1) + E10⊥vectore2
= E10bardbl(vectore2×
vectork1
k1) + E10⊥vectore2 vectore2=
vectore3×vectork1
|vectore3×vectork1|=
vectore3×vectork3
|vectore3×vectork3|
vectorE30 = vectorE30bardbl + vectorE30⊥
= E10bardbl(vectore1 cosθ1 +vectore3 sinθ1)
k2z
k2?
μ1k2
μ2k1 cosθ1
k2z
k2 +
μ1k2
μ2k1 cosθ1
+ E10⊥vectore21?
μ1k2z
μ2k1 cosθ1
1 + μ1k2zμ
2k1 cosθ1
= E10bardbl(vectore2×
vectork3
k3)
k2zk
2
+ μ1k2μ
2k1
cosθ1
k2z
k2 +
μ1k2
μ2k1 cosθ1
+ E10⊥vectore21?
μ1k2z
μ2k1 cosθ1
1 + μ1k2zμ
2k1 cosθ1
γ2=0,μ2=μ1======= E
10bardbl(vectore2×
vectork3
k3)
tan(θ1?θ2)
tan(θ1 +θ2) + E10⊥vectore2
sin(θ2?θ1)
sin(θ1 +θ2)
θ1=0==== n2?n1
n2 + n1[E10bardbl(vectore2×
vectork3
k3)?E10⊥vectore2] a109a94?a134a94
212/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a223a19a197a27a8a204～a152a132a156a185
vectork1 = k1(sinθ1vectore1 + cosθ1vectore3) vectork3 = k3(sinθ1vectore1?cosθ1vectore3) k1 = k3
vectorE10 = vectorE10bardbl + vectorE10⊥ = E10bardbl(vectore1 cosθ1?vectore3 sinθ1) + E10⊥vectore2
= E10bardbl(vectore2×
vectork1
k1) + E10⊥vectore2
vectorE20 = vectorE20bardbl + vectorE20⊥
= E10bardbl(vectore1k2zk
2
vectore3k2xk
2
) 2cosθ1k
2z
k2 +
μ1k2
μ2k1 cosθ1
+ E10⊥vectore2 21 + μ
1k2z
μ2k1 cosθ1
= E10bardbl(vectore2×
vectork2
k2)
2cosθ1
k2z
k2 +
μ1k2
μ2k1 cosθ1
+ E10⊥vectore2 21 + μ
1k2z
μ2k1 cosθ1
γ2=0,μ2=μ1======= E
10bardbl(vectore2×
vectork2
k2)
2cosθ1 sinθ2
sin(θ1+θ2)cos(θ1?θ2) + E10⊥vectore2
2sinθ2 cosθ1
sin(θ1+θ2)
θ1=0==== n1
n2 + n1[E10bardbl(vectore2×
vectork2
k2) + E10⊥vectore2] a134a94?a134a94a182 a109a94?a109a94
213/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a85a54
vectorS = 1
2μωRe[
vectork(vectorE·vectorE?)?vectorE(vectork·vectorE?)]
vectorS1 = 1
2μ1ωRe[
vectork1(vectorE10·vectorE?10)] = 1
2μ1ω
vectork1(vectorE10bardbl·vectorE?10bardbl + vectorE10⊥·vectorE?10⊥)
= vectorS1bardbl +vectorS1⊥
vectork1 = vectork?1 vectork1·vectorE10 = 0
vectorS1bardbl = vectork1
2μ1ω(
vectorE10bardbl·vectorE?10bardbl) = vectork1
2μ1ωE10bardblE
10bardbl
vectorS1⊥ = vectork1
2μ1ω(
vectorE10⊥·vectorE?10⊥) = vectork1
2μ1ωE10⊥E
10⊥
214/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a85a54
vectorS = 1
2μωRe[
vectork(vectorE·vectorE?)?vectorE(vectork·vectorE?)]
vectorS1 = vectorS1bardbl +vectorS1⊥
vectorS3 = 1
2μ1ωRe[
vectork3(vectorE30·vectorE?30)] = 1
2μ1ω
vectork3(vectorE30bardbl·vectorE?30bardbl + vectorE30⊥·vectorE?30⊥)
= vectorS3bardbl +vectorS3⊥
vectork3 = vectork?3 vectork3·vectorE30 = 0
vectorS3bardbl = vectork3
2μ1ω(
vectorE30bardbl·vectorE?30bardbl) = vectork3
2μ1ωE30bardblE
30bardbl
vectorS3⊥ = vectork3
2μ1ω(
vectorE30⊥·vectorE?30⊥) = vectork3
2μ1ωE30⊥E
30⊥
215/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a85a54
vectorS = 1
2μωRe[
vectork(vectorE·vectorE?)?vectorE(vectork·vectorE?)]
vectorS1 = vectorS1bardbl +vectorS1⊥ vectorS3 = vectorS3bardbl +vectorS3⊥
vectorS2 = 1
2μ2ωRe[
vectork2(vectorE2·vectorE?2)?vectorE2(vectork2·vectorE?2)]
= 12μ
2ω
Re[vectork2(vectorE2bardbl·vectorE?2bardbl + vectorE2⊥·vectorE?2⊥)?(vectorE2bardbl + vectorE2⊥)(vectork2·vectorE?2bardbl)]
= vectorS2bardbl +vectorS2⊥
vectorS2bardbl = 1
2μ2ω{
vectork2R(vectorE2bardbl·vectorE?2bardbl)?Re(vectorE2bardbl + vectorE2⊥)(vectork2·vectorE2bardbl)} a195a101a730
vectorS2⊥ = vectork2R
2μ2ω(
vectorE2⊥·vectorE?2⊥) a195a101a730
216/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a85a54
a18a46a161(z = 0)a254a152a135a2a161a3?vectorS =?Svectore3
D⊥≡
vectorS2⊥·?vectorS
vectorS1⊥·?vectorS
vextendsinglevextendsingle
z=0 =
S2⊥z
S1⊥z
vextendsinglevextendsingle
z=0 =
k2zR
2μ2ω(
vectorE2⊥·vectorE?2⊥)
k1z
2μ1ω(
vectorE10⊥·vectorE?10⊥)
= μ1k2zRμ
2k1 cosθ1
vectorE20⊥·vectorE?20⊥
E10⊥E?10⊥ e
2vectork2I·vectorrvextendsinglevextendsingle
z=0 =
μ1k2zR
μ2k1 cosθ1
vextenddoublevextenddouble 2
1 + μ1k2zμ
2k1 cosθ1
vextenddoublevextenddouble2
R⊥≡?
vectorS3⊥·?vectorS
vectorS1⊥·?vectorS
vextendsinglevextendsingle
z=0 =?
S3⊥z
S1⊥z
vextendsinglevextendsingle
z=0 =?
k3z
2μ1ω(
vectorE30⊥·vectorE?30⊥)
k1z
2μ1ω(
vectorE10⊥·vectorE?10⊥)
=
vectorE30⊥·vectorE?30⊥
E10⊥E?10⊥ =
vextenddoublevextenddouble1? μ1k2zμ
2k1 cosθ1
1 + μ1k2zμ
2k1 cosθ1
vextenddoublevextenddouble2
217/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a85a54
R⊥ + D⊥ =
4μ1k2zR
μ2k1 cosθ1 + (1?
μ1k2z
μ2k1 cosθ1)(1?
μ1k?2z
μ2k1 cosθ1)
bardbl1 + μ1k2zμ
2k1 cosθ1
bardbl2
=
2μ1
μ2k1 cosθ1(k2z + k
2z) + 1?
μ1
μ2k1 cosθ1(k2z + k
2z) +
μ21k2zk?2z
μ22k21 cos2 θ1
bardbl1 + μ1k2zμ
2k1 cosθ1
bardbl2
= 1
(vectorS2⊥?vectorS3⊥)·?vectorS = (D⊥ + R⊥)vectorS1⊥·?vectorS = vectorS1⊥·?vectorS
a108a92a19a197a54a92?Sa27⊥a62a94a85a254a31a117a108?Sa54a209a27 ⊥a27a135a19a197a218a223
a19a197a62a94a85a254a27a27a83a83a92.
218/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a85a54
D⊥ = μ1k2zRμ
2k1 cosθ1
vextenddoublevextenddouble 2
1 + μ1k2zμ
2k1 cosθ1
vextenddoublevextenddouble2 R
⊥ =
vextenddoublevextenddouble1? μ1k2zμ
2k1 cosθ1
1 + μ1k2zμ
2k1 cosθ1
vextenddoublevextenddouble2
Dbardbl =
4μ1 cosθ1Re(k2k?2zk?
2
)
μ2k1vextenddoublevextenddoublek2zk
2
+ k2μ1k
1μ2
cosθ1vextenddoublevextenddouble2
Rbardbl = vextenddoublevextenddouble
k2z
k2?
k2μ1
k1μ2 cosθ1
k2z
k2 +
k2μ1
k1μ2 cosθ1
vextenddoublevextenddouble
a126,γ2 = 0,μ1=μ2,k2=k1 sinθ1/sinθ2,k2z=k2 cosθ2=k1 sinθ1 cothθ2
Rωbardbl = vextenddoublevextenddouble?cosθ2 +
sinθ1
sinθ2 cosθ1
cosθ2 + sinθ1sinθ
2
cosθ1
vextenddoublevextenddouble2 = vextenddoublevextenddoublesin2θ1?sin2θ2
sin2θ1 + sin2θ2
vextenddoublevextenddouble2
= vextenddoublevextenddoublesin(θ1?θ2)cos(θ1 +θ2)sin(θ
1 +θ2)cos(θ1?θ2)
vextenddoublevextenddouble2 = tan2(θ1?θ2)
tan2(θ1 +θ2)
θ1+θ2=pi2===?0
a217a86a100a65a189a198
Rω⊥ = vextenddoublevextenddouble?sinθ1 cothθ2 + cosθ1sinθ
1 cothθ2 + cosθ1
vextenddoublevextenddouble2 = bracketleftbig?sinθ1 cosθ2 + cosθ1 sinθ2
sinθ1 cosθ2 + cosθ1 sinθ2
bracketrightbig2
= sin
2(θ1?θ2)
sin2(θ1 +θ2)
θ1+θ2=pi2===?cos22θ
1
Dbardbl = 1?Rbardbl D⊥ = 1?R⊥
219/384
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220/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a49a216
a62a94a197a108a253a152a19a20a212a159a76a161,a252a160a76a161a201a27a229a143
vectorf =?vectorn· arrowrighttophalfarrowrighttophalfJ =vectorfE +vectorfB
vectorfE = 1
2epsilon10E
2(vectorncos2θE +vectoreE sin2θE)
vectorfB = 1
2μ0B
2(vectorncos2θB +vectoreB sin2θB)
a92a19a197a62a94a124a27a144a149a216a40a189,a51a164a107a144a149a254a209a209a31a31a65a199a209a121:
cos2θ =
integraltext d?cos2θ
integraltext d? =
integraltext2pi
0 dφ
integraltextpi
0 dθsinθcos2θintegraltext
2pi
0 dφ
integraltextpi
0 dθsinθ
=?13
sin2θ =
integraltext d?sin2θ
integraltext d? =
integraltext2pi
0 dφ
integraltextpi
0 dθsinθsin2θintegraltext
2pi
0 dφ
integraltextpi
0 dθsinθ
= 0
vectorf =?vectorn
3(
1
2epsilon10E
2 + 1
2μ0H
2) =?vectorn
3W
221/384
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a20a8a110,a144a167a57a62a94a135
a251a19a78a8a164a27a152a189a62a46a83a27a27a62a62a94a197,a189a21a62a94a197a247a118a181
(?2 + k2)vectorE = 0?·vectorE = 0 vectorB =?iω?×vectorE
k2 = ω2μepsilon1
a62a46a254,γ >>1,a62a94a197a144a85a66a63a19a78a233a102,a51a108a76a161a65a135a66a223a29a221
a63,a162a83a254a174 a118a107a62a94a124
vectorn×vectorE = 0 vectorn×vectorH =vectoric vectorn·vectorD = σf vectorn·vectorB = 0
a51a19a78a76a161a254a62a124a114a221a82a134a117a76a161,a94a97a65a114a221a134a76a161a131a131.
222/384
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a20a8a110,a221a47a47a20a20a8a110
a139a73a6a58a192a51a221a47a20a8a110a27a152a135a14,a177a131a25a27a110a135a62a143a139a73a182.
u(x,y,z)a143Ex,Ey,Eza165a63a152a135
a0
a0a0a9
a45
a54
a0a0
a0a0
a0a0
z
x
y
L1 L2
L3
2u + k2u = 0 u(x,y,z) = X(x)Y(y)Z(z)
2u + k2u=YZ?
2X
x2 +XZ
2Y
y2 +XY
2Z
z2 +(k
2
x + k
2
y + k
2
z)XYZ= 0
d2X
dx2 +k
2
xX=0
d2Y
dy2 +k
2
yY=0
d2Z
dz2 +k
2
zZ=0 k
2=k2
x+k
2
y+k
2
z
u(x,y,z) = (C1 coskxx + D1 sinkxx)(C2 coskyy + D2 sinkyy)
×(C3 coskzz + D3 sinkzz)
x = 0,y = 0,z = 0 a63a62a46a94a135,
Ex y=0;z=0==== 0 Ex=(A1 coskxx+Aprime1 sinkxx)sinkyysinkzz
Ey x=0;z=0==== 0 Ey=sinkxx(A2 coskyy+Aprime2 sinkyy)sinkzz
Ez x=0;y=0==== 0 Ez=sinkxxsinkyy(A3 coskzz+Aprime3 sinkzz)
·vectorE = 0? Aprime1 = Aprime2 = Aprime3 = 0 kxA1 + kyA2 + kzA3 = 0
223/384
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a20a8a110,a221a47a47a20a20a8a110
a0
a0a0a9
a45
a54
a0a0
a0a0
a0a0
z
x
y
L1 L2
L3
Ex = A1 coskxxsinkyysinkzz Ey = A2 sinkxxcoskyysinkzz
Ez = A3 sinkxxsinkyycoskzz x = L1,y = L2,z = L3a63a27a62a94a135→
kx = mpiL
1
ky = npiL
2
kz = ppiL
3
m,n,p = 0,1,2,···
ω2μepsilon1 = k2 = k2x + k2y + k2z = pi2[(mL
1
)2 + ( nL
2
)2 + ( pL
3
)2]
·vectorE = 0 → kxA1 + kyA2 + kzA3 = 0
a137a189(m,n,p)a143a152a171a29a14a14a8a8a11,a252a135a213a225a160a8a197a47,a20a8a170a199a143
ωnmp = pi√μepsilon1
radicalbigg
( nL
1
)2 + (mL
2
)2 + ( pL
3
)2
λnmp = 2piω
mnp
√μepsilon1 = 2radicalBig
( nL
1
)2 + (mL
2
)2 + ( pL
3
)2
a62a94a124a160a117a110a83,a129a36a170a1540a62a94a197a143(110),(101),(011),a197a127a134a20a8a110a130a221a211a254a63.
a135a197a137a140,a230a94a228a107a55a225a57a161a27a20a8a110a23a41a112a170a8a11.
a49a198a137a140,a135a19a186a124a164a49a198a198a20a20a8a110a23a41a67a252a218a27a45a49a229.
224/384
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a62a94a197a27a27a189a189a149a68a194,a144a167a57a62a94a135braceleftbigg
×vectorE = iωvectorB
×vectorB =?iμepsilon1ωvectorE
braceleftbigg?·vectorE = 0
·vectorB = 0 vectorn×
vectorEvextendsinglevextendsingle
a62a46 = 0
(2)vectorE = [vectorE
z(x,y) + vectorEt(x,y)]eikzz?=?xy +
zvectorez
vectorB = [vectorBz(x,y) + vectorBt(x,y)]eikzz?xy =?
xvectorex +
yvectorey
braceleftbigg?iμepsilon1ωvectorE
z =?xy×vectorBt
iωvectorBz =?xy×vectorEt
braceleftbigg?iμepsilon1ωvectorE
t =?xy×vectorBz + ikzvectorez×vectorBt
iωvectorBt =?xy×vectorEz + ikzvectorez×vectorEtbraceleftbigg
xy·vectorEt + ikzEz = 0
xy·vectorBt + ikzBz = 0
a62a46(a43a57)a254a27a27a62a62a54:vectoric = vectorn×vectorH = 1μvectorn×vectorB = 1μvectorn×(vectorBt + vectorBz)ei(kzz?ωt)
a62a46(a43a57)a254a27a27a62a62a214,σf = vectorn·vectorD = epsilon1vectorn·vectorE = epsilon1vectorn·vectorEtei(kzz?ωt)
a41,Ez = Bz = 0 Ez = 0 Bz = 0
225/384
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a62a94a197a27a27a189a189a149a68a194,TEMa197braceleftbigg
xy·vectorEt = 0
xy×vectorEt = 0
braceleftbigg?
xy·vectorBt = 0
xy×vectorBt = 0
braceleftBigg vector
Et =?kzμepsilon1ωvectorez×vectorBt
ω
kz
vectorBt =vectorez×vectorEt
vectorn×vectorEtvextendsinglevextendsinglea62a46 = 0
vectorEt =xyφt?2xyφt = 0 φtvextendsinglevextendsingle
a62a46 = a126a234
a101a165a152,φt = a126a234a240a180a41→vectorEt = vectorBt = 0
a101a165a216a152,a101a83a9a62a46a62a179a131a211φt = a126a234a69a240a180a41.
a101a83a9a62a46a62a179a216a211,a19a145a183a62a175a75,za152a189 vectorE:a252a145a179a124,φta51a62a46a31a179
vectoric = 1
μvectorn×(
vectorBt + vectorBz)ei(kzz?ωt) = 1
μvectorn×(
kz
ωvectorez×
vectorEt)ei(kzz?ωt)
= kzμepsilon1vectorezvectorn·vectorEtei(kzz?ωt) =?kzμωvectorez?φt?nei(kzz?ωt)
σf = epsilon1vectorn·vectorEtei(kzz?ωt) = epsilon1?φt?nei(kzz?ωt) vectoric =? kzμepsilon1ωσfvectorez
226/384
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a62a94a197a27a27a189a189a149a68a194,TEa197
braceleftbigg?
xy·vectorEt = 0
xy×vectorEt = iωvectorBz
braceleftbigg?
xy·vectorBt =?ikzBz
xy×vectorBt = 0braceleftBigg
ω
kz
vectorBt =vectorez×vectorEt
vectorEt =?kz
μepsilon1ωvectorez×
vectorBt + i
μepsilon1ω?xy×
vectorBz vectorn×vectorEt
vextendsinglevextendsingle
a62a46 = 0
ω
kz
vectorBt =vectorez×(? kz
μepsilon1ωvectorez×
vectorBt + i
μepsilon1ω?xy×
vectorBz) = kz
μepsilon1ω
vectorBt + i
μepsilon1ω?xyBz
vectorBt = i
μepsilon1ω
1
ω
kz?
kz
μepsilon1ω
xyBz = ikzμepsilon1ω2?k2
z
xyBz
vectorEt =? kz
μepsilon1ω
ikz
μepsilon1ω2?k2zvectorez×?xyBz +
i
μepsilon1ω?xy×
vectorBz
=?iμepsilon1ω( k
2
z
μepsilon1ω2?k2z + 1)vectorez×?xyBz =
iω
μepsilon1ω2?k2zvectorez×?xyBz
ikz
μepsilon1ω2?k2z?
2
xyBz =?ikzBz → (?
2
xy +μepsilon1ω
2?k2
z)Bz = 0
227/384
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a62a94a197a27a27a189a189a149a68a194,TEa197braceleftbigg
xy·vectorEt = 0
xy×vectorEt = iωvectorBz
braceleftbigg?
xy·vectorBt =?ikzBz
xy×vectorBt = 0braceleftBigg
ω
kz
vectorBt =vectorez×vectorEt
vectorEt =?kz
μepsilon1ωvectorez×
vectorBt + i
μepsilon1ω?xy×
vectorBz vectorn×vectorEt
vextendsinglevextendsingle
a62a46 = 0
vectorBt = ikz
μepsilon1ω2?k2z?xyBz
vectorEt =?iω
μepsilon1ω2?k2zvectorez×?xyBz
(?2xy +μepsilon1ω2?k2z)Bz = 0
vectorn×vectorEtvextendsinglevextendsinglea62a46 =?iωμepsilon1ω2?k2
z
vectorn·?xyBzvextendsinglevextendsinglea62a46 = 0 →?Bz?n vextendsinglevextendsinglea62a46 = 0
kz =
radicalBig
μepsilon1ω2?k2x?k2y≥0 → ω≥k
2
x + k
2
y
μepsilon1 a31a142a170a199
vectoric = 1
μvectorn×(
vectorBt + vectorBz)ei(kzz?ωt) = 1
μ(
ikz
μepsilon1ω2?k2zvectorn×?xy +vectorez)Eze
i(kzz?ωt)
σf = epsilon1vectorn·vectorEtei(kzz?ωt) =?iωepsilon1μepsilon1ω2?k2
z
(vectorn×vectorez)·?xyBzei(kzz?ωt)
228/384
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a62a94a197a27a27a189a189a149a68a194,a221a47a197a19
a221a47a62a127a169a79a143a,b,a139a73a6a58a18a51a221a47a27a152a135a14,a252a131a25a27a62a169
a79a143xya182,a207a143a165a152,a195TEMa197.
a233TEa197:
(?2xy +μepsilon1ω2?k2z)Bz = 0
→ Bz = (a1 sinkxx + b1 coskxx)(a2 sinkyy + b2 coskyy)
Bz
n
vextendsinglevextendsingle
a62a46 = 0 →
Bz
x
vextendsinglevextendsingle
x=0,x=a =
Bz
y
vextendsinglevextendsingle
y=0,y=b = 0
Bz = Acoskxxcoskyy kxa = mpi kyb = npi m,n = 0,1,2,···
(mpia )2 + (npib )2 = ω2μepsilon1?k2z
ωc = 1√μepsilon1
radicalbigg
(mpia )2 + (npib )2 ω≥ωc λ≤λc = 2pi√μepsilon1ω
c
229/384
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a62a94a197a27a27a189a189a149a68a194,a221a47a197a19
a221a47a62a127a169a79a143a,b,a139a73a6a58a18a51a221a47a27a152a135a14,a252a131a25a27a62a169
a79a143xya182,a207a143a165a152,a195TEMa197.
a233TMa197:
(?2xy +μepsilon1ω2?k2z)Ez = 0
→ Ez = (a1 sinkxx + b1 coskxx)(a2 sinkyy + b2 coskyy)
Ezvextendsinglevextendsinglea62a46 = 0 → Ezvextendsinglevextendsinglex=0,x=a = Ezvextendsinglevextendsingley=0,y=b = 0
Ez = Asinkxxsinkyy kxa = mpi kyb = npi m,n = 0,1,2,···
(mpia )2 + (npib )2 = ω2μepsilon1?k2z
ωc = 1√μepsilon1
radicalbigg
(mpia )2 + (npib )2
a211a182a62a67a165a209a121a27a180a183a21a124; a197a19a43a165a209a121a27a180a55a197a124,a107a252a171a197a47TE,TMa197.
230/384
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a62a94a197a27a65a219a219a49a49a198a52a129,a197a196a144a167a27a65a219a219a49a49a198a67a113
a233a154a254a33a48a159a167a63a216epsilon1 = epsilon1(vectorr),a2μa69a143a126a234a27a156a185a181
2vectorE? 1v2?
2vectorE
t2 = (?lnμ)·?
vectorE?(?vectorE)·(?lnμ)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=0
[(?lnepsilon1)·vectorE]
×vectorE =
vectorB
t?·
vectorE =?(?lnepsilon1)·vectorE a154a254a33a253a14a48a159,v2≡ 1
μepsilon1 =
c2
n2(vectorr)
a152a132a27a27a189a189a21a197,vectorE(vectorr,t) = vectorC(vectorr)eiΦ(vectorr)?iωt vectorB(vectorr,t) = vectorCprime(vectorr)eiΦ(vectorr)?iωt
vectorC(vectorr),vectorCprime(vectorr),Φ(vectorr)a143a162a188a234,vectorC(vectorr),vectorCprime(vectorr)a180a8a204,Φ(vectorr)a180a160a131
iωvectorB(vectorr,t) =?×vectorE = [?×vectorC + i(?Φ)×vectorC]bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
iωvectorCprime
eiΦ(vectorr)?iωt
·vectorC + i(?Φ)·vectorC =?(?lnepsilon1)·vectorC
·vectorC =?(?lnepsilon1)·vectorC (?Φ)·vectorC = 0
231/384
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a62a94a197a27a65a219a219a49a49a198a52a129,a197a196a144a167a27a65a219a219a49a49a198a67a113
2vectorE? 1v2?
2vectorE
t2 =[(?lnepsilon1)·
vectorE] v2≡ 1
μepsilon1 =
c2
n2(vectorr)
vectorE(vectorr,t) = vectorC(vectorr)eiΦ(vectorr)?iωt vectorB(vectorr,t) = 1
ω[?i?×
vectorC + (?Φ)×vectorC]eiΦ(vectorr)?iωt
·vectorC =?(?lnepsilon1)·vectorC (?Φ)·vectorC = 0
(?2?1v2?
2
t2)
vectorE(vectorr,t) = [(?2vectorC)eiΦ+2?eiΦ·(?vectorC)+vectorC?2eiΦ+vectorCω2
v2e
iΦ]e?iωt
= eiΦ?iωt[?2vectorC?vectorC(?Φ)·?Φ + vectorCω
2n2
c2 + i(2?Φ·?
vectorC + vectorC?2Φ)]
[(?lnepsilon1)·vectorE] = [(vectorC·?lnepsilon1)?i(?Φ)(?lnepsilon1)·vectorC]eiΦ?iωt
2vectorC?vectorC(?Φ)·?Φ + vectorCω
2n2
c2 =(
vectorC·?lnepsilon1) =?(?·vectorC)
2?Φ·?vectorC + vectorC?2Φ =?(?Φ)(?lnepsilon1)·vectorC = (?Φ)?·vectorC
a144a167a252a62a58a166vectorC,a191a124a94vectorC ·?Φ = 0·(C2?Φ) =?Φ·?C2 + C2?2Φ = 0
232/384
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a62a94a197a27a65a219a219a49a49a198a52a129,a197a196a144a167a27a65a219a219a49a49a198a67a113
2vectorE? 1v2?
2vectorE
t2 =[(?lnepsilon1)·
vectorE] v2≡ 1
μepsilon1 =
c2
n2(vectorr)
vectorE(vectorr,t) = vectorC(vectorr)eiΦ(vectorr)?iωt vectorB(vectorr,t) = 1
ω[?i?×
vectorC + (?Φ)×vectorC]eiΦ(vectorr)?iωt
·vectorC =?(?lnepsilon1)·vectorC (?Φ)·vectorC = 0
2vectorC?vectorC(?Φ)·?Φ + vectorCω
2n2
c2 ==?(?·
vectorC)
2?Φ·?vectorC + vectorC?2Φ = (?Φ)?·vectorC·(C2?Φ) = 0
vectorS = 1
2μωRe(
vectorE×vectorB?) = 1
2μωRe{
vectorC×[i?×vectorC + (?Φ)×vectorC]}= C2?Φ
2μω
·(vectorS) = 0 a173a240a62a94a85a54
a65a219a219a49a49a198a67a113,vectorCa145a152a109a27a67a122a122a133a133a250?
Φ·?Φ = ω2n2c2
vectorC?2Φ = 0 a173a240a62a94a85a54
vectorC·?Φ = 0 vectorC⊥?Φ
vectorC·?epsilon1 = 0 vectorC⊥?epsilon1
233/384
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a62a94a197a27a65a219a219a49a49a198a52a129,a49a198a144a167
vectorE(vectorr,t) = vectorC(vectorr)eiΦ(vectorr)?iωt vectorB(vectorr,t) = 1
ω[?i?×
vectorC + (?Φ)×vectorC]eiΦ(vectorr)?iωt
a65a219a219a49a49a198a67a113,vectorCa145a152a109a27a67a122a122a133a133a250?
Φ·?Φ = ω2n2c2
vectorC?2Φ = 0 a173a240a62a94a85a54
vectorC·?Φ = 0 vectorC⊥?Φ
vectorC·?epsilon1 = 0 vectorC⊥?epsilon1
a49a130a27a59a44a189a194a143a100a31a131a161Φ =a126a234a27a123a130a27a59a44,a94a235a234sa157a54a27
a59a44vectorr(s)a163a227,a51a49a130a59a44a254,ds =√dx2 + dy2 + dz2 =√dr2.
Φ∝a31a131a161Φ =a126a234a27a123a130,dvectorr/ds∝a31a131a161Φ =a126a234a27a123a130.
Φ = advectorrds? a2 = ω2n2c2? a = ωncΦ = ωnc dvectorrds
vectorC⊥dvectorr
ds
vectorCprime= 1
ω?Φ×
vectorC = n
c
dvectorr
ds×
vectorC?a238a197:vectorE,vectorB,dvectorr
dsa164a109a195a39a88
a233a178a161a197,Φ = vectork·vectorr,k2 = ω2n2c2? dvectorrds = vectorkk
234/384
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a62a94a197a27a65a219a219a49a49a198a52a129,a49a198a144a167
vectorE(vectorr,t) = vectorC(vectorr)eiΦ(vectorr)?iωt vectorB(vectorr,t) = 1
ω[?i?×
vectorC + (?Φ)×vectorC]eiΦ(vectorr)?iωt
a65a219a219a49a49a198a67a113,vectorCa145a152a109a27a67a122a122a133a133a250?
Φ·?Φ = ω2n2c2
vectorC?2Φ = 0 a173a240a62a94a85a54
vectorC·?Φ = 0 vectorC⊥?Φ
vectorC·?epsilon1 = 0 vectorC⊥?epsilon1
a49a130a27a59a44a189a194a143a100a31a131a161Φ =a126a234a27a123a130a27a59a44,a94vectorr(s)a163a227.
Φ = ωnc dvectorrds ds =
radicalbig
dx2 + dy2 + dz2 =
√
dr2
a51a49a130a254sa63,dds(ndvectorrds) = cω dds?Φ = cωdvectorrds·Φ = cω
3summationdisplay
i=1
dxi
ds?
Φ
xi
=
3summationdisplay
i=1
dxi
ds?(n
dxi
ds ) = (?n)
dvectorr
ds·
dvectorr
ds+
1
2n?(
dvectorr
ds·
dvectorr
ds) =?n
a254a33a48a159==?dvectorr
ds=a126a234
a254a33a48a159a159a165a165a27a49a114a134a130! Φ(s)≡Φ(vectorr(s)) arrowtripleright dΦds =dvectorrds·?Φ=ωnc dvectorrds·dvectorrds=ωnc
235/384
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a62a94a197a27a65a219a219a49a49a198a52a129,a49a167a141a188a134a164a234a6a110
vectorE(vectorr,t) = vectorC(vectorr)eiΦ(vectorr)?iωt vectorB(vectorr,t) = 1
ω[?i?×
vectorC + (?Φ)×vectorC]eiΦ(vectorr)?iωt
a65a219a219a49a49a198a67a113,vectorCa145a152a109a27a67a122a122a133a133a250?
Φ·?Φ = ω2n2c2
vectorC?2Φ = 0 a173a240a62a94a85a54
vectorC·?Φ = 0 vectorC⊥?Φ
vectorC·?epsilon1 = 0 vectorC⊥?epsilon1
a49a130a27a59a44a189a194a143a100a31a131a161Φ =a126a234a27a123a130a27a59a44,a94vectorr(s)a163a227.
Φ = ωnc dvectorrds ds =
radicalbig
dx2 + dy2 + dz2 =
√
dr2 dds(ndvectorrds) =?n
Φ(s)≡Φ(vectorr(s)) arrowtripleright dΦds = ωnc arrowtripleright Φ(s1)?Φ(s2) = ωc
integraldisplay s2
s1
ds n(vectorr(s))
a49a167a141a188,S(vectorr1,vectorr2)≡
integraldisplay s2
s1
ds n(vectorr(s)) =
integraldisplay vectorr2
vectorr1
√
dr2n(vectorr) vectorr1=vectorr(s1),vectorr2=vectorr(s2)
a164a234a6a110,a162a83a27a49a130a59a44a166a49a167a141a188a18a52a138 !
236/384
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a62a94a197a27a65a219a219a49a49a198a52a129,a49a167a141a188a134a164a234a6a110
vectorE(vectorr,t) = vectorC(vectorr)eiΦ(vectorr)?iωt vectorB(vectorr,t) = 1
ω[?i?×
vectorC + (?Φ)×vectorC]eiΦ(vectorr)?iωt
a65a219a219a49a49a198a67a113,vectorCa145a152a109a27a67a122a122a133a133a250?
vectorC?2Φ = 0 a173a240a62a94a85a54
vectorC·?Φ = 0 vectorC⊥?Φ
vectorC·?epsilon1 = 0 vectorC⊥?epsilon1
Φ = ωnc dvectorrds ds =
radicalbig
dx2 + dy2 + dz2 =
√
dr2 dds(ndvectorrds) =?n
a49a167a141a188,S(vectorr1,vectorr2)≡
integraldisplay s2
s1
ds n(vectorr(s)) =
integraldisplay vectorr2
vectorr1
√
dr2n(vectorr) vectorr1=vectorr(s1),vectorr2=vectorr(s2)
a164a234a6a110,a162a83a27a49a130a59a44a166a49a167a141a188a18a52a138 !
0 = δS =
integraldisplay
[(δn)
√
dr2 + ndvectorr·δdvectorr√dr2 ] =
integraldisplay
[(?n)·δvectorr
√
dr2 + ndvectorr·dδvectorr√dr2 ]
=
integraldisplay
[(?n)·δvectorr
√
dr2?d( ndvectorr√dr2)·δvectorr] =
integraldisplay
[?n? d√dr2( ndvectorr√dr2)]·δvectorr
√
dr2
=
integraldisplay
[?n? dds(ndvectorrds )]·δvectorr
√
dr2 arrowtripleright dds(ndvectorrds ) =?n
237/384
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a207a165a127a193
238/384
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a207a165a127a193
a158a109,07a9912a271a70(a40a207a56) a101a2042:00-5:00
a47a47a58a58,a152a19205
a137a166,11a2730a70(a40a207a202)a20a2547:00-9:00 a47a47a58a58a110a137a1623408
12a271a70(a40a207a56)a254a2049:00-11:30 a47a47a58a58a110a137a1623408
a193a75,5a75a1924a75a182a122a7525a169a182a183a62a94a1243a75a33a68a1942a75
a127a123,a109a242(a140a145a214a33a249a194a33a41a80a33a138a146a29); a247a169100a169 a163a177a249a194a254a27a83a78a143a79a164
239/384
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a83a75a49a19a217a492a33a214a191a75(a):
a100a19a78a124a164a27a78a88,a174a127a49ia135a19a78a254a27a111a62a214qi,a133a19a78a9a140a177a107a48
a159,a2a195a103a100a62a214,a166a121a152a109a63a191a152a58vectorra63a27a62a179a134qia27a39a88a180a130a53
a224a103a27,a61,φ(vectorr,q1,q2,...,qn) =
nsummationdisplay
i=1
pi(vectorr)qi a217a165pi(vectorr)a134qia195a39.
a121a178:
a49a152a218,φ = φ1 +φ2 +···+φn,φia180a19a78ia145qi,a217a167a216a145a62a158a27a169
a217,a175a175a162a162a254,a122a152a135φka247a118
·[epsilon1?φk] = 0 φkvextendsinglevextendsingleS
k
= a126a234 φkvextendsinglevextendsingleS
i
= a126a234contintegraldisplay
Sk
dvectorS·epsilon1?φk =?qk
contintegraldisplay
Si
dvectorS·epsilon1?φk = 0
·[epsilon1?φ] =
nsummationdisplay
i=1
·[epsilon1?φi] = 0 φvextendsinglevextendsingleS
i
=
nsummationdisplay
j=1
φjvextendsinglevextendsingleS
i
= a126a234
contintegraldisplay
Si
dvectorS·epsilon1?φ =
nsummationdisplay
j=1
contintegraldisplay
Si
dvectorS·epsilon1?φj =?qi φa180a41
240/384
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a83a75a49a19a217a492a33a214a191a75(a):
a100a19a78a124a164a27a78a88,a174a127a49ia135a19a78a254a27a111a62a214qi,a133a19a78a9a140a177a107a48
a159,a2a195a103a100a62a214,a166a121a152a109a63a191a152a58vectorra63a27a62a179a134qia27a39a88a180a130a53
a224a103a27,a61,φ(vectorr,q1,q2,...,qn) =
nsummationdisplay
i=1
pi(vectorr)qi a217a165pi(vectorr)a134qia195a39.
a49a152a218,φ = φ1 +φ2 +···+φn,φia180a19a78ia145qi,a217a167a216a145a62a158a27a169
a217.
a49a19a218,a121a178φi = pi(vectorr)qia189a31a100a47a121 φi(vectorr,λqi) = λφi(vectorr,qi).
·[epsilon1?φi] = 0
φivextendsinglevextendsingleS
j
= a126a234
contintegraldisplay
Sj
dvectorS·epsilon1?φi =?qiδij
·[epsilon1?λφi] = 0
λφivextendsinglevextendsingleS
j
= a126a234
contintegraldisplay
Sj
dvectorS·epsilon1?λφi =?λqiδij
a61φi(vectorr,λqi) = λφi(vectorr,qi),a121a46.
241/384
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a83a752.19:
a140a187r0a27a165a161a167a51a165a139a730 < θ < pi/2a27a140a165a161a254a27a62a179a143φ0,a51
pi/2 <θ<pia27a140a165a161a254a27a62a179a143?φ0,a94a130a21a188a234a144a123a166a152a109a136a58
a27a27a62a62a179a169a217a186
φ(vectorr) = 14piepsilon1
0
integraldisplay
∞
dτprimeρ(vectorrprime)parenleftbig 1|vectorr?vectorrprime|?
r0
r
|vectorrprime?r20r2vectorr|
parenrightbig
14pi
integraldisplay
r0
dvectorSprimeφ(vectorrprime)·?primeparenleftbig 1|vectorr?vectorrprime|?
r0
r
|vectorrprime?r20r2vectorr|
parenrightbig
dvectorSprime =?vectorerprime r20d?prime,a166a41a171a27a9a123a130!
=
1
4pi
integraltext
r0 dS
primeφ(vectorrprime)·?
rprime
parenleftbig 1
|vectorr?vectorrprime|?
r0
r
|vectorrprime?r
20
r2vectorr|
parenrightbig r>r
0 (?dvectorSprime//vectorerprime)
14pi integraltextr
0
dSprimeφ(vectorrprime)·rprimeparenleftbig 1|vectorr?vectorrprime|? r0r
|vectorrprime?r
20
r2vectorr|
parenrightbig r<r
0 (dvectorSprime//vectorerprime)
242/384
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a83a752.19:
a140a187r0a27a165a161a167a51a165a139a730 < θ < pi/2a27a140a165a161a254a27a62a179a143φ0,a51
pi/2 <θ<pia27a140a165a161a254a27a62a179a143?φ0,a94a130a21a188a234a144a123a166a152a109a136a58
a27a27a62a62a179a169a217a186
Pn(cosΘ) =
nsummationdisplay
m=?n
(n?m)!
(n + m)!P
m
n (cosθ)P
m
n (cosθ
prime)eim(φ?φprime)
cosΘ = cosθcosθprime + sinθsinθprimecos(φ?φprime)
a1
a1
a1
a1
a1
a1
a1
a1
a1
a1a1a21
a0
a0
a0
a0
a0
a0
a0
a0a0a18
Θintegraldisplay
d?prime f(θ
prime)
|vectorr?vectorrprime| =
integraldisplay
d?primef(θprime)
∞summationdisplay
n=0
rn<
rn+1> Pn(cosΘ)
=
integraldisplay 2pi
0
dφprime
integraldisplay pi
0
dθprimesinθprimef(θprime)
∞summationdisplay
n=0
nsummationdisplay
m=?n
rn<
rn+1>
(n?m)!
(n + m)!P
m
n (cosθ)P
m
n (cosθ
prime)eim(φ?φprime)
a83a75a41a137a27a63a216a216a233a156 = 2pi
∞summationdisplay
n=0
rn<
rn+1> Pn(cosθ)
integraldisplay pi
0
dθprimesinθprimef(θprime)Pn(cosθprime)
243/384
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a83a75
14pi
integraldisplay
r0
dSprimeφ(vectorrprime) 1|vectorr?vectorrprime| Pn(1) = 1 Pn(0) n=a243==== (?1)n2 1·3·5···(n?1)2·4·6···n Pn(0) n=a219==== 0
=?r
2
0
4pi
integraldisplay
d?prime[θ(pi2?θprime)φ0?θ(θprime?pi2)θ(pi?θprime)φ0] 1|vectorr?vectorrprime|
=?r
2
0
2
∞summationdisplay
n=0
rn<
rn+1> Pn(cosθ)
integraldisplay pi
0
dθprimesinθprimePn(cosθprime)[θ(pi2?θprime)φ0?θ(θprime?pi2)θ(pi?θprime)φ0]
= r
2
0φ0
2
∞summationdisplay
n=0
rn<
rn+1> Pn(cosθ)[
integraldisplay 0
1
dx Pn(x)?
integraldisplay?1
0
dx Pn(x)] Pn(?x)=(?)nPn(x)
=?r20φ0
∞summationdisplay
n=1,3,···
rn<
rn+1> Pn(cosθ)
Pn+1(x)?Pn?1(x)
2n + 1
vextendsinglevextendsingle1
0
R 1
0 dxPn(x)=
Pn+1(x)?Pn?1(x)
2n+1
1
0
= r20φ0
∞summationdisplay
n=1,3,···
rn<Pn(cosθ)
(2n + 1)rn+1> [(?1)
n+1
2
1·3···n
2·4···(n+1)?(?1)
n?1
2
1·3···(n?2)
2·4···(n?1)]
= r20φ0
∞summationdisplay
n=1,3,···
rn<Pn(cosθ)
(2n + 1)rn+1> (?1)
n?1
2
1·3···(n?2)
2·4···(n + 1)[?n?n?1]
244/384
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a83a75
14pi
integraldisplay
r0
dSprimeφ(vectorrprime)[θ(r0?r)?θ(r?r0)]rprime 1|vectorr?vectorrprime|
=?r20φ0
∞summationdisplay
n=1,3,···
Pn(cosθ)[θ(r0?r)?θ(r?r0)]rprime r
n
<
rn+1> (?1)
n?1
2
1·3···(n?2)
2·4···(n + 1)
= r20φ0
∞summationdisplay
n=1,3,···
Pn(cosθ)(?1)n?12 1·3···(n?2)2·4···(n + 1)×
braceleftBigg (n+ 1) rn
rn+20 r<r0
nrn?10rn+1 r>r0
14pi
integraldisplay
r0
dSprimeφ(vectorrprime)[θ(r0?r)?θ(r?r0)]rprime
r0
r
|vectorrprime?r20r2vectorr|
=?r20φ0
∞summationdisplay
n=1,3,···
Pn(cosθ)(?1)n?12 1·3···(n?2)2·4···(n+1)[θ(r0?r)?θ(r?r0)]rprime
r0(r
20
r )
n
rrprime(n+1)
r20
r<r0
r0rprimen
r(r
20
r )n+1
r20
r>r0
=?r20φ0
∞summationdisplay
n=1,3,···
Pn(cosθ)(?1)n?12 1·3···(n?2)2·4···(n + 1)
braceleftBigg
(n+ 1)rn?10rn+1 r>r0
n rnrn+2
0
r<r0
245/384
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a83a75
14pi
integraldisplay
r0
dSprimeφ(vectorrprime)[θ(r0?r)?θ(r?r0)]rprime 1|vectorr?vectorrprime|
= r20φ0
∞summationdisplay
n=1,3,···
Pn(cosθ)(?1)n?12 1·3···(n?2)2·4···(n + 1)×
braceleftBigg (n+ 1) rn
rn+20 r<r0
nrn?10rn+1 r>r0
14pi
integraldisplay
r0
dSprimeφ(vectorrprime)[θ(r0?r)?θ(r?r0)]rprime
r0
r
|vectorrprime?r20r2vectorr|
=?r20φ0
∞summationdisplay
n=1,3,···
Pn(cosθ)(?1)n?12 1·3···(n?2)2·4···(n + 1)
braceleftBigg
(n+ 1)rn?10rn+1 r>r0
n rnrn+2
0
r<r0
φ(vectorr) =?14pi
integraldisplay
r0
dSprimeφ(vectorrprime)[θ(r0?r)?θ(r?r0)]rprimeparenleftbig 1|vectorr?vectorrprime|?
r0
r
|vectorrprime?r20r2vectorr|
parenrightbig
= φ0
∞summationdisplay
n=1,3,···
Pn(cosθ)(?1)n?12 1·3···(n?2)2·4···(n + 1)(2n + 1)×
rn
rn0 r<r0
rn+10
rn+1 r>r0
246/384
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a83a75a240a189a189a62a62a54a27a27a62a62a124a214a191a83a75:
a252a135a229a108a233a15a140a187a143aa27a55a225a140a165a238a92a62a19a199a143γa27a47a161,a166a140a165
a178a161a220 a176a134a47a161a178a224.a51a100a252a140a165a62a52a254a92a62a216V,a166a47a161a101a52
a161a78a67a62a54a169a217,a191a166a252a140a165a109 a27a62a123a218a122a135a140a165a27a26a47a62a123?
φ = cr
1
cr
2
V = (ca?cd)?(cd?ca) = 2ca?2cd →c = V2
a?
2
d
≈Va2
vectorj =?γ?( c
r1?
c
r2) =
γVa
2 (
vectorr1
r31?
vectorr2
r32)
I =
integraldisplay
dvectorS·vectorj≈γVa2a2 2pia2 = aVγpi a252a140a165a109a62a123 = VI = 1aγpi
φ = φ1 +φ2 φ1 = cr
1
a62a521a62a54a54a54a54a149a195a161a15 φ2 =?cr
1
a195a161a15a62a54a54a54a54a149a62a522
a122a135a140a165a27a26a47a47a62a62a123 = c/aI = V/2aVpi = 12aγpi
247/384
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a83a75a240a189a189a62a62a54a27a27a62a62a124a214a191a83a75:
a51a152a172a195a129a140a27a19a62a178a134a254a44a58pa63a207a92a62a54a191a149a195a161a15a54a209,a51a134a254a63a191a63a109 a152
a11a154,a216a157a185pa58.a193a121a51a100a11a154a62a14a254a63a191a252a58a27a62a179a11a131a138a143a216a109a11a154a158a84a252a58
a27a27a62a62a179a11a131a131a138a138a27a152a21
a38a37
a39a36
a114
vectord
p
a114
r20
d2
vectord
φ=cln|vectorr?vectord|+cprimeln|vectorr?a
2
d2
vectord|?c”lnr vectorE=?c(vectorr?vectord)
|vectorr?vectord|2?
cprime(vectorr?a2d2vectord)
|vectorr?a2d2vectord|2 +
c”vectorr
r2
vectorer·vectorEvextendsinglevextendsingler=a =vectorer·bracketleftbig c(vectorr?
vectord)
a2 + d2?2adcosθ +
cprime(vectorr?a2d2vectord)
a2 + a4d2?2aa2d cosθ?
c”vectorr
a2
bracketrightbig
= c(a?dcosθ) + c
primed2
a2(a?
a2
d cosθ)?
c”
a (d
2 + a2?2adcosθ)
d2 + a2?2adcosθ
= ca+c
primed2
a?
c”
a (d
2+a2)?(cd+cprimed?2c”d)cosθ
d2 + a2?2adcosθ
cprime=c=c”→0
cln|vectorr?vectord|= cln|vectorr1?
vectord|
|vectorr2?vectord| =
c
2 ln
a2 + d2?2adcosθ1
a2 + d2?2adcosθ2
cprimeln|vectorr?a
2
d2
vectord|= cln|vectorr1?a
2
d2
vectord|
|vectorr2?a2d2vectord| =
c
2 ln
a2 + a4d2?2aa2d cosθ1
a2 + a4d2?2aa2d cosθ2
248/384
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a83a753.2:
a254a33a195a161a127a134a11a206a47a218a130a43a167a122a252a160a127a221a130a23a42a234a143n,a62a54a114a221
a143I,a193a94a141a152a53a189a110a166a43a83a9a94a97a65a114a221vectorBa186
a43a83a9:?·vectorB = 0?×vectorB = 0
a43a57a254,vectorn·(vectorBa9?vectorBa83) = 0 μ0vectorer×(vectorBa9?vectorBa83) =vectoreθnI
a195a161a15,vectorB→0
a223a41a181 vectorB0a9 = 0 vectorB0a83 = nIμ
0
vectorez vectorer×vectorez =?vectoreθ
vectorB = vectorB0 + vectorBprime
a43a83a9:?·vectorBprime = 0?×vectorBprime = 0
a43a57a254,vectorn·(vectorBprimea9?vectorBprimea83) = 0 μ0vectorer×(vectorBprimea9?vectorBprimea83) = 0
a195a161a15,vectorBprime→ 1r2
a183a62a141a152a53a189a110→vectorBprime = 0
249/384
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a83a753.13:
a107a152a135a254a33a145a62a27a0a19a78a138,a217a140a187a143R0,a111a62a214a143Q,a56a166a165a138a55a103
a28a44a152a134a187a177a14a132a221ωa61a196,a166a166a165a165a83a9a27a94a124vectorB?
vectorBa83 =?μ0?φa83 vectorBa9 =?μ0?φa9
a165a83a9:?2φa83 = 0?2φa9 = 0 a195a161a15,vectorB→ 1r2
a165a138a254,?φa83?rφa9?r = 0
μ0vectorer×(?φa83φa9) = Q4piR2
0
ωvectorez×R0vectorer =? Q4piR
0
vectoreφ?cosθ?θ
vectorer×?φ =vectorer×(vectorer?φ?r +vectoreθ1r?φ?θ +vectoreφ 1rsinθ?φ?φ) =vectoreφ 1R
0
φ
θ
μ0(?φa83?θφa9?θ )vextendsinglevextendsingler=R
0
=?Q4pi?cosθ?θ
μ0(φa83?φa9)vextendsinglevextendsingler=R
0,θa157a54
=?Q4piP1(cosθ)
250/384
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a83a75a62a54a78a88a245a63a208a109a214a191a83a75:
W =
∞summationdisplay
n=0
Wn Wn = 1n
3summationdisplay
i1,···in
vectorJi
1···in·[
n
xi1···?xin
vectorA(vectorr)]
W0 = 0 W1 = vectorm·vectorB
vectorF =
∞summationdisplay
n=0
vectorFn vectorFn = 1
n
3summationdisplay
i1,···in
vectorJi
1···in×[
n
xi1···?xin
vectorB(vectorr)]
vectorF0 = 0 vectorF1 = (?vectorB)·vectorm
vectorL =
∞summationdisplay
n=0
vectorLn vectorJi
1···in =
3summationdisplay
i
vectoreiJii1···in
vectorLn = 1
n
3summationdisplay
i,i1,···in
braceleftbigvectorJ
ii1···in[
n
xi1···?xinBi(vectorr)]?J
i
ii1···in[
n
xi1···?xin
vectorB(vectorr)]bracerightbig
vectorL0 = vectorm×vectorB
251/384
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a83a75a62a54a78a88a245a63a208a109a214a191a83a75:
W =
integraldisplay
dτvectorj(vectorr)·vectorA(vectorr)
=
integraldisplay
dτvectorj(vectorr)·braceleftbigvectorA(0) +
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
xi1xi2···xinbracketleftbigx
i1
···x
in
vectorA(vectorr)bracketrightbig
vectorr=0
bracerightbig
=
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
vectorJi
1i2...in·
bracketleftbig?
xi1
xi2···
xin
vectorA(vectorr)bracketrightbig
vectorr=0 =
∞summationdisplay
n=0
Wn
W0 = 0 Wn = 1n!
3summationdisplay
i1i2...in=1
vectorJi
1i2...in·
bracketleftbig?
xi1
xi2···
xin
vectorA(vectorr)bracketrightbig
vectorr=0 nnegationslash= 0
W1 =
3summationdisplay
i=1
vectorJi·[?
xi
vectorA(vectorr)]bracketrightbig
vectorr=0 =
1
2
3summationdisplay
i,k=1
integraldisplay
dτprime(xprimeijk?xprimekji)[?Ak?x
i
]bracketrightbigvectorr=0
= 12
3summationdisplay
i,j,k,l,m=1
epsilon1jlmepsilon1jik
integraldisplay
dτprimexprimeljm[?Ak?x
i
]bracketrightbigvectorr=0 =
3summationdisplay
j=1
mjBj(0) = vectorm·vectorB(0)
252/384
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a83a75vectorF =
integraldisplay
dτvectorj(vectorr)×vectorB(vectorr)
=
integraldisplay
dτvectorj(vectorr)×
∞summationdisplay
n=0
1
n!
3summationdisplay
i1i2...in=1
xi1xi2···xinbracketleftbigx
i1
···x
in
vectorB(vectorr)bracketrightbig
vectorr=0
=
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
vectorJi
1i2...in×
bracketleftbig?
xi1
xi2···
xin
vectorB(vectorr)bracketrightbig
vectorr=0 =
∞summationdisplay
n=0
vectorFn
vectorFn = 1
n!
3summationdisplay
i1i2...in=1
vectorJi
1i2...in×
bracketleftbig?
xi1
xi2···
xin
vectorB(vectorr)bracketrightbig
vectorr=0
vectorF0 = 0
vectorF1 =
3summationdisplay
i=1
vectorJi×?
xi
vectorB(vectorr)vextendsinglevextendsingle
vectorr=0 =
1
2
3summationdisplay
i,k=1
integraldisplay
dτprime(xprimeijk?xprimekji)vectorek×?
vectorB(vectorr)
xi
vextendsinglevextendsingle
vectorr=0
= 12
3summationdisplay
i,k,l,n=1
epsilon1kln
integraldisplay
dτprime(xprimeijk?xprimekji)vectoren?Bl(vectorr)?x
i
vextendsinglevextendsingle
vectorr=0 =
3summationdisplay
i,j,k,l,n=1
mjepsilon1jikepsilon1klnvectoren?Bl(vectorr)?x
i
vextendsinglevextendsingle
vectorr=0
=
3summationdisplay
i,j,k,l,n=1
mj(δjlδin?δjnδil)vectoren?Bl?x
i
vextendsinglevextendsingle
vectorr=0=
3summationdisplay
i,j=1
mjvectorei?Bj?x
i
vextendsinglevextendsingle
vectorr=0=
3summationdisplay
i,j=1
mjvectorei?Bi?x
j
vextendsinglevextendsingle
vectorr=0=[vectorm·?
vectorB]vextendsinglevextendsingle
vectorr=0
253/384
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a83a75
vectorL =
integraldisplay
dτvectorr×[vectorj(vectorr)×vectorB(vectorr)] =
integraldisplay
dτbracketleftbig[vectorr·vectorB(vectorr)]vectorj(vectorr)?[vectorr·vectorj(vectorr)]vectorB(vectorr)]bracketrightbig
=
integraldisplay
dτ
bracketleftBig
xiBi(vectorr)vectorj(vectorr)?xiji(vectorr)vectorB(vectorr)
bracketrightBig
=
∞summationdisplay
n=0
vectorLn
vectorLn = 1
n!
3summationdisplay
i,i1i2...in=1
braceleftbigg
vectorJii
1...in
xi1···
xinBi(vectorr)?J
i
ii1...in
xi1···
xin
vectorB(vectorr)
bracerightbigg
vectorr=0
vectorL0 =
3summationdisplay
i=1
[vectorJiBi(0)?JiivectorB(0)] =
3summationdisplay
i,k=1
integraldisplay
dτprimexprimeijkvectorekBi(0)
Jii≡
integraldisplay
dτprimexprimeiji = 12
integraldisplay
dτprime?primel[xprimeixiprimejl] = 0 xprimeijk = 12(xprimeijk+xprimekji+xprimeijk?xprimekji)
= 12
3summationdisplay
i,k=1
integraldisplay
dτprime[(xprimeijk?xprimekji)vectorekBi(0)] =
3summationdisplay
i,k,n=1
mnepsilon1nikvectorekBi(0) = vectorm×vectorB(0)
254/384
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a126a75
a62a254a143qa27a27a58a58a58a62a62a214a26a152a51a229a195a161a140a26a47a47a19a19a78a134aa63a167a166a217a201a229a186
a177a58a58a62a62a214a134a217a186a148a62a214a27a235a130a165a58a143a139a73a6a58a181 φ= q4piepsilon1
0|vectorr?avectorex|
q4piepsilon1
0|vectorr+avectorex|
a114a
q
a114
q
a
vectorf = qvectorEa9vextendsinglevextendsingle
vectorr=avectorex =?q?[?
q
4piepsilon10|vectorr + avectorex|]
vextendsinglevextendsingle
vectorr=avectorex
=? q
2(vectorr + avectorex)
4piepsilon10|vectorr + avectorex|3
vextendsinglevextendsingle
vectorr=avectorex =?
q2
4piepsilon10(2a)2vectorex
fx =?qa?q4piepsilon1
0(2a)
=? q
2
4piepsilon10(2a2) a131a11a207a1022 a186
a177a186a148a62a214a27a160a152a143a139a73a6a58a181 φ = q4piepsilon1
0|vectorr?2avectorex|
q4piepsilon1
0r
vectorf=qvectorEa9vextendsinglevextendsingle
vectorr=2avectorex=?q?[?
q
4piepsilon10r]
vextendsinglevextendsingle
vectorr=2avectorex=?
q2vectorr
4piepsilon10r3
vextendsinglevextendsingle
vectorr=2avectorex=?
q2
4piepsilon10(2a)2vectorex
255/384
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a83a75a62a94a197a51a46a161a254a27a135a19a218a242a19a214a191a83a75(c):
Dbardbl≡
vectorS2bardbl·?vectorS
vectorS1bardbl·?vectorS
vextendsinglevextendsingle
z=0 =
S2bardblz
S1bardblz
vextendsinglevextendsingle
z=0
vectorE20bardbl=E10bardbl(vectore1k2z
k2?vectore3
k2x
k2 )
2cosθ1
k2z
k2 +
μ1k2
μ2k1cosθ1
=
1
2μ2ω[k2zR(
vectorE2bardbl·vectorE?2bardbl)?Re(E2bardblz + E2⊥z)(vectork2·vectorE?2bardbl)]
k1z
2μ1ω(E10bardblE
10bardbl)
vextendsinglevextendsingle
z=0
= μ1μ
2k1cosθ1
braceleftBigg
k2zR
4cos2θ1(k2zk
2
k?2z
k?2 +
k2x
k2
k2x
k?2)
bardblk2zk
2
+μ1k2μ
2k1
cosθ1bardbl2 +
Re[(k2xk?2zk?
2
k2xk2zk?
2
)k2xk
2
]4cos2θ1
bardblk2zk
2
+μ1k2μ
2k1
cosθ1bardbl2
bracerightBigg
= 4μ1 cosθ1μ
2k1
k2zR(k2zk?2z+k2xk?2xk
2k?2
) + Re[k2xk2xk
2k?2
(k?2z?k2z)]
bardblk2zk
2
+ μ1k2μ
2k1
cosθ1bardbl2
a129a0a249a145=0
= 4μ1 cosθ1μ
2k1
Re(k2zk2zk?2zk
2k?2
+ k?2zk2xk2xk
2k?2
)
bardblk2zk
2
+ μ1k2μ
2k1
cosθ1bardbl2 =
4μ1 cosθ1Re(k2k?2zk?
2
)
μ2k1bardblk2zk
2
+ μ1k2μ
2k1
cosθ1bardbl2
Rbardbl≡?
vectorS3bardbl·?vectorS
vectorS1bardbl·?vectorS
vextendsinglevextendsingle
z=0 =?
S3bardblz
S1bardblz
vextendsinglevextendsingle
z=0 =
vectorE30bardbl·vectorE?30bardbl
vectorE10bardbl·vectorE?10bardbl =
vextenddoublevextenddouble?k2zk
2
+ μ1k2μ
2k1
cosθ1
k2z
k2 +
μ1k2
μ2k1 cosθ1
vextenddoublevextenddouble2
256/384
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a83a75
a62a94a197a51a48a161a254a27a135a19a218a242a19a214a191a83a75(c):
Dbardbl =
4μ1 cosθ1Re(k2k?2zk?
2
)
μ2k1bardblk2zk
2
+ μ1k2μ
2k1
cosθ1bardbl2 Rbardbl =
vextenddoublevextenddouble?k2zk
2
+ μ1k2μ
2k1
cosθ1
k2z
k2 +
μ1k2
μ2k1 cosθ1
vextenddoublevextenddouble2
Rbardbl + Dbardbl
=
k2zk?2z
k2k?2?
μ1 cosθ1
μ2k1 (
k2k?2z
k?2 +
k?2k2z
k2 ) +
μ21 cos2 θ1
μ22k21 k2k
2 +
2μ1 cosθ1
μ2k1 (
k2k?2z
k?2 +
k?2k2z
k2 )
bardblk2zk
2
+ μ1k2μ
2k1
cosθ1bardbl2
= 1
(vectorS2bardbl?vectorS3bardbl)·?vectorS = (Dbardbl + Rbardbl)vectorS1bardbl·?vectorS = vectorS1bardbl·?vectorS
257/384
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a207a165a127a193
258/384
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a207a165a127a193
a138a146,a174a101a136a65a17a45 a156
a209a75a103a180,a67a67a88a88a204a135a212a110a83a78a182 a8a127a160a234a198a144a161a27a83a78a156
a202a192a111a27a83a78:
– a183a142a27a27a62a62a214a169a217a23a41a27a27a62a62a124
– a240a189a189a27a27a27a62a62a54a169a217a23a41a27a27a62a62a124
– a240a189a189a27a27a27a62a62a54a169a217a23a41a27a94a124
– a189a21a62a94a197a51a195a161a140a178a161a46a161a254a27a135a223a19
– a62a94a197a27a27a189a189a149a68a194
a141a152a53a189a110a216a252a213a127
a130a21a188a234a216a252a213a127
a245a52a208a109a216a252a213a127
259/384
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a62a94a197a27a203a19
260/384
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a62a94a124a27a165a165a179a179a218a73a179,a94a179a163a227a62a94a124; a53a137a67a134a218a218a53a53a137a216a216a67a67a53
·vectorE = ρepsilon1
0
·vectorB = 0?×vectorE =
vectorB
t?×
vectorB=μ0vectorj+μ0epsilon10?vectorE
t
×vectorE =t?×vectorA =×?
vectorA
t →?×(
vectorE +?vectorA
t ) = 0
vectorE +?vectorA
t =φ →
vectorE =φvectorA
t
braceleftbigg vectorE =φvectorA
tvector
B =?×vectorA
vectorA,φa216a141a152
braceleftbigg vectorA→vectorAprime = vectorA +?χ
φ→φprime = φχ?t
a122a152a124(vectorA,φ)a23a152a171a53a137,a216a211a53a137a233a65a211a152a124vectorE,vectorB,vectorE,vectorBa180a212a110a42a9a254.
a164a107a212a110a110a254a254a218a212a110a53a198a134a65a207a27a53a137a192a74a195a39—a53a137a216a216a67a67a53!
a51a234a198a254a152a171a216a67a53a210a233a65a152a171a233a161a53,a134a254a161a53a137a216a67a53a233a65a27a233a161a53a23U(1)a53
a137a233a161a53,a207a100,n a62a94a131a112a138a94a228a107U(1)a53a137a233a161a53.
a51a162a83a79a142a165,a192a74a152a189a27a94a135a53a114vectorA,φa164a228a107a27a216a40a189a27a53a137a103a100a221a129a155a52,a192
a74a152a171a53a137a27a189a94a135a23a192a152a171a53a137.
a165a212a53a137,?·vectorA = 0 a226a226a212a212a91a53a137,?·vectorA + 1c2?φ?t = 0
261/384
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a62a94a124a27a165a165a179a179a218a73a179,a136a75a203(d’Alembert)a144a167
×vectorB = μ0vectorj +μ0epsilon10?
vectorE
t →?×(?×
vectorA)=μ0vectorj+μ0epsilon10[φ
t?
2vectorA
t2 ]
·vectorE = ρepsilon1
0
→2φt?·vectorA = ρepsilon1
0
a165a212a53a137,?·vectorA = 0
2vectorA? 1c2?
2vectorA
t2 =?μ0
vectorj + 1
c2
t?φ =?μ0
vectorj2φ =?ρ
epsilon10
a226a226a212a212a91a53a137,?·vectorA + 1c2?φ?t = 0
2vectorA? 1c2?
2vectorA
t2 =?μ0
vectorj?2φ? 1
c2
2φ
t2 =?
ρ
epsilon10
c→∞a158,a163a20a183a62a94a156a47
262/384
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a237a180a179,a154a224a103(a107a13)a197a196a144a167a27a165a161a197a41
2φ(vectorr,t)? 1c2?
2
t2φ(vectorr,t) =?
ρ(vectorr,t)
epsilon10
2vectorrU(vectorr,vectorrprime,t)? 1c2?
2
t2U(vectorr,vectorr
prime,t) =?1
epsilon10δ(vectorr?vectorr
prime)ρ(vectorrprime,t)
φ(vectorr,t) =
integraldisplay
dτprime U(vectorr,vectorrprime,t)
U(vectorr,vectorrprime,t) = U(R,vectorrprime,t) vectorR =vectorr?vectorrprime =
3summationdisplay
i=1
Xivectorei
2vectorrU(R,vectorrprime,t) =
3summationdisplay
i=1
i?iU(R,vectorrprime,t) =
3summationdisplay
i=1
i[?U?R?iR] =
3summationdisplay
i=1
i[?U?RXiR]
=
3summationdisplay
i=1
parenleftbigXi
R
2U
R2?iR+
1
R
U
R+
U
RXi?i
1
R
parenrightbig=3summationdisplay
i=1
parenleftbigXiXi
R2
2U
R2+
1
R
U
R?
XiXi
R3
U
R
parenrightbig
=?
2U
R2 +
2
R
U
R =
1
R[
2U
R2R + 2
U
R] =
1
R
2(RU)
R2
263/384
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a237a180a179,a154a224a103(a107a13)a197a196a144a167a27a165a161a197a41
2φ(vectorr,t)? 1c2?
2
t2φ(vectorr,t) =?
ρ(vectorr,t)
epsilon10
2vectorrU(vectorr,vectorrprime,t)? 1c2?
2
t2U(vectorr,vectorr
prime,t) =?1
epsilon10δ(vectorr?vectorr
prime)ρ(vectorrprime,t)
φ(vectorr,t) =
integraldisplay
dτprime U(vectorr,vectorrprime,t) U(vectorr,vectorrprime,t) = U(R,vectorrprime,t)
vectorR =vectorr?vectorrprime =
3summationdisplay
i=1
Xivectorei?2vectorrU(R,vectorrprime,t) = 1R?
2(RU)
R2
1
R
2(UR)
R2?
1
c2
2U
t2 =?
1
epsilon10δ(
vectorR)ρ(vectorrprime,t)
2(UR)
R2?
1
c2
2(UR)
t2 = 0 RU(R,vectorr
prime,t) = f(t?R
c,vectorr
prime) + g(t + R
c,vectorr
prime)
U1(R,vectorrprime,t)≡1Rf(t?Rc,vectorrprime) a237a180a41 U2(R,vectorrprime,t)≡1Rg(t+Rc,vectorrprime) a135a99a41
264/384
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a237a180a179,a154a224a103(a107a13)a197a196a144a167a27a165a161a197a41
2vectorrU(vectorr,vectorrprime,t)? 1c2?
2
t2U(vectorr,vectorr
prime,t) =?1
epsilon10δ(vectorr?vectorr
prime)ρ(vectorrprime,t)
U1(R,vectorrprime,t)≡1Rf(t?Rc,vectorrprime) a237a180a41 U2(R,vectorrprime,t)≡1Rg(t+Rc,vectorrprime) a135a99a41
1epsilon1
0
δ(vectorR)ρ(vectorrprime,t) = (?2vectorr? 1c2?
2
t2)[
1
Rf(t?
R
c,vectorr
prime)]
= (?2vectorr 1R)f(t?Rc,vectorrprime) + 2?vectorrprime 1R·?vectorrf(t?Rc,vectorrprime)
+1R(?2vectorr? 1c2?
2
t2)f(t?
R
c,vectorr
prime)
=?4piδ(vectorR)f(t?Rc,vectorrprime) + 2c
vectorR
R3·
vectorR
R
tf(t?
R
c,vectorr
prime)
+1R(?
2
R2 +
2
R
R?
1
c2
2
t2)f(t?
R
c,vectorr
prime) =?4piδ(vectorR)f(t?R
c,vectorr
prime)
=?4piδ(vectorR)f(t) f(t,vectorrprime) = ρ(vectorr
prime,t)
4piepsilon10
U1(R,vectorrprime,t) = ρ(vectorr
prime,t?R
c )
4piepsilon10R U2(R,vectorr
prime,t) = ρ(vectorr
prime,t + R
c )
4piepsilon10R
265/384
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a237a180a179,a154a224a103(a107a13)a197a196a144a167a27a165a161a197a41
2φ(vectorr,t)? 1c2?
2
t2φ(vectorr,t) =?
ρ(vectorr,t)
epsilon10
2vectorrU(vectorr,vectorrprime,t)? 1c2?
2
t2U(vectorr,vectorr
prime,t) =?1
epsilon10δ(vectorr?vectorr
prime)ρ(vectorrprime,t)
φ(vectorr,t) =
integraldisplay
dτprime U(vectorr,vectorrprime,t) U(vectorr,vectorrprime,t) = U(R,vectorrprime,t)
U1(R,vectorrprime,t)=ρ(vectorr
prime,t?R
c )
4piepsilon10R U2(R,vectorr
prime,t)=ρ(vectorr
prime,t+R
c )
4piepsilon10R
vectorR=vectorr?vectorrprime=
3summationdisplay
i=1
Xivectorei
a233a137a189a158a143t,a31a138a161a143a177vectorrprimea143a165a37a27a165a161,t?Rc =a126a234→R =|vectorr?vectorrprime|=a126a234.
a31a138a161a177a132a199ca36a196,a31a138a161a27a138a27a140a2a851Ra67a122,a233a237a180a41,a31a138a161a42a220,a31a138a161a27a138
a160a2,a233a135a99a41,a31a138a161a194a160,a31a138a161a27a138a79a140,t2 >t1,a31a138a161a100vectorr1a163a20vectorr2
a237a180a41,R2=|vectorr2?vectorrprime|>R1=|vectorr1?vectorrprime| t1?R1c =t2?R2c R2?R1t
2?t1
=c
a135a99a41,R2=|vectorr2?vectorrprime|<R1=|vectorr1?vectorrprime| t1+R1c =t2+R2c R2?R1t
2?t1
=?c
266/384
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a237a180a179,a226a226a212a212a91a53a137a237a180a8a65,R/c a145 vectorE,vectorB a51a195a161a15a63a216a28a851/r2a80a126.
φ(vectorr,t) =
integraldisplay
dτprimeρ(vectorr
prime,t?R
c )
4piepsilon10R
vectorA(vectorr,t) =
integraldisplay
dτprimeμ0
vectorj(vectorrprime,t?R
c )
4piR
·vectorA = μ04pi?·
integraldisplay
dτprime
vectorj(vectorrprime,t?)
R =
μ0
4pi
integraldisplay
dτprime?·bracketleftbig
vectorj(vectorrprime,t?)
R
bracketrightbig
= μ04pi
integraldisplay
dτprime bracketleftbig(?1R)·vectorj(vectorrprime,t?) + 1R?·vectorj(vectorrprime,t?)bracketrightbig
= μ04pi
integraldisplay
dτprime bracketleftbig?(?prime1R)·vectorj(vectorrprime,t?) + 1R?
vectorj(vectorrprime,t?)
t? ·?t
bracketrightbig
= μ04pi
integraldisplay
dτprime bracketleftbigprime·[
vectorj(vectorrprime,t?)
R ] +
1
R?
prime·vectorj(vectorrprime,t?) + 1
R
vectorj(vectorrprime,t?)
t? ·?t
bracketrightbig
= μ04pi
integraldisplay
dτprime bracketleftbig1R?prime·vectorj(vectorrprime,t?)vextendsinglevextendsinglet? + 1R?
vectorj(vectorrprime,t?)
t? ·(?t
+?primet?)bracketrightbig
= μ04pi
integraldisplay
dτprime (?1R)t?ρ(vectorrprime,t?) = μ04pi
integraldisplay
dτprime (?1R)tρ(vectorrprime,t?)
=tμ04pi
integraldisplay
dτprime ρ(vectorr
prime,t?)
R =?μ0epsilon10
φ(vectorr,t)
t
267/384
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a237a180a179,a226a226a212a212a91a53a137;a165a212a53a137
a226a226a212a212a91a53a137:
φ(vectorr,t) =
integraldisplay
dτprimeρ(vectorr
prime,t?R
c )
4piepsilon10R
vectorA(vectorr,t) =
integraldisplay
dτprimeμ0
vectorj(vectorrprime,t?R
c )
4piR
a165a212a53a137:
φ(vectorr,t) =
integraldisplay
dτprimeρ(vectorr
prime,t)
4piepsilon10R
vectorA(vectorr,t) =
integraldisplay
dτprimeμ0
vectorj?(vectorrprime,t?R
c )
4piR
vectorj?(vectorr,t) =vectorj(vectorr,t)?epsilon10
tφ(vectorr,t)
a233a152a135a88a62a54a124,ρ(vectorr,t) = 0→φ(vectorr,t) = 0 a40a74a134a226a226a212a212a91a53a137a131a211.
268/384
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a237a180a179,a49a102a159a254a233a178a144a144a135a135a39a199a27a63a20
a88a74a74a49a49a102a107a154a34a183a142a142a159a159a254m2a49a102:
(?2? 1c2?
2
t2?
m2a49a102c2
planckover2pi12 )
φ(vectorr,t) =?ρ(vectorr,t)
epsilon10
(?2vectorr? 1c2?
2
t2?
m2a49a102c2
planckover2pi12 )
U(vectorr,vectorrprime,t) =?1
epsilon10δ(
vectorR)ρ(vectorrprime,t) vectorR=vectorr?vectorrprime
φ(vectorr,t) =
integraldisplay
dτprime?U(vectorr,vectorrprime,t)?U(vectorr,vectorrprime,t) =?U(R,vectorrprime,t)
U(R,vectorrprime,t) m2a49a102=0===== ρ(vectorrprime,t?Rc )
4piepsilon10R a195a159a254a254a226a226a102a152a189a189a19a19a151a178a144a144a135a135a39a199a156
a49a102a159a254a134a233a178a144a144a135a135a39a199a27a63a20a53a103a117?U(R,vectorrprime,t) a134
ρ(vectorrprime,t?Rc )
4piepsilon10R a27a11a79a156
a193a38,?U(R,vectorrprime,t) = ρ(vectorr
prime,t?R
c )
4piepsilon10R h(R)
269/384
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a237a180a179,a49a102a159a254a233a178a144a144a135a135a39a199a27a63a20
a88a74a74a49a49a102a107a154a34a183a142a142a159a159a254m2a49a102,?U(R,vectorrprime,t) = ρ(vectorrprime,t?Rc )4piepsilon1
0R
h(R)
1epsilon1
0
δ(vectorR)ρ(vectorrprime,t) = (?2vectorr? 1c2?
2
t2?
m2a49a102c2
planckover2pi12 )
U(vectorr,vectorrprime,t)
= h(R)(?2vectorr? 1c2?
2
t2?
m2a49a102c2
planckover2pi12 )
ρ(vectorrprime,t?Rc )
4piepsilon10R
+ρ(vectorr
prime,t?R
c )
4piepsilon10R?
2
vectorrh(R) + 2[?vectorr
ρ(vectorrprime,t?Rc )
4piepsilon10R ]·?vectorrh(R)
=?1epsilon1
0
δ(vectorR)ρ(vectorrprime,t)h(R) + ρ(vectorr
prime,t?R
c )
4piepsilon10R (
2
R2 +
2
R
R
m
2
a49a102c
2
planckover2pi12 )h(R) + 2[
R
ρ(vectorrprime,t?Rc )
4piepsilon10R ]
Rh(R)
=?h(0)epsilon1
0
δ(vectorR)ρ(vectorrprime,t) + ρ(vectorr
prime,t?R
c )
4piepsilon10R (
2
R2?
m2a49a102c2
planckover2pi12 )h(R)
+2R[Rρ(vectorr
prime,t?R
c )
4piepsilon10 ]
Rh(R)
270/384
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a237a180a179,a49a102a159a254a233a178a144a144a135a135a39a199a27a63a20
a88a74a74a49a49a102a107a154a34a183a142a142a159a159a254m2a49a102,?U(R,vectorrprime,t) = ρ(vectorrprime,t?Rc )4piepsilon1
0R
h(R)
1epsilon1
0
δ(vectorR)ρ(vectorrprime,t) =?h(0)epsilon1
0
δ(vectorR)ρ(vectorrprime,t) + ρ(vectorr
prime,t?R
c )
4piepsilon10R (
2
R2?
m2a49a102c2
planckover2pi12 )h(R)
+2R[Rρ(vectorr
prime,t?R
c )
4piepsilon10 ]
Rh(R)
h(0) = 1 {?
2
R2?
m2a49a102c2
planckover2pi12 + 2[
R lnρ(vectorr
prime,t?R
c )]
R}h(R) = 0
a144a107a62a214a151a221a216a145a158a109a67a122a226a103a84a181 h(R) = e±cplanckover2pi1 ma49a102R
a127a196a195a161a15a27a62a94a135,?U(R,vectorrprime,t) = ρ(vectorr
prime)
4piepsilon10Re
cplanckover2pi1 ma49a102R
a49a102a159a254a233a178a144a144a135a135a39a199a27a63a20,1R2? 1R2 e?cplanckover2pi1 ma49a102R
a77a77a94a94 1R2+epsilon1a28a28a91a91,epsilon1 = cplanckover2pi1 ma49a102 RlnR a15a167a131a112a138a94a67a164a67a167a131a112a138a94!
271/384
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a203a19a62a94a124,a152a132a53a159
φ(vectorr,t) =
integraldisplay
dτprimeρ(vectorr
prime,t?)
4piepsilon10R
vectorA(vectorr,t) =
integraldisplay
dτprimeμ0
vectorj(vectorrprime,t?)
4piR t
= t?R
c
vectorB =?×vectorA vectorE =φvectorA
tcontintegraldisplay
∞
dvectorσ·vectorS =
contintegraldisplay
∞
dvectorσ·(vectorE×vectorH) =? 0
vectorn≡
vectorR
R =
vectorr
r + O(
1
r)
t?
t = 1?t
=R
c =?
vectorn
c
ψ(t?) =?tψ?t =?vectornc?ψ?t
·vectorf(t?) =?t?·?
vectorf
t =?
vectorn
c·
vectorf
t
×vectorf(t?) =?t?×?
vectorf
t =?
vectorn
c×
vectorf
t
272/384
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a203a19a62a94a124,a152a132a53a159
vectorBa203 = (?×vectorA)vextendsinglevextendsingle1
r
=
integraldisplay
dτprime μ04piR?×vectorj(vectorrprime,t?) =
integraldisplay
dτprime μ04picR?
vectorj(vectorrprime,t?)
t ×vectorn
vectorEa203 = (φvectorA
t )
vextendsinglevextendsingle
1
r
=
integraldisplay
dτprimebracketleftbig? 14piRepsilon1
0
ρ(vectorrprime,t?)? μ04piR?
vectorj(vectorrprime,t?)
t
bracketrightbig
=
integraldisplay
dτprimebracketleftbig 14piepsilon1
0cR
ρ(vectorrprime,t?)
t vectorn?
μ0
4piR
vectorj(vectorrprime,t?)
t
bracketrightbig
=?φ?tvectornc?
integraldisplay
dτprime μ04piR?
vectorj(vectorrprime,t?)
t =?cvectorn?·
vectorA?
integraldisplay
dτprime μ04piR?
vectorj(vectorrprime,t?)
t
=
integraldisplay
dτprime μ04piRbracketleftbig?cvectorn?·vectorj(vectorrprime,t?)
vectorj(vectorrprime,t?)
t
bracketrightbig
=
integraldisplay
dτprime μ04piRbracketleftbigvectorn?
vectorj(vectorrprime,t?)
t ·vectorn?
vectorj(vectorrprime,t?)
t
bracketrightbig
vectorn×vectorEa203 = cvectorBa203 cvectorBa203×vectorn = vectorEa203
273/384
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a203a19a62a94a124,a152a132a53a159
vectorBa203 =
integraldisplay
dτprime μ04picR?
vectorj(vectorrprime,t?)
t ×vectorn
vectorn×vectorEa203 = cvectorBa203 cvectorBa203×vectorn = vectorEa203
We = 12epsilon10E2 Wm = 12μ0H2 = B
2
2μ0 =
E2
2c2μ0 =
1
2epsilon10E
2 = We
vectorS = vectorE×vectorH = cB2
μ0vectorn = 2Wmcvectorn = (We + Wm)cvectorn
a203a19a245a199:
I =
integraldisplay
a165
dvectorσ·vectorS =
integraldisplay
a165
dσ Svectorn·vectorn =
integraldisplay
a165
dσ S =
integraldisplay
d? Sr2 =
integraldisplay
d? dId?
dI
d? = Sr
2
274/384
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a203a19a62a94a124,a245a52a208a109
R =|vectorr?vectorrprime|= (r2+rprime2?2vectorr·vectorrprime)12 = r(1?vectorn·vectorr
prime
r +···) = r?vectorn·vectorr
prime+···
vectorA(vectorr,t)vextendsinglevextendsinglea203 =
integraldisplay
dτprime μ04piRvectorj(vectorrprime,t?Rc ) =
integraldisplay
dτprime μ04piRvectorj(vectorrprime,t?rc+vectorn·vectorr
prime
c?···)
= μ04pir
integraldisplay
dτprime bracketleftbig
∞summationdisplay
k=0
1
k!
kvectorj(vectorrprime,t?rc)
tk (
vectorn·vectorrprime
c?···)
kbracketrightbig
= μ04pir
integraldisplay
dτprime
∞summationdisplay
k=0
1
k!
kvectorj(vectorrprime,t?rc)
tk (
vectorn·vectorrprime
c )
k
= μ04pir
∞summationdisplay
k=0
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck
k
tk
integraldisplay
dτprimevectorj(vectorrprime,t?rc)xprimei1···xprimeik
= μ04pir
∞summationdisplay
k=0
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck
k
tk
vectorJi
1···ik(t?
r
c)
vectorJi
1···ik(t) =
integraldisplay
dτprimevectorj(vectorrprime,t)xprimei1···xprimeik
275/384
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a203a19a62a94a124,a245a52a208a109
vectorJi
1···ik(t) =
integraldisplay
dτprimevectorj(vectorrprime,t)xprimei1···xprimeik
vectorA(vectorr,t)vextendsinglevextendsinglea203 = μ0
4pir
∞summationdisplay
k=0
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck
k
tk
vectorJi
1···ik(t?
r
c)
vectorB(vectorr,t)vextendsinglevextendsinglea203 = (?×vectorA)vextendsinglevextendsingle1
r
= μ04pir
∞summationdisplay
k=0
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck?×
k
tk
vectorJi
1···ik(t?
r
c)
= μ04pir
∞summationdisplay
k=0
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck (?
vectorn
c×
t)
k
tk
vectorJi
1···ik(t?
r
c)
= μ04pir
∞summationdisplay
k=0
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck+1
k+1
tk+1
vectorJi
1···ik(t?
r
c)×vectorn =
∞summationdisplay
k=0
vectorBk
vectorBk = μ0
4pir
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck+1
k+1
tk+1
vectorJi
1···ik(t?
r
c)×vectorn
vectorBk+1a39vectorBka245 1cxprimet ～ lprimeωc ～ lprimeλ,a135a208a109a208a55a76lprimelessmuchλ
276/384
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a203a19a62a94a124,a245a52a208a109
vectorB(vectorr,t)vextendsinglevextendsinglea203=
∞summationdisplay
k=0
vectorBk vectorBk= μ0
4pir
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck+1
k+1
tk+1
vectorJi
1···ik(t?
r
c)×vectorn
vectorB0 = μ0
4picr
t
vectorJ(t?r
c)×vectorn =
μ0
4picr
t
integraldisplay
dτprimevectorj(vectorrprime,t?rc)×vectorn
= μ04picr
3summationdisplay
i,j,k=1
tepsilon1ijk
integraldisplay
dτprime ji(vectorrprime,t?rc)njvectorek
= μ04picr
3summationdisplay
i,j,k=1
epsilon1ijkt
integraldisplay
dτprime bracketleftbig?primel[jl(vectorrprime,t?rc)xprimei]?xprimei?primeljl(vectorrprime,t?rc)bracketrightbignjvectorek
= μ04picr
3summationdisplay
i,j,k=1
epsilon1ijkt
integraldisplay
dτprime xprimei?ρ(vectorr
prime,t?r
c)
t njvectorek
= μ04picr
3summationdisplay
i,j,k=1
epsilon1ijk?
2
t2qi(t?
r
c)njvectorek =
μ0
4picr
¨vectorP(t?r
c)×vectorn
vectorP(t) = summationdisplayqi(t)vectorei qi(t) =
integraldisplay
dτprime xprimeiρ(vectorrprime,t)
277/384
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a203a19a62a94a124,a245a52a208a109
vectorB(vectorr,t)vextendsinglevextendsinglea203=
∞summationdisplay
k=0
vectorBk vectorBk= μ0
4pir
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck+1
k+1
tk+1
vectorJi
1···ik(t?
r
c)×vectorn
vectorB1 = μ0
4pic2r
3summationdisplay
i=1
ni?
2
t2
vectorJi(t?r
c)×vectorn
= μ04pic2r
3summationdisplay
i=1
2
t2
integraldisplay
dτprime xprimeivectorj(vectorrprime,t?rc)×vectorn
= μ04pic2r
3summationdisplay
i,j,k,l=1
ni?
2
t2epsilon1jkl
integraldisplay
dτprime xprimeijj(vectorrprime,t?rc)nkvectorel
= μ08pic2r
3summationdisplay
i,j,k,l=1
ni?
2
t2epsilon1jkl
integraldisplay
dτprime [xprimeijj + xprimejji + xprimeijj?xprimejji]nkvectorel
= μ08pic2r
3summationdisplay
i,j,k,l=1
ni?
2
t2epsilon1jkl
integraldisplay
dτprimebracketleftbig
3summationdisplay
m=1
[?primem(jmxprimeixprimej)?xprimeixprimej(?primemjm)]+
3summationdisplay
m,n,p=1
epsilon1ijpepsilon1nmpxprimenjmbracketrightbignkvectorel
278/384
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a203a19a62a94a124,a245a52a208a109
vectorB(vectorr,t)vextendsinglevextendsinglea203=
∞summationdisplay
k=0
vectorBk vectorBk= μ0
4pir
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck+1
k+1
tk+1
vectorJi
1···ik(t?
r
c)×vectorn
vectorB1 = μ0
4pic2r
3summationdisplay
i=1
ni?
2
t2
vectorJi(t?r
c)×vectorn
= μ08pic2r
3summationdisplay
i,j,k,l=1
ni?
2
t2epsilon1jkl
integraldisplay
dτprime bracketleftbig
3summationdisplay
m=1
[?primem(jmxprimeixprimej)?xprimeixprimej(?primemjm)]
+
3summationdisplay
m,n,p=1
epsilon1ijpepsilon1nmpxprimenjmbracketrightbignkvectorel
= μ08pic2r
3summationdisplay
i,j,k,l=1
ni?
2
t2
integraldisplay
dτprime bracketleftbigepsilon1jklxprimeixprimejtρ(vectorrprime,t?rc)
+
3summationdisplay
m,n,p=1
epsilon1jklepsilon1ijpepsilon1nmpxprimenjm(vectorrprime,t?rc)bracketrightbignkvectorel
279/384
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a203a19a62a94a124,a245a52a208a109
vectorB1 = μ0
8pic2r
3summationdisplay
i,j,k,l=1
ni?
2
t2
integraldisplay
dτprime bracketleftbigepsilon1jklxprimeixprimejtρ(vectorrprime,t?rc)
+
3summationdisplay
m,n,p=1
epsilon1jklepsilon1ijpepsilon1nmpxprimenjm(vectorrprime,t?rc)bracketrightbignkvectorel
= μ08pic2r
3summationdisplay
i,j,k,l=1
nibracketleftbig?
3
t3epsilon1jklqij(t?
r
c) + 2
2
t2
3summationdisplay
p=1
epsilon1jklepsilon1ijpmp(t?rc)bracketrightbignkvectorel
= μ08pic2r
3summationdisplay
i,j,k,l=1
nibracketleftbig?
3
t3epsilon1jkl
1
3Qij(t?
r
c) + 2
2
t2
3summationdisplay
p=1
epsilon1jklepsilon1ijpmp(t?rc)bracketrightbignkvectorel
= μ04pic2rbracketleftbig16vectorn·
···arrowrighttophalfarrowrighttophalf
Q ×vectorn + (¨vectorm×vectorn)×vectornbracketrightbig
arrowrighttophalfarrowrighttophalfQ(t?r
c)=
integraldisplay
dτprime(3vectorrprimevectorrprime?arrowrighttophalfarrowrighttophalfIrprime2)ρ(vectorrprime,t?rc) vectorm(t?rc)=12
integraldisplay
dτprimevectorrprime×vectorj(vectorrprime,t?rc)
280/384
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a203a19a62a94a124,a245a52a208a109,a62a243a52a203a19
a152a225a85a130a100A,Ba252a135a2a165a124a164,(vectorla233a225,a100Ba141a149A),A,Ba254a136a145a131a135a62a214,a191a62a0a51A
Ba109a27a27a19a19a130a254a23a41a112a170a62a54
a0
a0
a0
a0a0a18
a115
a115 r
vectornA
B
J = dQAdt = J0 cosωt QA = J0ω sinωt
l→0,J0ω →0,lJ0ωa27a189 vectorP = QAvectorl = J0ωvectorlsinωt
vectorB(vectorr,t) = μ0
4picr
¨vectorP(t?r
c)×vectorn =?
μ0J0ω
4picr
vectorlsinω(t?r
c)×vectorn
= μ0
vectorl
4picr
˙J(t?r
c)×vectorn
vectorS(vectorr,t) = c
μ0B
2vectorn = μ0l
2
(4pi)2cr2
˙J2(t?r
c)sin
2θvectorn
dI
d? =
μ0l2
(4pi)2c
˙J2(t?r
c)sin
2θ
281/384
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a203a19a62a94a124,a245a52a208a109,a62a243a52a203a19
a152a225a85a130a100A,Ba252a135a2a165a124a164,(vectorla233a225,a100Ba141a149A),A,Ba254a136a145a131a135a62a214,a191a62a0a51A
Ba109a27a27a19a19a130a254a23a41a112a170a62a54
a0
a0
a0
a0a0a18
a115
a115 r
vectornA
B
dI
d? =
μ0l2
(4pi)2c
˙J2(t?r
c)sin
2θ
I(t) =
integraldisplay
d? dId? =
integraldisplay 2pi
0
dφ
integraldisplay pi
0
dθsinθ dId?
= 2pi
integraldisplay pi
0
dθ μ0l
2
(4pi)2c
˙J2(t?r
c)sin
3θ = μ0l
2
6pic
˙J2(t?r
c) =
μ0l2ω2J20
6pic sin
2ω(t?r
c)
I = lim
T→∞
1
T
integraldisplay T
0
dt I(t) = lim
T→∞
1
T
integraldisplay T
0
dt μ0l
2ω2J2
0
6pic
1
2[1?cos2ω(t?
r
c)]
= μ0l
2ω2J2
0
12pic limT→∞
1
T[t?
1
2ω sin2ω(t?
r
c)]
vextendsinglevextendsingleT
0 =
μ0l2ω2J20
12pic
= μ0l
2ω2
6picR(
1
2J
2
0R) =
Ra203
R (
1
2J
2
0R)
Ra203 = μ0l
2ω2
6pic
1
2J
2
0R,a57a245a199 R,a85a130a62a123
282/384
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a203a19a62a94a124,a245a52a208a109,a62a111a52a203a19
a62a254a143q,q,?2qa27a110a135a2a165a169a79a160a117a127a1432la27a225a85a130a27a252a224a218a165a109,vectorla143 a108a85a130a165a58
a141a149a152a224a224a27a27a165a254,q = q0 cosωt,q0→∞,l→0a2q0l2a27a189.
a0
a0
a0
a0a0a18
a115
a118
a115
r
vectorn
q
2q
q J = ˙q =?q0ωsinωt = J0 sinωt
vectorP =
integraldisplay
dτprimevectorrprimeρ = qvectorl + q(?vectorl) = 0
vectorm = 12
integraldisplay
dτprimevectorrprime×vectorj = 12
integraldisplay
vectorrprime×Jdvectorlprime = 0
arrowrighttophalfarrowrighttophalfQ= integraldisplay dτprime3vectorrprimevectorrprimeρ = 3vectorlvectorlq + 3(?vectorl)(?vectorl)q = 6qvectorlvectorl
vectorB(vectorr,t) = μ0
6c2r4pivectorn·
···arrowrighttophalfarrowrighttophalf
Q ×vectorn = μ06c2r4pi6 ···qvectorn·vectorlvectorl×vectorn
= l
2μ0q0ω3
4pic2r sinω(t?
r
c)cosθsinθ
vectorS(vectorr,t) = c
μ0B
2vectorn = μ0l
4q2
0ω
6
c3r2(4pi)2 sin
2ω(t?r
c)sin
2θcos2θvectorn
283/384
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a203a19a62a94a124,a245a52a208a109,a62a111a52a203a19
a62a254a143q,q,?2qa27a110a135a2a165a169a79a160a117a127a1432la27a225a85a130a27a252a224a218a165a109,vectorla143 a108a85a130a165a58
a141a149a152a224a224a27a27a165a254,q = q0 cosωt,q0→∞,l→0a2q0l2a27a189.
a0
a0
a0
a0a0a18
a115
a118
a115
r
vectorn
q
2q
q vectorS(vectorr,t) = c
μ0B
2vectorn = μ0l
4q2
0ω
6
c3r2(4pi)2 sin
2ω(t?r
c)sin
2θcos2θvectorn
dI
d? =
μ0l4q20ω6
c3(4pi)2 sin
2θcos2θsin2ω(t?r
c)
I = μ0l
4q2
0ω
6
c3(4pi)2 sin
2ω(t?r
c)2pi
integraldisplay pi
0
dθ sin3θcos2θ
= μ0l
4q2
0ω
6
30c3pi sin
2ω(t?r
c)
I = μ0l
4q2
0ω
6
60c3pi =
μ0l4J20ω2
60c3pi =
Ra203
R (
1
2J
2
0R)
Ra203 = μ0l
4ω4
30pic
284/384
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a203a19a62a94a124,a245a52a208a109,a94a243a52a203a19
a62a54a23a140a187a143a,I = I0 cosωt,vectorm = Ipia2vectorez
vectorj(vectorr,t) = jθ(r,z,t)vectoreθ jr = jz = 0
˙ρ(vectorr,t) =·vectorj =?1r?jθ(r,z,t)?θ = 0 ˙arrowrighttophalfarrowrighttophalfQ = 0
vectorB(vectorr,t) = μ0
4pic2r
¨vectorm×vectorn = ω2μ0pia2
4pic2r I0 cosωt
vectorez×vectorn
vectorS(vectorr,t) = c
μ0B
2vectorn = ω
4μ0a4I2
0
16c3r2 cos
2ωt?(vectorez×vectorn)·(vectorez×vectorn)vectorn
= ω
4μ0a4I2
0
16c3r2 cos
2ωt?sin2θvectorn
dI
d? =
ω4μ0a4I20
16c3 cos
2ωt?sin2θ I(t) = pi
3ω4μ0a4I2
0
6c3 cos
2ωt?
I = pi
3ω4μ0a4I2
0
12c3 =
Ra203
R (
1
2J
2
0R) Ra203 =
μ0a4ω4pi3
6c3
285/384
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a203a19a62a94a124,a62a94a197a27a251a19
a168a141a100a6a110,a51a197a68a194a27a197a10a161a254a27a122a152a58a209a180a103a63a197a13,a122a152a135
a197a13a117a209a2a27a165a161a102a197,a165a161a102a197a27a157a161a47a164a35a27a197a10a161.
u?Ex,Ey,Ez,Bx,By,Bz
2u(vectorr,t)? 1c2?
2
t2u(vectorr,t) = 0
vectorR =vectorrprime?vectorr vectorr?τ
u(vectorr,t) =
integraldisplay
dτprimeδ(vectorrprime?vectorr)u(vectorrprime,t?Rc ) =
integraldisplay
dτprime(?14pi)(?prime2 1R)u(vectorrprime,t?Rc )
=
integraldisplay
dτprime(?14pi)braceleftbig?prime·[(?prime1R)u(vectorrprime,t?Rc )]?(?prime1R)·?primeu(vectorrprime,t?Rc )bracerightbig
=
integraldisplay
dτprime(?14pi)braceleftbig?prime·[(?prime1R)u(vectorrprime,t?Rc )?1R?primeu(vectorrprime,t?Rc )]+ 1R?prime2u(vectorrprime,t?Rc )bracerightbig
primet? =?1c?primeR =?
vectorR
cR
286/384
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a203a19a62a94a124,a62a94a197a27a251a19
1
R?
prime2u(vectorrprime,t?R
c ) =
1
R?
prime·[?primeu(vectorrprime,t?)vextendsinglevextendsingle
t
u(vectorrprime,t?)
t?
vectorR
cR]
= 1R?prime·[?primeu(vectorrprime,t?)vextendsinglevextendsinglet?]?
vectorR
cR2·?
prime?u(vectorr
prime,t?)
t
1
cR
u(vectorrprime,t?)
t
prime·vectorR
R
= 1R?prime2u(vectorrprime,t?)vextendsinglevextendsinglet
vectorR
cR2·
t
primeu(vectorrprime,t?)vextendsinglevextendsingle
t
vectorR
cR2·?
prime?
t?u(vectorr
prime,t?)vextendsinglevextendsingle
t?
+ 1c2R?
2
t?2u(vectorr
prime,t)u(vectorr
prime,t?)
t?
1
cR[
3
R?
1
R]
= 1R[?prime2u(vectorrprime,t?)vextendsinglevextendsinglet? + 1c2?
2
t?2u(vectorr
prime,t?)]? 2vectorR
cR2·?
prime?
t?u(vectorr
prime,t?)vextendsinglevextendsingle
t?
2cR2?u(vectorr
prime,t)
t?
= 2Rc2?
2
t?2u(vectorr
prime,t?)? 2vectorR
cR2·?
prime?
t?u(vectorr
prime,t?)vextendsinglevextendsingle
t
2
cR2
u(vectorrprime,t)
t?
= 2cbraceleftbig
vectorR
cR·
vectorR
R2
2
t?2u(vectorr
prime,t?)? vectorR
R2·?
prime?
t?u(vectorr
prime,t?)vextendsinglevextendsingle
t(?
prime· vectorR
R2)
u(vectorrprime,t?)
t?
bracerightbig
287/384
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a203a19a62a94a124,a62a94a197a27a251a19
1
R?
prime2u(vectorrprime,t?R
c )
= 2cbraceleftbig
vectorR
cR·
vectorR
R2
2
t?2u(vectorr
prime,t?)? vectorR
R2·?
prime?
t?u(vectorr
prime,t?)vextendsinglevextendsingle
t(?
prime· vectorR
R2)
u(vectorrprime,t?)
t?
bracerightbig
=?2cbraceleftbig
vectorR
R2·[?
vectorR
cR
t? +?
primevextendsinglevextendsingle
t?]
t?u(vectorr
prime,t?) + (?prime· vectorR
R2)
u(vectorrprime,t?)
t?
bracerightbig
=?2cbraceleftbig
vectorR
R2·?
prime?
t?u(vectorr
prime,t?)+(?prime· vectorR
R2)
u(vectorrprime,t?)
t?
bracerightbig =?2
c?
prime·braceleftbig vectorR
R2
t?u(vectorr
prime,t?)bracerightbig
u(vectorr,t) = (?14pi)
integraldisplay
τ
dτprime?prime·braceleftbig(?prime1R)u(vectorrprime,t?Rc )? 1R?primeu(vectorrprime,t?Rc )
2c
vectorR
R2
t?u(vectorr
prime,t?)bracerightbig
=(?14pi)
integraldisplay
τ
dτprime?prime·braceleftbig?
vectorR
R3u(vectorr
prime,t?R
c )?
1
R?
primeu(vectorrprime,t?)vextendsinglevextendsingle
t
vectorR
cR2
t?u(vectorr
prime,t?)bracerightbig
= 14pi
integraldisplay
τ
dvectorSprime ·braceleftbig1R?primeu(vectorrprime,t?) +
vectorR
R3u(vectorr
prime,t?) + vectorR
cR2
t?u(vectorr
prime,t?)bracerightbigvextendsinglevextendsingle
t?=t?Rc
288/384
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a203a19a62a94a124,a62a94a197a27a251a19～a196a16a191a197a98a23
a127a196a178a161a197u(vectorr,t) a19a149a109a107a2a154a27a195a161a140a178a161Sa254.a138a88a101a196a16a191a197a98a23:
a216a154a83a9,u,?ua51Sa161a254a63a63a63a63a1430.
a154a83a27u,?ua27a138a31a117a118a107a63a219a182a189a230a78a212a158,a92a19a197a27a138.
u(vectorr,t) =
braceleftbigg 0
a154a9
Aei(ωt?ωzc ) a154a83
u(vectorr,t) = 14pi
integraldisplay
τ
dvectorSprime·braceleftbig1R?primeu(vectorrprime,t?) +
vectorR
R3u(vectorr
prime,t?) + vectorR
cR2
t?u(vectorr
prime,t?)bracerightbigvextendsinglevextendsingle
t?=t?Rc
primeu(vectorrprime,t?) =?iωcvectorezu(vectorrprime,t?)t?u(vectorrprime,t?) = iωu(vectorrprime,t?)
u(vectorr,t) = A4pi
integraldisplay
a154
dvectorSprime·braceleftbig1R(?iωcvectorez) +
vectorR
R3 +
vectorR
cR2iω
bracerightbigu(vectorrprime,t?)vextendsinglevextendsingle
t?=t?Rc
= A4pi
integraldisplay
a154
dvectorSprime·braceleftbig1R(?iωcvectorez) +
vectorR
R3 +
vectorR
cR2iω
bracerightbigeiωt?vextendsinglevextendsingle
t?=t?Rc
289/384
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a203a19a62a94a124,a62a94a197a27a251a19～a196a16a191a197a98a23
a127a196a178a161a197u(vectorr,t) a19a149a109a107a2a154a27a195a161a140a178a161Sa254.a138a88a101a196a16a191a197a98a23:
a216a154a83a9,u,?ua51Sa161a254a63a63a63a63a1430.
a154a83a27u,?ua27a138a31a117a118a107a63a219a182a189a230a78a212a158,a92a19a197a27a138.
u(vectorr,t) =
braceleftbigg 0
a154a9
Aei(ωt?ωzc ) a154a83
u(vectorr,t) = A4pi
integraldisplay
a154
dvectorSprime·braceleftbig1R(?iωcvectorez) +
vectorR
R3 +
vectorR
cR2iω
bracerightbigeiωt?vextendsinglevextendsingle
t?=t?Rc
dvectorSprime=?dSprimevectorez=====?A
4pie
iωt
integraldisplay
a154
dSprimebraceleftbig?iωcR +?cosθR2? iωcR cosθbracerightbige?iωRc
= A4pieiωt
integraldisplay
a154
dSprimebraceleftbigiωcR(1 + cosθ) + cosθR2 bracerightbige?iωRc
a8a124a58a108a182a22a15a158,a204a135a49a152a145a0a122,a23a197a70a218a164a251a19
a8a124a58a108a182a22a22a67a67a158,a204a135a49a19a145a0a122,a23a153a114a16a251a19
290/384
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a203a19a62a94a124,a62a94a197a27a251a19～a197a70a218a164a251a19
a233a152a135a127a176a169a79a143a,ba27a221a47a47a2a2a154:
a1
a1
a1
a1
a1a1a1
a1
a1
a1
a1a1
a45
a54
a1
a1
a1
a1a1a21
a8a8
a8a8
a8a8
a8a8a8a42
a3
a3a3a23
a24a24a24a24
a24a24a24a24a58
vectorez
vectorey vectorex
vectorr
vectorRvectorρ
b a
vectorρ = xvectorex + yvectorey ρ2 = x2 + y2
vectorR =vectorr?vectorρ vectorr·vectorez = rcosθ3
vectorr·vectorex = rcosθ1 vectorr·vectorey = rcosθ2
R = (r2 +ρ2?2vectorr·vectorρ)12 = [r2 +ρ2?2r(xcosθ1 + ycosθ2)]12
cosθ =?
vectorR·vectorez
R =
rcosθ3
R =
rcosθ3
[r2 +ρ2?2r(xcosθ1 + ycosθ2)]12
ρlessmuchr
R = r[1 + ρ
2
r2?
2
r(xcosθ1 + ycosθ2)]
1
2 = r?xcosθ1?ycosθ2 +···
cosθ = cosθ3[1 + xr cosθ1 + yr cosθ2 +···]
u(vectorr,t) = A4pieiωt
integraldisplay
a154
dSprimebraceleftbigiωcR(1 + cosθ) + cosθR2 bracerightbige?iωRc
= A4pieiωt
integraldisplay
a154
dSprimeiωcr(1 + cosθ3)e?iωc[r?xcosθ1?ycosθ2]
291/384
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a203a19a62a94a124,a62a94a197a27a251a19～a197a70a218a164a251a19
a233a152a135a127a176a169a79a143a,ba27a221a47a47a2a2a154:
a1
a1
a1
a1
a1a1a1
a1
a1
a1
a1a1
a45
a54
a1
a1
a1
a1a1a21
a8a8
a8a8
a8a8
a8a8a8a42
a3
a3a3a23
a24a24a24a24
a24a24a24a24a58
vectorez
vectorey vectorex
vectorr
vectorRvectorρ
b a
u(vectorr,t) = A4pieiωt
integraldisplay
a154
dSprimebraceleftbigiωcR(1+cosθ)+cosθR2 bracerightbige?iωRc
= A4pieiωt
integraldisplay
a154
dSprimeiωcr(1 + cosθ3)e?iωc[r?xcosθ1?ycosθ2]
= A4pieiωtiωcr(1 + cosθ3)
integraldisplay a
a
dx
integraldisplay b
b
dyeiωc cosθ1x+iωc cosθ2ye?iωcr
= A4pieiωt?iωcriωcrc
2
ω2(1 + cosθ3)
2isin(ωac cosθ1)
cosθ1
2isin(ωbc cosθ2)
cosθ2
= A4piei(ωt?ωcr)4icωr(1 + cosθ3)sin(
ω
cacosθ1)
cosθ1
sin(ωcbcosθ2)
cosθ2
a251a19a197a143a165a161a197,a8a204a51ωcacosθ1 = mpi,ωcbcosθ2 = npi,(m,n,1,2,3)a63a1430(a86a58).
a252a191a251a19a27a28a28a91a91
292/384
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a100a194a131a233a216
293/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a100a194a131a233a216,a196a58～a6a207
a143a159a111a51a62a62a196a196a229a198a145a165a63a216a100a194a131a233a216,a167a180a62a62a196a196a229a198a117a208a27a23a212
a240a142a100a137a27a27a62a62a94a110a216a1631865a164a18a26a10a227a140a27a164a245
a167a143a107a101a90a134a162a8a216a20a218a27a27a47a47a144
–a220a169a175a175a162a162a163a231a78a203a19a167a6a102a117a49a164?a254a102a216
–a220a169a175a175a162a162a163a132a0a164?a131a233a216
a254a102a216a218a131a233a216a8a164a19a155a173a86a212a110a198a129a149a140a27a252a145a117a121a182
a143a180a19a155a155a173a173a86a212a110a198a26a177a37a199a117a208a27a196a58
a100a194a131a233a216a180a51a233a62a94a110a216a63a49a10a28a161a48a11(a183a62a94a33a68
a194a33a203a19)a0a167a233a101a90a216a218a20a144a161a27a29a92a239a196
294/384
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a100a194a131a233a216,a196a58～a131a233a53a6a110
a216a37a181a152a131a46a53a235a127a88a51a212a110a198a254a180a17a28a31a100a27
a162a8a181a51a216a211a46a53a88a137a211a24a27a162a8a55a107a211a24a27a40a74a156
a110a216a181a63a219a196a29a212a110a53a198a51a216a211a46a53a88a165a55a107a131a211a27a144a167a47a170!
a131a233a53a181a132a221a180a131a233a233a27a27a88a51a196a186a195a253a233a235a127a88a182a92a132a221a216a180a131a233a233a27a27!
a179a124a209a158a147a1631564-1642a164a27a131a233a53a6a110a181
a70a37a96a163a120a120a901473-1563a164a134a47a37a96a27a216a212(a19a172a167a70a126a178a8)
a47a165a131a233a20a19a27a36a196a132a221,30a250a112/a166?a154a74,a8a158a118a107a110a216a167a144a85a157a227a121a121a150a150
a233a148a31a218a156a106a64a24a85a248a108a47a161a13a51a152a165a202a51a152a227a27a192a220a167a167a130a119a53a216a140
a85a139a254a47a165a27a132a221a34a183a130a65a84a250a26a167a130a180a156a175a47a149a220a144a36a196a34
a106a14a78a111a139a26a254a249a135a63a167a81a186a183a130a119a20a27a180a167a106a14a149a192a189a246a149a220a167a189a246
a149a79a27a144a149a156a49a167a118a107a63a219a11a201a34
a88a74a183a130a228a234a27a182a158a250a26a78a51a242a254a27a186a131a8a136a17a167a51a47a165a249a24a152a135a95a152
a237a13a49a27a156a132a63a167a165a167a183a130a84a97a250a152a24a245a114a167a27a192a186a66a156a44a13a37a250a9a216
a20a249a24a27a75a143a34
295/384
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a100a194a131a233a216,a196a58～a131a233a53a6a110
a216a37a181a152a131a46a53a235a127a88a51a212a110a198a254a180a17a28a31a100a27
a162a8a181a51a216a211a46a53a88a137a211a24a27a162a8a55a107a211a24a27a40a74a156
a110a216a181a63a219a196a29a212a110a53a198a51a216a211a46a53a88a165a55a107a131a211a27a144a167a47a170!
a131a233a53a181a132a221a180a131a233a233a27a27a88a51a196a186a195a253a233a235a127a88a182a92a132a221a216a180a131a233a233a27a27!
a179a124a209a158a147a1631564-1642a164a27a131a233a53a6a110a181
a70a37a96a163a120a120a901473-1563a164a134a47a37a96a27a216a212(a19a172a167a70a126a178a8)
a47a165a131a233a20a19a27a36a196a132a221,30a250a112/a166?a154a74,a140a237,a156a106
a39a117a247a86a151a218a120a120a90a252a140a173a46a78a88a27a27a233a233a123(1632):
a233a154a74a27a216a121a180a131a233a53a6a110a129a64a27a76a227:
a114a92a218a152a10a42a108a39a51a152a94a140a69a96a134a101a27a204a242a112a167a50a52a92a130a145a65a144a241a71a33
a8a82a218a217a167a2a156a193a34a242a83a152a152a144a140a89a18a167a217a165a152a65a94a126a34a44a0a33a254a152a135
a89a180a167a52a89a152a37a152a37a37a47a47a47a37a37a37a20a20a101a161a27a152a135a176a176a157a157a45a112a34
a69a202a88a216a196a158a181a2a193a209a177a31a132a149a242a83a136a144a149a156a49a182a126a149a136a135a144a149a145a66a105
a196a182a89a37a37a63a101a161a27a45a102a165a182a92a114a63a219a192a220a68a137a92a27a42a108a158a167a144a135a229a108
a131a31a167a149a249a152a144a149a216a55a39a44a152a144a149a94a141a245a27a229a182a92a86a12a224a97a167a195a216a149a61
a135a144a149a97a76a27a229a108a209a131a31a34
a50a166a69a177a63a219a132a221a99a63a167a144a135a36a196a180a33a132a27a167a143a216a3a134a3a109a47a123a196a34
296/384
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a100a194a131a233a216,a196a58～a131a233a53a6a110
a50a166a69a177a63a219a132a221a99a63a167a144a135a36a196a180a33a132a27a167a143a216a3a134a3a109a47a123a196a34
a92a242a117a121a167a164a107a254a227a121a150a106a206a118a107a67a122a167a92a195a123a108a217a165a63a219a152a135a121a150a53
a40a189a167a69a180a51a51a36a36a196a132a180a202a88a216a196a196a27a27a167a61a166a69a36a196a196a27a27a131a8a175a34
a51a97a23a158a167a92a242a218a177a99a152a24a51a69a46a134a254a97a76a131a211a27a229a108a167a92a97a149a69a151a143
a216a172a39a97a149a69a222a53a27a15a167a143a44a92a97a20a152a165a158a167a12a101a27a69a46a134a149a88a92a97a27
a131a135a135a144a144a149a163a196a34
a92a114a216a216a159a111a192a220a68a137a92a27a211a138a158a167a216a216a166a180a51a69a222a132a180a51a69a151a167a144a135
a92a103a67a213a213a51a51a233a161a167a92a143a216a73a135a94a141a245a245a27a27a229a34
a89a37a242a148a107a99a152a24a167a37a63a101a161a27a45a102a167a152a37a143a216a172a37a149a69a151a167a143a44a89a37
a51a152a165a158a167a69a174a49a168a10a78a245a221a34
a126a51a89a165a105a149a89a18a99a220a164a94a27a229a167a216a39a105a149a89a18a0a220a53a27a140a34a166a130a152a24
a97a115a47a105a149a152a51a89a18a62a14a63a219a47a144a27a160a17a34
a8a82a218a241a71a242a85a89a145a66a47a20a63a156a49a167a166a130a251a216a172a149a69a151a56a165a34a191a216a207a143
a166a130a127a158a109a51a51a152a165a167a248a108a10a69a27a36a196a167a143a96a254a69a27 a36a196a119a209a92a27a24
a102a34
a88a74a58a45a134a235a167a92a242a119a20a235a148a152a249a31a152a24a149a254a44a229a167a216a149a63a219a152a62a163
a196a34
a212a78a228a107a46a53a34a143a44a64a158a132a118a107a218a238a238a27a27a46a53a189a198
297/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a100a194a131a233a216,a196a58～a131a233a53a6a110
a216a37a181a152a131a46a53a235a127a88a51a212a110a198a254a180a17a28a31a100a27
a162a8a181a51a216a211a46a53a88a137a211a24a27a162a8a55a107a211a24a27a40a74a156
a110a216a181a63a219a196a29a212a110a53a198a51a216a211a46a53a88a165a55a107a131a211a27a144a167a47a170!
a131a233a53a181a132a221a180a131a233a233a27a27a88a51a196a186a195a253a233a235a127a88a182a92a132a221a216a180a131a233a233a27a27!
a179a124a209a158a147a1631564-1642a164a27a131a233a53a6a110a181
a70a37a96a163a120a120a901473-1563a164a134a47a37a96a27a216a212(a19a172a167a70a126a178a8)
a47a165a131a233a20a19a27a36a196a132a221,30a250a112/a166?a154a74,a140a237,a156a106
a39a117a247a86a151a218a120a120a90a252a140a173a46a78a88a27a27a233a233a123(1632)
a233a154a74a27a216a121a180a131a233a53a6a110a129a64a27a76a227 a137a63a219a162a8a195a123a40a189a69a180a196a180a51a51a36a36a196a196a27a27a156
a218a238a158a147a1631642-1727a164a27a131a233a53a6a110a181 a218a238a239a225a10a110a216
a103a44a243a198a27a234a198a6a110(1685),a218a238a110a189a198,a25a107a218a229a189a198,a19a209a109a202a86a189a198,...
a163a137vectorF = mvectoraa177a159a111a143a235a236a212?
– a127a51a253a233a152a109a134a253a233a158a109,a167a130a166 vectorF = mvectora a164a225
– a131a233a167a130a137a33a132a36a196a196a27a27a152a109a134a158a109a143a166 vectorF = mvectora a164a225
a131a233a53a6a110a144a129a117a229a198a27a137a198!
298/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a100a194a131a233a216,a196a58～a131a233a53a6a110
a178a59a229a198,a229a198a196a29a36a196a196a189a189a198a233a164a107a46a53a88a209a164a225,a46a53a88a131a109a207a76a179a124a209a67a134a233a88
a45
a54
a8a8
a8a25
a54
a8a8
a8a25
a45 vectorV
z zprime
x xprime
y yprime
xprime = x?Vt
yprime = y
zprime = z
tprime = t
a178a8a27a132a221a220a164a39a88:?
vprimex = vx?V
vprimey = vy
vprimez = vz
vectoraprime =vectora
a218a238a238a189a189a198a51a179a124a209a67a134a118a107a237a19a101a180a216a216a67a67a27a156
a62a94a121a121a150a150a27a196a29a53a198a183a94a94a117a117a159a111a235a127a88? a240a142a100a137a118a107a163a137!
a179a124a209a27a158a152a42a103,a101a212a159a36a196a132a221a131a233a44a235a127a88a143c,a51a44a152a235a127a88a217a132a221a216a140a85
a247a136a136a135a135a144a149a209a180c.
a6a107a27a179a124a209a27a131a233a53a6a110a233a233a62a62a62a196a196a229a198a216a164a225!
a62a94a121a121a150a150a40a189a152a135a65a207a235a127a88.a131a233a84a65a207a235a127a88a27a36a196a143a253a233a36a196.
a8a158a27a110a41,a64a143a62a94a197a51”a177a20”a48a159a159a165a165a68a194,a131a233a53a6a110a216a164a225a180a103a44a27! a40a197,a89a197
299/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a100a194a131a233a216,a196a58～a162a8a196a58
a36a196a48a159a27a49a27a68a194a175a75—a220a162a162a8,a55a76a98a23a177a20a26a89a220a169a86a218
a207a233a20a216a211a235a127a88a165a49a132a27a11a201:
a114 a0a0 l1
l2
a240a142a16a154-a35a88a162a8,a23a164a236a131a233a253a233a183a142a88a177a89a178a132a221va36a196
T1= l1c?v+ l1c+v = 2cl1c2?v2 T2=
2
radicalBig
l22+(vT22 )2
c =
2l2√
c2?v2
a1
a1
a1
a1a1a21a65a65
a65
a65a65a85
T2
2 cl2
T2v?T
(1) = T1?T2 = 2cl1
c2?v2?
2l2√
c2?v2
a242a164a236a61 90?,
T(2) = 2l1√c2?v2? 2cl2c2?v2 → v = 0T
(1) =?T(2)
vnegationslash= 0T(1)negationslash=?T(2)
a101vnegationslash= 0,a90a21a94a171a51a61a196a76a167a165a152a189a135a117a41a67a122(v > 5Km/s?a163a196a152a138a94a171).a162a8
a40a216a180 v = 0,a51a152a99a111a71a209a118a255a209a47a165a254a49a132a51a216a211a144a149a254a27a11a201,a40a74a134a20a19a132a221a195a39
a86a40a162a8,a196a189a10a49a13a134a49a132a107a39.
a164a107a162a8a209a141a209,a49a132a216a157a54a117a42a9a246a164a51a27a235a127a88,a133a134a49a13a13a36a36a196a132a221a195a39.
300/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a131a233a216a196a29a6a110,a226a226a212a212a91a67a134,a196a29a6a110
a226a226a213a213a91a74a209a194a160a98a23,a247a177a20a36a196a144a149a107a127a221a194a160radicalbig1?v2/c2
a79a207a100a34a34a74a74a209:
a131a233a53a6a110,a164a107a46a53a235a127a88a209a180a31a100a27
a49a132a216a216a67a67a6a110,a253a152a165a49a132a51a63a219a219a46a46a53a88a247a63a152a144a149a240a143c,a134a49a13a195a39.
a131a233a53a6a110a119a138a183a130,a51a216a211a46a53a88a137a211a24a27a162a8a55a107a211a24a27a40a74a156
a63a219a196a29a212a110a53a198a51a216a211a46a53a88a165a55a107a131a211a27a144a167a47a170!
a132a221a131a233a53-a195a253a233a183a142:a207a76a229a198,a62a94a189a217a167a121a121a150a150a195a123a255a209a253a233a36a196.
a131a233a53a6a110a180a26a201,a167a177a110a53a27a229a229a254a254a93a212a10a177a99200a245a99a27a110a216a218a162a8a156
a49a132a216a216a67a67a6a110a119a138a183a130,a49a197a134a217a167a197a196a216a211a167a217a68a194a132a221a134a235a127a88a195a39.
a49a132a216a216a67a67a6a110a45a60a164a41a186 a118a107a177a20! a62a94a197a197a216a216a180a48a159a27a8a11!
301/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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DavidMattingly,UniversityofCaliforniaatDavis,gr-qc/0502097(2005.2.22)?1a102a146/a166
302/384
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trianglerightsldtrianglerightsld
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trianglerightsld
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a131a233a216a196a29a6a110,a226a226a212a212a91a67a134,a179a124a209a67a134
a98a23,a152a109a180a254a33a134a136a149a211a53a27a167a158a109a180a254a33a27!
(t,x,y,z)a218(tprime,xprime,yprime,zprime)a131a109a27a39a88a55a76a180a130a53a27,a143a159a111?
a140a177a192a74:a252a135a131a233a36a196a132a221a143vectorVa27a46a53a88Sa218S’,a252a88a27a139a73a88a139a73a182
a144a149a131a211,xa182a18a51vectorVa144a149,a191a23t = 0,tprime = 0a158a252a139a88a173a220.
a45
a54
a8a8
a8a25
a54
a8a8
a8a25
a45 vectorVS S
prime
z zprime
x xprime
y yprime
xprime = αx +βt yprime = y zprime = z tprime = γx +δt
a98a23,a36a196a196a27a27a131a233a53:
a108Sa119,S’a107a36a196a132a221V? dxprime = 0 dxdt = V?β =?αV
S’a119,Sa107a36a196a132a221-V? dx = 0 dxprimedtprime =?V?β =?δV
xprime = α(x?Vt) yprime = y zprime = z tprime = γx +αt
a179a124a209a67a134,tprime = t? γ = 0,α = 1 a253a233a158a109! a196a189a179a124a209a67a134a55a76a196a189a253a233a158a109
303/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a131a233a216a196a29a6a110,a226a226a212a212a91a67a134,a196a29a226a226a212a212a91a67a134
a252a135a131a233a36a196a132a221a143vectorVa27a46a53a88Sa218S’,a252a88a27a139a73a88a139a73a182
a144a149a131a211,xa182a18a51vectorVa144a149,a191a23t = 0,tprime = 0a158a252a139a88a173a220.
a45
a54
a8a8
a8a25
a54
a8a8
a8a25
a45 vectorVS S
prime
z zprime
x xprime
y yprime
a152a135a175a135a51Sa218S’a88a165a27a158a152a139a73a169a79a143(t,x,y,z)a218 (tprime,xprime,yprime,zprime)
xprime = αx +βt yprime = y zprime = z tprime = γx +δt
a51t = 0,tprime = 0a158a6a58a117a209a152a135a165a161a62a94a197,a51ta158a143a62a94a197a136a20a177a6a58Oa143a165a37a27a44
a135a165a197a10a161a254,a131a65a51tprimea158a143a62a94a197a136a20a177a6a58O’a143a165a37a27a44a135a165a197a10a161a254,
braceleftbigg x2 + y2 + z2 = c2t2
xprime2 + yprime2 + zprime2 = c2tprime2 →
braceleftbiggα2(x?Vt)2 + y2 + z2 = c2(γx +αt)2
xprime2 + yprime2 + zprime2 = c2tprime2
→
α2?c2γ2 = 1
2Vα2?2c2γα = 0
c2α2?V2α2 = c2
→ α = 1radicalBig
1?V2c2
γ =?
V
c2radicalBig
1?V2c2
xprime = x?VtradicalBig
1?V2c2
yprime = y zprime = z tprime = t?
V
c2xradicalBig
1?V2c2
V ≤c Vlessmuchc==?a179a124a209a67a134
304/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a131a233a216a196a29a6a110,a226a226a212a212a91a67a134,a196a29a226a226a212a212a91a67a134～a63a191a36a196a144a149a139a73a132a221a67a134?
vectorrprime·vectorV
V =
vectorr·vectorV
V?VtradicalBig
1?V2c2
vectorrprime?vectorrprime·vectorVV2 vectorV =vectorr?vectorr·vectorVV2 vectorV
tprime = t?vectorr·
vectorV
c2radicalBig
1?V2c2
→
vectorrprime =vectorr?vectorr·vectorVV2 vectorV + vectorr·vectorVV?VtradicalBig
1?V2c2
vectorV
V
tprime = t?vectorr·
vectorV
c2radicalBig
1?V2c2
vectoru = dvectorrdt vectoruprime≡dvectorr
prime
dtprime =
dvectorrprime
dt
dtprime
dt
=
vectoru?vectoru·vectorVV2 vectorV + vectoru·vectorVV?VradicalBig
1?V2c2
vectorV
V
1?vectorV·vectoruc2radicalBig
1?V2c2
vectoruprime =
radicalBig
1?V2c2 (vectoru?vectoru·vectorVV2 vectorV) + (vectoru·vectorVV2?1)vectorV
1? vectorV·vectoruc2 a131a233a216a165a27a132a221a220a164a39a88a156
Uα≡dx
α
dτ =
1radicalBig
1?V2c2
dxα
dt →
Ui= viradicalBig
1?v2c2
x1=x,x2=y,x3=z
U4= icradicalBig
1?v2c2
x4 = ict
305/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a131a233a216a196a29a6a110,a226a226a212a212a91a67a134,a196a29a226a226a212a212a91a67a134～a63a191a36a196a144a149a139a73a132a221a67a134
vectorrprime =vectorr?vectorr·
vectorV
V2
vectorV + vectorr·
vectorV
V?VtradicalBig
1?V2c2
vectorV
V
vectoruprime =
radicalBig
1?V2c2 (vectoru?vectoru·vectorVV2 vectorV) + (vectoru·vectorVV2?1)vectorV
1? vectorV·vectoruc2 |
vectoruprime| |vectoru|=c==?c |vectoru|<c?|vectoruprime|<c
vectora≡dvectorudt = d
2vectorr
dt2 vectora
prime≡dvectoru
prime
dtprime =
d2vectorrprime
dtprime2 =
dvectoruprime
dt
dtprime
dt
vectoraprime = 1
(1? vectorV·vectoruc2 )2
braceleftbig(1?V2
c2 )(vectora?
vectora·vectorV
V2
vectorV) +
radicalbigg
1?V
2
c2
vectora·vectorV
V2
vectorV
+vectora·vectorV
(1?V2c2 )(vectoru?vectoru·vectorVV2 vectorV) + (vectoru·vectorVV2?1)vectorV
radicalBig
1?V2c2
(1? vectorV·vectoruc2 )c2
bracerightbig
306/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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trianglerightsld
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a131a233a216a27a158a152a110a216,a211a158a53a167a36a196a158a168a134a186a102
a252a135a175a135P1a218P2a51a252a235a127a88a165a136a107a139a73(x1,t1),(xprime1,tprime1)a218 (x2,t2),(xprime2,tprime2)
a211a158a27a131a233a53:
tprime =?t?
V
c?xradicalBig
1?V2c2
t=0,?xnegationslash=0→?tprimenegationslash= 0
a51a152a135a46a53a88a165a216a211a47a47a58a58a211a158a117a41a27a252a175a135a167
a51a44a9a27a46a53a88a119a191a191a216a216a211a158a117a41a156
a211a158a131a233a53a27a28a28a91a91
a51a44a9a27a46a53a88a167a38a210a114a27a180a167a216a152a152a24a24a181
– a88a74a74a49a49a132a216a216a67a67a210a242a19a151a216a211a158a156
– a88a74a74a49a49a132a85a179a124a209a67a134a67a167a242a45a158a180a167a27a11a79a167a69a206a2a177a211a158a156
307/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
Back
Close
a131a233a216a27a158a152a110a216,a211a158a53a167a36a196a158a168a134a186a102
a252a135a175a135P1a218P2a51a252a235a127a88a165a136a107a139a73(x1,t1),(xprime1,tprime1)a218 (x2,t2),(xprime2,tprime2)
a36a196a158a168a67a250:
xprime = 0→?t =?t
prime + V
c2?x
prime
radicalBig
1?V2c2
=?t
prime
radicalBig
1?V2c2
tprimea180a36a196a168a103a67a27a27a214a214a234a163S’a88a252a175a135a135a109a109a27a158a109a11a164
ta180a183a142a142a168a168a27a27a214a214a234a163Sa88a252a175a135a135a109a109a27a158a109a11a164?t≥?tprime
a158a109a109a109a109a133a131a233a53a27a28a28a91a91
a216a211a47a47a58a58a58a27a27a183a142a142a168a168a55a76a31a158a233a79a217a167a88a119a118a233a79
Sa119S’a168a67a250! S’a119Sa168a143a67a250a156 (a86a41a102a15a184)
308/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
Back
Close
a131a233a216a27a158a152a110a216,a211a158a53a167a36a196a158a168a134a186a102
a252a135a175a135P1a218P2a51a252a235a127a88a165a136a107a139a73(x1,t1),(xprime1,tprime1)a218 (x2,t2),(xprime2,tprime2)
a36a196a186a102a67a225:
t = 0→?xprime =?x?V?tradicalBig
1?V2c2
=?xradicalBig
1?V2c2
a36a196a186a102a127a221a221a27a27a255a254a162a83a254a180a252a135a186a102a252a224a224a27a27a31a158a39a22a34
a252a135a88a27a27a31a31a158a158a180a180a216a152a151a27a34 a196a127a221?x≤?xprime
a127a221a131a233a53a27a28a28a91a91
Sa119S’a186a67a225,S’a119Sa186a143a67a225a156(a187a144a76a236a201; a186a102a75a150a112)
a164a107a249a10a209a180a36a196a198a167a13a216a180a196a229a198a198a8a8a65a156
309/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
Back
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a131a233a216a27a158a152a110a216,a39a117a158a109a27a181a53
a64a10a168a254a141a171a27a158a109a180a216a180a47a212a110a27a158a109a48?
a218a238a238a27a27a42a58a181 a253a233a233a27a27a158a109a103a28a254a33a47a54a178
a226a226a212a212a91a27a42a58a181a218a92a131a233a253a233a158a109a47a88a47a170a76a136a48a27a35a27a219a141a158a109
a117a178a226a226a212a212a91a67a134a167a2a118a107a152a239a253a233a158a109! a226a226a213a213a91a195a10a234a198a167a2a118a195a212a110 a13a8a119
a36a92a52a27a42a58a181 a242a219a141a158a109a41a186a143a131a233a177a20a27a36a196a88a165a183a142a27
a207a76a76a49a49a38a210a4a233a233a27a27a158a168a27a27a214a214a234a208a148a49a132a51a136a46a53a88a209a152a152a24a24
a171a64a53a198a131a233a53a167a2a118a107a152a239a253a233a158a109! a36a92a52a195a10a243a198? a2a118a195a212a110 a13a8a119
a79a207a100a34a27a42a58a181a118a107a253a233a158a109a156a219a141a158a109a210a180a46a53a88a27a212a110a158a109a156
310/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a131a233a216a27a158a152a110a216,a109a133a27a216a216a67a67a53
ca107a129,a168a218a186a102a255a254a27a40a74a134a235a127 a88a27a192a74a107a39,a167a130a216a85a50a138a143a253a233a73a79a53a239a254
a175a135a131a109a27a158a152a39a88,a73a135a207a233a35a27a134 a235a127a88a192a74a195a39a27a254,a61a51a226a212a91a67a134a101a216
a67a27a254a254a53a53a138a143a253a233a73a79.
a233a152a135a235a127a88a165a27a252a135a175a135(x1,y1,z1,t1),(x2,y2,z2,t2),a189a194a167a130a131a109a27a109a133a143:
s2≡c2(t1?t2)2?(x1?x2)2?(y1?y2)2?(z1?z2)2
= c2(?t)2?(?x)2?(?y)2?(?z)2
sprime2≡c2(tprime1?tprime2)2?(xprime1?xprime2)2?(yprime1?yprime2)2?(zprime1?zprime2)2
=c2[t1?t2?
V
c2(x1?x2)radicalBig
1?V2c2
]2?[x1?x2?V(t1?t2)radicalBig
1?V2c2
]2?(y1?y2)2?(z1?z2)2
=(t1?t2)2[ c
2
1?V2c2?
V2
1?V2c2 ]+(x1?x2)
2[
V2
c2
1?V2c2?
1
1?V2c2 ]?(y1?y2)
2?(z1?z2)2
=?s2
a152a132a27a226a226a212a212a91a67a134a51a234a198a254a189a194a143a2a177a109a133a216a216a67a67a27 (x,y,z,t)a27a224a103a130a53a67a134.
311/384
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a131a233a216a27a158a152a110a216,a109a133a27a216a216a67a67a53
a233a195a161a2a109a133:
ds2 = c2dt2?dvectorr·dvectorr = c2dt2(1?v
2
c2) = c
2dτ2 =?dl2
0 vectorv =
dvectorv
dt
a233ds2 > 0,dτ ≡ 1c√ds2a180a168a27a27a107a158(a6a158),a167a180a100a109a133
a218a209a27a228a107a158a109a254a106a27a216a216a67a67a254.
a233ds2 < 0,dl0≡√?ds2a180a186a27a27a107a127a221,a167a180a100a109a133a218a209
a27a228a107a127a221a254a106a27a216a216a67a67a254.
a252a175a135a39a88a27a169a97:
s2 >0,a97a158a158a175a175a135,a97a158a109a133
s2 <0,a97a152a175a135,a97a152a109a133
s2 = 0,a97a49a175a135,a97a49a109a133
312/384
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a131a233a216a27a158a152a110a216,a97a158a109a133?s2 >0
a38a210a132a221,ˉv =
radicalbigg
x2
t2 +
y2
t2 +
z2
t2
0<?s2 = c2?t2x2y2z2 =?t2(c2?ˉv2)→ˉv <c
a252a175a135a131a109a140a177a94a49a38a210a239a225a225a233a233a88,a207a143a239a225a225a233a233a88a164a73a27a38a210a132a221a2a117c.
a158a109a103a83,?tprime =?t?
V
c2?xradicalBig
1?V2c2
=?tradicalBig
1?V2c2
[1?Vc2?x?t]
x
t ≤ˉv<cv <c
bracerightbiggv?x
t <c
2→ v?x
c2?t<1→1?
V
c2
x
t>0→?t
primea134?ta211a210
0<?s2=x2y2z2+c2?t2=xprime2yprime2zprime2+c2?tprime2
→?tnegationslash= 0?tprimenegationslash= 0
a252a175a135a216a211a158a117a41a167a167a130a131a109a27a158a109a103a83a216a172a54a54a16a16,a207a74a53a2a177!
a152a109a103a83,?y =?z = 0?xprime =?x?V?tradicalBig
1?V2c2
=?x 1?
V
ˉVradicalBig
1?V2c2
313/384
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a131a233a216a27a158a152a110a216,a97a158a109a133?s2 >0
a38a210a132a221,a252a175a135a131a109a140a177a94a49a38a210a239a225a225a233a233a88.
a158a109a103a83,a252a175a135a131a109a27a158a109a103a83a216a172a54a54a16a16,a207a74a53a2a177!
a152a109a103a83,?y =?z = 0?xprime =?x?V?tradicalBig
1?V2c2
=?x 1?
V
ˉVradicalBig
1?V2c2
x a134?xprime a216a152a189a211a210,a192 V = ˉV =||?x?t||
xprime = 0?yprime = 0?zprime = 0?tprime =?t?
V
c2?xradicalBig
1?V2c2
=?t?
x2
c2?t2radicalBig
1x2c2?t2
a175a135a131a109a27a152a109a103a83a145a235a127a88a192a74a13a67,a127a51a152a135a235a127a88,a252a175a135a51a152a109a211a152a47a47a58a58a117a41!
314/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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a131a233a216a27a158a152a110a216,a97a152a109a133?s2 <0
a38a210a132a221,0>?s2=c2?t2x2y2z2=?t2(c2?ˉv2)→ˉv >c
a252a175a135a131a109a195a123a94a49a38a210a239a225a225a233a233a88,a207a143a239a225a225a233a233a88a164a73a27a38a210a132a221a140a117c.
a158a109a103a83,?tprime =?t?
V
c2?xradicalBig
1?V2c2
=?tradicalBig
1?V2c2
[1?Vc2?x?t]
V
c <1?x
c?ta61a140a140a1171a143a140a2a1171
bracerightbiggV?x
c2?ta140a140a1171a189a2a1171→1?
V?x
c2?ta140a140a1170a189a2a1170
tprimea216a152a189a134?ta211a210.
a192?y = 0,?z = 0
V = c
2?t
x = c
c
x
t
= ccˉv <c?tprime =?t?
V
c2?xradicalBig
1?V2c2
= 0
a175a135a131a109a27a158a109a103a83a180a216a40a189a189a27a27,a145a235a127a88a192a74a13a67.
a127a51a152a135a235a127a88,a51a217a165a252a175a135a211a158a117a41,a207a74a53a216a216a2a2a177.
315/384
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a131a233a216a27a158a152a110a216,a97a152a109a133?s2 <0
a38a210a132a221,a252a175a135a131a109a195a123a94a49a38a210a239a225a225a233a233a88.
a158a109a103a83,a175a135a131a109a27a158a109a103a83a180a216a40a189a189a27a27,a145a235a127a88a192a74a13a67.
a127a51a152a135a235a127a88,a51a217a165a252a175a135a211a158a117a41,a207a74a53a216a216a2a2a177.
a152a109a103a83,?y =?z = 0
xprime =?x?V?tradicalBig
1?V2c2
=?x 1?
V
ˉVradicalBig
1?V2c2
V <c
ˉV >c
bracerightbigg
V < ˉV→1?VˉV >0
xa134?xprimea211a210,a65a79a47,
0>?s2=x2y2z2+c2?t2=xprime2yprime2zprime2+c2?tprime2
→a216a127a51?xprime =?yprime =?zprime = 0
a51a63a219a235a127a88a119,a252a175a135a131a109a27a152a109a160a152a216a211,a133a160a152a131a233a103a83a216a172a54a54a16a16
a216a127a51a51a249a249a24a24a152a152a135a235a127a88,a51a217a165a252a175a135a51a211a152a47a47a58a58a117a41.
316/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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a131a233a216a27a158a152a110a216,a97a49a109a133?s2 = 0
a38a210a132a221,0=?s2=c2?t2x2y2z2=?t2(c2?ˉv2)→ˉv=c
a252a175a135a131a109a140a177a94a49a38a210a239a225a225a233a233a88,a207a143a239a225a225a233a233a88a164a73a27a38a210a132a221a221a31a31a117c.
a158a109a103a83,?tprime =?t?
V
c2?xradicalBig
1?V2c2
=?tradicalBig
1?V2c2
[1?Vc2?x?t]
x
t ≤ˉv = cv <c
bracerightbiggv?x
t <c
2→ v?x
c2?t <1→1?
V
c2
x
t >0
tprimea134?ta211a210.
y = 0,?z = 0 →?x = c?t
tprime =?t?
V
c2?xradicalBig
1?V2c2
=?t 1?
V
cradicalBig
1?V2c2
=?t
radicalbiggc?v
c + v
v→c→0
a252a175a135a131a109a27a158a109a103a83a69a216a172a54a54a16a16.a207a74a53a69a69a44a44a2a177.
317/384
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trianglerightsldtrianglerightsld
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a131a233a216a27a158a152a110a216,a97a49a109a133?s2 = 0
a38a210a132a221,a252a175a135a131a109a140a177a94a49a38a210a239a225a225a233a233a88.
a158a109a103a83,a252a175a135a131a109a27a158a109a103a83a69a216a172a54a54a16a16.a207a74a53a69a69a44a44a2a177.
a127a51a51a249a249a24a24a152a152a135a235a127a88,a51a217a165a252a175a135a211a158a117a41,a249a135a235a127a88a177a49a132a131a233a6a235a127a88a36a196.
a152a109a103a83,?y =?z = 0→?x = c?t
xprime =?x?V?tradicalBig
1?V2c2
=?x 1?
V
cradicalBig
1?V2c2
=?x
radicalbiggc?v
c + v
v→c→0
a252a175a135a131a109a27a152a109a103a83a216a172a54a54a16a16
a127a51a152a135a235a127a88,a51a217a165a252a175a135a51a211a152a47a47a58a58a117a41,a249a135a235a127a88a177a49a132a131a233a6a235a127a88a36a196.
a178a59a229a198a131a8a117a18c→∞,a249a158a144a107a97a158a158a175a175a135.
a177(x,y,z,t)a143a139a73a228a107?s2a216a216a67a67a53a27a152a109a161a143Minkowskia152a109.
a111a145a158a152a51a110a145a102a152a109(a152a145a158a109,a252a145a152a109)a27a221a75a171a165,a242a121a51(t = 0,
x = y = z = 0)a18a143a152a175a135,a75a75a249a249a158a109a133a169a97a177’a49a73’a143a46,a49a73a144a167a143
c2t2 = x2 +y2,a49a73a83a143a97a158a109a133,a49a73a254a143a97a49a109a133,a49a73a9a9a143a143a97a152a109
a133,a97a158a171a26a121a51(t = 0,x = y = z = 0)a169a143a253a233a76a22a218a253a233a153a53.
318/384
triangleleftsldtriangleleftsld
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a131a233a216a27a158a152a110a216,a207a74a53a134a129a140a38a210a68a52a132a221
a107a207a74a53a27a175a135(a97a158a189a97a49)a131a109a27a38a210a68a52a132a221≤c
a195a207a74a53a27a175a135a163a97a152a164a131a109a27a38a210a68a52a132a221>c
a129a140a27a212a110a38a210a68a52a132a221a143c a135a49a132a226a102(tachyon)a19a151a187a128a207a74a53
a164a107a27a131a112a138a94a143a129a245a144a85a177a132a221ca68a52a182a195a135a229a138a94
a110a41a49a132a216a216a67a67a6a110a181
– a49a132a81a180a129a140a27a212a110a38a210a68a52a132a221
– a131a233a53a6a110a135a166a129a140a212a110a38a210a68a52a132a221a134a235a127a88a192a74a195a39
a39a117a38a210a181
– a159a58a138a143a38a210a132a221a78a180a40a189
– a197a138a143a38a210a132a221a216a78a180a40a189
319/384
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trianglerightsldtrianglerightsld
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a131a233a216a110a216a27a14a67a47a170,a111a145a158a152a139a73a73a67a67a134
a152a132a226a226a212a212a91a67a134a107a56a135a213a225a103a100a221,a110a135a131a233a36a196,a110a135a139a73a182a131a233a61a196.
c2t2?(x2+y2+z2)=c2tprime2?(xprime2+yprime2+zprime2) x1=x,x2=y,x3=z,x4=ict→
x21+x22+x23+x24=xprime12+xprime22+xprime32+xprime42 a189
4summationdisplay
μ=1
xμxμ =
4summationdisplay
μ=1
xprimeμxprimeμ
a247a118xμ = 0a134xprimeμ = 0a233a65a27a152a132a130a53a67a134a143,xprimeμ =
4summationdisplay
μ=1
aμνxν
4summationdisplay
μ=1
xprimeμxprimeμ =
4summationdisplay
μ,ν,λ=1
aμνxνaμλxλ =
4summationdisplay
μ,ν,λ=1
aμνaμλxνxλ =
4summationdisplay
μ=1
xμxμ
4summationdisplay
μ=1
aμνaμλ = δνλ a137a20910a135a21a229a144a167 xμ=
4summationdisplay
ν=1
δμνxν=
4summationdisplay
ν,λ=1
aλμaλνxν=
4summationdisplay
λ=1
aλμxprimeλ
a44a152a171a76a136a47a170,
4summationdisplay
μ=1
aνμaλμ = δνλ
320/384
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trianglerightsldtrianglerightsld
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a131a233a216a110a216a27a14a67a47a170,a111a145a158a152a139a73a73a67a67a134
a152a132a226a226a212a212a91a67a134a107a56a135a213a225a103a100a221,a110a135a131a233a36a196,a110a135a139a73a182a131a233a61a196.
xprimeμ =
4summationdisplay
μ=1
aμνxν
4summationdisplay
μ=1
aμνaμλ = δνλ xμ =
4summationdisplay
λ=1
aλμxprimeλ
4summationdisplay
μ=1
aνμaλμ = δνλ
X =
x1
x2
x3
x4
A =
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
Xprime = AX X = ATXprime ATA = I AAT = I
A =
1radicalBig
1?V2c2
0 0 iVcradicalBig
1?V2c2
0 1 0 0
0 0 1 0
iVcradicalBig
1?V2c2
0 0 1radicalBig
1?V2c2
1 = detI = det(ATA) = (detAT)(detA) = (detA)2→(detA) =±1
detA = 1a100a88a61a196a8a164 detA =?1a100a88a61a196a196a83a83a92a254a152a109a135a19a189a158a109a135a252a8a164
321/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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trianglerightsld
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a131a233a216a110a216a27a14a67a47a170,a212a110a110a254a254a85a158a152a67a134a53a159a169a97
a131a233a53a6a110a27a234a198a76a136a181 3+1a145a158a152a220a254a144a167
0a30a220a254:(a73a254,a226a226a212a212a91a216a216a67a67a254) φprime = φ
1a30a220a254:(a165a254) Aprimeμ =
4summationdisplay
ν=1
aμνAν
2a30a220a254:Bprimeμν =
4summationdisplay
μprime,νprime=1
aμμprimeaννprimeBμprimeνprime
···
na30a220a254,Tprimeμ1···μn =
4summationdisplay
ν1,···,νn=1
aμ1ν1···aμnνnTν1···νn
na30a220a254a27a65a58a180a1074na135a169a254,a51a226a226a212a212a91a67a134a101,a85na135a139a73a166a200a27a67a134a144a170a67a134.
322/384
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trianglerightsldtrianglerightsld
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a131a233a216a110a216a27a14a67a47a170,a212a110a110a254a254a85a158a152a67a134a53a159a169a97
Tprimeμ1···μn =
4summationdisplay
ν1,···,νn=1
aμ1ν1···aμnνnTν1···νn
Tμ1···μn = Fμ1···μn
4summationdisplay
ν1,·,νn=1
aν1μ1···aνnμnTμ1···μn =
4summationdisplay
ν1,·,νn=1
aν1μ1···aνnμnFμ1···μn →Tprimeμ1···μn =Fprimeμ1···μn
a51a35a139a73a88a165a27a144a167a170a47a170a254a134a206a139a73a88a165a27a144a167a170a152a152a24a24—a131a233a53a6a110
a124a94a220a254,a140a242a163a227a212a110a53a198a27a144a167a21a164a247a118a131a233a53a6a110a27a47a170—a212a110a53a198a27a14a67a67a76a76a136.
a242a164a107a212a110a53a198a209a94a220a254a76a136,a73a127a23:
a220a254a131a109a27a36a142a123a75.
a17a23a27a220a254a56a107a245a8a171a220a254.
a228a78a27a212a110a53a198a88a219a94a220a254a76a136.
323/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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trianglerightsld
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a131a233a216a110a216a27a14a67a47a170,a220a254a36a142
a92a92a126a126a123,a211a30a220a254a92a92a126a126a0a69a180a249a30a220a254
Fprimeμ1···μn =
4summationdisplay
ν1,···,νn=1
aμ1ν1···aμnνnFν1···νn Gprimeμ1···μn =
4summationdisplay
ν1,···,νn=1
aμ1ν1···aμnνnGν1···νn
Fprimeμ1···μn±Gprimeμ1···μn =
4summationdisplay
ν1,···,νn=1
aμ1ν1···aμnνn[Fν1···νn±Gμ1···μn]
a166a123,ma30a220a254a134na30a220a254a27a166a200a143m + na30a220a254
Aprimeμ1···μm =
4summationdisplay
ν1,···,νm=1
aμ1ν1···aμmνmAν1···νm Bprimeμ1···μn =
4summationdisplay
ν1,···,νn=1
aμ1ν1···aμnνnBν1···νn
Tμ1···μm+n ≡Aμ1···μmBμm+1···μm+n
Tprimeμ1···μm+n = Aprimeμ1···μmBprimeμm+1···μm+n
=
4summationdisplay
ν1,···,νm=1
aμ1ν1···aμmνmAν1···νm
4summationdisplay
νm+1,···,νm+n=1
aμm+1νm+1···aμm+nνm+nBνm+1···νm+n
324/384
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trianglerightsldtrianglerightsld
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a131a233a216a110a216a27a14a67a47a170,a220a254a36a142
a92a92a126a126a123,a211a30a220a254a92a92a126a126a0a69a180a249a30a220a254
a166a123,ma30a220a254a134na30a220a254a27a166a200a143m + na30a220a254
Aprimeμ1···μm =
4summationdisplay
ν1,···,νm=1
aμ1ν1···aμmνmAν1···νm Bprimeμ1···μn =
4summationdisplay
ν1,···,νn=1
aμ1ν1···aμnνnBν1···νn
Tμ1···μm+n ≡Aμ1···μmBμm+1···μm+n
Tprimeμ1···μm+n = Aprimeμ1···μmBprimeμm+1···μm+n
=
4summationdisplay
ν1,···,νm=1
aμ1ν1···aμmνmAν1···νm
4summationdisplay
νm+1,···,νm+n=1
aμm+1νm+1···aμm+nνm+nBνm+1···νm+n
=
4summationdisplay
ν1,···,νm+n=1
aμ1ν1···aμm+nνm+nAν1···νmBνm+1···νm+n =
4summationdisplay
ν1,···,νm+n=1
aμ1ν1···aμm+nνm+nTν1···νm+n
a160a191,na30a220a254a194a160a152a103a67a164n-2a30a220a254
4summationdisplay
μi,μj=1
δμiμjTμ1···μi···μj···μna161a143Tμ1···μi···μj···μna233a141a73μi,μja27a160a191.
325/384
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a131a233a216a110a216a27a14a67a47a170,a220a254a27a126a102
c,h,a62a214,a183a142a142a159a159a254···a143a226a226a212a212a91a73a254.
a111a145a78a200a3d4x≡dx1dx2dx3dx4 = icdVdta27a67a134a39a88a143:
d4xprime =||detA||d4x = d4x→a226a226a212a212a91a73a254dVprimedtprime = dVdt
a111a145a132a221uμ = dxμdτ a143a111a165a254,a207dxμa143a111a165a254,dτa180a216a216a67a67a254.
a14a67a135a251x
μ
a180a152a30a220a254
xprimeμ =
4summationdisplay
ν=1
xν
xprimeμ
xν =
4summationdisplay
ν=1
aμνx
ν
4summationdisplay
μ=1
xμ
xμ =
x
x +
y
y +
z
z?
1
c2
t
t =?
2? 1
c2
2
t2
δμνa180a19a30a252a160a220a254,δprimeμν = δμν =
4summationdisplay
λ=1
aμλaνλ =
4summationdisplay
λ,λprime=1
aμλaνλprimeδλλprime
326/384
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a131a233a216a53a212a110a198
a174a127a127a51a152a64a220a254a254a110a110a216a140a177a108a234a198a254a76a136a131a233a53a6a110
a144a135a114a212a110a144a167a21a164a180a111a145a158a152a165a27a220a254a144a167
a183a130a78a111a127a23a198a76a27a212a110a144a167a85a196a21a164a220a254a144a167?
a216a139a73a9a167a183a130a216a127a23a217a167a167a164a164a107a212a110a110a254a254a218a111a145a220a254a107a159a111a39a88a186
a173a35a34a34a192a192a183a130a27a229a198a218a62a62a196a196a229a198a110a216
a135a166a110a216a51a152a109a169a210a247a118a131a233a53a6a110a19a209a212a110a110a254a254a218a111a145a220a254a27a39a88a156
a173a35a237a19a209a229a198a218a62a62a196a196a229a198a110a216
a191a239a225a94a111a145a220a254a76a136a27a212a110a144a167
a66a23a172a181a216a108a162a8a167a13a108a110a216a6a6a75a75a19a209a10a229a198a218a62a62a196a196a229a198a27a164a107a144a167
327/384
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a131a233a216a229a198,a129a2a138a94a254a6a110
a233a117a122a152a135a229a198a78a88,a127a51a152a135a23a138a94a254a27a200a169S,a167a233a117a162a83a36a196a107a129a2a138.
a233a152a135a103a100a221a221a27a27a78a88,S =
integraldisplay β
α
dt L(q.˙q)
0 = δS =
integraldisplay β
α
dt[?L?˙qδ˙q +?L?qδq] =
integraldisplay β
α
dt[?L?˙q ddtδq +?L?qδq]
=
integraldisplay β
α
dt [ ddt(?L?˙qδq) + (?L?q? ddt?L?˙q)δq] →L?q + ddt?L?˙q = 0
p =?L?˙q H = p˙q?L
dp
dt =
L
q
dH
dt =
dp
dt ˙q + p
d˙q
dt?
L
q
dq
dt?
L
˙q
d˙q
dt = 0
328/384
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a131a233a216a229a198,a58a226a102a229a198
S =?m0c
integraldisplay b
a
ds ds = c
radicalbigg
1?v
2
c2dt→S =?m0c
2
integraldisplay b
a
dt
radicalbigg
1?v
2
c2
L =?m0c2
radicalbigg
1?v
2
c2
c→∞→?m
0c2 +
1
2m0v
2
L?x
i
+ ddt?L?v
i
= 0 → ddt?L?v
i
= 0
pi=?L?v
i
= m0viradicalBig
1?v2c2
vectorp= m0vectorvradicalBig
1?v2c2
=m0vectoru=mvectorv m= m0radicalBig
1?v2c2
H = vectorp·vectorv?L = m0v
2
radicalBig
1?v2c2
+ m0c2
radicalbigg
1?v
2
c2 =
m0c2radicalBig
1?v2c2
= mc2
dpi
dt = 0
dvectorp
dt = 0
dH
dt = 0
329/384
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a131a233a216a229a198,a58a226a102a229a198
S =?m0c2
integraldisplay b
a
dt
radicalbigg
1?v
2
c2 m =
m0radicalBig
1?v2c2
vectorp = m0vectorvradicalBig
1?v2c2
= mvectorv dvectorpdt = 0 H = mc2 dHdt = 0
uμ = dx
μ
dτ =
1radicalBig
1?v2c2
dxμ
dt = (
vectorvradicalBig
1?v2c2
,icradicalBig
1?v2c2
)
pμ = m0uμ = ( m0vectorvradicalBig
1?v2c2
,im0cradicalBig
1?v2c2
) = (vectorp,iHc ) dp
μ
dτ = 0
summationdisplay
μ
pμpμ = m
2
0v
2?m2
0c
2
radicalBig
1?v2c2
=?m20c2 → H
2
c2?p
2 = m2
0c
2
330/384
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a131a233a216a229a198,a145a62a62a58a58a226a102a57a62a214a169a217a51a9a62a94a124a165
a62a94a124a88a219a163a227a186
a212a110a110a254a254a140a85a217a158a152a67a134a53a159a169a97,a216a211a220a254a147a76a76a216a216a211a212a110
a62a94a124a65a84a225a117a44a152a220a254,a98a23a217a143a111a165a254Aμ = (vectorA,icφ)
a233a39a158a152a139a73a165a254xμ = (x,y,z,ict)a57a217a226a226a212a212a91a67a134a39a88
xprime = x?vtradicalBig
1?v2c2
yprime = y zprime = z tprime = t?
v
c2xradicalBig
1?v2c2
a140a26(vectorA,φ)a27a226a226a212a212a91a67a134a39a88
Aprimex = Ax?
v
c2φradicalBig
1?v2c2
Aprimey = Ay Aprimez = Az φprime = φ?vAxradicalBig
1?v2c2
a143a159a111a135a94a111a221a165a254a124a53a163a227a62a94a124a186
331/384
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a131a233a216a229a198,a145a62a62a58a58a226a102a57a62a214a169a217a51a9a62a94a124a165
a98a23a23a165a165a254a124Aμ = (vectorA,icφ)a163a227a62a94a124,a53a189a10 (vectorA,φ)a27a226a226a212a212a91a67a134a39a88,
Aprimex = Ax?
v
c2φradicalBig
1?v2c2
Aprimey = Ay Aprimez = Az φprime = φ?vAxradicalBig
1?v2c2
a143a163a227a145a62a58a226a102a51a9a62a94a124a27a49a143,a55a76a51a103a100a58a226a102a138a94a254a92a254a152a135a135a65a62a94
a124a134a145a62a226a102a117a41a131a112a138a94a27a145a34 a249a152a145a150a8a65a157a185a58a226a102a27a53a159a218a62a94a124a27a53
a159a167a191a133a180a226a226a212a212a91a67a134a73a254a181
S =?m0c
integraldisplay b
a
ds + e
4summationdisplay
μ=1
integraldisplay b
a
dxμ Aμ a129a2a205a220
a249a24a27a27a62a62a94a131a112a138a94a228a107a53a137a216a216a67a67a53a181Aμ→Aprimeμ = Aμ +?μχ
S=
integraldisplay b
a
(?m0cds+evectorA·dvectorr?eφdt)=
integraldisplay b
a
dt (?m0c2
radicalbigg
1?v
2
c2 +e
vectorA·vectorv?eφ)
332/384
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a131a233a216a229a198,a145a62a62a58a58a226a102a57a62a214a169a217a51a9a62a94a124a165
S =
integraldisplay b
a
dt(?m0c2
radicalbigg
1?v
2
c2 +e
vectorA·vectorv?eφ) L=?m0c2
radicalbigg
1?v
2
c2 +e
vectorA·vectorv?eφ
vectorp = m0vectorvradicalBig
1?v2c2
+ evectorA a20a75a196a254negationslash= a197a22a196a254 H = m0c
2
radicalBig
1?v2c2
+ eφ
(vectora·vectorb) =vectora·?vectorb +vectorb·?vectora +vectorb×(?×vectora) +vectora×(?×vectorb)
3summationdisplay
i=1
L
xivectorei=?L=e?(
vectorA·vectorv)?e?φ=evectorv·?vectorA + evectorv×(?×vectorA)?e?φ
d
dt
m0vectorvradicalBig
1?v2c2
= dvectorpdt?ed
vectorA
dt
L?x
i
+ ddt?L?v
i
= 0=========?L?edvectorA
dt
= evectorv·?vectorA + evectorv×(?×vectorA)?e?φ?e(?
vectorA
t +
vectorr
t·?
vectorA)
= e(
vectorA
tφ) + evectorv×(?×
vectorA)
333/384
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a131a233a216a229a198,a145a62a62a58a58a226a102a57a62a214a169a217a51a9a62a94a124a165
S =
integraldisplay b
a
dt(?m0c2
radicalbigg
1?v
2
c2 +e
vectorA·vectorv?eφ) L=?m0c2
radicalbigg
1?v
2
c2 +e
vectorA·vectorv?eφ
vectorp = m0vectorvradicalBig
1?v2c2
+ evectorA a20a75a196a254negationslash= a197a22a196a254 H = m0c
2
radicalBig
1?v2c2
+ eφ
d
dt
m0vectorvradicalBig
1?v2c2
= e(
vectorA
tφ) + evectorv×(?×
vectorA)≡vectorF = evectorE + evectorv×vectorB
vectorE =vectorA
tφ
vectorB =?×vectorA
Eprimex = Ex Eprimey = Ey?vBzradicalBig
1?v2c2
Eprimez = Ez + vByradicalBig
1?v2c2
Bprimex = Bx Bprimey = By +
v
c2EzradicalBig
1?v2c2
Bprimez = Bz?
v
c2EyradicalBig
1?v2c2
334/384
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a131a233a216a229a198,a62a214a169a217a51a9a62a94a124a165
a58a58a62a62a214a57a217a54a196a180a235a89a62a214a169a217a57a217a54a196a196a27a27a65a126
ρ(vectorr,vectorrprime) = eδ(vectorr?vectorrprime) vectorj(vectorr,vectorrprime) = ρ(vectorr,vectorrprime)vectorv
edxμ = eδ(vectorr?vectorrprime)dτdxμ = ρdτdxμ = ρdxμdt dtdτ = jμdtdτ
jμ≡ρdxμdt = (vectorj,icρ)
jprimex = jx?vρradicalBig
1?v2c2
jprimey = jy jprimez = jz ρprime = ρ?
v
c2jxradicalBig
1?v2c2
4summationdisplay
μ=1
integraldisplay b
a
dxμ eAμ =
4summationdisplay
μ=1
integraldisplay
dtdτAμjμ =
integraldisplay
dtdτ(vectorA·vectorj?φρ)
0 =
4summationdisplay
μ=1
integraldisplay
dtdτjμ?μχ =?
4summationdisplay
μ=1
integraldisplay
dtdτ(?μjμ)χ
4summationdisplay
μ=1
μjμ = 0 a189?ρ?t +?·vectorj = 0
335/384
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a131a233a216a229a198,a14a67a67a76a76a136P
μ = pμ + eAμ pμ = (
m0vectorvradicalBig
1?v2c2
,im0cradicalBig
1?v2c2
)
Fμν≡?μAννAμ=
0 B3?B2?icE1
B3 0 B1?icE2
B2?B1 0?icE3
i
cE1
i
cE2
i
cE3 0
Fprimeμν =
4summationdisplay
λ,λprime=1
aμλFλλprimeaνλprime
Fprime = AFAT?
Fij =
3summationdisplay
k=1
epsilon1ijkBk
F4i =?Fi4 = icEi
Bi =
3summationdisplay
j,k=1
1
2epsilon1ijkFjk
Ei =?icF4i
fμ≡
4summationdisplay
ν=1
Fμνjν = (ρvectorE +vectorj×vectorB,icvectorj·vectorE)
fi = Fi4j4 +
3summationdisplay
k=1
Fikjk =?icEiicρ+
3summationdisplay
k,l=1
epsilon1ikljkBl = (ρvectorE +vectorj×vectorB)i
f4 =
3summationdisplay
k=1
F4kjk = icEkjk = icvectorE·vectorj
336/384
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a131a233a216a229a198,a14a67a67a76a76a136
fμ≡
4summationdisplay
ν=1
Fμνjν = (ρvectorE +vectorj×vectorB,icvectorj·vectorE)
a233a233a58a58a58a62a62a214,jμ = eδ(vectorr?vectorrprime)dxμdt =
integraldisplay
dtprime eδ(vectorr?vectorrprime)δ(t?tprime)dxμdtprime
=
integraldisplay
dτprimeeδ(4)(x?xprime)dxμdτprime =
integraldisplay
dτprimeeδ(4)(x?xprime)uμ
dpμ
dτ = Kμ
a138a146===?K
μ = (vectorK,
i
c
vectorK·vectorv) =
4summationdisplay
ν=1
eFμνuν fμ =
integraldisplay
dτδ(4)(x?xprime)Kμ
= parenleftbig eradicalBig
1?v2c2
(vectorE +vectorv×vectorB),
ie
c
vectorE·vectorv
radicalBig
1?v2c2
parenrightbig
a189,vectorF = vectorK
radicalbigg
1?v
2
c2
vectorF = dvectorP
dt
vectorF·vectorv = dH
dt
337/384
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a131a233a216a62a62a196a196a229a198,a138a94a254
a143a163a227a62a94a124a36a196,a65a51a138a94a254a165a92a152a145a88a135a65a62a94a124a27a145,a247a118a53a137a216a216a67a67a6a110.
4summationdisplay
μ,ν=1
FμνFμν = 2(B2? 1c2E2)
4summationdisplay
μ,ν,σ,ρ=1
epsilon1μνσρFμνFσρ = icvectorE·vectorB =
4summationdisplay
μ,ν,σ,ρ=1
4epsilon1μνσρ?μ(Aν?σAρ)
S =
integraldisplay
dtdτbracketleftbig(? 14μ
0
)
4summationdisplay
μ,nu=1
FμνFμν +
4summationdisplay
μ=1
jμAμbracketrightbig
0 = δS =
integraldisplay
dtdτbracketleftbig(? 14μ
0
)
4summationdisplay
μ,nu=1
δ(FμνFμν) +
4summationdisplay
μ=1
jμδAμbracketrightbig
=
integraldisplay
dtdτbracketleftbig(? 12μ
0
)
4summationdisplay
μ,ν=1
FμνδFμν +
4summationdisplay
μ=1
jμδAμbracketrightbig
338/384
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a131a233a216a62a62a196a196a229a198,a240a142a100a137a144a167a124
0 = δS =
integraldisplay
dtdτbracketleftbig(? 14μ
0
)
4summationdisplay
μ,nu=1
δ(FμνFμν) +
4summationdisplay
μ=1
jμδAμbracketrightbig
=
integraldisplay
dtdτbracketleftbig(? 12μ
0
)
4summationdisplay
μ,ν=1
FμνδFμν +
4summationdisplay
μ=1
jμδAμbracketrightbig
=
integraldisplay
dtdτbracketleftbig(?1μ
0
)
4summationdisplay
μ,ν=1
Fμνδ(?μAν) +
4summationdisplay
μ=1
jμδAμbracketrightbig
=
integraldisplay
dtdτbracketleftbig(?1μ
0
)
4summationdisplay
μ,ν=1
Fμν(?μδAν) +
4summationdisplay
μ=1
jμδAμbracketrightbig
=
integraldisplay
dtdτbracketleftbig(?1μ
0
)
4summationdisplay
μ,ν=1
[?μ(FμνδAν)?(?μFμν)δAν] +
4summationdisplay
μ=1
jμδAμbracketrightbig
4summationdisplay
μ=1
μFμν =?μ0jν?μFνλ +?νFλμ +?λFμν = 0
339/384
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a131a233a216a62a62a196a196a229a198,a240a142a100a137a144a167a1244summationdisplay
μ=1
μFμν =?μ0jν?μFνλ +?νFλμ +?λFμν = 0
3summationdisplay
j=1
jFji =?μ0ji4F4i→
3summationdisplay
j,k=1
jepsilon1jikBk =?μ0ji? 1c2?Ei?t
→?×vectorB=μ0vectorj+μ0epsilon10?
vectorE
t
3summationdisplay
i=1
iFi4 =?μ0j4 →
3summationdisplay
i=1
iEi(?ic) =?μ0icρ →?·vectorE = ρepsilon1
0
1F23+?2F31+?3F12=0→
3summationdisplay
i,j,k=1
iepsilon1jkiFjk=0→
3summationdisplay
i=1
iBi=0→?·vectorB=0
jFk4 +?kF4j +?4Fjk = 0→
3summationdisplay
j,k=1
epsilon1ijk?j(icFk4) =?ic?4(12
3summationdisplay
j,k=1
epsilon1jkiFjk)
→
3summationdisplay
j,k=1
epsilon1ijk?jEk =tBi→?×vectorE =
vectorB
t
340/384
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a131a233a216a62a62a196a196a229a198,a165a254a179a134a240a142a100a137a144a167a124
a88a219a110a41a240a142a100a137a144a167a124a27a53a13a186 a162a8a189a198a182a46a130a75a70a254a182
4summationdisplay
μ=1
μFμν =?μ0jν?μFνλ +?νFλμ +?λFμν = 0
a144a135a189a194a10 a165a254a179Aμ
a210a140a177a189a194a124a114 Fμν ≡?μAννAμ
a124a114a27a40a8a166a144a167?μFνλ +?νFλμ +?λFμν = 0 a103a44a164a225
a124a114a27a40a8a132a166a144a167 summationtextμ,ν?μ?νFμν = 0 a103a44a164a225
a167a191a155a88a140a177a189a194a54 jν ≡?1μ
0
summationtext
μ?μFμν
a54a54a247a247a118a54a197a240a144a167 summationtextν?νjν = 0
341/384
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a131a233a216a62a62a196a196a229a198,a85a196a254a197a240
4summationdisplay
μ=1
μFμν =?μ0jν?μFνλ+?νFλμ+?λFμν=0 fμ≡
4summationdisplay
ν=1
Fμνjν
fμ=
4summationdisplay
ν=1
Fμνjν =?1μ
0
4summationdisplay
ν,νprime=1
Fμν?νprimeFνprimeν =?1μ
0
4summationdisplay
ν,νprime=1
[?νprime(FμνFνprimeν)?(?νprimeFμν)Fνprimeν]
=?1μ
0
4summationdisplay
ν,νprime=1
[?νprime(FμνFνprimeν)?12(?νprimeFμν)Fνprimeν?12(?νFμνprime)Fννprime]
=?1μ
0
4summationdisplay
ν,νprime=1
[?νprime(FμνFνprimeν) + 12(?νprimeFμν +?νFνprimeμ)Fννprime]
=?1μ
0
4summationdisplay
ν,νprime=1
[?νprime(FμνFνprimeν)?12(?μFννprime)Fννprime]
=?1μ
0
4summationdisplay
ν,νprime=1
νprime[FμνFνprimeν?14δμνprime
4summationdisplay
ν”=1
Fνν”Fνν”]≡?
4summationdisplay
νprime=1
νprimeTμνprime
342/384
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a131a233a216a62a62a196a196a229a198,a85a196a254a197a240
fμ =?
4summationdisplay
ν=1
νTμν Tμν ≡ 1μ
0
[
4summationdisplay
νprime=1
FμνprimeFννprime?14δμν
4summationdisplay
μprime,νprime=1
FμprimeνprimeFμprimeνprime] = Tνμ
4summationdisplay
μprime,νprime=1
FμprimeνprimeFμprimeνprime = 2B2? 2c2E2 Fij =
3summationdisplay
k=1
epsilon1ijkBk F4i =?Fi4 = icEi
Tij = 1μ
0
[
4summationdisplay
k=1
FikFjk + Fi4Fj4?12δij(B2? 1c2E2)
= 1μ
0
[
4summationdisplay
k,l,lprime=1
epsilon1iklepsilon1jklprimeBlBlprime? 1c2EiEj?12δij(B2? 1c2E2)]
= 1μ
0
[δijB2?BiBj? 1c2EiEj?12δij(B2? 1c2E2)]
= 1μ
0
[?BiBj? 1c2EiEj + 12δij(B2 + 1c2E2)]
= 1μ
0
(?BiBj + 12δijB2) +epsilon10(?EiEj + 12δijE2) =Jij
343/384
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a131a233a216a62a62a196a196a229a198,a85a196a254a197a240
(ρvectorE +vectorj×vectorB,icvectorj·vectorE) = fμ =?
4summationdisplay
ν=1
νTμν
Tμν ≡ 1μ
0
[
4summationdisplay
νprime=1
FμνprimeFννprime?14δμν
4summationdisplay
μprime,νprime=1
FμprimeνprimeFμprimeνprime] = Tνμ
4summationdisplay
μprime,νprime=1
FμprimeνprimeFμprimeνprime = 2B2? 2c2E2 Fij =
3summationdisplay
k=1
epsilon1ijkBk F4i =?Fi4 = icEi
Tij= 1μ
0
(?BiBj + 12δijB2) +epsilon10(?EiEj + 12δijE2) =Jij
T44= 1μ
0
[
4summationdisplay
i=1
F24i?12(B2?1c2E2)]=?12μ
0
(B2+ 1c2E2) =?12μ
0
B2?epsilon102E2=?W
Ti4=T4i = 1μ
0
4summationdisplay
j=1
FijF4j = iμ
0c
4summationdisplay
j,k=1
epsilon1ijkBkEj = iμ
0c
(vectorE×vectorB)i = icSi
344/384
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a131a233a216a62a62a196a196a229a198,a85a196a254a197a240
(ρvectorE +vectorj×vectorB,icvectorj·vectorE) = fμ =?
4summationdisplay
ν=1
νTμν
Tμν ≡ 1μ
0
[
4summationdisplay
νprime=1
FμνprimeFννprime?14δμν
4summationdisplay
μprime,νprime=1
FμprimeνprimeFμprimeνprime] = Tνμ
Tij= 1μ
0
(?BiBj + 12δijB2) +epsilon10(?EiEj + 12δijE2) =Jij
T44=? 12μ
0
B2?epsilon102E2=?W Ti4=T4i = iμ
0c
(vectorE×vectorB)i = icSi = icgi
fi =?
4summationdisplay
ν=1
νTiν =?
4summationdisplay
j=1
jTijTi4ic?t arrowtripleright vectorf =· arrowrighttophalfarrowrighttophalfJvectorg?t
f4 =?
4summationdisplay
ν=1
νT4ν =?
4summationdisplay
i=1
iT4iT44ic?t arrowtripleright vectorf·vectorv =·vectorSW?t
vectorf·vectorv = ρvectorv·vectorE =vectorj·vectorE
345/384
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a145a62a226a102
a218a62a94a124a27a131a112a138a94
346/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a36a196a145a62a226a102a27a163a227,a237a180a8a65
ρ(vectorr,t) = qδ(vectorr?vectorr0) vectorj(vectorr,t) = ρ(vectorr,t)vectorv0(t) = qvectorv0(t)δ(vectorr?vectorr0)
φ(vectorr,t) =
integraldisplay
d3xprimeρ(vectorr
prime,t?R
c )
4piepsilon10R =
integraldisplay
d3xprimeqδ(vectorr
prime?vectorr0(t?|vectorr?vectorrprime|
c ))
4piepsilon10|vectorr?vectorrprime|
vectorA(vectorr,t)=
integraldisplay
d3xprimeμ0
vectorj(vectorrprime,t?R
c )
4piR =
integraldisplay
d3xprimeμ0qvectorv0(t?
|vectorr?vectorrprime|
c )δ(vectorr
prime?vectorr0(t?|vectorr?vectorrprime|
c ))
4pi|vectorr?vectorrprime|
δa188a234a144a51vectorrprime?vectorr0(t?|vectorr?vectorr
prime|
c ) = 0a63a216a1430,a23a249a135a144a167a27a41a143:vectorr
braceleftbiggvectorrvectorr
0(t?
|vectorr?vectorr?|
c ) = 0
t?≡t?|vectorr?vectorr?|c →
braceleftbiggvectorr? =vectorr
0(t?) vectorv?≡vectorv0(t?)
|vectorr?vectorr?|= c(t?t?)
φ(vectorr,t) = q4piepsilon1
0|vectorr?vectorr?|
integraldisplay
d3xprimeδ(vectorrprime?vectorr0(t?|vectorr?vectorr
prime|
c ))
vectorA(vectorr,t) = μ0qvectorv?
4pi|vectorr?vectorr?|
integraldisplay
d3xprimeδ(vectorrprime?vectorr0(t?|vectorr?vectorr
prime|
c ))
347/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a111a66-a145a28a16a179 (Li′enard-Wiechert)
φ(vectorr,t) = q4piepsilon1
0|vectorr?vectorr?|
integraldisplay
d3xprimeδ(vectorrprime?vectorr0(t?|vectorr?vectorr
prime|
c ))
vectorA(vectorr,t) = μ0qvectorv?
4pi|vectorr?vectorr?|
integraldisplay
d3xprimeδ(vectorrprime?vectorr0(t?|vectorr?vectorr
prime|
c ))integraldisplay
d3xprimeδ(vectorrprime?vectorr0(t?|vectorr?vectorr
prime|
c )) =
integraldisplay
d3x?prime 1J(vectorr,vectorrprime)δ(vectorr?prime) = 1J(vectorr,vectorr?)
vectorr?prime =vectorrprime?vectorr0(t?|vectorr?vectorr
prime|
c )
J=||?x
i
prime
xprimej||
vextendsinglevextendsingle
vectorrprime=vectorr? =||
xprimej[x
prime
i?x0i(t?
|vectorr?vectorrprime|
c )]||
vextendsinglevextendsingle
vectorrprime=vectorr? =1?
(vectorr?vectorr?)·vectorv?
|vectorr?vectorr?|c
φ(vectorr,t) = q4piepsilon1
0|vectorr?vectorr?|
1
1?(vectorr?vectorr?)·vectorv?|vectorr?vectorr?|c
vectorA(vectorr,t) = μ0qvectorv?
4pi|vectorr?vectorr?|
1
1?(vectorr?vectorr?)·vectorv?|vectorr?vectorr?|c
348/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a62a94a124
braceleftbigg vectorE =φvectorA
tvector
B =?×vectorA
braceleftBiggφ(vectorr,t) = q
4piepsilon10R?
1
1?vectorR?·vectorv?R?c
vectorA(vectorr,t) = μ0qvectorv?
4piR?
1
1?vectorR?·vectorv?R?c
braceleftbigg vector
R?=vectorr?vectorr?
vectorr?=vectorr?(vectorr,t)=vectorr0(t?)
vectorR? =vectorr?vectorr? vectorr? =vectorr0(t?),
R?
t =
1
2R?
R?2
t =
1
2R?
vectorR?·vectorR?
t =
vectorR?
R?·(?
vectorr?
t?)
t?
t =?vectorv
·vectorR
R?
t?
t
R? = 12RR?2 = 12R(vectorR?·vectorR?) = 1R?(?vectorR?)·vectorR?
= 1R?(arrowrighttophalfarrowrighttophalfItvectorr
t?)·
vectorR? = 1
R?(
vectorRt?vectorv?·vectorR?)
R? = c(t?t?)→?R
t = c?
t?
t?R
=?c?t?
t?
t =
1
1?vectorR?·vectorv?R?c
R?
t =
vectorR?R? ·vectorv?
1?vectorR?·vectorv?R?c?t
=?
vectorR?
cR?
1?vectorR?·vectorv?R?c?R
=
vectorR?
R?
1?vectorR?·vectorv?R?c
vectorR?
t =?
vectorr?
t?
t?
t =
vectorv?
1? vectorR?·vectorv?R?c?
vectorR? =arrowrighttophalfarrowrighttophalfItvectorr?
t? =
arrowrighttophalfarrowrighttophalfI? vectorR?vectorv?R?c
1? vectorR?·vectorv?R?c
349/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a62a94a124
braceleftbigg vectorE =φvectorA
tvector
B =?×vectorA
braceleftBiggφ(vectorr,t) = q
4piepsilon10R?
1
1?vectorR?·vectorv?R?c
vectorA(vectorr,t) = μ0qvectorv?
4piR?
1
1?vectorR?·vectorv?R?c
braceleftbigg vector
R?=vectorr?vectorr?
vectorr?=vectorr?(vectorr,t)=vectorr0(t?)
t?
t =
1
1?vectorR?·vectorv?R?c
R?
t =
vectorR?R? ·vectorv?
1?vectorR?·vectorv?R?c?t
=?
vectorR?
cR?
1?vectorR?·vectorv?R?c?R
=
vectorR?
R?
1?vectorR?·vectorv?R?c
vectorR?
t =
vectorv?
1? vectorR?·vectorv?R?c?
vectorR? =arrowrighttophalfarrowrighttophalfI?
vectorR?vectorv?
R?c
1? vectorR?·vectorv?R?c
vectorE(vectorr,t)= q
4piepsilon10S?3(
vectorRR?vectorv?
c )(1?
v?2
c2)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
vectorEa154a154a203a203
+ q4piepsilon1
0c2S?3
bracketleftbigvectorR?×[(vectorRR?vectorv?
c )×vectora
]bracketrightbig
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
vectorEa203
vectorB(vectorr,t) = vectorR?
cR?×
vectorE(vectorR,t)
vectorR? =vectorr?vectorr? vectora?≡d2vectorr0(t?)
dt?2 S
≡R?(1?vectorR
·vectorv?
R?c )
350/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a62a94a124
vectorE = vectorEa154a154a203a203 + vectorEa203 vectorB = vectorBa154a154a203a203 + vectorBa203
vectorEa154a154a203a203 = q
4piepsilon10S?3(
vectorRR?vectorv?
c )(1?
v?2
c2 )
vectorBa154a154a203a203 = vectorR?
cR?×
vectorEa154a154a203a203
vectorEa203 = q
4piepsilon10c2S?3
bracketleftbigvectorR?×[(vectorRR?vectorv?
c )×vectora
]bracketrightbig vectorB
a203 =
vectorR?
cR?×
vectorEa203
vectorR = vectorRR?vectorv?
c
vectorR·vectorv? = vectorR?·vectorvR?v?2
c
R2 = R?2(1 + v
2
c2 )?2R
vectorR
·vectorv?
c = R
2(1?v
2
c2 )?R
2vectorR·vectorv
c
→R?2(1?v
2
c2 ) = R
2 + R?2vectorR·vectorv
c
S? = RvectorR?·vectorv
c = R
1
c(
vectorR·vectorv? + R?v?2
c ) = R
(1?v
2
c2 )?
1
c
vectorR·vectorv?
S?2 = R?2(1?v
2
c2 )
2?(1?v
2
c2 )R
2vectorR·vectorv
c +
1
c2(
vectorR·vectorv?)2
351/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a62a94a124
vectorEa154a154a203a203 = q
4piepsilon10S?3(
vectorRR?vectorv?
c )(1?
v?2
c2 )
vectorBa154a154a203a203 = vectorR?
cR?×
vectorEa154a154a203a203
R?2(1?v
2
c2 ) = R
2 + R?2vectorR·vectorv
c S
= R?(1?v
2
c2 )?
1
c
vectorR·vectorv?
S?2 = R?2(1?v
2
c2 )
2?(1?v
2
c2 )R
2vectorR·vectorv
c +
1
c2(
vectorR·vectorv?)2
= (1?v
2
c2 )[R
2+R?2vectorR·vectorv
c ]?(1?
v?2
c2 )R
2vectorR·vectorv
c +
1
c2(
vectorR·vectorv?)2
= (1?v
2
c2 )R
2 + (vectorR·vectorv
)2
c2 ≡S
2 vectorR = vectorRR?vectorv
c
vectorEa154a154a203a203 = qvectorR
4piepsilon10S3(1?
v?2
c2 )
vectorBa154a154a203a203 = vectorv?
c2 ×
vectorEa154a154a203a203
352/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a62a94a124
vectorEa154a154a203a203 = qvectorR
4piepsilon10S3(1?
v?2
c2 )
vectorBa154a154a203a203 = vectorv?
c2 ×
vectorEa154a154a203a203
S2 = (1?v
2
c2 )R
2 + (vectorR·vectorv
)2
c2
vectorR = vectorRR?vectorv?
c
vectorr0(t) a33a132=== vectorVt
vectorv? = vectorV
vectorr? = vectorVt?
c(t?t?)=R?=|vectorr?vectorr?|
vectorR=vectorr?vectorr(t?t?)vectorv?=vectorr?vectorVt
a233vectorR//vectorv?
braceleftBigg vector
Ea154a154a203a203//= qvectorR4piepsilon1
0R3
(1?v?2c2 )
vectorBa154a154a203a203//=0
a233vectorR⊥vectorv?
vectorEa154a154a203a203⊥= qvectorR
4piepsilon10R3
1radicalBig
1?v?2c2
vectorBa154a154a203a203⊥= μ0qvectorv?×vectorR
4piR3
1radicalBig
1?v?2c2
a8v?lessmuchca158,Ea154a154a203a203a136a144a149a211a234a138; a8v?→ca158,Ea154a154a203a203//→0,Ea154a154a203a203⊥→∞.
353/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a62a94a124
vectorE = vectorEa154a154a203a203 + vectorEa203 vectorB = vectorBa154a154a203a203 + vectorBa203 S? = RvectorR?·vectorv?
c
vectorEa203 = q
4piepsilon10c2S?3
bracketleftbigvectorR?×[(vectorRR?vectorv?
c )×vectora
]bracketrightbig vectorB
a203 =
vectorR?
cR?×
vectorEa203
vectorR?·vectorEa203 = 0 vectorn? = vectorR?R? vectorn?·vectorEa203 = 0
a92a132a36a196a196a27a27a226a102a172a23a41a203a19a156
a33a132a36a196a196a27a27a226a102a152a189a216a172a23a41a203a19a237a186a144a8v? <ca158
a233a33a132a36a196a196a27a27a226a102a167a8vectorn?·vectorv?c = 1a158S? = 0a167vectorEa203a165a209a1210/0?Cerenkova203a19
vectorS= 1
μ0
vectorEa203×vectorBa203= 1
μ0c
vectorEa203×(vectorn?×vectorEa203)= 1
μ0c[
vectorE2a203vectorn(vectorn?·vectorEa203)vectorEa203]
= q
2
16pi2epsilon120c5μ0S?6|
vectorR?×[(vectorRR?vectorv?
c )×vectora
]|2vectorn?
= q
2vectorn?
16pi2epsilon10c3R?2
|vectorn?×[(vectornvectorv?c )×vectora?]|2
(1?vectorv?·vectorn?c )6
354/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a203a19a245a199a57a14a169a217
vectorS= 1
μ0
vectorEa203×vectorBa203 = q2vectorn?
16pi2epsilon10c3R?2
|vectorn?×[(vectornvectorv?c )×vectora?]|2
(1?vectorv?·vectorn?c )6
I?≡
integraldisplay
S?
d dI
d =
integraldisplay
S?
dvectorσ?·(vectorSdtdt?) =
integraldisplay
S?
d R?2vectorn?·(vectorSdtdt?)
dI?
d=R
2vectorn?·(vectorSdt
dt?)=
q2vectorn?
16pi2epsilon10c3R?2·vectorn
R?2|vectorn
×[(vectornvectorv?
c )×vectora
]|2
(1?vectorv?·vectorn?c )6 (1?
vectorv?·vectorn?
c )
= q
2
16pi2epsilon10c3
|vectorn?×[(vectornvectorv?c )×vectora?]|2
(1?vectorv?·vectorn?c )5
|vectorn?×[(vectornvectorv
c )×vectora
]|2 =|vectorn?·vectora?(vectornvectorv
c )?(1?
vectorn?·vectorv?
c )vectora
|2
=(vectorn?·vectora?)2(1+vectorv
2
c2?2
vectorn?·vectorv?
c )?2vectorn
·vectora?(vectorn?·vectoravectorv
·vectora?
c )(1?
vectorn?·vectorv?
c )+(1?
vectorn?·vectorv?
c )
2vectora?2
= 2(vectorn
·vectora?)(vectorv?·vectora?)
c (1?
vectorn?·vectorv?
c ) + a
2(1?vectorn
·vectorv?
c )
2?(1?v
2
c2 )(vectorn
·vectora?)2
I? =
integraldisplay
d ( dI
d)
a177vectorv?a143za182=====? q2
6piepsilon10c3
[a?2?(vectora?×vectorv?)2c2 ]
(1?v?2c2 )3
355/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a203a19a245a199a57a14a169a217
|vectorn?×[(vectornvectorv
c )×vectora
]|2
= 2(vectorn
·vectora?)(vectorv?·vectora?)
c (1?
vectorn?·vectorv?
c ) + a
2(1?vectorn
·vectorv?
c )
2?(1?v
2
c2 )(vectorn
·vectora?)2
vectora?bardblvectorv?,a18vectorv?a134vectorn?a27a89a14a143θ?
|vectorn?×[(vectornvectorv
c )×vectora
]|2
= 2ca?2v?cosθ?(1?v
c cosθ
)+a?2(1?v
c cosθ
)2?(1?v
2
c2 )a
2 cos2θ?
= a?2(1?cos2θ?) = a?2 sin2θ?
dI?
d =
q2a?2 sin2θ?
16pi2epsilon10c3(1?v?c cosθ?)5 I
= q
2
6piepsilon10c3
a?2
(1?v?2c2 )3
a111a245a199a39a154a131a233a216a156a47a245a10a207a102(1?v?2c2 )?3
a203a19a129a114a27a144a149θ?≤pi/2a167a133a8a22a112a132a167
a22a26a26a67a67θ? = 0
356/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a203a19a245a199a57a14a169a217
|vectorn?×[(vectornvectorv
c )×vectora
]|2
= 2(vectorn
·vectora?)(vectorv?·vectora?)
c (1?
vectorn?·vectorv?
c ) + a
2(1?vectorn
·vectorv?
c )
2?(1?v
2
c2 )(vectorn
·vectora?)2
vectora?⊥vectorv?,a177vectorv?a143za182,vectora?a143xa182,vectorn?·vectorv? = v? cosθ?,vectorn?·vectora? = a? sinθ? cosφ?
|vectorn?×[(vectornvectorv
c )×vectora
]|2 = a?2[(1?v
c cosθ
)2?(1?v
2
c2 )sin
2θ?cos2φ?]
dI?
d =
q2a?2
16pi2epsilon10c3
1
(1?v?c cosθ?)3
bracketleftbig1?(1?v?2c2 )sin2θ?cos2φ?
(1?v?c cosθ?)2
bracketrightbig
I?= q
2
6piepsilon10c3(1?v?2c2 )2 a233xza178a161,φ
=0 dI
d=
q2a?2
16pi2epsilon10c3
(v?c?cosθ?)2
(1?v?c cosθ?)5
a111a245a199a39a154a131a233a216a156a47a245a10a207a102(1?v?2c2 )?2,
a2a39//a156a47a8a10(1?v?2c2 )?1
a203a19a169a217a154a182a233a161a167a134φ? = 0,pia129a114a167φ? = pi/2a129a102
a112a132a36a196a158a203a19a129a114a27a144a149θ? = 0
357/384
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a203a19a170a170a204a204a169a219
vectorEa203(vectorr,t) =
integraldisplay ∞
∞
dω vectorEω(vectorr)e?iωt vectorEω(vectorr) = 12pi
integraldisplay ∞
∞
dt vectorEa203(vectorr,t)eiωt
vectorEω(vectorr) = 1
2pi
integraldisplay ∞
∞
dt eiωt q4piepsilon1
0c2R?
vectorn?×[(vectornvectorv?c )×vectora?]
(1?vectorn?·vectorv?c )3
= 12pi
integraldisplay ∞
∞
dt? dtdt?eiω(t?+R?c ) q4piepsilon1
0c2R?
vectorn?×[(vectornvectorv?c )×vectora?]
(1?vectorn?·vectorv?c )3
= 12pi
integraldisplay ∞
∞
dt? eiωt? qe
iωR?c
4piepsilon10c2R?
vectorn?×[(vectornvectorv?c )×vectora?]
(1?vectorn?·vectorv?c )2?
ω→∞→ 0
ω→0,v?lessmuchc,R?,vectorn?a134t?a195a39→ qeiωR?c
8piepsilon10c2R?vectorn
×(vectorn?×?vectorv?)
a220a151a203a19a181a145a62a226a102a92a19a20a212a159a113a254a167a51a45a69a76a167a165a126a132a23a41a27a203a19a34
358/384
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a203a19a170a170a204a204a169a219
(dWd? )?≡
integraldisplay ∞
∞
dt? ( dI
d) =
integraldisplay ∞
∞
dt?epsilon10cvectorEa203(vectorr,t)·vectorEa203(vectorr,t)R?2( dtdt?)
=
integraldisplay ∞
∞
dtepsilon10cR?2
integraldisplay ∞
∞
dωdωprimee?iωte?iωprimetvectorEω(vectorr)·vectorEωprime(vectorr)
= 2piepsilon10c
integraldisplay ∞
∞
dωR?2vectorEω(vectorr)·vectorE?ω(vectorr)
= 4piepsilon10c
integraldisplay ∞
0
dωR?2vectorEω(vectorr)·vectorE?ω(vectorr)≡
integraldisplay ∞
0
dω (dW
ω
d? )
dW?ω
d? = 4piepsilon10cR
2vectorEω(vectorr)·vectorE?ω(vectorr)
ω→∞→ 0?vectorv?·vectorn?≡?v?cosΘ
ω→0,v?lessmuchc,R?,vectorn?a134t?a195a39→ q2
16pi3epsilon10c3[(vectorn
·?vectorv?)vectornvectorv?]2 = q2?v?2 sin2 Θ
16pi3epsilon10c3
W?ω =
integraldisplay
d? 4piepsilon10cR?2vectorEω(vectorr)·vectorE?ω(vectorr)
ω→∞→ 0
ω→0,v?lessmuchc,R?,vectorn?a134t?a195a39→ q2?v?2
6pi2epsilon10c3?a36a170a170a117a126a234?a112a170a170a117a34
359/384
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a131a212a133a197(Cerenkov)a203a19
a195a161a140a254a33a48a159(μ,epsilon1)a165a27a203a19a124:
vectorj(vectorr,t) =
integraldisplay ∞
∞
dωvectorjω(vectorr)e?iωt vectorjω(vectorr) = 12pi
integraldisplay ∞
∞
dtvectorj(vectorr,t)eiωt
vectorA(vectorr,t) =
integraldisplay ∞
∞
dω vectorAω(vectorr)e?iωt vectorAω(vectorr) = 12pi
integraldisplay ∞
∞
dt vectorA(vectorr,t)eiωt
vectorA(vectorr,t) = μ
4pi
integraldisplay
dτprime
vectorj(vectorrprime,t?nR
c )
r =
μ
4pir
integraldisplay
dτprime
integraldisplay ∞
∞
dωvectorjω(vectorrprime)e?iω(t?nRc )
vectorAω(vectorr) = μ
4pir
integraldisplay
dτprimevectorjω(vectorrprime)eiωnRc = μ8pi2r
integraldisplay
dτprime
integraldisplay ∞
∞
dtvectorj(vectorrprime,t)eiω(t+nRc )
= μ8pi2r
integraldisplay
dτprime
integraldisplay ∞
∞
dt evectorV0(t)δ(vectorrprime?vectorr0(t))eiω(t+nRc )
≈ eμ8pi2reiωrnc
integraldisplay
dτprime
integraldisplay ∞
∞
dtvectorV0(t)δ(vectorrprime?vectorr0(t))eiω(t?ncvectorn·vectorrprime)
= eμ8pi2reiωrnc
integraldisplay ∞
∞
dtvectorV0(t)eiω(t?ncvectorn·vectorr0(t))
360/384
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a131a212a133a197(Cerenkov)a203a19
vectorAω(vectorr) = eμ
8pi2re
iωrnc
integraldisplay ∞
∞
dtvectorV0(t)eiω(t?ncvectorn·vectorr0(t))
a63a216a33a132a36a196a196a27a27a156a47,vectorV0(t) = vectorv0 vectorr0(t) =vectorr0 +vectorv0t nc = 1v
vectorAω(vectorr)=eμvectorv0
8pi2re
iωv(r?vectorn·vectorr0)
integraldisplay ∞
∞
dt e?iωt(1?vectorn·vectorv0v )=eμvectorv04pir eiωv(r?vectorn·vectorr0)δ[ω(1?vectorn·vectorv0v )]
a233a253a152,v = c,v0 <c,vectorn·vectorv0v <1,δa188a234a240a1430.
a233a48a159,v <c,v0 <c,a107a140a85a127a51vectorn·vectorv0v = 1.
a38a37
a39a36
a108a114a114a45v0 < v
a38a37
a39a36
a22a21
a23a20
a114a114a45v0 = v
a38a37
a39a36
θa114 a45v0 > v a80a80a80
a80a80a80
a2
a2
cosθ = v/v0 a38a37
a39a36
a30a29
a31a28
a114a114 a45a80a80a80
a80a80a80
a2
a2
a2a2
a100a158a27a203a19a124a23Cerenkova203a19,a167a180a100a117a145a62a226a102a27a132a221a135a76 a48a159a165a27a49a132a158,a48
a159a83a23a41a112a19a19a62a62a54a45a117a27a103a197a134a6a53a53a226a226a102a27a27a62a62a94a124a131a112a90a21a47a164a27.
361/384
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a131a212a133a197(Cerenkov)a203a19
vectorB(vectorr,t) =
integraldisplay ∞
∞
dω vectorBω(vectorr)e?iωt
vectorBω(vectorr) = 1
2pi
integraldisplay ∞
∞
dt vectorB(vectorr,t)eiωt = 12pi
integraldisplay ∞
∞
dt?×vectorA(vectorr,t)eiωt
= 12pi
integraldisplay ∞
∞
dt
integraldisplay ∞
∞
dωprime?×e?iωprimetvectorAωprime(vectorr)eiωt =?×vectorAω(vectorr)
=?×bracketleftbigeμvectorv04pir δ[(1?vectorn·vectorv0v )ω]eiωv(vectorr?vectorn·vectorr0)bracketrightbig = ieμωvectorn×vectorv04pivr δ[(1?vectorn·vectorv0v )ω]
dWω
d? =
4piv
μ r
2vectorBω·vectorB?ω = (vectorn×vectorv0)·(vectorn×vectorv0)ω
2e2μ
4pi δ
2[(1?vectorn·vectorv0
v )ω]
= v
2
0 sin
2θω2e2μ
8pi2 δ[(1?
vectorn·vectorv0
v )ω]
integraldisplay ∞
∞
dt eiω(1?vectorn·vectorv0v )t
= v0 sin
2θω2e2μ
8pi2 δ[(1?
vectorn·vectorv0
v )ω]
integraldisplay ∞
∞
dx ei ωv0(1?vectorn·vectorv0v )x
= sin
2θω2e2μ
8pi2 δ[(1?
vectorn·vectorv0
v )
ω
v0]
integraldisplay ∞
∞
dx
362/384
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a131a212a133a197(Cerenkov)a203a19
dWω
d? =
sin2θω2e2μ
8pi2 δ[(1?
vectorn·vectorv0
v )
ω
v0]
integraldisplay ∞
∞
dx
= Lsin
2θω2e2μ
8pi2 δ[(1?
vectorn·vectorv0
v )
ω
v0] vectorn·vectorv0≡v0 cosθ
dWω
dL d? =
sin2θω2e2μ
8pi2 δ[(1?
vectorn·vectorv0
v )
ω
v0]=
ω2e2μ
8pi2 (1?
v2
v20)δ[(1?
vectorn·vectorv0
v )
ω
v0]
dWω
dL =
ω2e2μ
8pi2 (1?
v2
v20)
integraldisplay pi
0
d?δ[ωv( vv
0
cosθ)] = ωe
2μv
4pi [1?
v2
v20]
a217a165,v = v(ω),Wωa144a51v(ω) <v0a164a137a209a27a170a227a107a1540a0a122.
a100a117cosθ = v(ω)v0,a216a211a170a199a62a94a197a27a203a19a14a189a216a131a211,a94a200a197a236a192a74a152a189a27a170
a145,a140a177a26a26a20a20a40a189a189a27a27θa138,a207a13a255a189θa210a140a177a189a209a226a102a27a132a221.
a121a51a131a212a133a197a203a19a50a141a65a94a117a226a102a79a234a236a165,a167a27a96a58a180a144a80a185a140a117a152a189a132a221a27
a226a102,a207a13a59a157a10a36a132a226a102a27a90a54,a13a133a140a177a79a40a47a255a209a226a102a27a36a196a132a221.
363/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94
a107a152a9a154a62a94a229a138a94a94a117a117a62a102,a166a217a51dta158a109a83a36a196a229a108dvectorl
vectorFa154a62a94 + vectorFa62a94 = mvectora
vectorFa62a94a147a76a62a102a23a41a27a27a62a62a94a124a233a103a67a23a41a27a140a85a27a138a94a156
a107a60a60a64a64a143a62a102a23a41a27a27a62a62a94a124a216a65a84a233a103a67a107a138a94
a249a144a180a183a142a218a33a132a36a196a158a27a40a216
a61a166a127a196a107a103a103a138a138a94a167a167a143a180a195a161a140a156
vectorFa62a94 =?e[vectorE +vectorv×vectorB]
a107a60a226a100a96a178a62a102a23a41a27a27a62a62a94a124a216a65a84a233a103a67a107a138a94!
a40a220a85a254a61a122a218a197a240a189a198a53a63a216a140a85a27a103a103a138a138a94
364/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94,a85a254a61a122a134a197a240a189a198; a218a238a238a189a189a198
a85a254a61a122a134a197a240a189a198,a107a152a9a154a62a94a229a138a94a94a117a117a62a102,a166a217a51dta158a109a83a36a196a229a108dvectorl,
vectorFa154a62a94 + vectorFa62a94 = mvectora?
contintegraldisplay
V
dvectorσ·vectorS =
integraldisplay
V
dV vectorfa62a94·vectorv + ddt
integraldisplay
V
dVw
(vectorFa154a62a94?mvectora)·dvectorl =?vectorFa62a94·dvectorl =?
integraldisplay
dV vectorfa62a94·dvectorl = dWa103+
contintegraldisplay
dvectorσ·vectorSdt
dWa103 =
integraldisplay
∞
dV 12(epsilon10E2 + 1μ
0
B2) dWa203≡
contintegraldisplay
∞
dσ·vectorSdt =
integraldisplay t2
t1
dt I
dWa62a94≡dWa103 +dWa203a180a145a62a62a27a27a27a62a62a102a36a196a23a41a27a27a62a62a94a124a69a164a27a85a254a67a122.
a63a216a157a185a10a145a62a226a102a27a103a131a112a138a94a163a103a67a201a103a67a23a41a27a124a27a138a94a229a156a156a164a164
a233a216a145a62a226a102,dWa62a94 = 0 →(vectorFa154a62a94?mvectora)·dvectorl = 0→vectorFa154a62a94 = mvectora
a233a145a62a226a102,dWa62a94negationslash= 0 →(vectorFa154a62a94?mvectora)·dvectorl = dWa62a94 = dWa103+dWa203
365/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94,a62a102a27a178a59a36a196a144a167
a28a46,a62a214a254a33a169a217a51a140a187a143r0a27a165a165,r0a161a143a62a102a27a27a62a62a94a140a187,a62a214a151a221ρ =?e4
3pir
30
4pir2E =
braceleftbigg ρ
epsilon10
4
3pir
3 r≤r0
eepsilon1
0
r>r0 →vectorE =
braceleftbigg? er
4piepsilon10r30vectorer r≤r0
e4piepsilon1
0r2
vectorer r>r0
vectorB=
integraldisplay
dτprimeμ04piρvectorv×
vectorR
R3 =
μ0
4pivectorv×
integraldisplay
dτprimeρ
vectorR
R3 =μ0epsilon10vectorv×
integraldisplay
dτprime ρ
vectorR
4piepsilon10R3 =
vectorv
c2×
vectorE
Wa103 = We + Wm
We =
integraldisplay
dτ epsilon102E2 = epsilon1024pi[
integraldisplay r0
0
dr r2( er4piepsilon1
0r30
)2 +
integraldisplay ∞
r0
dr r2( e4piepsilon1
0r2
)2]
= e
2
8piepsilon10[
integraldisplay r0
0
dr r
4
r60 +
integraldisplay ∞
r0
dr 1r2] = e
2
8piepsilon10(
1
5r0 +
1
r0) =
3
20piepsilon10
e2
r0
Wm=
integraldisplay
dτprime 12μ
0
B2=
integraldisplay
dτ 12μ
0c4
(vectorv×vectorE)·(vectorv×vectorE)=
integraldisplay
dτ 12μ
0c4
[v2E2?(vectorv·vectorE)2]
=
integraldisplay
dτ E2 v
2
2μ0c4(1?
1
3) =
v2
3μ0epsilon10c4We =
2v2
3c2We
366/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94,a62a102a27a178a59a36a196a144a167
integraldisplay
dτEiEj=13δij
integraldisplay
dτE2
integraldisplay
dτ(vectorv·vectorE)2=
3summationdisplay
i,j=1
vivj
integraldisplay
dτEiEj=13v2
integraldisplay
dτE2
vectorB = vectorv
c2×
vectorE Wa103 = We + Wm We = 3
20piepsilon10
e2
r0 Wm =
2v2
3c2We
Wa103 = We + Wm = We(1 + 2v
2
3c2) dWa103 =
4
3c2Wevectorv·vectoradt
Wa203 =
integraldisplay t2
t1
dt μ06pice2a2 =
integraldisplay t2
t1
μ0
6pice
2vectora·dvectorv = μ0e
2
6pic[vectora·vectorv
vextendsinglevextendsinglet2
t1?
integraldisplay t2
t1
dtvectorv·˙vectora]
a63a216vectorv·vectoravextendsinglevextendsinglet
1
= vectorv·vectoravextendsinglevextendsinglet
2
a27a156a47:
Wa203 =?μ0e
2
6pic
integraldisplay t2
t1
dtvectorv·˙vectora dWa203 =?μ0e
2
6picvectorv·
˙vectoradt
(vectorFa154a62a94?mvectora)·dvectorl = 43c2Wevectorv·vectoradt?μ0e
2
6picvectorv·
˙vectoradt
vectorFa154a62a94 + vectorFs = (m + 4
3c2We)vectora
vectorFs = μ0e2
6pic
˙vectora
367/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94,a62a94a159a254; a203a19a123a90a229
vectorF + vectorFs = (m + 4
3c2We)vectora
vectorFs = μ0e2
6pic
˙vectora
a107a8a159a254me = m + 43c2Wea165 43c2Wea53a103a62a94a131a112a138a94a27a94a124a85a233a159a254a27a0a122,a88a219a110
a41a186 a23a62a102a27a62a94a159a254,a167a100a64a220a169a195a123a134a62a102a169a109a27a62a94a124a47a164,a249a220a169a85a254a195a123
a108a62a102a27a36a196a85a254a165a169a108a209a22.
a127a196a20a162a8a254a42a9a20a20a27a27a27a62a62a102a159a254a65a65a107a107a131a8a220a169a180a62a94a159a254
me～Wec2 = 3μ020pi e
2
r0 →re～
μ0e2
pime ～10
15a146
a233a39a229a216a102a27a186a221a14310?13a146,a6a102a27Bohra140a187a1430.5×10?10a146,a62a102a180a154a126a2a27a226a102.
a62a102a203a19a62a94a124,a103a28a65a201a20a135a192,a135a192a229a143vectorFs.
μ0e2
6pic
˙vectora = mevectora→vectorr(t) =vectorr(0) +vectorv(0)t + μ0e2
6picmevectora(0)e
6picme
μ0e2 t
a62a102a242a195a129a92a132a36a196!
368/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94,a204a130a27a103a44a176a221
a140a178a59a59a27a27a6a102a117a49a28a46,a62a102a254a33a169a217a117a6a102a83,a6a102a216a63a117a62a102a31a165a37a160
a152,a8 a6a102a201a9a46a138a94a45a117a0,a6a102a216a160a108a62a102a31a165a37a160a152vectorr0(t),a6a102a216a134a62a102a31a165
a37a131a109a23a41a152a6a53a161a69a229
vectorf =?evectorE =?eρvectorr0(t)
3epsilon10 ≡?meω
2
0vectorr0(t) ω
2
0 =
eρ
3epsilon10me =
e2
4epsilon10pimer20
a166a62a102a138a92a132a36a196(a20a8a196)a13a203a19a117a49,a162a8a254a119a20a27a117a49a216a180a252a170a199,a13a180a107a152a189
a27a170a199a169a217a176a221,a51a49a204a165a76a121a143a6a102a117a49a27a204a130a107a152a189a103a44a176a221.
meω20vectorr0(t) + μ0e
2
6pic
···
vectorr0 (t) = me¨vectorr0(t)
a193a38a41,vectorr0(t) =vectorr0(0)e?iωt→?meω20 + μ0e
2
6pic(?i)
3ω3 =?meω2
ω2 = ω20? iμ0e
2
6picmeω
3 ω =
braceleftbiggω
0 a216a127a196a20a203a19a135a192
ω0?iΓ2 a127a196a20a203a19a135a192
Γ≡μ0e
2ω2
0
6picme,a203a19a135a192a8a65
369/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94,a204a130a27a103a44a176a221
meω20vectorr0(t) + μ0e
2
6pic
···
vectorr0 (t) = me¨vectorr0(t)
a193a38a41,vectorr0(t) =vectorr0(0)e?iωt ω =
braceleftbiggω
0 a216a127a196a20a203a19a135a192
ω0?iΓ2 a127a196a20a203a19a135a192
Γ≡μ0e
2ω2
0
6picme,a203a19a135a192a8a65
vectorr0(t) =vectorr0(0)e?Γ2te?iω0t vectora(t) =vectora(0)e?Γ2te?iω0t vectora(0) =?ω2vectorr0(0)
vectorB(vectorr,t) = eμ0
4picrvectora(t
)×vectorn
vectorE(vectorr,t) = cvectorB(vectorr,t)×vectorn = eμ0
4pir[vectora(t
)×vectorn]×vectorn = eμ0
4pir[vectora(t
)·vectornvectorn?vectora(t?)]
= eμ04pir[vectora(0)·vectornvectorn?vectora(0)]eiωrc e?iωt = vectorE(vectorr,0)e?iωt
vectorE(vectorr,0)≡ eμ0
4pir[vectora(0)·vectornvectorn?vectora(0)]e
iωrc = eμ0ω
2
4pir [?vectorr0(0)·vectornvectorn +vectorr0(0)]
vectorE(vectorr,t) =
braceleftbigg 0 t<0
vectorE(vectorr,0)E?iωt t≥0 =
integraldisplay ∞
∞
dωprime vectorEωprimee?iωprimet
370/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94,a204a130a27a103a44a176a221
vectorEωprime = 1
2pi
integraldisplay ∞
∞
dt vectorE(vectorr,t)eiωprimet = 12pivectorE(vectorr,0)
integraldisplay ∞
0
dt ei(ωprime?ω)t = i
vectorE(vectorr,0)
2pi(ωprime?ω)
a152a135a6a102a117a209a27a54a76a252a160a31a161a27a49a85a254 =
integraldisplay ∞
∞
dt S =12epsilon10c
integraldisplay ∞
∞
dt vectorE(vectorr,t)·vectorE?(vectorr,t)
= 12epsilon10c
integraldisplay ∞
∞
dt
integraldisplay ∞
∞
dωprime
integraldisplay ∞
∞
dω” vectorEωprimee?iωprimet·(vectorEω”e?iω”t)?
= piepsilon10c
integraldisplay ∞
∞
dωprime
integraldisplay ∞
∞
dω” δ(ωprime?ω”)vectorEωprime·vectorE?ω” = piepsilon10c
integraldisplay ∞
∞
dωprime vectorEωprime·vectorE?ωprime
a49a114 = a252a160a158a109a252a160a31a161a254a54a76a27a49a85a254 = n0
integraldisplay ∞
∞
dt S≡
integraldisplay ∞
∞
dωprime Iωprime
n0,a252a160a158a109a117a49a27a6a102a234
Iωprime = pin0cepsilon10vectorEωprime·vectorE?ωprime = n0cepsilon10
vectorE(vectorr,0)·vectorE(vectorr,0)?
4pi(ωprime?ω)(ωprime?ω?) =
n0cepsilon10vectorE(vectorr,0)·vectorE(vectorr,0)?
4pi[(ωprime?ω0)2+Γ2]
λ =||?(2picωprime )||= 2picω2
0
ωprime = 2picω2
0
Γ = μ0e
2
3me ～10
4?A
371/384
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a62a94a197a27a209a19a134a225a194,a48a159a27a218a218a209a209,a103a100a62a102a233a233a62a62a94a197a27a209a19
a152a189a170a199a27a9a62a94a197a221a19a20a62a102a254a166a62a102a177a131a211a170a199a137a114a189a8a196a191a149a9a203a19a209a62
a94a197a23a62a94a197a27a209a19.
a63a216a62a102vlessmuchca36a196a27a156a47,a62a102a36a196a27a130a221～a8a204～vTlessmuchcT = λ,Ta143a92a19a197a177a207,
λa143a92a19a197a27a197a127.a183a130a140a67a113a242a9a62a124a119a164a134vectorra195a39,a62a102a201a27a94a229～evB～evEca131
a233a233a62a62a229a143a140a3a209.
m¨vectorr0(t) = μ0e
2
6pic
···
vectorr0 (t) + evectorE0e?iωt
a193a38a41,vectorr0(t) =vectorr0(0)e?iωt→?mω2vectorr0(0) = μ0eiω
3
6pic vectorr0(0) + e
vectorE0
vectorr0(0) =?e
vectorE0
mω2?iμ0e2ω36pic =
evectorE0
mω2(1?2iree2ω3pic )
re<vTlessmuchλ→ωrec lessmuch2pi
→?e
vectorE0
mω2
vectora(t) =vectora(0)e?iωt vectora(0) =?ω2vectorr0(0) = e
vectorE0
m
vectorE(vectorr,t) = vectorE(vectorr,0)e?iωt vectorE(vectorr,0) = eμ0ω2
4pir e
iωcr[vectorr0(0)?vectorr0(0)·vectornvectorn]
372/384
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a62a94a197a27a209a19a134a225a194,a48a159a27a218a218a209a209,a103a100a62a102a233a233a62a62a94a197a27a209a19
vectorE(vectorr,t) = vectorE(vectorr,0)e?iωt vectorE(vectorr,0) = eμ0ω2
4pir e
iωcr[vectorr0(0)?vectorr0(0)·vectornvectorn]
vectorS(vectorr,t) = epsilon10c
2
vectorE(vectorr,t)·vectorE?(vectorr,t)vectorn = μ0e2ω4
32pi2r2c[r
2
0(0)?(vectorn·vectorr0(0))
2]
= μ0e
4E2
0
32pi2m2er2c(1?cos
2α) = I0r
2
e
r2 sin
2α
vectorn·vectorE0 = E0 cosα I0 = E
2
0
2μ0c re =
μ0e2
4pime
dI
d? = I0r
2
e sin
2α
I =
contintegraldisplay
dvectorσ·vectorS =
integraldisplay
d? r2vectorS·vectorn = 2piI0r2e
integraldisplay pi
0
dαsin2α = 83pir2eI0 = σI0
σ = 83pir2ea180Thomsona209a19a31a161.
373/384
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a62a94a197a27a209a19a134a225a194,a48a159a27a218a218a209a209,a103a100a62a102a233a233a62a62a94a197a27a209a19
vectorn·vectorE0 = E0 cosα I0 = E
2
0
2μ0c re =
μ0e2
4pime
dI
d? = I0r
2
e sin
2α I = σI0 σ = 8
3pir
2
e
a45
a54
a8a8
a8a25
a8a8
a8a8a8a42
a72a72a72a89
y
z
x
vectorE0 vectorrφ
θ
a101a101a92a92a19a197a8a204a145a197a169a217,a75a140a177a233a217a18a178a254,a23a92a19a144a149
a143za144a149,vectorra160a117 xza178a161vectorE0a134xa182a89a14φ,vectorra134za144a149a27a89a14θ,
nEx = cosφ nEz = 0 nrx = sinθ nry = 0
sin2α = 1?cos2α = 1?(vectornE·vectornr)2 = 1?(nExnrx)2 = 1?cos2φsin2θ
sin2α = 12pi
integraldisplay 2pi
0
dφsin2α= 12pi
integraldisplay 2pi
0
dφ[1?12 sin2θ(1+cos2φ)]=12(1+cos2θ)
dI
d? =
1
2I0r
2
e(1 + cos
2θ) dσ
d? =
d(I/I0)
d? =
1
2r
2
e(1 + cos
2θ)
Thomsona135a169a31a161.
374/384
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a62a94a197a27a209a19a134a225a194,a48a159a27a218a218a209a209,a229a80a62a102a233a233a62a62a94a197a27a209a19
m¨vectorr0(t) =?mγ˙vectorr0(t)?ω20mvectorr0(t) + evectorE0e?iωt?mγ˙vectorr0(t),a202a162a229a145
a193a38a41,vectorr0(t)=vectorr0(0)e?iωt→?mω2vectorr0(0)=iωmγvectorr0(0)?ω20mvectorr0(0)+evectorE0
vectorr0(0) = e
vectorE0
(ω20?ω2)m?iωmγ =
evectorE0
mradicalbig(ω20?ω2)2 +ω2γ2e
iδ =vectorr0eiδ
vectora(t) =vectora(0)e?iωt vectora(0) =?ω2vectorr0(0)
vectorr0 = e
vectorE0
mradicalbig(ω20?ω2)2 +ω2γ2 tanδ =
ωγ
ω20?ω2
vectorE(vectorr,t) = vectorE(vectorr,0)e?iωt vectorE(vectorr,0) = eμ0ω2
4pir e
iωcr[vectorr0(0)?vectorr0(0)·vectornvectorn]
vectorS(vectorr,t) = epsilon10c
2
vectorE(vectorr,t)·vectorE?(vectorr,t)vectorn = μ0e2ω4
32pi2r2c[r
2
0?(vectorn·vectorr0)
2]
= μ0e
2ω4
32pi2r2c(
e
m)
2 E
2
0 sin
2α
(ω20?ω2)2 +ω2γ2 =
μ0e4E20
32pi2m2r2c
ω4 sin2α
(ω20?ω2)2 +ω2γ2
= I0r
2
e
r2 sin
2α ω
4
(ω20?ω2)2 +ω2γ2
375/384
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a62a94a197a27a209a19a134a225a194,a48a159a27a218a218a209a209,a229a80a62a102a233a233a62a62a94a197a27a209a19
vectorS(vectorr,t) = I0r2e
r2 sin
2α ω
4
(ω20?ω2)2 +ω2γ2 I0 =
E20
2μ0c re =
μ0e2
4pim
dσ
d? =
1
2r
2
e(1 + cos
2θ) ω
4
(ω20?ω2)2 +ω2γ2 I = σI0
σ = 83pir2e ω
4
(ω20?ω2)2 +ω2γ2
ωlessmuchω0→ dσd? = 12r2e(1 + cos2θ)(ωω
0
)4 σ = 83pir2e(ωω
0
)4
Rayleigha209a19,a51a140a132a49a204a165,a249a249a49a49a209a19a129a8,a98a49a209a19a129a245,a160a108a92a19a19a229a229a144a149a26a194a27
a49,a217a112a170(a55)a220a169a211a27a39a173a140a117a92a19a229a49a204a169a217a165 a27a112a170a164a169,a13a51a223a19a229a49a204
a164a169a165a36a170(a249)a220a169a27a39a173a79a92a10,a211a158a223a19a19a229a229a111a114a221a126a8a10.a85a152a165a55a218.a70a209a189a73
a19a165a249a218.a103a85a20a204a78a180a232a231.a193a85a49a130a216a118a209a180a100a167a218a229a27.
ωgreatermuchω0→ dσd? = 12r2e(1+cos2θ) σ = 83pir2e
a103a100a62a102a209a19.
376/384
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a62a94a197a27a209a19a134a225a194,a48a159a27a218a218a209a209,a229a80a62a102a233a233a62a62a94a197a27a209a19
dσ
d? =
1
2r
2
e(1+cos
2θ) ω
4
(ω20?ω2)2+ω2γ2 σ =
8
3pir
2
e
ω4
(ω20?ω2)2+ω2γ2
ω = ω0→ dσd? = 12r2e(1 + cos2θ)(ω0γ )2 σ = 83pir2e(ω0γ )2
a1a8,a92a19a197a85a254a26a114a17a27a225a194a50a186a152,a23a92a19a197a252a160a170a199a109a133a92a19a20a252a160a161a200a27
a85a254a143 I0(ω),a75a252a160a158a109a229a80a62a102a203a19a27a111a85a254
W = 83pir2e
integraldisplay ∞
∞
dω ω
4
(ω20?ω2)2 +ω2γ2I0(ω)
≈ 83pir2eI0(ω0)
integraldisplay ∞
∞
dω ω
4
0
4ω20(ω0?ω)2 +ω20γ2
= 23pir2eω20I0(ω0)
integraldisplay ∞
∞
dω ω
4
0
(ω0?ω)2 + (γ2)2
= 4pi3γr2eω20I0(ω0)
integraldisplay ∞
∞
du 1u2 + 1 = 4pi
2
3γ r
2
eω
2
0I0(ω0)
a100a100a117a117a85a254a197a240,a167a143a180a62a102a252a160a158a109a108a92a19a197a225a194a27a111a85a254.
377/384
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a62a94a197a27a209a19a134a225a194,a48a159a27a218a218a209a209,a48a159a27a218a218a209a209
a23a48a159a159a165a165a252a160a78a200a62a102a234a143N.a122a135a62a102a177a27a107a170a199ω0a8a196,a48a159a159a165a165a27a27a62a62a124a143
vectorE = vectorE0e?iωt vectorP = epsilon10χevectorE = Nevectorr0(t) = Ne evectorE0e?iωt
(ω20?ω2)m?iωm?γ
χe = Ne
2
mepsilon10[ω20?ω2?iω?γ] →epsilon1 = epsilon10(1 +χe) = epsilon10 +
Ne2
m[ω20?ω2?iω?γ]
epsilon1r = epsilon1primer + iepsilon1”r epsilon1primer = 1 + Ne
2
epsilon10m
ω20?ω2
(ω20?ω2)2 +ω2?γ2
epsilon1”r = Ne
2
epsilon10m
ω?γ
(ω20?ω2)2 +ω2?γ2?γ,a202a162a88a234 γ,a62a62a19a19a199
k2?k2R?k2I + 2ivectorkR·vectorkI = ω2μepsilon1+ iωμγ a23a181μ,γ,ωa143a162
k2R?k2I =ω2μepsilon10epsilon1primer 2vectorkR·vectorkI=ω2μepsilon10epsilon1”r+ωμγ=ωμ(γ+ωepsilon10epsilon1”r)≡ωμγprime
epsilon1a27a74a220a229a27a138a94a180a242a6a53a27γa67a143γprime,a69a164a62a94a197a27a225a194.
a101γprime = 0,a75epsilon1a162a220a134ωa107a39a191a155a88a242a19a199a134ωa107a39,a61a216a211
a170a199a27a27a62a62a94a197a27a242a19a14a221a216a211,a249a135a53a159a23a23a48a48a159a27a218a218a209a209.
a88a88a92a92a19a49a180a120a49,a218a218a209a209a69a164a209a19a49a143a245a218a49.
378/384
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a62a196a229a198
a51a121a147a212a110a198a165a27a27a47a47a160
379/384
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a111a140a229a198
a169a219a229a198
a218a79a229a198
a254a102a229a198
a62a62a196a196a229a198
380/384
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a229a198a78a88
a178a59a229a198
a178a59a218a79a229a198
a254a102a229a198
a254a102a218a79a229a198
381/384
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a131a112a138a94a229
a62a229
a94a229
a62a94a229a127a127a167a167a229α～
1
137
a89a169a1023×10?7m 0.4eV
a149a6a1025×10?11m 0.2keV
a102a229<10?18m G～ 1300GeV2
a62a102< 10?18m 1√G～300GeV
a62a102a218a152
a114a229< 10?15m gs～1?10
a159a102,a165a102,pia48a102～10?15m 100MeV
a140a218a152
······
······
a218a229 a127a127a167a167a229 g～ 15×1019GeV2
10?35 m 1√g～5×1019GeV
...,..,..
TOE(a135a117?)···
TOE,Theory Of Everything !
382/384
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a136a171a229a198
a131a112a138a94 a178a59(a218a79) a254a102(a218a79)
======== ========== ================
a218a229 a50a194a131a233a216 a254a102a218a229(a107a129a167a221)
a62a94
a62
a94
a62a62a196a196a229a198 a62a62a196a196a229a198 (a107a129a167a221)
a102 × a155a196a229a198(a107a129a167a221)
a114 × a218a196a229a198(a107a129a167a221)
383/384
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a19a155a155a173a173a86a212a110a198a252a140a226a187
a212a159:
a226a102 a254a102
a197 a178a59a124
→ a254a102a124 a228a107a197a226a19a148a53
a143a159a111a135a107a124a186
a254a102a229a198a254a102a21
a131a233a216a196a254a21a169a97
a226a102a23a41a7a171a78a121a143a254a102a124→a131a233a216a254a102a124a216
a123a161a254a102a124a216
384/384
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a62a94a138a94
a62a94a138a94a180a134a60a97a70a126a41a185a129a151a131a131a39a27a196a29a131a112a138a94
a62a94a138a94a180a60a97a131a233a110a41a129a208a27a196a29a131a112a138a94
a2a167a51a212a159a173a46a27a196a29a8a164a165a53a189a144a85a229a57a207a138a94a156
a207a143a144a130a62a214a47a216a164a152a135a17a23a33a103a84a27a196a29a173a189a40a8:
–a58a58a62a62a214a209a121a195a161a140
–a107a129a169a217a62a214a195a123a130a62a94a138a94a242a62a214a118a224a51a152a229
a62a102a28a46a111a180a45a20a40a74
a103a44a46a73a135a217a167a27a85a47a164a196a29a173a189a40a8a27a196a29a131a112a138a94!
–a140a186a221,a218a229a131a112a138a94
–a2a186a221,a114a131a112a138a94
–a88a219a110a41a102a131a112a138a94a27a138a94?
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a178 a59 a62 a196 a229 a198
a152a117a140a198a212a110a88
a28 a147
2/384
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a86a135
a163a227a62a94a131a112a138a94a27a178a59a110a216
a51a247a247a42a42a186a221a221a27a27a154a218a238a229a198a110a216 a197,a226a102mnegationslash= 0
a178a59a124a216 (a62a94a124,a218a229a124)
a67a147a212a110a27a86a103,a78a88a218a144a144a123a123 a124,a218a152,a14a67
3/384
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a19a225a134a214
a103a63a249a194(pdfa130a170)a218a19a89(pdfa130a170)a249a76a27a220a169a169a152a152a51a31a228a198a44
〈a62a62a196a196a229a198〉a167 a72a97a245a167 a112a31a19a152a209a135a22a167 a49a19a135
〈a62a62a196a196a229a198〉a167 a220a220a76a76a146a167a235a254a205a167 a152a117a140a198a209a135a22
〈a178a59a59a62a62a62a196a196a229a198〉a254a101a254a167[a123]J.D a35a142a214a205a167a193a23a254a200
a60a172a19a152a209a135a22
〈Classical Electrodynamics〉a167 Third Edition
John David Jackson,U.C.Berkeleya167John Wiley & Sons,Inc.
〈a62a62a196a196a229a198a123a178a19a167〉,a124a35a114a205a167 a16a174a140a198a209a135a22
〈a62a62a196a196a229a198〉a167 a217a253a205a167 a72a174a140a198a209a135a22
〈a62a62a196a196a229a198〉a167 a52a250a178a205a167 a201a199a140a198a209a135a22
〈a178a59a124a216〉a167 a220a233a59a205a167 a137a198a209a135a22
5/384
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a19a198a140a106
a234a198a79a23,a165a254a169a219;a124a216;a173a130a139a73a182δa188a234;Helmholtza189a110
a62a94a121a121a150a150a27a202a72a53a198,a62a94a131a112a138a94a27a13; a62a94a131a112a138a94a27a124a134a253
a152a165a27a196a29a162a8a189a198; a253a152a165a62a94a131a112a138a94a27a124a144a167; a240
a142a100a137a144a167a124a27a200a169a47a170a134a135a169a47a170; a48a159a165a62a94a131a112a138
a94a27a124a144a167; a62a94a124a27a62a138a39a88; a62a94a131a112a138a94a85a254a196a254a27
a61a122a134a197a240
a183a62a124a218a173a240a62a54a27a27a62a62a94a124,a183a62a179a57a217a135a169a144a167; a141a152a53a189a110
a57a65a65a94a94; a46a202a46a100a144a167,a169a108a67a254a123; a186a148a123;a130a21a188a234; a183a62
a124a27a85a254; a240a189a189a62a62a54a27a27a62a62a124; a173a240a62a54a27a94a124,a165a165a179a179; a94a124a175
a75a27a152a132a41a123; a245a52a208a109
a62a94a197a27a68a194,a110a142a253a14a48a159a165a27a197a196a144a167a57a178a161a62a94a197a41; a189a21
a197a196a144a167a57a178a161a197a41; a62a94a197a51a46a161a254a27a135a19a218a242a19; a20a8
a110; a62a94a197a27a27a189a189a149a68a194; a62a94a197a27a65a219a219a49a49a198a52a129
a207a165a127a193 a156
6/384
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a19a198a140a106
a62a94a197a27a203a19,a62a94a124a27a165a179a218a73a179; a237a180a179; a203a19a62a94a124; a62a94
a197a27a251a19
a100a194a131a233a216,a131a233a216a27a162a8a196a58a182 a131a233a216a196a29a6a110a167a226a212a91a67
a134; a131a233a216a27a158a152a110a216; a131a233a216a110a216a27a14a67a47a170; a131a233a216a229
a198; a131a233a216a62a62a196a196a229a198
a145a62a226a102a218a62a94a124a27a131a112a138a94,a36a196a145a62a226a102a27a62a94a124; a112a132a36a196
a145a62a226a102a27a203a19; a203a19a170a204a169a219; a131a212a133a197a203a19; a145a62a226a102
a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94
a40a229a138,a62a62a196a196a229a198a51a121a147a212a110a198a165a27a27a47a47a160
a207a34a127a193 a156
7/384
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a138a146a134a127a193
a207a19a181a28a94a163 a141a250a191a51a110a137a1622412,a62a12362772794
wsz04@mails.tsinghua.edu.cn
a122a177a2a152a103a138a146a167a80a164a49; a138a146a8510%a80a92a111a164a49
a207a165a127a193a83a78a134a207a34a127a193a83a78a216a173a69
a207a165a127a193a109a242a186a207a34a127a193a52a242a186a136a8545%a80a92a111a164a49
a178a158a137a166a207a76e-maila189a62a123a233a88a21a158a109
wangq@mail.tsinghua.edu.cn
62772791(O); 62783615(H); a110a137a1623408
a83a75a145a216a252a213a254a167a250a64a27a74a75a140a145a254a249
8/384
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a239a198a134a167a119
a55a76a135a107a158a109a218a176a229a27a221a92
a145a167a129a208a135a253a83a182 a152a189a135a107a69a83; a135a103a67a193a88a22a237a19
a183a65a245a120a78a101a27a252a204a182 a116a152a250a170a170a180a180a212a110a39a88a132a180a180a234a234a198a39a88
a39a53a86a103a218a144a144a123a123; a227a229a108a234a198a165a214a212a110,a108a212a110a165a21a234a198!
a216a135a249a75a137a216a209a53; a138a146a103a67a137; a29a177a138a146a216a135a246a20a101a177
a213a225a103a127a167 a125a193a88a74a175a75a167 a207a166a216a211a27a110a41a218a252a204a27a144a144a123a123
a8a92a117a121a102a216a195a189a139a216a254a63a221a158a167a152a189a135a134a80a147a233a88 a156
a107a175a75a53a134a80a147a3a3a3a3a167a159a111a209a140a177a33 a90a25a216a135a31a31a20a20a127a193a131a0 !
9/384
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a202a207a212a110a145a68a199a134a92a130a10
a56a66a123
a70a34a51a100a145a254a92a130a85a198a20
a252a204a123
10/384
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trianglerightsld
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a234a198a79a23
11/384
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trianglerightsld
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a165a254a169a219,a165a254a27a27a189a189a194
a110a145a152a109a27a110a135a213a225a144a149a167a110a135a112a131a82a134a27a252a160a165a254
vectore1 vectore2 vectore3 a189 vectorex vectorey vectorez a189 vectori vectorj vectork
a63a191a191a152a152a135a110a145a152a109a27a165a254 vectorAa94a110a135a196a165a254a208a109a181
vectorA = A1vectore1 + A2vectore2 + A3vectore3 =
3summationdisplay
i=1
Aivectorei
a165a254a27a28a181
|vectorA|=
radicalbig
A21 + A22 + A33 =
radicaltpradicalvertex
radicalvertexradicalbt 3summationdisplay
i=1
AiAi
a252a160a165a254a181
|vectore1|=|vectore2|=|vectore3|= 1 a189|vectorei|= 1
12/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
a92a92a126a126a123a181
vectorA±vectorB = (A1vectore1 + A2vectore2 + A3vectore3)±(B1vectore1 + B2vectore2 + B3vectore3)
= (A1±B1)vectore1 + (A2±B2)vectore2 + (A3±B3)vectore3
=
3summationdisplay
i=1
(Ai±Bi)vectorei
a58a166a181 (a58a200a167a73a200a167a83a200)
a196a165a27a27a58a58a166a181
vectore1·vectore1 =vectore2·vectore2 =vectore3·vectore3 = 1
vectore1·vectore2 =vectore2·vectore1 =vectore1·vectore3 =vectore3·vectore1 =vectore2·vectore3 =vectore3·vectore2 = 0
a189
vectorei·vectorej = δij≡
braceleftbigg 1 i = j
0 inegationslash= j
13/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
vectorA·vectorB = (A1vectore1 + A2vectore2 + A3vectore3)·(B1vectore1 + B2vectore2 + B3vectore3)
= A1B1vectore1·vectore1 + A1B2vectore1·vectore2 + A1B3vectore1·vectore3
+A2B1vectore2·vectore1 + A2B2vectore2·vectore2 + A2B3vectore2·vectore3
+A3B1vectore3·vectore1 + A3B2vectore3·vectore2 + A3B3vectore3·vectore3
= A1B1 + A2B2 + A3B3 =
3summationdisplay
i=1
AiBi = vectorB·vectorA
δija228a107a88a101a53a159a181
δij = δji
3summationdisplay
j=1
Ajδij =
3summationdisplay
j=1
Ajδji = Ai
3summationdisplay
i=1
δii = 3
a124a94δija27a53a159a167
vectorA·vectorB =
3summationdisplay
i=1
Aivectorei·
3summationdisplay
j=1
Bjvectorej =
3summationdisplay
i,j=1
AiBjvectorei·vectorej =
3summationdisplay
i,j=1
AiBjδij =
3summationdisplay
i=1
AiBi
14/384
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triangleleftsld
trianglerightsld
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
a58a166a27a191a191a194a194a181
a253a233a127a221 (a28),vectorA·vectorA =
3summationdisplay
i=1
AiAi =|vectorA|2
a252a160a165a254a181
vectorA
|vectorA| a28a1431,a169a254a143a144a149a14
a89a14a181
vectorA·vectorB
|vectorA||vectorB|≡cosθ
vectorA·vectorB = 0vectorA⊥vectorB
a8a8
a8a8
a8a8
a8a8a8a42
a45θ vectorB
vectorA
a144a149a14a181 cosθi≡ Ai|vectorA| a169a254Aia180a28a27a221a75
15/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
a3a3a166a166a181 (a3a200a167a165a200a167a9a200)
a196a165a27a3a3a166a166a181
vectore1×vectore1 =vectore2×vectore2 =vectore3×vectore3 = 0
vectore1×vectore2 =?vectore2×vectore1 =vectore3
vectore2×vectore3 =?vectore3×vectore2 =vectore1
vectore3×vectore1 =?vectore1×vectore3 =vectore2
a189
vectorei×vectorej =
3summationdisplay
k=1
epsilon1ijkvectorek
epsilon1ijk≡
1 1,2,3a27a243a152a134 a126a181 epsilon1123,epsilon1231,epsilon1312
1 1,2,3a27a219a152a134 a126a181 epsilon1132,epsilon1213,epsilon1321
0 a217a167 a126a181 epsilon1111,epsilon1223,epsilon1313
16/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
vectorA×vectorB = (A1vectore1 + A2vectore2 + A3vectore3)×(B1vectore1 + B2vectore2 + B3vectore3)
= A1B1vectore1×vectore1 + A1B2vectore1×vectore2 + A1B3vectore1×vectore3
+A2B1vectore2×vectore1 + A2B2vectore2×vectore2 + A2B3vectore2×vectore3
+A3B1vectore3×vectore1 + A3B2vectore3×vectore2 + A3B3vectore3×vectore3
= (A1B2?A2B1)vectore3 + (A3B1?A1B3)vectore2 + (A2B3?A3B2)vectore1
=?vectorB×vectorA =
3summationdisplay
i,j,k=1
epsilon1ijkAiBjvectorek
epsilon1ijka228a107a88a101a53a159a181
epsilon1ijka2a134a63a252a141a73a152a103a67a210 epsilon1ijk =?epsilon1jik =?epsilon1kji =?epsilon1ikj
3summationdisplay
k=1
epsilon1ijkepsilon1lmk = δilδjm?δimδjl
17/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
3summationdisplay
j,k=1
epsilon1ijkepsilon1ljk =
3summationdisplay
j=1
(δilδjj?δijδjl) = 3δil?δil = 2δil
3summationdisplay
i,j,k=1
epsilon1ijkepsilon1ijk =
3summationdisplay
i,j=1
(δiiδjj?δijδji) = 9?3 = 6
vectorA×vectorBa27a79a142a76a167a140a123a21a143a181
vectorA×vectorB =
3summationdisplay
i=1
Aivectorei×
3summationdisplay
j=1
Bjvectorej =
3summationdisplay
i,j=1
AiBjvectorei×vectorej=
3summationdisplay
i,j,k=1
AiBjepsilon1ijkvectorek
vectorA×vectorA = 0
(vectorA×vectorB)·vectorC = (
3summationdisplay
i,j,k=1
epsilon1ijkAiBjvectorek)·
3summationdisplay
n=1
Cnvectoren =
3summationdisplay
i,j,k,n=1
epsilon1ijkAiBjCnvectorek·vectoren
=
3summationdisplay
i,j,k,n=1
epsilon1ijkAiBjCnδkn =
3summationdisplay
i,j,k=1
epsilon1ijkAiBjCk a49a15a170;a204a130
18/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
a3a3a166a166a27a191a191a194a194a181
a253a233a127a221(a28),|vectorA×vectorB|=|vectorA||vectorB|sinθ
a144a149a181 vectorA×vectorB⊥vectorA,vectorB vectorAbardblvectorBvectorA×vectorB = 0
a24a24a24a24
a24a24a58
a80a80a80
a80a80a80
a80a80a80a113
a54vectorA×vectorB
θ
vectorA
vectorB
|vectorA×vectorB|2 = (vectorA×vectorB)·(vectorA×vectorB) = (vectorA·vectorA)(vectorB·vectorB)?(vectorA·vectorB)(vectorA·vectorB)
= |vectorA|2|vectorB|2?(|vectorA||vectorB|cosθ)2 =|vectorA|2|vectorB|2(1?cos2θ)
= |vectorA|2|vectorB|2 sin2θ
(vectorA×vectorB)·vectorA = 0 (vectorA×vectorB)·vectorB = 0
19/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
(vectorA×vectorB)·(vectorC×vectorD)
= [(A1B2?A2B1)vectore3 + (A3B1?A1B3)vectore2 + (A2B3?A3B2)vectore1]
·[(C1D2?C2D1)vectore3 + (C3D1?C1D3)vectore2 + (C2D3?C3D2)vectore1]
= (A1B2?A2B1)(C1D2?C2D1) + (A3B1?A1B3)(C3D1?C1D3)
+(A2B3?A3B2)(C2D3?C3D2)
= A1C1B2D2 + A2C2B1D1 + A3C3B1D1 + A1C1B3D3
+A2C2B3D3 + A3C3B2D2?A1D1B2C2?A2D2B1C1
A3D3B1C1?A1D1B3C3?A2D2B3C3?A3D3B2C2
= (A1C1 + A2C2 + A3C3)(B1D1 + B2D2 + B3D3)
(A1D1 + A2D2 + A3D3)(B1C1 + B2C2 + B3C3)
= (vectorA·vectorC)(vectorB·vectorD)?(vectorA·vectorD)(vectorB·vectorC)
(vectorA×vectorB)·(vectorA×vectorB) = (vectorA·vectorA)(vectorB·vectorB)?(vectorA·vectorB)(vectorB·vectorA)
a52a165a254a181vectorA a152a109a135a19→?vectorA a182a165a254a181vectorA a152a109a135a19→ vectorA
a52a165a254×a52a165a254=a182a165a254
20/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
a152a135a14a160a117a139a73a6a58a58a27a27a110a14a47a167a217a108a6a58a209a117a27a252a94a62a8a164a165a254
vectorr =
3summationdisplay
i=1
xivectorei cosθi = xir r≡|vectorr|=
radicalbig
x21 + x22 + x23
vectorrprime =
3summationdisplay
i=1
xprimeivectorei cosθprimei = x
prime
i
rprime r
prime≡|vectorrprime|=
radicalBig
xprime12 + xprime22 + xprime32
a49a110a94a62a8a164a27a165a254
vectorR =vectorr?vectorrprime =
3summationdisplay
i=1
(xi?xprimei)vectorei a1a1
a1
a1a1a21
a16a16a16
a16a16a16
a16a16a16a49
a80a80a80
a80a80a80a105vectorr
vectorr prime
vectorR
cosφi =
vectorR·vectorei
R =
xi?xprimei
R R =
radicaltpradicalvertex
radicalvertexradicalbt 3summationdisplay
i=1
(xi?xprimei)(xi?xprimei)
21/384
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a165a254a169a219,a165a254a131a109a36a142a27a27a189a189a194
a110a135a83a14a169a79a80a143α(vectorr,vectorrprime);β(vectorrprime,vectorR);γ(vectorr,vectorR)
a1
a1
a1
a1a1a21
a16a16a16
a16a16a16
a16a16a16a49
a80a80a80
a80a80a80a105vectorr
vectorr prime
vectorR
α
γ β
sinα = |vectorr×vectorr
prime|
rrprime
sin(pi?β) = |
vectorR×vectorrprime|
Rrprime =
|vectorr×vectorrprime|
Rrprime sinγ =
|vectorR×vectorr|
Rr =
|vectorr×vectorrprime|
Rr
sinα
R =
sinβ
r =
sinγ
rprime =
|vectorr×vectorrprime|
rrprimeR
R2 = (vectorr?vectorrprime)·(vectorr?vectorrprime) =vectorr·vectorr +vectorrprime·vectorrprime?2vectorr·vectorrprime= r2 + rprime2?2rrprimecosα
r2 = (vectorR +vectorrprime)·(vectorR +vectorrprime) = vectorR·vectorR+vectorrprime·vectorrprime+2vectorR·vectorrprime= R2+rprime2?2rrprimecosβ
rprime2 = (vectorr?vectorR)·(vectorr?vectorR) =vectorr·vectorr + vectorR·vectorR?2vectorr·vectorR= r2 + R2?2rRcosγ
22/384
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a165a254a169a219,a220a254a237a50
a110a145a152a109a165a27a19a30a220a254,arrowrighttophalfarrowrighttophalfT≡
3summationdisplay
i,j=1
Tijvectoreivectorej
arrowrighttophalfarrowrighttophalfI ≡ 3summationdisplay
i,j=1
δijvectoreivectorej =
3summationdisplay
i=1
vectoreivectorei
arrowrighttophalfarrowrighttophalfI ·vectorA = vectorA·arrowrighttophalfarrowrighttophalfI = vectorA
na30a220a254:
3summationdisplay
i1,i2,...,in=1
Ti1,i2,...,invectorei1vectorei2···vectorein
23/384
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a220a254a237a50arrowrighttophalfarrowrighttophalfS ±arrowrighttophalfarrowrighttophalfT =
3summationdisplay
i,j=1
Sijvectoreivectorej±
3summationdisplay
i,j=1
Tijvectoreivectorej =
3summationdisplay
i,j=1
(Sij±Tij)vectoreivectorej
arrowrighttophalfarrowrighttophalfT ·vectorA = ( 3summationdisplay
i,j=1
Tijvectoreivectorej)·(
3summationdisplay
k=1
Akvectorek) =
3summationdisplay
i,j,k=1
TijAkvectorei(vectorej·vectorek) =
3summationdisplay
i,j=1
TijAjvectorei
vectorA·arrowrighttophalfarrowrighttophalfT = (
3summationdisplay
k=1
Akvectorek)·(
3summationdisplay
i,j=1
Tijvectoreivectorej) =
3summationdisplay
i,j,k=1
TijAk(vectorek·vectorei)vectorej =
3summationdisplay
i,j=1
AiTijvectorej
arrowrighttophalfarrowrighttophalfT ×vectorA = ( 3summationdisplay
i,j=1
Tijvectoreivectorej)×(
3summationdisplay
k=1
Akvectorek) =
3summationdisplay
i,j,k=1
TijAkvectorei(vectorej×vectorek)
=
3summationdisplay
i,j,k,l=1
TijAkepsilon1jklvectoreivectorel
vectorA×arrowrighttophalfarrowrighttophalfT = (
3summationdisplay
k=1
Akvectorek)×(
3summationdisplay
i,j=1
Tijvectoreivectorej) =
3summationdisplay
i,j,k=1
TijAk(vectorek×vectorei)vectorej
=
3summationdisplay
i,j,k,l=1
AkTijepsilon1kilvectorelvectorej
24/384
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a124a216,a124a27a27a189a189a194
a73a254a124,a152a109a139a73 x,y,z a27a73a254a188a234a169 φ(vectorr) = φ(x,y,z)
a165a254a124,a152a109a139a73 x,y,za27a165a254a188a234a169 vectorA(vectorr) =
3summationdisplay
i=1
Ai(x,y,z)vectorei
a19a30a220a254a124,a152a109a139a73 x,y,za27a19a30a220a254a188a234a169
arrowrighttophalfarrowrighttophalfT (vectorr) = 3summationdisplay
i=1
Tij(x,y,z)vectoreivectorej
......
a124a207a126a61a157a54a152a109a139a73a167a143a143a157a157a54a158a109a139a73a34
a178a126a94a229a130a163a227a152a109a27a165a254a124a167 a229a130a27a144a149a134a165a254a124a27a144a149a131
a211a167a229a130a27a151a221a20a39a117a165a254a124a28a27a140a2a169
25/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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26/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a124a216,a124a27a135a169a53a159
a135a169a142a206a27a27a189a189a194:
a110a145a152a109a165a140a177a169a79a233a110a135a139a73a63a49a135a251
1≡x?2≡y?3≡z?ivectorej =vectorej?i
≡vectore1x +vectore2y +vectore3z =
3summationdisplay
i=1
vectorei?i
2≡?·?=
3summationdisplay
i=1
i?i =x2 +y2 +z2?×?= 0
a144a107a233a134a14a139a73a88a167a135a169a142a206a226a134a196a165a233a180a34
a107a86a173a14a218a167a61a107a135a169a142a206a27a53a159(?i),a113a180a180a165a165a254(vectorei)a34
a185?a27a144a167a252a62a167a76a166?a27a135a169a142a206a218a165a254a53a159a211a158a26a177
a247a118a226a85a178a239a156
27/384
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trianglerightsld
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a124a216,a124a27a135a169a53a159
a73a254a124φa27a70a221a181?φ =
3summationdisplay
i=1
vectorei?iφ=vectore1?φ?x +vectore2?φ?y +vectore3?φ?z
a165a254a124vectorAa27a209a221a181
·vectorA = (
3summationdisplay
i=1
vectorei?i)·(
3summationdisplay
j=1
Ajvectorej) =
3summationdisplay
i,j=1
vectorei·vectorej?iAj =
3summationdisplay
i=1
iAi
=?A1?x +?A2?y +?A3?z
a165a254a124vectorAa27a94a221a181
×vectorA = (
3summationdisplay
i=1
vectorei?i)×(
3summationdisplay
j=1
Ajvectorej) =
3summationdisplay
i,j=1
vectorei×vectorej?iAj =
3summationdisplay
i,j,k=1
epsilon1ijk(?iAj)vectorek
=vectore3(?A2?xA1?y ) +vectore2(?A1?zA3?x ) +vectore1(?A3?yA2?z )
28/384
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a124a216,a124a27a135a169a53a159
a135a169a142a206a233a252a135a124a27a138a94
·(vectorf×vectorg)
=
3summationdisplay
i,j,k,l=1
ivectorei·(fjgkepsilon1jklvectorel) =
3summationdisplay
i,j,k,l=1
vectorei·vectorelepsilon1jkl?i(fjgk)
=
3summationdisplay
i,j,k,l=1
δilepsilon1jkl(gk?ifj + fj?igk) =
3summationdisplay
i,j,k=1
epsilon1jki(gk?ifj + fj?igk)
=
3summationdisplay
i,j,k=1
epsilon1ijkgk?ifj?epsilon1jikfj?igk
=vectorg·(?×vectorf)?vectorf·(?×vectorg)
29/384
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a124a216,a124a27a135a169a53a159
×(vectorf×vectorg)
=
3summationdisplay
i,j,k,l=1
ivectorei×(fjgkepsilon1jklvectorel) =
3summationdisplay
i,j,k,l=1
vectorei×vectorelepsilon1jkl?i(fjgk)
=
3summationdisplay
i,j,k,l,m=1
epsilon1ilmvectoremepsilon1jkl(gk?ifj + fj?igk)
=
3summationdisplay
i,j,k,m=1
vectorem(?δijδmk +δikδmj)(gk?ifj + fj?igk)
=
3summationdisplay
i,j,k,m=1
vectorem(?δijδmkgk?ifj?δijδmkfj?igk +δikδmjgk?ifj +δikδmjfj?igk)
=?
3summationdisplay
i,k=1
(vectorekgk?ifi + fi?igkvectorek) +
3summationdisplay
i,j=1
(gi?ifjvectorej +vectorejfj?igi)
=?vectorg?·vectorf?vectorf·?vectorg +vectorg·?vectorf +vectorf?·vectorg
30/384
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a124a216,a124a27a135a169a53a159
a135a169a142a206a233a69a220a124a27a138a94
φ(u) =
3summationdisplay
i=1
vectorei?iφ(u) =
3summationdisplay
i=1
vectorei?φ?u?iu= φprime(u)?u
·vectorf(u) =
3summationdisplay
i=1
ifi(u) =
3summationdisplay
i=1
fi
u?iu=
vectorfprime(u)·?u
×vectorf(u) =
3summationdisplay
i,j,k=1
epsilon1ijkvectorek?ifj(u) =
3summationdisplay
i,j,k=1
epsilon1ijkvectorek?fj?u?iu=?u×vectorfprime(u)
a135a169a142a206a233vectorRa27a138a94
vectorr =
3summationdisplay
i=1
xivectorei?=
3summationdisplay
i=1
vectorei?i?i =x
i
vectorrprime =
3summationdisplay
i=1
xprimeivectorei?prime =
3summationdisplay
i=1
vectorei?primei?primei =xprime
i
31/384
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a124a216,a124a27a135a169a53a159vectorR =vectorr?vectorrprime = 3summationdisplay
i=1
(xi?xprimei)vectorei
ixj = δij?primeixprimej = δij?ixprimej =?primeixj = 0
·vectorR =
3summationdisplay
i=1
i(xi?xprimei)= 3 =prime·vectorR
×vectorR =
3summationdisplay
i,j,k=1
epsilon1ijk?i(xj?xprimej)vectorek =
3summationdisplay
i,j,k=1
epsilon1ijkδijvectorek= 0 =prime×vectorR
R =
3summationdisplay
i=1
vectorei?iR = 12R
3summationdisplay
i,j=1
vectorei?i[(xj?xprimej)(xj?xprimej)]
= 1R
3summationdisplay
i,j=1
vectorei(xj?xprimej)?i(xj?xprimej) = 1R
3summationdisplay
i,j=1
vectorei(xj?xprimej)δij
= 1R
3summationdisplay
i=1
vectorei(xi?xprimei)=
vectorR
R =
primeR
32/384
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a124a216,a124a27a200a169a53a159
a70a221a221a27a27a130a200a169:
integraldisplay B
A
dvectorl·?φ =
3summationdisplay
i,j=1
integraldisplay B
A
vectoreidxi·?jφvectorej =
3summationdisplay
i,j=1
integraldisplay B
A
dxi?jφvectorei·vectorej
=
3summationdisplay
i,j=1
integraldisplay B
A
dxi?jφδij =
3summationdisplay
i=1
integraldisplay B
A
dxi?iφ =
integraldisplay B
A
dφ
= φB?φA
a45a27 a54
a63
a0a0a18a64a64a73
a64a64a82a0a0a9
a115?φ
a27a191a191a194a194a181
vectorl·?φ = φB?φA a189 |?φ|cosθ = φB?φA|?vectorl| a144a149a19a234====?φ|?vectorl|
φa27a140a2a143φa247a136a144a149a27a19a234a165a129a140a27a19a234a138a167a144a149a247a228a107 a129a140
a19a234a27a144a149a34a63a191a144a149a27a144a149a19a234a61a180?φ a51a100a144a149a254a27a221a75a34
φa27a144a149a180a31a138a161a27a123a130a144a149a169
33/384
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a124a216,a124a27a200a169a53a159a209a221a221a27a27a78a200a169:
a78a200 τ a165a27a78a200a169
integraldisplay
τ
dτ?·vectorf =
contintegraldisplay
τ
dvectorS·vectorf a78a200 τ a76a161a27a161a200a169
a192za144a149a166a121,τa76a161a169a143a254a101a252a76a161:SB～zB(x,y)a218SA～zA(x,y)
integraldisplay
τ
dτzf3(x,y,z)
=
integraldisplay
dxdy
integraldisplay zB(x,y)
zA(x,y)
dzzf3(x,y,z)
=
integraldisplay
dxdy[f3(x,y,zB(x,y))?f3(x,y,zA(x,y))]
=
integraldisplay
SB
dS3f3(x,y,zB(x,y))?
integraldisplay
SA
dS3f3(x,y,zA(x,y))
=
contintegraldisplay
τ
dS3 f3(x,y,z) a250a170a144a183a94a94a117a117a78a200a165a195a154a201a201a27a27a156a185
34/384
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a124a216,a124a27a200a169a53a159
integraldisplay
τ
dτ?=
contintegraldisplay
τ
dvectorS
→
integraldisplay
τ
dτ?φ =
contintegraldisplay
τ
dvectorSφ
integraldisplay
τ
dτ?×vectorf =
contintegraldisplay
τ
dvectorS×vectorf
·vectorfa27a191a191a194a194a181
contintegraldisplay
τ
dvectorS·vectorf a189a194a143a181vectorfa108τa54a209a27a207a254
τ?·vectorf =
contintegraldisplay
τ
dvectorS·vectorf
·vectorf = lim
τ→0
contintegraltext
τ d
vectorS·vectorf
τ = a252a160a78a200a108a124a58a54a209a27a207a254
·vectorf = 0a195a192a207a254a9a196a167a195a13a182?·vectorf negationslash= 0a107a192a207a254a9a196a167a107a13!
a45a27 a54
a63
a0a0a18a64a64a73
a64a64a82a0a0a9
a115 a27a0a9a63a64a64a82a45
a0a18a54a64a64a73
a115 a252a135a31a254a20a75a58a58a62a62a214a13a27a27a62a62a124a130a28a28a91a91
35/384
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a124a216,a124a27a200a169a53a159
a94a221a221a27a27a161a200a169,a173a161 S a165a27a161a200a169
integraldisplay
S
dvectorS ·?×vectorf =
contintegraldisplay
S
dvectorl·vectorf a173a161 S a62a46a27a173a130a200a169
xa144a149,a173a161a144a167:z = g(x,y),a123a130:vectorn = n1vectore1 + n2vectore2 + n3vectore3
dz?gprimexdx?gprimeydy = 0 dvectorr = dxvectore1 + dyvectore2 + dzvectore3
vectorn·dvectorr = 0? vectorn∝?gprimexvectorex?gprimeyvectorey +vectorez
vectorn = c(?gprimexvectorex?gprimeyvectorey +vectorez) vectorn·vectorn=1====?c2(1 + (gprimex)2 + (gprimey)2) = 1
nx =? g
prime
xradicalBig
1 + gprimex2 + gprimey2
ny =? g
prime
yradicalBig
1 + gprimex2 + gprimey2
nz = 1radicalBig
1 + gprimex2 + gprimey2
36/384
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a124a216,a124a27a200a169a53a159a94a221a221a27a27a161a200a169:
integraldisplay
S
dvectorS ·?×vectorf =
contintegraldisplay
S
dvectorl·vectorf
xa144a149,a173a161a144a167:z = g(x,y),a123a130:vectorn = n1vectore1 + n2vectore2 + n3vectore3
nx,y =? g
prime
x,yradicalBig
1 + gprimex2 + gprimey2; nz = 1radicalBig
1 + gprimex2 + gprimey2
dS2
dS3 =
n2dS
n3dS =?g
prime
y
a173a161a62a46a51 xy a178a161a221a75a171a141a173a130a62a46a169a143:y = yB(x),y = yA(x)integraldisplay
S
(dS2?3f1?dS3?2f1) =?
integraldisplay
S
dS3 (gprimey?3f1 +?2f1)
=?
integraldisplay
S
dS3 ddyf1(x,y,g(x,y))
=?
integraldisplay
dx
integraldisplay yB(x)
yA(x)
dy ddyf1(x,y,g(x,y))
=?
integraldisplay
dx [f1(x,yB(x),g(x,yB(x)))?f1(x,yA(x),g(x,yA(x)))]
=
contintegraldisplay
S
dx f1(x,y,g(x,y)) a173a161a9a123a130a85a109a195a39a88a53a189xya178a161a27a173a130a27a20a149a143a95a158a2
a250a170a144a183a94a94a117a117a173a161a165a195a154a201a201a27a27a156a185!
37/384
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a124a216,a124a27a200a169a53a159
integraldisplay
S
dvectorS ·?×vectorf =
integraldisplay
S
dvectorS×?·vectorf =
contintegraldisplay
S
dvectorl·vectorf?
integraldisplay
S
dvectorS×?=
contintegraldisplay
S
dvectorl
integraldisplay
S
dvectorS×?φ =
contintegraldisplay
S
dvectorlφ
integraldisplay
S
(dvectorS×?)×vectorf =
contintegraldisplay
S
dvectorl×vectorf
×vectorfa27a191a191a194a194a181
vectorS ·?×vectorf =
contintegraldisplay
S
dvectorl·vectorf ≡vectorfa55Sa130a55a27a130a254
|?×vectorf|cosθ = lim
S→0
contintegraltext
Sd
vectorl·vectorf
S
×vectorf a140a2a143a124a58a63a129a140a140a252a252a160a161a200a200a130a130a254a138a167a144a149a247
a228a107a129a140a130a254a27a173a161a144a149a169a63a191a144a149a173a161a27a130a254
a180?×vectorfa51a173a161a144a149a254a221a75a34
×vectorf = 0a195a192a130a254,a195a94;?×vectorf negationslash= 0a192a130a254,a107a94a34
a98a54a105a54a54a14a13a15a12a54
a22a21
a23a20
a30a29
a31a28
38/384
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a173a130a139a73,a206a139a73
a139a73a3a181 r =
radicalbig
x2 + y2,θ = arctan yx,z a196a165a181 vectorer,vectoreθ,vectorez
a38a37
a39a36
a101a114a0
a0
a0a0a18a54
a64
a64
a64a64a73 a45
vectorervectoreyvectoreθ
vectorexθθ
vectorez
vectorer·vectorex = cosθ vectorer·vectorey = sinθ vectorer·vectorez = 0
vectoreθ·vectorex =?sinθ vectoreθ·vectorey = cosθ vectoreθ·vectorez = 0
vectorez·vectorex = 0 vectorez·vectorey = 0 vectorez·vectorez = 1
vectorer = cosθvectorex + sinθvectorey vectoreθ =?sinθvectorex + cosθvectorey
vectorex = cosθvectorer?sinθvectoreθ vectorey = sinθvectorer + cosθvectoreθ
vectorer
r =0
vectorer
θ =vectoreθ
vectorer
z =
vectoreθ
r =0
vectoreθ
θ =?vectorer
vectoreθ
z =
vectorez
r =
vectorez
θ =
vectorez
z =0
a247a196a165a144a149a27a195a161a2a130a3a181 dl1 = dr,dl2 = rdθ,dl3 = dz
a135a169a142a206a181?=vectorerr+vectoreθ1rθ+vectorezz a165a254a124a181vectorf = frvectorer+fθvectoreθ+fzvectorez
a73a254a124a27a70a221a181?φ =vectorer?φ?r+vectoreθ1r?φ?θ+vectorez?φ?z
a165a254a124a27a209a221a181?·vectorf = [vectorerr+vectoreθ1rθ+vectorezz]·(frvectorer+fθvectoreθ+fzvectorez)
39/384
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a173a130a139a73,a206a139a73
a139a73a3a181 r =
radicalbig
x2 + y2,θ = arctan yx,z a196a165a181 vectorer,vectoreθ,vectorez
vectorer = cosθvectorex + sinθvectorey vectoreθ =?sinθvectorex + cosθvectorey
vectorer
r =0
vectorer
θ =vectoreθ
vectorer
z =
vectoreθ
r =0
vectoreθ
θ =?vectorer
vectoreθ
z =
vectorez
r =
vectorez
θ =
vectorez
z =0
a135a169a142a206a181?=vectorerr+vectoreθ1rθ+vectorezz a165a254a124a181vectorf = frvectorer+fθvectoreθ+fzvectorez
a73a254a124a27a70a221a181?φ =vectorer?φ?r+vectoreθ1r?φ?θ+vectorez?φ?z
a165a254a124a27a209a221a181?·vectorf = [vectorerr+vectoreθ1rθ+vectorezz]·(frvectorer+fθvectoreθ+fzvectorez)
=?fr?r +1r?fθ?θ +?fz?z +vectoreθ1r·(fr?vectorer?θ +fθ?vectoreθ?θ )
=?fr?r +1r?fθ?θ +?fz?z +frr
vectorf →vector2 = vector?·vector?=?2
r2+
1
r
2
θ2+
2
z2+
1
r
r
40/384
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a173a130a139a73,a206a139a73
vectorer
r =0
vectorer
θ =vectoreθ
vectorer
z =
vectoreθ
r =0
vectoreθ
θ =?vectorer
vectoreθ
z =
vectorez
r =
vectorez
θ =
vectorez
z =0
a135a169a142a206a181?=vectorerr+vectoreθ1rθ+vectorezz?2=?
2
r2+
1
r
2
θ2+
2
z2+
1
r
r
a73a254a124a27a70a221a181?φ =vectorer?φ?r+vectoreθ1r?φ?θ+vectorez?φ?z
a165a254a124a27a209a221a181?·vectorf =?fr?r +1r?fθ?θ +?fz?z +frr
a165a254a124a27a94a221a181?×vectorf = [vectorerr+vectoreθ1rθ+vectorezz]×(frvectorer+fθvectoreθ+fzvectorez)
=vectorez?fθ?r?vectoreθ?fz?r?vectorez?frr?θ+vectorer?fzr?θ+vectoreθ?fr?z?vectorer?fθ?z
+frrvectoreθ×?vectorer?θ +fθrvectoreθ×?vectoreθ?θ
=vectorez[?fθ?rfrr?θ+fθr]+vectoreθ[?fr?zfz?r]+vectorer[?fzr?θfθ?z]
41/384
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a173a130a139a73,a165a139a73
a0
a0a0a9
a45
a54
a0
a0a0a18
a64
a64a64a82
z
θ
φ
vectorr
x
y
vectorρ
a38a37
a39a36
a0
a0
a0a0a18a54
a64
a64
a64a64a73 a45
vectorervectorez?vectoreθ
vectoreρθ
θ
a38a37
a39a36
a101a114a0
a0
a0a0a18a54
a64
a64
a64a64a73 a45
vectoreρvectoreyvectoreφ
vectorexφφ
vectorez
a139a73a3a181 r=
radicalbig
x2+y2+z2,θ=arctan(
radicalbig
x2+y2/z),φ=arctan(y/x)
a196a165a181 vectorer,vectoreθ,vectoreφ x,ya178a161a196a165a181 vectoreρ
vectoreρ= cosφvectorex+sinφvectorey vectoreφ=?sinφvectorex+cosφvectorey
vectorex= cosφvectoreρ?sinφvectoreφ vectorey= sinφvectoreρ+cosφvectoreφ
vectorer·vectoreρ = sinθ vectorer·vectorez = cosθ vectorer·vectoreφ = 0
vectorer·vectorex = cosφvectorer·vectoreρ = cosφsinθ vectorer·vectorey = sinφvectorer·vectoreρ = sinφsinθ
vectoreθ·vectoreρ = cosθ vectoreθ·vectorez =?sinθ vectoreθ·vectoreφ = 0
vectoreθ·vectorex = cosφvectoreθ·vectoreρ =cosφcosθ vectoreθ·vectorey = sinφvectoreθ·vectoreρ =sinφcosθ
vectoreφ·vectorex =?sinφ vectoreφ·vectorey = cosφ vectoreφ·vectorez = 0?
vectorer
vectoreθ
vectoreφ
=
cosφsinθ sinφsinθ cosθ
cosφcosθ sinφcosθ?sinθ
sinφ cosφ 0
vectorex
vectorey
vectorez
vectorex
vectorey
vectorez
=
cosφsinθ cosφcosθ?sinφ
sinφsinθ sinφcosθ cosφ
cosθ?sinθ 0
vectorer
vectoreθ
vectoreφ
42/384
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a173a130a139a73,a165a139a73
a38a37
a39a36
a0
a0
a0a0a18a54
a64
a64
a64a64a73 a45
vectorervectorez?vectoreθ
vectoreρθ
θ
a38a37
a39a36
a101a114a0
a0
a0a0a18a54
a64
a64
a64a64a73 a45
vectoreρvectoreyvectoreφ
vectorexφφ
vectorez
vectorer
vectoreθ
vectoreφ
=
cosφsinθ sinφsinθ cosθ
cosφcosθ sinφcosθ?sinθ
sinφ cosφ 0
vectorex
vectorey
vectorez
vectorex
vectorey
vectorez
=
cosφsinθ cosφcosθ?sinφ
sinφsinθ sinφcosθ cosφ
cosθ?sinθ 0
vectorer
vectoreθ
vectoreφ
vectorer
r =
vectoreθ
r =
vectoreφ
r = 0
vectorer
θ =vectoreθ
vectoreθ
θ =?vectorer
vectoreφ
θ = 0
vectorer
φ = sinθvectoreφ
vectoreθ
φ = cosθvectoreφ
vectoreφ
φ =?sinθvectorer?cosθvectoreθ
a247a196a165a144a149a27a195a161a2a130a3a181 dl1 = dr,dl2 = rdθ,dl3 = rsinθdφ
a135a169a142a206a181?=vectorerr+vectoreθ1rθ+vectoreφ 1rsinθφ a165a254a124a181vectorf=frvectorer+fθvectoreθ+fφvectoreφ
a73a254a124a27a70a221a181?φ =vectorer?φ?r+vectoreθ1r?φ?θ+vectoreφ 1rsinθ?φ?φ
43/384
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a173a130a139a73,a165a139a73
vectorer
r =
vectoreθ
r =
vectoreφ
r = 0
vectorer
θ =vectoreθ
vectoreθ
θ =?vectorer
vectoreφ
θ = 0
vectorer
φ = sinθvectoreφ
vectoreθ
φ = cosθvectoreφ
vectoreφ
φ =?sinθvectorer?cosθvectoreθ
a135a169a142a206a181?=vectorerr+vectoreθ1rθ+vectoreφ 1rsinθφ a165a254a124a181vectorf=frvectorer+fθvectoreθ+fφvectoreφ
a73a254a124a27a70a221a181?φ =vectorer?φ?r+vectoreθ1r?φ?θ+vectoreφ 1rsinθ?φ?φ
a165a254a124a27a209a221a181?·vectorf=[vectorerr+vectoreθ1rθ+vectoreφ 1rsinθφ]·(frvectorer+fθvectoreθ+fφvectoreφ)
=?fr?r +1r?fθ?θ + 1rsinθ?fφ?φ+vectoreθ1r·(fr?vectorer?θ +fθ?vectoreθ?θ )+vectoreφ 1rsinθ·(fr?vectorer?φ +fθ?vectoreθ?φ)
=?fr?r +1r?fθ?θ + 1rsinθ?fφ?φ+2frr +fθ cosθrsinθ
vectorf →vector2=vector?·vector?=?2
r2+
1
r2
2
θ2+
1
r2 sin2θ
2
φ2+
2
r
r+
cosθ
r2 sinθ
θ
44/384
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a173a130a139a73,a165a139a73
vectorer
r =
vectoreθ
r =
vectoreφ
r = 0
vectorer
θ =vectoreθ
vectoreθ
θ =?vectorer
vectoreφ
θ = 0
vectorer
φ = sinθvectoreφ
vectoreθ
φ = cosθvectoreφ
vectoreφ
φ =?sinθvectorer?cosθvectoreθ
a73a254a124a27a70a221a181?φ =vectorer?φ?r+vectoreθ1r?φ?θ+vectoreφ 1rsinθ?φ?φ
a165a254a124a27a209a221a181?·vectorf=?fr?r +1r?fθ?θ + 1rsinθ?fφ?φ+2frr +fθ cosθrsinθ
2=?
2
r2+
1
r2
2
θ2+
1
r2 sin2θ
2
φ2+
2
r
r+
cosθ
r2 sinθ
θ
a165a254a124a27a94a221a181?×vectorf=[vectorerr+vectoreθ1rθ+vectoreφ 1rsinθφ]×(frvectorer+fθvectoreθ+fφvectoreφ)
=vectoreφ?fθ?r?vectoreθ?fφ?r?vectoreφ?frr?θ+vectorer?fφr?θ+vectoreθ?frrsinθ?φ?vectorer?fθrsinθ?φ
+1rvectoreθ×[fr?vectorer?θ +fθ?vectoreθ?θ ]+ 1rsinθvectoreφ×[fr?vectorer?φ +fθ?vectoreθ?φ +fφ?vectoreφ?φ]
45/384
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a173a130a139a73,a165a139a73
vectorer
r =
vectoreθ
r =
vectoreφ
r = 0
vectorer
θ =vectoreθ
vectoreθ
θ =?vectorer
vectoreφ
θ = 0
vectorer
φ = sinθvectoreφ
vectoreθ
φ = cosθvectoreφ
vectoreφ
φ =?sinθvectorer?cosθvectoreθ
a165a254a124a27a94a221a181?×vectorf=[vectorerr+vectoreθ1rθ+vectoreφ 1rsinθφ]×(frvectorer+fθvectoreθ+fφvectoreφ)
=vectoreφ?fθ?r?vectoreθ?fφ?r?vectoreφ?frr?θ+vectorer?fφr?θ+vectoreθ?frrsinθ?φ?vectorer?fθrsinθ?φ
+1rvectoreθ×[fr?vectorer?θ +fθ?vectoreθ?θ ]+ 1rsinθvectoreφ×[fr?vectorer?φ +fθ?vectoreθ?φ +fφ?vectoreφ?φ]
=vectoreφ?fθ?r?vectoreθ?fφ?r?vectoreφ?frr?θ+vectorer?fφr?θ+vectoreθ?frrsinθ?φ?vectorer?fθrsinθ?φ
+1rvectoreθ×[frvectoreθ?fθvectorer]? fφrsinθvectoreφ×[sinθvectorer + cosθvectoreθ]
=vectorer[?fφr?θfθrsinθ?φ+fφcosθrsinθ]+vectoreθ[?frrsinθ?φfφ?r?fφr]+vectoreφ[?fθ?rfrr?θ+fθr]
46/384
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δ a188a234 a152a145a152a109a189a194a134a53a159
δ(x?x0) =
braceleftbigg 0 xnegationslash= x
0
∞ x = x0
integraldisplay
a157a41x0
dx δ(x?x0) = 1
δ(x?x0)≥0 δ(x?x0) = δ(x0?x) f(x)δ(x?x0) = f(x0)δ(x?x0)
δ(f(x)) =
summationdisplay
i
1
|fprime(xi)|δ(x?xi) xia180f(x) = 0a27a27a49a49 i a135a135a138a138
δepsilon1(x?xprime)≡1pi epsilon1(x?xprime)2 +epsilon12 = Re1pi ix?xprime + iepsilon1 epsilon1→0
+
→
braceleftbigg 0 xnegationslash= xprime
∞ x = xprimeintegraldisplay
∞
∞
dx 1pi epsilon1(x?xprime)2 +epsilon12 = 1pi
integraldisplay ∞
∞
dt 1t2 + 1 = 1
δ(x?xprime) = lim
epsilon1→0+
δepsilon1(x?xprime) = Re
integraldisplay ∞
0
dk
pi e
ik(x?xprime+iepsilon1)
= 12pi[
integraldisplay ∞
0
dk eik(x?xprime)?|k|epsilon1?
integraldisplay?∞
0
d(?k) ei(?k)(x?xprime)?|?k|epsilon1] =
integraldisplay ∞
∞
dk
2pie
ik(x?xprime)?epsilon1|k|
47/384
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δ a188a234 a110a145a152a109a189a194a134a53a159
δ(vectorr?vectorrprime) = δ(x?xprime)δ(y?yprime)δ(z?zprime)
δ(vectorr?vectorrprime)≥0 δ(vectorr?vectorrprime) = δ(vectorrprime?vectorr)
f(vectorr)δ(vectorr?vectorrprime) = f(vectorrprime)δ(vectorr?vectorrprime) δ(a(vectorr?vectorrprime)) = 1|a|δ(vectorr?vectorrprime)
a41a219a76a136,δ(vectorr?vectorrprime) =?14pi?2 1|vectorr?vectorrprime|
48/384
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a124a216,Helmholtza189a110
a63a152a171a141a165a27a165a254a124a100a217a209a221a218a94a221a57a51a171a141a254a27a62a46a94a135a17a28a251a189!
H.V.Helmholtz
U(vectorr)≡
integraldisplay
dτprime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
vectorW(vectorr)≡
integraldisplay
dτprime?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
U(vectorr) =?
integraldisplay
dτprime[? 1|vectorr?vectorrprime|]?
prime·vectorF(vectorrprime)
4pi =
integraldisplay
dτprime[?prime 1|vectorr?vectorrprime|]?
prime·vectorF(vectorrprime)
4pi
=
integraldisplay
dτprime{?prime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|?
prime[?prime·vectorF(vectorrprime)]
4pi|vectorr?vectorrprime| }
×vectorW(vectorr) =
integraldisplay
dτprime? 1|vectorr?vectorrprime|×?
prime×vectorF(vectorrprime)
4pi =?
integraldisplay
dτprime?prime 1|vectorr?vectorrprime|×?
prime×vectorF(vectorrprime)
4pi
=
integraldisplay
dτprime{prime×?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime| +
prime×[?prime×vectorF(vectorrprime)]
4pi|vectorr?vectorrprime| }
=
integraldisplay
dτprime{prime×?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime|+
prime[?prime·vectorF(vectorrprime)]prime2vectorF(vectorrprime)
4pi|vectorr?vectorrprime| }
49/384
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a124a216,Helmholtza189a110
a63a152a171a141a165a27a165a254a124a100a217a209a221a218a94a221a57a51a171a141a254a27a62a46a94a135a17a28a251a189!
U(vectorr) =
integraldisplay
dτprime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
vectorW(vectorr) =
integraldisplay
dτprime?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
U(vectorr) =
integraldisplay
dτprime{?prime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|?
prime[?prime·vectorF(vectorrprime)]
4pi|vectorr?vectorrprime| }
×vectorW(vectorr) =
integraldisplay
dτprime{prime×?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime| +
prime[?prime·vectorF(vectorrprime)]prime2vectorF(vectorrprime)
4pi|vectorr?vectorrprime| }
integraldisplay
dτprime?
prime2vectorF(vectorrprime)
4pi|vectorr?vectorrprime| =?
integraldisplay
dτprime[?prime·?
primevectorF(vectorrprime)
4pi|vectorr?vectorrprime|
prime 1
4pi|vectorr?vectorrprime|·?
primevectorF(vectorrprime)]
=?
integraldisplay
dτprime{?prime·?
primevectorF(vectorrprime)
4pi|vectorr?vectorrprime|
prime[vectorF(vectorrprime)·?prime 1
4pi|vectorr?vectorrprime|] +
vectorF(vectorrprime)?prime2 1
4pi|vectorr?vectorrprime|}
=?
integraldisplay
dτprime{?prime·?
primevectorF(vectorrprime)
4pi|vectorr?vectorrprime|
prime[vectorF(vectorrprime)·?prime 1
4pi|vectorr?vectorrprime|]}+
vectorF(vectorr)
50/384
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a124a216,Helmholtza189a110
a63a152a171a141a165a27a165a254a124a100a217a209a221a218a94a221a57a51a171a141a254a27a62a46a94a135a17a28a251a189!
U(vectorr) =
integraldisplay
dτprime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
vectorW(vectorr) =
integraldisplay
dτprime?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
U(vectorr)?
integraldisplay
dτprime?prime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime| =?
integraldisplay
dτprime?
prime[?prime·vectorF(vectorrprime)]
4pi|vectorr?vectorrprime|
×vectorW(vectorr) +
integraldisplay
dτprime?prime×?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime| =
integraldisplay
dτprime?
prime[?prime·vectorF(vectorrprime)]prime2vectorF(vectorrprime)
4pi|vectorr?vectorrprime|integraldisplay
dτprime{?prime·?
primevectorF(vectorrprime)
4pi|vectorr?vectorrprime|
prime[vectorF(vectorrprime)·?prime 1
4pi|vectorr?vectorrprime|]}=
integraldisplay
dτprime?
prime2vectorF(vectorrprime)
4pi|vectorr?vectorrprime|+
vectorF(vectorr)
vectorF(vectorr) =U(vectorr)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
a112a124
+?×vectorW(vectorr)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
a238a124
+
integraldisplay
dτprime{prime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
+?prime×?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime| +?
prime·?
primevectorF(vectorrprime)
4pi|vectorr?vectorrprime|
prime[vectorF(vectorrprime)·?prime 1
4pi|vectorr?vectorrprime|]}
a233a195a46a152a109,a101vectorF(vectorrprime) r
prime→∞
→0,a75a62a46a145a143a34!
51/384
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a124a216,Helmholtza189a110
U(vectorr) =
integraldisplay
dτprime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
vectorW(vectorr) =
integraldisplay
dτprime?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
vectorF(vectorr) =U(vectorr)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
a112a124
+?×vectorW(vectorr)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
a238a124
+
integraldisplay
dτprime{prime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
+?prime×?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime| +?
prime·?
primevectorF(vectorrprime)
4pi|vectorr?vectorrprime|
prime[vectorF(vectorrprime)·?prime 1
4pi|vectorr?vectorrprime|]}
a165a254a124a100a217a209a221a218a94a221a57a62a46a94a135a17a28a251a189!
a209a221a180a23a41a41a165a165a254a124a229a130a117a209a189a224a241a27a47a13a48a48a34a34
a94a221a180a23a41a41a165a165a254a124a229a130a94a61a27a47a13a48a48a34a34
a40a189a165a254a124a73a239a225a163a227a249a252a171a47a13a48a27a212a110a94a135a156
52/384
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a62a94a121a150a27a202a72a53a198
53/384
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a62a94a131a112a138a94a27a13,a62a214,a62a214a151a221
a62a214a180a62a131a112a138a94a131a13,a212a159a27a62a131a112a138a94a114a221a134a217a16a145a27a62a214a164
a20a39,a62a214a140a20a140a75,a143a140a143a34(a61a216a145a62),a62a214a27a229a13? a31a222a41a62
a58a58a62a62a214,a195a78a200,a145a62a254,a180a196a29a62a254a252a160a27a18a234a21
(1.6021917×10?19a165a212)a34a169a225a53a51a247a247a42a42a212a110a165a140a3a209a34
a62a214a151a221,a62a214a169a217a51a152a109a78a200a165.
ρ = lim
τ→0
q
τ = a124a58a63a63a252a252a160a78a200a27a27a62a62a254
σ = lim
S→0
q
S = a124a58a63a63a252a252a160a161a200a27a27a62a62a254
η = lim
l→0
q
l = a124a58a63a63a252a252a160a127a221a221a27a27a27a62a62a254
a58a58a62a62a214a27a27a62a62a214a151a221,a160a117vectorr0a63a62a254a143q a27a58a62a214a51a152a109a27a62
a214a151a221a169a217a143ρ(vectorr,vectorr0)
integraldisplay
τ
dτ ρ(vectorr,vectorr0) = q →ρ(vectorr,vectorr0) =
braceleftbigg 0 vectorrnegationslash=vectorr
0
∞ vectorr =vectorr0 = qδ(vectorr?vectorr0)
54/384
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a62a94a131a112a138a94a27a13,a62a54,a62a54a151a221
a36a196a196a62a62a214a47a164a62a54,a107a144a149a218a62a254a140a2a34
a62a54a151a221a165a254vectorja27a140a2a143a252a160a158a109a54a76a82a134a117a62a54a144a149a252a160a161a200a27
a62a254a167a144a149a143a62a54a144a149a34
a233a63a191a152a135a216a152a189a82a134a117a62a54a144a149a27a161a3?vectorS,a252a160a158a109a207a76a27a62
a254a143j?S⊥ =vectorj·?vectorS,a233a107a129a140a2a161a3,a252a160a158a109a54a76a27a62a254a23a62a54
a114a221 a16a16a16
a16a16a16a45
a45a45 a45a45
a45?S⊥
vectorj J =
integraltext
Sd
vectorS·vectorj
a177a132a221vectorva36a196a196a27a27a27a62a62a214a228a107a62a54a151a221 vectorj = ρvectorv
a62a54a151a221a94a227a120a209a143a62a54a130,a167a144a149a143vectorja144a149,
a151a221a143vectorja140a2a34a51a62a54a130a227a254a247vectorja144a149a138a152a2a62a54a43:
Jdl = j?Sdvectorl =vectorjdτ = ρvectorvdτ a245a171a132a221====
summationdisplay
i
ρivectorvidτ =
summationdisplay
i
qivectorvi
ρ? =?ρ+→ρ = ρ? +ρ+ = 0 vectorj = ρ+vectorv+ +ρ?vectorv? = ρ+(vectorv+?vectorv?)
55/384
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a62a94a131a112a138a94a27a13,a62a54,a62a54a151a221; a94a94a252a252a52
a62a54a169a217a51a76a161a254a94a161a62a54a151a221vectori,a144a149a143a62a54a144a149,a140a2a143 a252a160a158a109a54a76a82a134a117a62
a54a144a149a252a160a127a221a221a27a27a27a62a62a254a34a233a63a152a130a3?vectorl,a252a160a158a109a207a76a27a27a62a62a254vectori·?vectorla34
a233a177a132a221vectorvia36a196a196a27a27a161a62a214 σi,vectori =
summationdisplay
i
σivectorvia161a62a54a151a221a134a78a62a54a151a221a107a39,vectorj = vectoriδ,
Jdl = i?ldvectorl =vectoridS =
summationdisplay
i
σivectorvidS =
summationdisplay
i
qivectorvi
a94a94a252a252a52:
a94a131a112a138a94a23a41a13a27a64a163a123a164a181
a94a99?a94a214(a94a94a252a252a52) Qm; a94a214a151a221ρm; a94a54a151a221vectorjm
a94a214a131a109a132a108a97a113a117a165a213a189a198a27a178a144a144a135a135a39a199
a62a54a143a140a177a23a41a94a138a94
a83a23a74a209a169a102a62a54a103a94a98a96?a94a131a112a138a94a94a100a100a62a54a36a196a196a62a62a214a23a41
a132a107a118a107a94a94a252a252a52?
56/384
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57/384
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a62a94a131a112a138a94a27a124a134a253a152a165a27a196a29a162a8a189a198,a62a214a197a240; a85a92a6a110
a62a214a197a240a189a198:
a62a214a216a172a103a196a23a41a189a7a171,a216a154a45a254a217a167a62a214
contintegraldisplay
τ
dvectorS·vectorj =?ddt
integraldisplay
τ
dτρ
a233a152a135a63a191a78a200τ,a101a51a44a152a227a158a109a83a217a62a254a126a8,a75a126a8a27a62a254a152
a189a180 a108a108a100a100a78a200a27a76a161Sa54a209(a189a54a63a31a254a27a75a62a214)a34
a85a92a6a110:
a62a94a131a112a138a94a229a228a107a140a83a92a53a159
vectorf = summationdisplay
i
vectorfi
58/384
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a62a94a131a112a138a94a27a124a134a253a152a165a27a196a29a162a8a189a198,a165a213a189a198
a165a213a189a198:
a1
a1
a1
a1a1a21
a16a16a16
a16a16a16
a16a16a16a49
a80a80a80
a80a80a80a105
a115
a115
vectorrprime
q1
vectorr primeprime
vectorR q
2
a233a183a142a58a58a62a62a214,a211a210a131a189,a201a210a131a225,a131a112a138a94a229a247a62a214
a235a130a144a149,a140a2a20a39a117a62a214 a27a27a62a62a254,a135a39a117a62a214a131a109a229a108a178a144
vectorf21 = k1q1q2
R3
vectorR k1 = 1
4piepsilon10 epsilon10 = 8.854×10
12a165a2122/a218a238·a1462
vectorf21a180q2a233q1a27a138a94a229,q1a233q2a27a138a94a229a143vectorf12 =?vectorf21,vectorRa180a108q2a141a149q1a27a165a254a34
a178a144a144a135a135a39a1863→3+epsilon1 epsilon1<3×10?16?a144a149?a233a161a53a182a52a165a254?vectorR = 0?
a233a121a162a162a173a173a46a127a51a27a252a172a145a62a78,τ2a233τ1a27a138a94a229vectorF21
vectorF21 =
integraldisplay
dvectorf21 =
integraldisplay
τ1
integraldisplay
τ2
k1ρ1(vectorrprime)dτprimeρ2(vectorrprimeprime)dτprimeprime
R3
vectorR vectorR =vectorrprime?vectorrprimeprime
vectorf21 =
integraldisplay
τ1
dτprime
integraldisplay
τ2
dτprimeprimek1q1δ(vectorr
prime?vectorr1)q2δ(vectorrprimeprime?vectorr2)
|vectorrprime?vectorrprimeprime|3 (vectorr
prime?vectorrprimeprime) = k1q1q2
|vectorr1?vectorr2|3(vectorr1?vectorr2)
59/384
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a165a213a189a198a134a25a107a218a229a189a198a27a39a22
a229 a189a198 a214 a8a65
a218a229 vectorf = Gm1m2R3 vectorR a159a254 a131a112a225a218
a62a229 vectorf = k1q1q2R3 vectorR a62a214 a211a53a131a189a163a189a229a164a164a182a182a201a53a131a225a163a218a229a164
a218a229a166a212a159a207a143a131a112a225a218a224a56a20a152a229a167a212a159a245a27a47a144a172a22a53a22a245a156 a218a238a238a27a27a42a58=====?
a137a187a187a165a165a23a41a40a8a27a212a110a196a58
a211a24a27a145a62a212a159a62a229a69a164a212a159a109a27a131a112a252a189a167a126a102a211a171a212a159a27a118a224a8a65!
a252a159a102a109a27a27a62a62a229a39a217a218a229a1141036a21!
a131a2291a49a99a27a159a102a109a27a27a62a62a229a134a131a2291a102a146a27a159a102a109a27a218a229a131a31!
60/384
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a62a94a131a112a138a94a27a124a134a253a152a165a27a196a29a162a8a189a198,a62a54a3a27a131a112a138a94
a62a54a3a27a131a112a138a94a181
a240a189a189a62a62a54a3J2dvectorl2a233J1dvectorl1a27a138a94a229a143
a80a80a80
a80a80a80a113
a0a0a18
a54J2d
vectorl2 vector
R J1dvectorl1
d2vectorF21 = k2J1dvectorl1×(J2d
vectorl2×vectorR
R3 ) =
k2J1J2
R3 [(
vectorR·dvectorl1)dvectorl2?(dvectorl1·dvectorl2)vectorR]
a62a54a3J1dvectorl1a233 J2dvectorl2a27a138a94a229a143
d2vectorF12 = k2J1J2R3 [?(vectorR·dvectorl2)dvectorl1 + (dvectorl1·dvectorl2)vectorR] a52a165a254
a217a165,a83a23a129a64a27a243a138a255a254a27a216a180a62a54a3a167a13a180a62a54a23a131a109a27a131a112a138a94a229a156
vectorR =vectorr1?vectorr2 k2 = μ0
4pi μ0 = 4pi×10
7a250a54·a146/a165a2122
a252a62a54a3a131a109a27a138a94a229a134a135a138a94a229a216a152a152a24a24,vectorF1 + vectorF2negationslash= 0
(a126:a252a36a196a196a58a58a58a62a62a214vectorv2bardblvectorR⊥vectorv1)→ dvectorP1dt + dvectorP2dt negationslash= 0a189 vectorP1 + vectorP2negationslash= 0,
a196a254a197a240a216a50a164a225a34
61/384
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a62a94a131a112a138a94a27a124a134a253a152a165a27a196a29a162a8a189a198,a124
a124a181
a252a135a62a54a3a27a131a112a138a94a196a254a197a240a216a164a225a27a121a150a233a171a252a135a62a54a3a8a164
a27 a78a88a191a191a216a216a180a152a135a181a52a78a88,a132a107a217a167a16a145a196a254a27a212a159a162a51a34
a124a27a86a103a129a64a180a123a46a49a74a209a27a181a124a180a219a141a27a182a145a251a117a18a135a152a109a34a13
a23a41a124a167 a124a135a138a94a94a117a117a13a34a124a180a68a52a13a131a109a131a112a138a94a27a49a78a34
vectorF21 =
integraldisplay
τ1
dτprime [ ρ1(vectorrprime)
integraldisplay
τ2
dτprimeprimek1ρ2(vectorr
primeprime)
R3
vectorR + vectorj1(vectorrprime)×
integraldisplay
τ2
dτprimeprimek2
vectorj2(vectorrprimeprime)×vectorR
R3 ]
=
integraldisplay
τ1
dτprime [ ρ1(vectorrprime)vectorE1(vectorrprime) + vectorj1(vectorrprime)×vectorB1(vectorrprime) ] vectorR =vectorrprime?vectorrprimeprime
vectorE1(vectorrprime) = k1
integraldisplay
dτprimeprimeρ2(vectorr
primeprime)
R3
vectorR vectorB1(vectorrprime) = k2
integraldisplay
dτprimeprime
vectorj2(vectorrprimeprime)
R3
vectorR
a165a212a189a198a218a83a23a39a105a189a198a165a13a201a229a27a250a170a169a41a143 a13a23a41a124a250a170a218a13a13a51a51a124a165a201a229a250a170a27a40a220a156
a62a94a229a131a109a27a27a83a83a92a6a110a61a61a122a143a124a27a27a83a83a92a6a110,?a124a144a167a180a130a53a27 !
a52a165a254 a171a169a52a33a182a165a254a27a126a102 a182a165a254 Biot-Savarta189a198!
62/384
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a62a94a131a112a138a94a27a124a134a253a152a165a27a196a29a162a8a189a198,a124
a39a117a226a226a213a213a91a229a181
vectorF =
integraldisplay
dτ[ρ(vectorr)vectorE(vectorr) +vectorj(vectorr)×vectorB(vectorr)] =
integraldisplay
dτρ(vectorr)vectorE(vectorr) +
integraldisplay
Jdvectorl×vectorB(vectorr)
a233a177a132a221vectorva36a196,a62a254a143qa27a27a58a58a58a62a62a214,Jdvectorl = qvectorv,ρdτ = q,
vectorf = q(vectorE +vectorv×vectorB) a226a226a212a212a91a250a170
a62a94a124a140a169a143a201a229a13a23a41a27a27a62a62a94a124vectorEs,vectorBsa218a9a62a94a124vectorEe,vectorBe
vectorE = vectorEs + vectorEe vectorB = vectorBs + vectorBe
a103a62a94a124a6a29a29a216a216a209a121a51a250a170a165a167a2a127a196a13a144a85a97a201a111a124a167a195a123a171a169a124a27a209a28a156
a13a164a201a27a103a103a138a138a94a229a27a63a110a191a191a216a216a152a189a103a84! a58a13a33a67a122a62a94a124....
vectorFs≡
integraldisplay
dτ[ρ(vectorr)vectorEs(vectorr) +vectorj(vectorr)×vectorBs(vectorr)] =
integraldisplay
τ
integraldisplay
τ
dτprimedτ”[k1ρ(vectorrprime)ρ(vectorr”) +k2vectorj(vectorrprime)×vectorj(vectorr”)×] (
vectorrprime?vectorr”)
|vectorrprime?vectorr”|3= 0
a252a252a226a226a102a2a134 a52a220a163a180
63/384
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a62a94a131a112a138a94a27a124a134a253a152a165a27a196a29a162a8a189a198,a123a46a49a49a62a62a94a97a65a189a198
a62a124a100a62a214a23a41,a94a124a100a62a54a61a36a196a27a62a214a23a41,a207a13a62a124a134a94a124a27
a39a88a152a189a134a36a196a61a158a109a27a67a122a107a39.
a152a138a52a220a19a130a164a157a140a27a94a207a27a85a67,a242a51a100a52a220a19a130a254a23a41a97a65a62
a196a179,a217a234a138 a31a117a252a160a158a109a83a94a207a27a67a122a199,a100a100a23a41a27a97a65a62a196a179
a164a251a189a189a27a27a97a65a62a54a144a149 a123a142a19a130a83a94a207a27a67a122.
ε =?dΦdt ε =
contintegraldisplay
S
dvectorl·vectorE Φ =
integraldisplay
S
dvectorS·vectorB
vectorSa144a149a134vectorla144a149a164a109a195a218a94a39a88a34 a62a94a97a65a28a28a91a91
64/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a200a169a47a170
a62a214a197a240:
contintegraldisplay
τ
dvectorS·vectorj =?
integraldisplay
τ
dτ?ρ?t a165a212a189a198:
vectorE = k1
integraldisplay
dτprimeρ(vectorr
prime)
R3
vectorR =?k1
integraldisplay
dτprimeρ(vectorrprime)?1R =bracketleftbig
integraldisplay
dτprimeρ(vectorr
prime)
R
bracketrightbig
φ(vectorr) = k1
integraldisplay
dτprimeρ(vectorr
prime)
R +a126a234
vectorE(vectorr) =φ(vectorr)
contintegraldisplay
S
dvectorl·vectorE =
integraldisplay
S
dvectorS·(?×vectorE) =?
integraldisplay
S
dvectorS·[?×(?φ)]= 0
contintegraldisplay
τ
dvectorS·vectorE =
integraldisplay
τ
dτ?·vectorE =?
integraldisplay
τ
dτ?·(?φ) =?
integraldisplay
τ
dτ?2
integraldisplay
dτprimek1ρ(vectorr
prime)
R
=?
integraldisplay
τ
dτ
integraldisplay
dτprimek1ρ(vectorrprime)?2 1R =?
integraldisplay
τ
dτ
integraldisplay
dτprimek1ρ(vectorrprime)(?4pi)δ(vectorr?vectorrprime)
= 1epsilon1
0
integraldisplay
τ
dτρ(vectorr)= 1epsilon1
0
Qa83
65/384
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a233a112a100a189a110a27a95a149a110a41contintegraldisplay
τ
dvectorS·vectorE = 1epsilon1
0
Qa83
a98a189a152a109a165a27a164a107a62a214a209a169a217a117a139a73a6a58a78a67a27a152a135a2a171a141a165,
a51a177a139a73a6a58a143a11a37a140a187a143ra27a165a161a254
a101a62a214a154a165a233a161a169a217,a211a165a161a254a216a211a58a58a27a27a207a254a140a85a112a216a131a211
E = a207a254/?S
a178a254a96a53:
E =a152a135a27a189a165a161a254a252a160a161a200a254a27a178a254a207a254= a111a207a2544pir2
a216a211a140a187a27a165a161a254a27a111a207a254= Q/epsilon10a209a131a211,a134a140a187ra195a39
arrowtripleright a178a144a144a135a135a39a199,E = Q/(4piepsilon10r2)
66/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a200a169a47a170
a39a105a189a198,vectorB =
integraldisplay k
2J(vectorrprime)dvectorlprime×vectorR
R3 =?
integraldisplay
k2J(vectorrprime)dvectorlprime×?1R
=?×
integraldisplay k
2J(vectorrprime)dvectorlprime
R
vectorA(vectorr) = k2
integraldisplay
dvectorlJ(vectorr
prime)
R +?χ =
integraldisplay
dτprime
vectorj(vectorrprime)
R +?χ
vectorB(vectorr) =?×vectorA(vectorr)
contintegraldisplay
τ
dvectorS·vectorB =
integraldisplay
τ
dτ?·vectorB =
integraldisplay
τ
dτ?·(?×vectorA)= 0
67/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a200a169a47a170contintegraldisplay
S
dvectorl·vectorB =
integraldisplay
S
dvectorS·(?×vectorB) =?
integraldisplay
S
dvectorS·?×[
integraldisplay
dvectorlprime×k2J(vectorrprime)?1R]
=?
integraldisplay
S
dvectorS·
integraldisplay
∞
dτprimek2?×[vectorj(vectorrprime)×?1R]
= k2
integraldisplay
S
dvectorS·
integraldisplay
∞
dτprime[vectorj(vectorrprime)·vectorj(vectorrprime)?2]1R
integraldisplay
∞
dτprimevectorj(vectorrprime)·1R =?
integraldisplay
∞
dτprimevectorj(vectorrprime)·?prime?1R
=
integraldisplay
∞
dτprimebracketleftbig?prime·[?vectorj(vectorrprime)?1R] + [?prime·vectorj(vectorrprime)]?1Rbracketrightbig
=?
integraldisplay
∞
dvectorSprime·vectorj(vectorrprime)?1R +
integraldisplay
∞
dτprime[?prime·vectorj(vectorrprime)]?1R
= 0
0 =
contintegraldisplay
τ
dvectorS·vectorj =
integraldisplay
τ
dτ?·vectorj→?·vectorj = 0
68/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a200a169a47a170
contintegraldisplay
S
dvectorl·vectorB = k2
integraldisplay
S
dvectorS·
integraldisplay
dτprime[vectorj(vectorrprime)·vectorj(vectorrprime)?2]1R
=?k2
integraldisplay
S
dvectorS·
integraldisplay
dτprimevectorj(vectorrprime)?2 1R
= 4pik2
integraldisplay
S
dvectorS·
integraldisplay
dτprimevectorj(vectorrprime)δ(vectorr?vectorrprime)
= 4pik2
integraldisplay
S
dvectorS·vectorj(vectorr)= μ0J
a123a46a49a49a62a62a94a97a65a189a198:
contintegraldisplay
S
dvectorl·vectorE =?
integraldisplay
S
dvectorS·?
vectorB
t
a165a212a189a198
contintegraldisplay
S
dvectorl·vectorE = 0a27a237a50!
69/384
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a233a83a23a130a180a189a110a27a95a149a110a41contintegraldisplay
S
dvectorl·vectorB = μ0J
a98a189a152a109a165a27a164a107a62a54a209a247a169a217a117za182a78a67a27a152a135a2a171a141a165,
a51a177a182a143a11a37a140a187a143ra27a9a130a180a254
a101a62a54a154a206a233a161a169a217,a211a130a180a254a216a211a58a27a94a97a65a114a221a140a85a112
a216a131a211
a178a254a96a53:
B =a152a135a27a189a130a180a254a252a160a127a221a254a27a178a254a130a254= a130a2542pir
a216a211a140a187a27a130a180a254a27a130a254= μ0Ja209a131a211,a134a140a187ra195a39
arrowtripleright a134a130a62a54a94a124a27a229a108a135a39a199,B = μ0J/(2pir)
70/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a135a169a47a170
contintegraldisplay
τ
dvectorS·vectorj =?
integraldisplay
τ
dτ?ρ?t →
integraldisplay
τ
dτ(?·vectorj +?ρ?t) = 0 →?ρ?t +?·vectorj = 0
contintegraldisplay
τ
dvectorS·vectorE = 1epsilon1
0
integraldisplay
τ
dτρ →
integraldisplay
τ
dτ(?·vectorE? 1epsilon1
0
ρ) = 0 →?·vectorE = 1epsilon1
0
ρ
contintegraldisplay
S
dvectorl·vectorB = μ0
integraldisplay
S
dvectorS·vectorj→
integraldisplay
S
dvectorS·(?×vectorB?μ0vectorj) = 0→?×vectorB = μ0vectorj
contintegraldisplay
τ
dvectorS·vectorB = 0 →
integraldisplay
τ
dτ?·vectorB = 0 →?·vectorB = 0
contintegraldisplay
τ
dvectorl·vectorE =?
integraldisplay
τ
dvectorS·?
vectorB
t →
integraldisplay
τ
dvectorS·(?×vectorE +?
vectorB
t ) = 0
→?×vectorE =
vectorB
t
vectorF =
integraldisplay
τ
dτ[ρ(vectorr)vectorE(vectorr) +vectorj×vectorB(vectorr)] →
integraldisplay
τ
dτ(vectorf?ρvectorE?vectorj×vectorB) = 0
→ vectorf = ρvectorE +vectorj×vectorB
71/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a135a169a47a170
ρ
t +?·
vectorj = 0?·vectorE = 1
epsilon10ρ?×
vectorB = μ0vectorj
ρ
t = epsilon10
t?·
vectorE = epsilon10?·?vectorE
t
·vectorj = 1μ
0
·(?×vectorB) = 1μ
0
(?×?)·vectorB = 0
×vectorB →?×vectorB = μ0vectorj +μ0epsilon10?
vectorE
t ←a160a163a62a54a145(a62a41a94)
72/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a135a169a47a170
·vectorE = 1epsilon1
0
ρ
3summationdisplay
i=1
iEi = 1epsilon1
0
ρ a62a214a180a62a124a117a209a224a241a27a13
×vectorE =
vectorB
t
3summationdisplay
i,j=1
epsilon1ijk?iEj =Bk?t a94a124a27a158a109a67a122a199a180a62a124a94a61a27a13
·vectorB = 0
3summationdisplay
i=1
iBi = 0 a94a124a216a117a209a218a224a241
×vectorB = μ0vectorj +μ0epsilon10?
vectorE
t
3summationdisplay
i,j=1
epsilon1ijk?iBj = μ0jk +μ0epsilon10?Ek?t
a62a54a151a221a218a62a124a27a158a109a67a122a199a180a94a124a94a61a27a13
vectorf = ρvectorE +vectorj×vectorB fk = ρEk +
3summationdisplay
i,j=1
epsilon1ijkjiBj
73/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a135a169a47a170
a108 Helmholtz a189a110a119a240a142a100a137a144a167a124:
vectorF(vectorr) =U(vectorr)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
a112a124
+?×vectorW(vectorr)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
a238a124
+
integraldisplay
dτprime{prime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
+?prime×?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime| +?
prime·?
primevectorF(vectorrprime)
4pi|vectorr?vectorrprime|
prime[vectorF(vectorrprime)·?prime 1
4pi|vectorr?vectorrprime|]}
U(vectorr) =
integraldisplay
dτprime?
prime·vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
vectorW(vectorr) =
integraldisplay
dτprime?
prime×vectorF(vectorrprime)
4pi|vectorr?vectorrprime|
a195a46a152a109a181 vectorF(vectorr) = 14pi
integraldisplay
dτprime?
prime·vectorF(vectorrprime)
|vectorr?vectorrprime|3 (vectorr?vectorr
prime) + 1
4pi
integraldisplay
dτprime[?
prime×vectorF(vectorrprime)]
|vectorr?vectorrprime|3 ×(vectorr?vectorr
prime)
vectorE(vectorr) = 1
4pi
integraldisplay
dτprime
1
epsilon10ρ(vectorr
prime)
|vectorr?vectorrprime|3(vectorr?vectorr
prime) vectorB(vectorr) = 1
4pi
integraldisplay
dτprimeμ0
vectorj(vectorrprime)
|vectorr?vectorrprime|3×(vectorr?vectorr
prime)
·vectorE = 1epsilon1
0
ρ?×vectorE =
vectorB
t?·
vectorB = 0?×vectorB = μ0vectorj +μ0epsilon10?vectorE
t
74/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a240a142a100a137a144a167a124a27a135a169a47a170
a92a92a94a94a252a252a52a151a221ρma218a94a94a252a252a52a54a151a221vectorjma27a0a122:
·vectorE = 1epsilon1
0
ρ
×vectorE =?vectorjm
vectorB
t
·vectorB = ρm
×vectorB = μ0vectorj +μ0epsilon10?
vectorE
t
a62a94a233a243a181
ρ→
radicalbiggepsilon1
0
μ0ρm
vectorj→
radicalbiggepsilon1
0
μ0
vectorjm vectorE→ 1√
epsilon10μ0
vectorB
ρm→?
radicalbiggμ
0
epsilon10ρ
vectorjm→?
radicalbiggμ
0
epsilon10
vectorj vectorB→?√epsilon10μ0vectorE
75/384
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a253a152a165a62a94a131a112a138a94a27a124a144a167,a62a94a233a243
·vectorE = 1epsilon1
0
ρ?×vectorE =?vectorjm
vectorB
t
·vectorB = ρm?×vectorB = μ0vectorj +μ0epsilon10?
vectorE
t
·(vectorE + i√epsilon1
0μ0
vectorB) = 1√
epsilon10(
1√
epsilon10ρ+
i√
μ0ρm)
×(vectorE + i√epsilon1
0μ0
vectorB) =?vectorjm + i
radicalbiggμ
0
epsilon10
vectorj + i√μ0epsilon10?
t(
vectorE + i√
epsilon10μ0
vectorB)
a62a94a233a243a181 θa134a158a152a139a73a195a39
vectorEprime + i√
epsilon10μ0
vectorBprime = eiθ(vectorE + i√
epsilon10μ0
vectorB)
1√
epsilon10ρ
prime + i√
μ0ρ
prime
m = e
iθ( 1√
epsilon10ρ+
i√
μ0ρm)
vectorjprimem + i
radicalbiggμ
0
epsilon10
vectorjprime = eiθ(?vectorjm + i
radicalbiggμ
0
epsilon10
vectorj)
76/384
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a212a159a27a27a62a62a94a53a159
a121a162a173a46a191a216a180a253a152,a180a100a212a159a8a164a27a34a247a42a137a140a27a212a159a209a100a6
a102a124a164,a6a102a180a152a135a177a145a20a62a27a6a102a216a143a165a37,a177a140a55a88a145a75a62a27a62
a102a31a94a61a27a88a218a34a229a124a27a138a94a27a180a62a94a131a112a138a94,a61a6a102a134a62a102,a62
a102a134a62a102a131a109a27a62a94a131a112a138a94a229,a167a251a189a10a18a135a6a102a27a49a143a34a122a135
a6a102a209a180a152a135a2a27a62a94a88a218,a175a245a27a2a62a94a88a218a92a200a229a53 a8a164a140a27
a62a94a88a218a34a23a41a136a171a136a24a69a44a27a69a220a62a94a8a65,a47a164a195a119a245a231a27a121
a162a162a173a173a46a34
a6a75a254a122a135a6a102a27a62a94a131a112a138a94a49a143a79a142a152a217a10,a247a42a212a159a27a62
a94a49a143a210a127a23a10a34a207a143a212a78a165a6a102a27a234a56a150a8a1801023a254a63,1023a254a63
a27 a83a92a51a56a99a27a79a142a85a229a89a178a254a180a195a123a162a121a27,a191a133a122a135a6a102a88a218
a27a141a91a49a143a143a180a195a123a127a23a27a34a207a100,a212a159a27a62a94a49a143 a216a140a85a207a76a114
a217a165a27a122a135a6a102a27a62a94a49a143a99a91a239a196a152a217a50a63a216,a13a144a85a56a209a228a107
a1a53a27 a65a58,a191a124a94a249a10a65a58a63a216a34a252a135a6a102a27a176a91a49a143a51a247a42a254
a207a126a180a118 a107a8a74a27,a144a107a64a10a140a220a169a6a102a209a228a107a27a53a159a226a172a107a247a42
a8a65a34
77/384
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a212a159a27a27a62a62a94a53a159
a108a62a214a14a221a119a181 a106+
vectorEnegationslash=0
→ a106+
a45a45
a45
vectorE
a233a154a52a53a27a6a102a181a122a135a6a102a88a218a209a180a152a135a165a37a143a20a62a214(a6a102a162)a177a140a143a75a62a214a27
a88a218a34a195a9a46a75a143a158,a75a62a214 a27a169a217a180a233a161a27,a18a135a6a102a180a62a165a53,a212a78a18a78a143a210a180
a62a165a53a27a34a101a127a51a9a62a124,a108a229a198a254a119a167a20a62a214a165a37a134a107a8a75a62a214a165a37a117a41a160a108,
a47a164a152a62a243a52a102vectorp = qvectorl,a212a159a159a165a165a27a2a62a243a52a102a247a9a62a124a252a15,a52a122a47a164a247a247a42a42a62a124a34
a233a52a53a27a6a102a181 a122a135a6a102a88a218a29a28a210a180a67a113a27a62a243a52a102a34 a108a218a79a254a119a167a62a243
a52a102a51a195a9a124a158a100a57a222a225a145a197a252a15a167a9a62a124a27a209a121a166a243a52a102a107a247a9a124a144a149a252a15a27
a170a179W =?vectorp·vectorE,a166a26a209a207a252a15a27a27a62a62a243a52a102a109a169a211a149a252a15a34
a189a194a52a122a114a221:
vectorP = lim
τ→0
summationdisplay
τ
vectorp
τ = npvectorp np,a252a160a78a200a27a27a62a62a243a52a229
vectorP = vectorP(vectorE,vectorB) a212a159a27a27a62a62a94a53a159a144a167
78/384
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a212a159a27a27a62a62a94a53a159
a108a62a54a14a221a119:
a122a135a6a102a88a218a165a55a6a102a216a94a61a27a62a102a47a164a152a2a62a54a23,a8a164a152a2a94a243a52a102a34a195a9
a46a75a143a158,a2a94a243a52a102a144a149a44a207,a233a9a46a27a75a143a131a112a45a158,a216a23a41a247a42a8a65a34a8a107a9
a124a158,a108a229a198a14a221a119a167a2a94a243a52a102a201a229a221 vectorL = vectorm× vectorB = dvectorJdt,vectorJ bardbl vectorm a47a164Larmora63
a196a250a218a63a196a167a13a23a41a31a8a27a135a94a124a144a149a61a196a31a100a135a94a124a94a83a252a15a27a94a221a163a95a94a164,a94
a122a47a164a247a42a94a124a34 a108a218a79a14a221a119a167a94a243a52a102a51a9a94a124a107a247a9a124a144a149a252a15a27a170a179
W=?vectorm·vectorB,a166a26a100a57a222a225a209a207a252a15a27a94a243a52a102a247a94a124a211a149a252a15a163a94a94a164a207a126a140a117a95a94a34
a189a194a94a122a114a221:
vectorM = lim
τ→0
summationdisplay
τ
vectorm
τ = nmvectorm nm,a252a160a78a200a27a94a243a52a229
vectorM = vectorM(vectorE,vectorB) a212a159a27a27a62a62a94a53a159a144a167
79/384
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a52a122a62a214a134a94a122a62a54
a52a122a62a214a134a94a122a62a54(a154a19a78a156a185):
a62a94a138a94a144a163a79a62a214,a62a54,a233a62a165a53a27a212a78a180a223a178a27a34a62a94a124a233
a212a159a27a138a94a162a83a254 a180a62a94a124a233a212a159a165a16a145a51a6a102a216,a62a102a254a27a62a214
a57a217a36a196a47a164a27a62a54a27a138a94a34a249a10 a62a214a218a62a54a78a88a51a6a102a254,a233a27
a21a212a159a6a102a113a26a27a189a51a172a130a254,a207a100a167a130a180a26a229a80 a52a27a34a23a41a52a122
a8a65a27a62a214a161a143a52a122a62a214,a217a62a214a151a221a80a143ρprime,a23a41a94a122a8a65a27a62
a54a161a143a94a122a62a54,a217a62a54a151a221a80a143vectorjprimea34a233a65a27a103a100a62a214a27a62a214a151a221a80
a143ρf,a68a19a19a62a62a54a27a27a62a62a54a151a221a80a143vectorjca34
a100a100a117a117ρprimea180a100a100a117a117a52a122a69a164a27,a175a75a122a143a152a230a100a52a122a114a221vectorPa163a227a27 a2a243
a52a102a23a41a27ρprimea180a245a8?
a100a117vectorjprimea180a100a117a94a122a69a164a27,a175a75a122a143a152a230a100 a52a122a114a221vectorPa163a227a27a2a62
a243a52a102a218a94a122a114a221vectorMa163a227a27a2a94a243a52a102 a23a41a27vectorjprimea180a245a8?
80/384
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a52a122a62a214a134a94a122a62a54
φ(vectorr) =
integraldisplay
dτprimek1ρ
prime(vectorrprime)
R +
integraldisplay
dSprimek1σ
prime(vectorrprime)
R +a126a234
vectorR =vectorr?vectorrprime
φ(vectorr) =
integraldisplay k
1vectorR·vectorP(vectorrprime)dτprime
R3 +a126a234 =
integraldisplay
dτprimek1(?prime1R)·vectorP +a126a234
=
integraldisplay
dτprimek1[?prime·(
vectorP
R)?
prime·vectorP
R ] +a126a234
=
integraldisplay
dvectorSprime·k1
vectorP
R +
integraldisplay
dτprimek1(
prime·vectorP)
R +a126a234
=
integraldisplay
dτprimek1(
prime·vectorP)
R +
integraldisplay
dSprimek1vectorn
prime·vectorP
R +a126a234
ρprime(vectorr) =·vectorP(vectorr) σprime(vectorr) = vectorn·vectorP(vectorr)
a254a33a52a122a158a195a78a229a80a62a214; a52a122a51a212a159a76a161a23a41a41a229a229a80a161a62a214a34
81/384
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a52a122a62a214a134a94a122a62a54
vectorjprime =vectorj1 +vectorj2?ρprime
t +?·
vectorj1 = 0 ρprime =·vectorP vectorj1 =?vectorP
t
vectorA(vectorr) =
integraldisplay
dτprimek2
vectorj2(vectorrprime)
R +
integraldisplay
dSprimek2
vectoriprime(vectorrprime)
R +a126a234
vectorR =vectorr?vectorrprime
vectorA(vectorr) =
integraldisplay k
2 vectorM(vectorrprime)dτprime×vectorR
R3 +a126a234 =
integraldisplay
dτprimek2 vectorM×?prime1R +a126a234
=?
integraldisplay
dτprimek2(?prime1R)× vectorM =?k2
integraldisplay
dτprime[?prime×(
vectorM
R)?
1
R(?
prime× vectorM)]
=
integraldisplay
dτprimek2(?
prime× vectorM)
R +
integraldisplay
dSprimek2
vectorMprime×vectornprime
R +a126a234
vectorj2(vectorr) =?× vectorM(vectorr) vectoriprime(vectorr) = vectorM(vectorr)×vectorn vectorjprime(vectorr) =?vectorP(vectorr)
t +?×
vectorM(vectorr)
a254a33a94a122a191a133a52a122a216a145a158a109a67a122a158a195a78a94a122a62a54; a94a122a51a212a159a76a161
a23a41a94a122a161a62a54a34
82/384
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a212a159a159a165a165a27a27a62a62a94a124a144a167
·vectorE = 1epsilon1
0
(ρf +ρprime)?×vectorE =
vectorB
t
·vectorB = 0?×vectorB = μ0(vectorjc +vectorjprime) +μ0epsilon10?
vectorE
t
·vectorE = 1epsilon1
0
(ρf·vectorP) →?·(epsilon10vectorE + vectorP) = ρf
×vectorB = μ0(vectorjc +?× vectorM +?
vectorP
t ) +μ0epsilon10
vectorE
t
→?×( 1μ
0
vectorB? vectorM) =vectorjc +?
t(epsilon10
vectorE + vectorP)
a62a160a163a165a254,vectorD = epsilon10vectorE + vectorP a94a124a114a221a165a254,vectorH = 1μ
0
vectorB? vectorM
·vectorD = ρf?×vectorH =vectorjc +?
vectorD
t
83/384
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a212a159a159a165a165a27a27a62a62a94a53a159a144a167
a62a48a159,a162a8a137a209
vectorP = χeepsilon10vectorE χea181a52a122a199 a195a254a106a181 P a134 epsilon1
0E a211a254a106· vectorE = ρ/epsilon10
vectorD = epsilon10vectorE + vectorP = epsilon10(1 +χe)vectorE = epsilon10epsilon1rvectorE= epsilon1vectorE
epsilon1r = 1 +χe a181a131a233a48a62a126a234 epsilon1 = epsilon10epsilon1r a181a253a233a48a62a126a234
a130a53a48a159,χea134vectorEa195a39a34 a20a75a62a214a165a37a160a108a229a108a20a39a117a62a124vectorEa34
a154a130a53a48a159a181
Di =
3summationdisplay
j=1
εijEj +
3summationdisplay
j,k=1
εijkEjEk +
3summationdisplay
j,k,l=1
εijklEjEkEl +···
a102a124a67a113a156
84/384
triangleleftsldtriangleleftsld
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a212a159a159a165a165a27a27a62a62a94a53a159a144a167
a19a78,a238a238a48a48a189a198
J =?φR R = λlS
R,a62a123 λ,a62a123a199 γ = 1λ a62a62a19a19a199
j = JS =?φRS = Sλl?φS = Eλ = γE
vectorj = γvectorE vectorE,a218a229a62a54a27a27a62a62a124
85/384
triangleleftsldtriangleleftsld
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a48a159a159a165a165a62a94a131a112a138a94a27a124a144a167,a212a159a159a165a165a27a27a62a62a94a53a159a144a167
a94a48a159a181
a94a94,a95a94a48a159:
vectorM = χmvectorH χma94a122a199
vectorB = μ0(vectorH + vectorM) = μ0(1 +χm)vectorH = μ0μrvectorH= μvectorH
μr = 1 +χm a131a233a94a19a199 μ = μ0μr a253a233a94a19a199
μr >1,a94a94a48a159a182 μr <1,a95a94a48a159
a99a94a225a14:
a51Ha2a27a171a141,Ba134Ha180a130a53a39a88,μra67a113a143a126a234,a2a233a140a34
a8Ha140a20a152a189a167a221,μr～1,a136a20a4a218a71a21a34
a77a94a225a14a181
μra134a94a122a123a164a107a39a34 a8Ha108a195a20a107,
a113a108a107a20a195a158,a209a121a144a123a94a221a34
a91a94a225a14a181
a94a221a174a28a220a136a20a4a218,χm = 0,vectorM = vectorM0,vectorB = μ0vectorH +μ0 vectorM0
a206a71a94a99a94a124a27a28a28a91a91 a47a165a94a124a27a28a28a91a91 a102a124a67a113a156
86/384
triangleleftsldtriangleleftsld
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trianglerightsld
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a62a94a124a27a62a138a39a88,a123a149a169a254
·vectorD = ρf?
contintegraldisplay
dvectorS·vectorD =
integraldisplay
dτρf
a152a207a76a46a161a27a161a200a143?Sa27a2a11a206a78τ
vectorna143a108a48a1591a141a149a48a1592a27a46a161a123a149a252a160a165a254?vectorS = vectorn?S
σf =
integraltext dτρ
f
S =
1
S
contintegraldisplay
τ
dvectorS·vectorD = 1?S[vectorD1·(vectorS) + vectorD2·?vectorS]
= vectorD1·(?vectorn) + vectorD2·vectorn =?D1n + D2n
vectorn·(vectorD2?vectorD1) = σf
contintegraldisplay
dvectorS·vectorB = 0 vectorD→vectorB; ρf →0,σf →0
vectorn·(vectorB2?vectorB1) = 0
a46a161a254a161a62a214a69a164vectorDa51a46a161a252a224a107a30a23,a46a161a254vectorBa123a149a169a254a254a235a235a89a34
87/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a62a94a124a27a62a138a39a88,a131a149a169a254
×vectorH =vectorjc +?
vectorD
t?
contintegraldisplay
dvectorl·vectorH =
integraldisplay
dvectorS·(vectorjc +?
vectorD
t )
a152a207a76a46a161a27a163a180,a51a48a1591a2182a252a62a27a247a131a149a144a149a27
a62a127a143?l,vectorna143a108a48a1591a141a149a48a1592a27a46a161a123a130a252a160
a165a254,vectorta143a163a180a157a140a76a161a27a123a149(a46a161a27a131a149)a252a160a165a254
vectort·vectoric?l =vectort·[vectoric?l +?vectorD?t?lδ]vextendsinglevextendsingle
δ→0 =
integraldisplay
dvectorS·(vectorjc +?
vectorD
t ) =
contintegraldisplay
dvectorl·vectorH
=?l[vectorH1·(?vectort×vectorn) + vectorH2·(vectort×vectorn)] =?lvectort·[vectorn×(vectorH2?vectorH1)]
vectorta51a46a161a254a180a63a191;vectoric,vectorn×(vectorH2?vectorH1)a195a82a134a46a161a27a169a254
vectorn×(vectorH2?vectorH1) =vectoriccontintegraldisplay
dvectorl·vectorE =?
integraldisplay
dvectorS·?
vectorB
t
vectorH→vectorE vectorD→?vectorB vectorjc→0 vectoric→0
vectorn×(vectorE2?vectorE1) = 0
a46a161a254vectorEa131a149a169a254a254a235a235a89,a46a161a254a161a62a54 a69a164vectorHa51a46a161a252a224a107a30a23a34
88/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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trianglerightsld
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a85a196a254a27a61a122a134a197a240,a124a218a62a214a88a218a85a196a254a61a122a218a197a240a189a198a27a152a132a47a170
a124a27a85a254a151a221w,a252a160a78a200a27a124a164a145a27a85a254,a180a158a152a139a73a27
a188a234 w = w(vectorr,t)
a124a27a196a254a151a221vectorg,a252a160a78a200a27a124a164a145a27a196a254,a180a158a152a139a73a27a188
a234 vectorg =vectorg(vectorr,t)
a124a27a85a54a151a221vectorS,a163a227a85a254a27a68a194,a140a2a31a117a252a160a158a109a82a134a54
a76a252a160a238 a31a161a27a85a254,a144a149a141a149a85a254a68a194a27a144a149,a180a158a152a139
a73a27a188a234 vectorS = vectorS(vectorr,t)
a124a27a196a254a54a151a221
arrowrighttophalfarrowrighttophalfJ,
a167a163a227a196a254a27a68a194,?vectorS· arrowrighttophalfarrowrighttophalfJ =?vectorpa189
a194a143a252a160a158a109a207a76a161a3?vectorSa54a209a27a196a254,a180a158a152a139a73a27a188a234
arrowrighttophalfarrowrighttophalfJ =arrowrighttophalfarrowrighttophalfJ (vectorr,t)
89/384
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trianglerightsldtrianglerightsld
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trianglerightsld
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a85a196a254a27a61a122a134a197a240,a124a218a62a214a88a218a85a196a254a61a122a218a197a240a189a198a27a152a132a47a170
a85a254a61a122a134a197a240a189a198,a252a160a158a109a207a76a76a161a54a92Va27a85a254a31a117
a252a160a158a109a124a233 Va83a62a214a164a137a27a245(a61a245a199)a134Va83a252a160a158a109
a62a94a124a85a254a27a79a92a131a218a34
contintegraldisplay
V
dvectorσ·vectorS =
integraldisplay
V
dVvectorf·vectorv + ddt
integraldisplay
V
dVw
vectorf,a124a233a233a62a62a214a138a94a229a151a221 vectorv,a62a214a36a196a132a221
·vectorS =vectorf·vectorv +?w?t
a196a254a61a122a134a197a240a189a198,a252a160a158a109a207a76a76a161a54a92Va27a196a254a31
a117V a83a62a214a164a201a27a229a134Va83a252a160a158a109a62a94a124a196a254a27a79a92a131
a218a34
contintegraldisplay
V
dvectorσ· arrowrighttophalfarrowrighttophalfJ =
integraldisplay
V
dVvectorf + ddt
integraldisplay
V
dVvectorg
· arrowrighttophalfarrowrighttophalfJ =vectorf +?vectorg?t
90/384
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trianglerightsld
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a85a196a254a27a61a122a134a197a240,a124a218a62a214a88a218a85a196a254a61a122a218a197a240a189a198a27a152a132a47a170
a212a110a110a254a254 a151a221 a54a151a221 a197a240a189a198 a197a240a240a214a214
a62a214 ρ vectorj?·vectorj +?ρ?t = 0 Q = integraltext dVρ
a85a254 w vectorS?·vectorS +?w?t +vectorf·vectorv = 0 Ea85a254 = integraltext dVw
a196a254 vectorg arrowrighttophalfarrowrighttophalfJ?· arrowrighttophalfarrowrighttophalfJ +?vectorg?t +vectorf = 0 vectorPa196a254 = integraltext dVvectorg
91/384
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a62a94a131a112a138a94a85a254a196a254a27a61a122a134a197a240,a85a254a151a221a134a85a54a151a221
vectorf·vectorv = (ρfvectorE +ρfvectorv×vectorB)·vectorv = ρfvectorv·vectorE =vectorjc·vectorE = vectorE·(?×vectorHvectorD
t )
= vectorE·(?×vectorH)?vectorE·?
vectorD
t =·(
vectorE×vectorH) + vectorH·(?×vectorE)?vectorE·?vectorD
t
=·(vectorE×vectorH)?vectorE·?
vectorD
t?
vectorH·?vectorB
t =·
vectorSw
t
vectorS = vectorE×vectorH?w
t =
vectorE·?vectorD
t +
vectorH·?vectorB
t
w
t = epsilon1
vectorE·?vectorE
t +
1
μ
vectorB·?vectorB
t =
t
1
2(epsilon1E
2 + 1
μB
2) =?
t
1
2(
vectorE·vectorD + vectorH·vectorB)
w = 12(vectorE·vectorD + vectorH·vectorB) a124a228a107a85a254a96a178a124a180a212a159a27a152a171a47a21a156
92/384
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a62a94a131a112a138a94a85a254a196a254a27a61a122a134a197a240,a196a254a151a221a134a196a254a54a151a221
vectorf = ρfvectorE +vectorjc×vectorB = (?·epsilon10vectorE)vectorE + (?× 1
μ0
vectorBepsilon10vectorE
t )×
vectorB
= epsilon10(?·vectorE)vectorE + 1μ
0
(?×vectorB)×vectorB?epsilon10?
vectorE
t ×
vectorB
= epsilon10(?·vectorE)vectorE + 1μ
0
[(?·vectorB)vectorB + (?×vectorB)×vectorB]
epsilon10[?
vectorE
t ×
vectorB + vectorE×?vectorB
t +
vectorE×(?×vectorE)]
= epsilon10[(?·vectorE)vectorE + (?×vectorE)×vectorE] + 1μ
0
[(?·vectorB)vectorB + (?×vectorB)×vectorB]
epsilon10t(vectorE×vectorB)
= epsilon10[(?·vectorE)vectorE + (vectorE·?)vectorE?(?vectorE)·vectorE]
+ 1μ
0
[(?·vectorB)vectorB + (vectorB·?)vectorB?(?vectorB)·vectorB]?epsilon10t(vectorE×vectorB)
93/384
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a62a94a131a112a138a94a85a254a196a254a27a61a122a134a197a240,a196a254a151a221a134a196a254a54a151a221
vectorf = epsilon10[(?·vectorE)vectorE + (vectorE·?)vectorE?(?vectorE)·vectorE]
+ 1μ
0
[(?·vectorB)vectorB + (vectorB·?)vectorB?(?vectorB)·vectorB]?epsilon10t(vectorE×vectorB)
= epsilon10[?·(vectorEvectorE)?12?E2] + 1μ
0
[?·(vectorBvectorB)?12?B2]?epsilon10t(vectorE×vectorB)
= epsilon10?·(vectorEvectorE?12 arrowrighttophalfarrowrighttophalfI E2) + 1μ
0
·(vectorBvectorB?12 arrowrighttophalfarrowrighttophalfI B2)?epsilon10t(vectorE×vectorB)
=· arrowrighttophalfarrowrighttophalfJvectorg?t
vectorg = epsilon10vectorE×vectorB = epsilon10μ0vectorS
arrowrighttophalfarrowrighttophalfJ =?epsilon1
0vectorEvectorE?
1
μ0
vectorBvectorB + 1
2
arrowrighttophalfarrowrighttophalfI (epsilon1
0E2 +
1
μ0B
2)
a252a145a62a78a131a109a27a218a238a238a49a49a110a189a198a152a132a216a164a225a156 vectorf21 = dP1dt vectorf12 = dP2dt
94/384
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a62a94a131a112a138a94a85a254a196a254a27a61a122a134a197a240,a62a94a85a254a68a209
a152a152a152a152a152a227a54a107a62a54a27a27a19a19a130,a165a165a182a182a143za182,a252a224a31a161a169a79a143Aa218B,a253a253a76a76a161C,A,Ba254a62a124a247za182a144a149a134a62a54a144a149a149a131a131a211,a85a54
a151a221a195za144a149a169a254,a85a254a216a85a108Aa190Ba161a54a92a189a54a209a19a78.a233a253a161C,
vectorE = Etvectornz + Envectornr epsilon10En = σ Eta143a19a78a83a27za144a149a62a124
vectorH = J
2piavectornφ vectornφ,a253a253a76a76a161a82a134a117za144a149a27a131a165a254
vectorS = vectorE×vectorH = J
2pia(?Etvectornr + Envectornz) =
J
2pia(?Etvectornr +
σ
epsilon10vectornz)
a247za144a149a27a85a54a195a123a63a209a19a78,a247vectornra144a149a85a54a134Eta134Ja144a149a152a151
(JEt > 0)a111a180a54a92a19a78a27,a85a254a108a19a130a253a161a54a92a19a78a83,a216a180a247a62
a54a144a149a94a19a130a54a92.a252a160a158a109a108a253a161C(a140a187a143a,a127a143l)a54a92a27a85a254
a143:
Sr2pial =?vectorS·vectornr2pial = JEt2pia2pial = JEtl = JVAB = J2RAB
VAB,RAB,a19a130a252a224a224a27a27a27a62a62a179a11,a62a123
95/384
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a62a94a131a112a138a94a85a254a196a254a27a61a122a134a197a240,a62a94a85a254a68a209
a252a160a158a109a108a253a161C(a140a187a143a,a127a143l)a54a92a27a85a254|vectorS|2pial = J2RABa20
a208a31a117a252a160a158a109a19a130a254a164a158a209a27a57a85a34a88a74a164a63a216a27a180a62a179,A,Ba161
a254a211a24a195a85a254a54a209a92,a85a254a144a85a100a253a161a54a209a92,a100a117a51a62a179a83
a220Ea134Ja135a149,vectorSr =? J2piaEtvectornra238a143a20,a61a85a254a240a180a108a62a179a253a161a54a209a34
a152a135a248a62a163a180,a85a254a216a180a247a19a130a83a220a68a52a137a75a49,a13a180a108a62a179a76a161a54
a20a152a109,a50a108a152a109(a204a135a180a19a130a76a161a78a67)a54a92a19a130a218a75a49.
a64a143a85a254a180a108a68a209a19a130a83a62a102a36a196a196a85a247a19a130a68a53a27a42a58a180a134a216
a27,(a196a75,a233a2a54a62,a62a54a145a158a85a67a144a149,a85a254a75a152a158a54a53,a152a158a54a22)
a175a162a254,a101a108vectorj = nevectorva15a79,a18j =1a83/a206a1462=1a165a212/a166·a206a1462,e = 1.6×10?19a165a212,n =
1023/a102a1463 = 1020/a206a1463,a26|vectorv|= j/ne = 1/16a206a146/a166a34
a88a100a2a27a27a62a62a102a178a254a164a163a132a221(a2a167a152a166a168a174a170a76a101020/3/16≈3×105a135a6a102)
a216a140a85a248a248a137a137a137a75a75a49 a254a158a209a27a85a254a34
96/384
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a62a94a131a112a138a94a85a254a196a254a27a61a122a134a197a240,a220a254a254a229a229
contintegraldisplay
V
dvectorσ· arrowrighttophalfarrowrighttophalfJ =
integraldisplay
V
dVvectorf + ddt
integraldisplay
V
dVvectorg
vectorF≡
integraldisplay
dVvectorf = d
vectorPa197a22
dt
vectorPa62a94≡
integraldisplay
dVvectorg → ddt[vectorPa197a22+vectorPa62a94] =
contintegraldisplay
dvectorS·(?arrowrighttophalfarrowrighttophalfJ )
arrowrighttophalfarrowrighttophalfJa180a252a160a76a161a164a201a27a229.a51a63a152a76a161a78a67,a101a107a62a94a124,a75a249a135a76a161
a210a135a201a229.
a152a76a161,a123a149a144a149a252a160a165a254vectorn,a62a124a134vectorna27a89a14θE,a62a124a221a75a164a51a27
a131a149a144a149a252a160a165a254vectoreE,a94a124a134vectorna27a89a14θB,a94a124a221a75a164a51a27a131a149a144a149
a252a160a165a254vectoreB,a100a76a161a252a160a76a161a164a201a27a229a143
vectorfa76a161 =?vectorn· arrowrighttophalfarrowrighttophalfJ = epsilon10(vectorn·vectorEvectorE?1
2vectornE
2) + 1
μ0(vectorn·
vectorBvectorB?1
2vectornB
2)
a64
a64
a64
a64
a64a64a73
a0
a0a18a54
a12
a12a12
vectorn
θE
vectorEvectorfEvectoreE
= (EcosθEvectorn + EsinθEvectoreE)epsilon10EcosθE?12epsilon10vectornE2
+(BcosθBvectorn + BsinθBvectoreB)Bμ
0
cosθB? 12μ
0
B2 =vectorfE +vectorfB
vectorfE = 1
2epsilon10E
2(vectorncos2θE +vectoreE sin2θE) vectorfB = 1
2μ0B
2(vectorncos2θB +vectoreB sin2θB)
97/384
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trianglerightsld
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a183a62a124a218a173a240a62a54a27a27a62a62a94a124
98/384
triangleleftsldtriangleleftsld
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triangleleftsld
trianglerightsld
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a183a62a179a57a217a135a169a169a144a144a167
·vectorD = ρf?×vectorE =
vectorB
t
·vectorB = 0?×vectorH =vectorjc +?
vectorD
t
a183a21a62a94a124a94a135:
vectorE
t = 0
vectorB
t = 0
a183a21a62a94a124a247a118:
braceleftbigg?·vectorD = ρ
f
×vectorE = 0
braceleftbigg?·vectorB = 0
×vectorH =vectorjc
a183a62a124a134a183a94a124a112a216a39a233,a140a177a169a79a63a49a63a216!
99/384
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a183a62a179a57a217a135a169a169a144a144a167,a62a179a57a217a247a118a27a135a169a169a144a144a167
braceleftbigg?·vectorD = ρ
f
×vectorE = 0
braceleftbigg vectorD = epsilon1vectorE
vectorf = ρvectorE a62a94a135:
braceleftbiggvectorn·(vectorD
2?vectorD1) = σf
vectorn×(vectorE2?vectorE1) = 0 vectorn,1→2
contintegraldisplay
dvectorl·vectorE =
integraldisplay
dvectorS·(?×vectorE) = 0 →
integraldisplay 2
1
dvectorl·vectorEa134a180a187a195a39
a189a194a62a179a188a234,φ2?φ1 =?
integraldisplay 2
1
dvectorl·vectorE vectorE =φ
a195a94a124a152a189a140a177a76a143a73a254a124a27a70a221 !
φ|∞ = 0 → φ1 =
integraldisplay ∞
1
dvectorl·vectorE =
integraldisplay ∞
1
dvectorl·vectorfa252a160a62a214
φa27a191a191a194a194,a252a160a20a62a214a108a124a58a163a20a195a161a15 (a189a114a252a160a75a62a214a108a195a161a15
a163a20a124a58)a62a124a229a164a137a27a245a34
a62a124a180a62a179a27a129a140a152a109a67a122a199a27a75a138,a144a149a82a134a117a31a179a161(a62a229a130a134
a31a179a161a82a134,a144a149a100a112a62a179a141a149a36a36a62a62a179)a34
100/384
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a183a62a179a57a217a135a169a169a144a144a167,a62a179a57a217a247a118a27a135a169a169a144a144a167
·vectorD = ρf →?(epsilon1?φ) =?ρf
vectorD2n?vectorD1n = σf → epsilon12?φ2
n?epsilon11
φ1
n =?σf
φ2?φ1 =?
integraldisplay 2
1
dvectorl·E → φ2?φ1 =?1epsilon1vectorn·vectorτf
epsilon1a180a152a109a139a73a27a188a234?φ?n = vectorn·?φ φ(vectorr) =
integraldisplay
dτprime ρ(vectorr
prime)
4piepsilon10|vectorr?vectorrprime|+a126a234
a1081a202a180a187a192a143a20a208a207a76a46a161,a144a149a247a46a161a123a149vectorn,dvectorl =vectorndl
++
++
++
+
1 2σf
vectorn
dl
a45
a46a161a27a243a52a0a151a221,τf = σfdl E = σfepsilon1
0
= τfepsilon1
0dl
vectorE = vectorτf
epsilon10dl
vectorE·dvectorl = vectorτf
epsilon10dl·vectorndl =
1
epsilon10vectorτf·vectorn
101/384
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a183a62a179a57a217a135a169a169a144a144a167,a154a52a138a189a110
a254a33a48a159a159a165a165a195a62a214a169a217a27a171a141a83a27a63a219a152a58a58a27a27a183a62a179
a209a216a140a85a18a52a140a189a52a2a138 a156
a98a23a100a189a110a216a164a225a34
a75a51a254a33a48a159a165a195a62a214a169a217a27a171a141a83a127a51a152a58aa27a183a62a179φaa18a52
a140a163a2a164a34
a207a13a127a51a157a140a217a27a118a10a2a181a52a173a161sa167a51a217a83a195a62a214a167a62a46a254a27
a122a152a58a27a62a179a27a9a123a149a135a251a240a143a75a163a20a164a167a191a133a216a85a28a143a34a167 a177
a2a121a108a62a46a112a149a9a62a179a111a180a126a2a163a79a92a164a34
a249a134sa83a62a214a143a34a131a112a103a241a1810 = Q =
contintegraldisplay
dvectors·vectorD =?epsilon1
contintegraldisplay
ds?φ?n > (<)0
a183a62a179a27a52a138a138a144a144a140a85a160a117a62a46a189a107a62a214a63 a156
102/384
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a83a75
2.14 a120a209a188a234dδ(x)dx a27a227,a96a178ρ =?(vectorp·?)δ(vectorr) a180a152a135a160a117a6a58a27
a243a52a102a27a27a62a62a214a151a221.
dδ(x)
dx = lim?x→0
δ(x +?x)?δ(x)
x a45
a54
x =x
x = 0
integraldisplay
dV ρ =?
integraldisplay
dV pi?iδ(vectorr) = 0
integraldisplay
dVxiρ =?
integraldisplay
dVxipj?jδ(vectorr) =?
integraldisplay
dVbraceleftbig?j[xipjδ(vectorr)]?piδ(vectorr)bracerightbig = pi
integraldisplay
dV xixjρ =?
integraldisplay
dV xixjpk?kδ(vectorr)
=?
integraldisplay
dV braceleftbig?k[xixjpkδ(vectorr)]?(pixj + pjxi)δ(vectorr)bracerightbig = 0
integraldisplay
dV xi1xi2···xinρ = 0 n≥2
103/384
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a83a75
1.5 a121a178a62a243a52a221a27a67a122a199dvectorpdt = integraltext dVvectorj(vectorr,t)
dvectorp
dt =
integraldisplay
dV vectorrtρ(vectorr,t) =?
integraldisplay
dV vectorr?·vectorj(vectorr,t)
=?vectorei
integraldisplay
dV xi?kjk(vectorr,t) =vectorei
integraldisplay
dV braceleftbigk[xijk(vectorr,t)] + ji(vectorr,t)bracerightbig
=
integraldisplay
dVvectorj
=
integraldisplay
dV braceleftbig·(vectorjvectorr) +vectorj·?vectorrbracerightbig
negationslash=
integraldisplay
dV braceleftbig(vectorj·vectorr) + (?vectorr)·vectorjbracerightbig
negationslash=
integraldisplay
dV [vectorrtρ(vectorr,t) +vectorvρ(vectorr,t)]
=·vectorjvectorr=0 vectorj
104/384
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a83a75
1.10 a121a178a252a135a62a54a130a23a27a131a112a138a94a140a2a131a31a167a144a149a149a131a131a135a34
a240a189a189a62a62a54a3J2dvectorl2a233J1dvectorl1a27a138a94a229a143 a80a80a80a80
a80a80a105
a0a0a18
a54J1d
vectorl1 vector
R J2dvectorl2
vectorF21 = k2
contintegraldisplay
2
contintegraldisplay
1
J1dvectorl1×(J2d
vectorl2×vectorR
R3 )
vectorR =vectorr1?vectorr2
=
contintegraldisplay
2
contintegraldisplay
1
J1dvectorl1×J2dvectorl2×
vectorR
R3
=?
contintegraldisplay
2
contintegraldisplay
1
J2dvectorl2×J1dvectorl1×
vectorR
R3
=?k2
contintegraldisplay
2
contintegraldisplay
1
J2dvectorl2×(J1d
vectorl1×vectorR
R3 ) =?
vectorF12
vectorA×(vectorB×vectorC)negationslash=?vectorB×(vectorA×vectorC)
105/384
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a83a75
1.10 a121a178a252a135a62a54a130a23a27a131a112a138a94a140a2a131a31a167a144a149a149a131a131a135a34
a240a189a189a62a62a54a3J2dvectorl2a233J1dvectorl1a27a138a94a229a143 a80a80a80a80
a80a80a105
a0a0a18
a54J1d
vectorl1 vector
R J2dvectorl2
vectorF21 = k2
contintegraldisplay
2
contintegraldisplay
1
J1dvectorl1×(J2d
vectorl2×vectorR
R3 )
vectorR =vectorr1?vectorr2
=
contintegraldisplay
2
contintegraldisplay
1
k2J1J2
R3 [(
vectorR·dvectorl1)dvectorl2?(dvectorl1·dvectorl2)vectorR]
a62a54a3J1dvectorl1a233 J2dvectorl2a27a138a94a229a143:1?2; vectorRvectorR
contintegraldisplay
2
contintegraldisplay
1
k2J1J2
R3 (
vectorR·dvectorl1)dvectorl2 =
contintegraldisplay
2prime
contintegraldisplay
1prime
k2J1J2
R3 (
vectorR·dvectorl1prime)dvectorl2prime
1prime=2,2prime=1======
contintegraldisplay
1
contintegraldisplay
2
k2J1J2
R3 (
vectorR·dvectorl2)dvectorl1 =?
contintegraldisplay
1
contintegraldisplay
2
k2J1J2
R3 (?
vectorR·dvectorl2)dvectorl1
106/384
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a83a751.10 a121a178a252a135a62a54a130a23a27a131a112a138a94a140a2a131a31a167a144a149a149a131a131a135a34
a240a189a189a62a62a54a3J2dvectorl2a233J1dvectorl1a27a138a94a229a143 a80a80a80a80
a80a80a105
a0a0a18
a54J1d
vectorl1 vector
R J2dvectorl2
vectorF21 = k2
contintegraldisplay
2
contintegraldisplay
1
J1dvectorl1×(J2d
vectorl2×vectorR
R3 )
vectorR =vectorr1?vectorr2
=
contintegraldisplay
2
contintegraldisplay
1
k2J1J2
R3 [(
vectorR·dvectorl1)dvectorl2?(dvectorl1·dvectorl2)vectorR]
a62a54a3J1dvectorl1a233 J2dvectorl2a27a138a94a229a143:1?2; vectorRvectorR
contintegraldisplay
2
contintegraldisplay
1
k2J1J2
R3 (
vectorR·dvectorl1)dvectorl2 =k2
contintegraldisplay
1
J2dvectorl2
contintegraldisplay
1
J1dvectorl1·?1 1R
=?k2
integraldisplay
dV2 vectorj2
integraldisplay
dV1(?1 1R)·vectorj1
=?k2
integraldisplay
dV2 vectorj2
integraldisplay
dV1braceleftbig[?1·
vectorj1
R]?
1
R?1·
vectorj1bracerightbig = 0
a52a220a163a180
107/384
triangleleftsldtriangleleftsld
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a83a75
a49a152a217a49a110a33a214a191a138a146(b),a166a121a94a243a52a102(a62a54a23a161a200S → 0,a62
a54J→∞,JSa27a189)a23a41a27a165a254a179a143,
vectorA = μ0
4pi
vectorm×vectorr
r3 +a126a234
vectorS = 1
2
contintegraldisplay
vectora×dvectorl vectorm = JvectorS
(1)vector
A = k2
contintegraldisplay
dvectorl J|vectorr?vectora| = k2
integraldisplay
dvectorS×?a J|vectorr?vectora| = k2
integraldisplay
dvectorS×Jvectorrr3
vectorA = k2
contintegraldisplay
dvectorl J|vectorr?vectora| = k2
contintegraldisplay
dvectorl J√r2?2vectorr·vectora = k2
contintegraldisplay
dvectorl Jrradicalbig1?2vectorr·vectora/r
= k2
contintegraldisplay
dvectorlJr[1 +vectorr·vectorar ] = k2Jr2
contintegraldisplay
dvectorlvectorr·vectora = k2J2r2
contintegraldisplay
[dvectorlvectorr·vectora?vectoradvectorl·vectorr]
= k2J2r2
contintegraldisplay
(vectora×dvectorl)×vectorr = k2Jr2 vectorS×vectorr = μ04pi vectorm×vectorrr3
contintegraldisplay
J[dvectorlvectorr·vectora +vectora dvectorl·vectorr] =
integraldisplay
dτa[vectorj(vectora) vectorr·vectora +vectoravectorj(vectora)·vectorr] =
integraldisplay
dτa?a·[vectorj(vectora)vectoravectorr·vectora]
= 0
108/384
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a141a152a53a189a110a57a65a65a94a94,a183a62a175a75a141a152a53a189a110
a152a135a171a141τ,a62a46S0,S1,S2,···,S0a180a129a9a157a140a161,S1,S2,···a180a83
a220a152a201a157a140a161a34
a38a37
a39a36
a105
a102
S0
S2S1τa83a48a159a180a169a171a235a89a27(epsilon1
i(vectorr)a180a235a89a188a234,a133epsilon1i >0),
τa83ρf,σfa169a217a174a127,a133a101a15a94a135a131a152a164a225:
(1),φvextendsinglevextendsingleS
i
a174a127
(2),?φ?nvextendsinglevextendsingleS
i
=?EnvextendsinglevextendsingleS
i
a174a127
a101S0a51a195a161a15,a75a230a94a195a161a15a103a44a62a94a135a181
φvextendsinglevextendsingler→∞→1r a189 Evextendsinglevextendsingler→∞→ 1r2
a75,a100a183a62a124a135a169a169a144a144a167a57a62a94a135a251a189a189a27a27τa165a27a27a62a62a124a141a152a34
109/384
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a141a152a53a189a110a57a65a65a94a94,a183a62a175a75a141a152a53a189a110
a38a37
a39a36S
02S01
21
Sprime?·[epsilon11?φ
prime
1] =?ρf1?·[epsilon12?φ
prime
2] =?ρf2
·[epsilon11?φprimeprime1] =?ρf1?·[epsilon12?φprimeprime2] =?ρf2
φprime1vextendsinglevextendsingleSprime = φprime2vextendsinglevextendsingleSprime (epsilon11?φ
prime
1
n?epsilon12
φprime2
n )
vextendsinglevextendsingle
Sprime = σf
φprimeprime1vextendsinglevextendsingleSprime = φprimeprime2vextendsinglevextendsingleSprime (epsilon11?φ
primeprime
1
n?epsilon12
φprimeprime2
n )
vextendsinglevextendsingle
Sprime = σf
φprime1vextendsinglevextendsingleS
01
= φprimeprime1vextendsinglevextendsingleS
01
= a174a127 a189nφprime1vextendsinglevextendsingleS
01
=nφprimeprime1vextendsinglevextendsingleS
01
= a174a127
φprime2vextendsinglevextendsingleS
02
= φprimeprime2vextendsinglevextendsingleS
02
= a174a127 a189nφprime2
vextendsinglevextendsingle
S02 =
nφ
primeprime
2
vextendsinglevextendsingle
S02 = a174a127
·[epsilon11?(φprime1?φprimeprime1)] = 0 (φprime1?φprimeprime1)vextendsinglevextendsingleSprime = (φprime2?φprimeprime2)vextendsinglevextendsingleSprime
·[epsilon12?(φprime2?φprimeprime2)] = 0 epsilon11n(φprime1?φprimeprime1)vextendsinglevextendsingleSprime = epsilon12n(φprime2?φprimeprime2)vextendsinglevextendsingleSprime
(φprime1?φprimeprime1)vextendsinglevextendsingleS
01
= 0 a189n(φprime1?φprimeprime1)
vextendsinglevextendsingle
S01 = 0
(φprime2?φprimeprime2)vextendsinglevextendsingleS
02
= 0 a189n(φprime2?φprimeprime2)vextendsinglevextendsingleS
02
= 0
110/384
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a141a152a53a189a110a57a65a65a94a94,a183a62a175a75a141a152a53a189a110integraldisplay
τ1
dτepsilon11|vectorEprime1?vectorEprimeprime1|2 +
integraldisplay
τ2
dτepsilon12|vectorEprime2?vectorEprimeprime2|2=? 0 → vectorEprime1 = vectorEprimeprime1 vectorEprime2 = vectorEprimeprime2
=
integraldisplay
τ1
dτepsilon11|?(φprime1?φprimeprime1)|2 +
integraldisplay
τ2
dτepsilon12|?(φprime2?φprimeprime2)|2
=
integraldisplay
τ1
dτepsilon11?(φprime1?φprimeprime1)·?(φprime1?φprimeprime1) +
integraldisplay
τ2
dτepsilon12?(φprime2?φprimeprime2)·?(φprime2?φprimeprime2)
=
integraldisplay
τ1
dτ{?·[epsilon11(φprime1?φprimeprime1)?(φprime1?φprimeprime1)]?(φprime1?φprimeprime1)?·[epsilon11?(φprime1?φprimeprime1)]}
+
integraldisplay
τ2
dτ{?·[epsilon12(φprime2?φprimeprime2)?(φprime2?φprimeprime2)]?(φprime2?φprimeprime2)?·[epsilon12?(φprime2?φprimeprime2)]}
=
contintegraldisplay
τ1
dvectorS1·[epsilon11(φprime1?φprimeprime1)?(φprime1?φprimeprime1)] +
contintegraldisplay
τ2
dvectorS2·[epsilon12(φprime2?φprimeprime2)?(φprime2?φprimeprime2)]
=
integraldisplay
S01
dvectorS1·[epsilon11(φprime1?φprimeprime1)?(φprime1?φprimeprime1)] +
integraldisplay
S02
dvectorS2·[epsilon12(φprime2?φprimeprime2)?(φprime2?φprimeprime2)]
+
integraldisplay
Sprime
dvectorS·[epsilon11(φprime1?φprimeprime1)?(φprime1?φprimeprime1)]?
integraldisplay
Sprime
dvectorS·[epsilon12(φprime2?φprimeprime2)?(φprime2?φprimeprime2)]
111/384
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a141a152a53a189a110a57a65a65a94a94,a107a19a78a127a51a158a27a141a152a53a189a110
a19a78a83a220a107a233a245a140a177a103a100a185a196a27a62a102.a242a19a78a152a92a174a127a27a9a62a124
a165,a19a78a254a62a214a201a62a124a229a138a94a135a173a35a169a217,a134a20a167a23a41a27a97a65a62a124a218
a9a62a124a140a2a152a24,a144a149a131a135,a166a19a78a83a111 a62a124a143a34a167a103a100a62a102a66a216a50
a189a149a36a196a136a20a178a239a71a21,a19a78a57a217a51a9a62a124a101a136a20a178a239a27a28a28a91a91 a249a158
a19a78a180a31a179a78 φ1?φ2 =
integraldisplay 2
1
dvectorl·vectorE a19a78a76a161a180a31a179a161
a83a220a220a216a216a145a62,a62a214a169a217a51a76a161 vectorEa83 = 0→ρf =?·vectorD =?·(epsilon1vectorE) = 0
a19a78a9a76a161a181 vectorEt = 0 vectorEn = σfepsilon1vectorn
vectorn×(vectorE2?vectorE1) = 0 vectorn·(epsilon12vectorE2?epsilon11vectorE1) = σf a38a37
a39a36
a106 a101
S02S01
21
Sprime
a134a254a33a211a24a27a171a141,a2S1,S2,...a165a107a152a10a180a19a78,a75a101τa169a171a235a89,
epsilon1i >0,τa83ρf,σfa174a127,a133a233a233a19a19a78a62a46,a101a15a94a135a131a152 a164a225:
(1).a19a78a254a62a179a174a127 (2).a19a78a254a111a62a214a174a127
a75a41a141a152 a34
112/384
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a141a152a53a189a110a57a65a65a94a94,a107a19a78a127a51a158a27a141a152a53a189a110
a192a144a107a152a135a9a62a46S0a218a152a135a83a62a46S1,a83a62a46a157a140a27a180a19a78
a38a37
a39a36
a109
S0
S1?·[epsilon1?φprime] =?ρf
contintegraldisplay
S1
dSepsilon1?φ
prime
n =?Q a189 φ
primevextendsinglevextendsingle
S1 = a126a234
·[epsilon1?φprimeprime] =?ρf
contintegraldisplay
S1
dSepsilon1?φ
primeprime
n =?Q a189 φ
primeprimevextendsinglevextendsingle
S1 = a126a234
a217a165S0a254a27a62a94a135a211a99a161a152a152a24a24.
·[epsilon1?(φprime?φprimeprime)] = 0
contintegraldisplay
S1
dSepsilon1n(φprime?φprimeprime) = 0 a189 (φprime?φprimeprime)vextendsinglevextendsingleS
1
= 0
integraldisplay
dτepsilon1|vectorEprime?vectorEprimeprime|2 =
integraldisplay
dτepsilon1?(φprime?φprimeprime)·?(φprime?φprimeprime)
=
integraldisplay
dτ{?[epsilon1(φprime?φprimeprime)?(φprime?φprimeprime)]?(φprime?φprimeprime)?·[epsilon1?(φprime?φprimeprime)]}
=
contintegraldisplay
S0
dvectorS·epsilon1(φprime?φprimeprime)?(φprime?φprimeprime) +
contintegraldisplay
S1
dvectorS·epsilon1(φprime?φprimeprime)?(φprime?φprimeprime)
= epsilon1(φprime?φprimeprime)vextendsinglevextendsingleS
1
contintegraldisplay
S1
dvectorS·epsilon1?(φprime?φprimeprime) = 0 → vectorEprime = vectorEprimeprimea41a141a152!
113/384
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a141a152a53a189a110a57a65a65a94a94,a107a19a78a127a51a158a27a141a152a53a189a110
a126,a152a135a152a37a19a78a110,a19a78a254a140a177a145a143a140a177a216a145a62a214
a101a110a83a195a62a214:
a19a78a83a76a161a254a62a207a254a131a218a1430,a110a83?2φ = 0
→a110a83φ =a126a234,vectorE = 0,σf = epsilon1En,a110a83a76a161a195a62a214,a62a214a144a85a169
a217a51a9a76a161(a101a19a78a254a164a145a27a111a62a254Qa189a9a62a124vectorEa85a67,a110a83vectorE a69a145
a1770,a83a76a161a69a145a177a195a62a214,a2φa27a253a233a234a138a85a67)a100a121a121a150a150a23a183a62a182a45a34
a101a110a83a107a62a214,a110a83a111a62a254a143q:
a19a78a9a76a161a62a207a254a131a218a1431epsilon1(Q + q),qa143a19a78a110a254a27a111a62a214.a100
a252a135a94a135a216a145a110a83a62a214a169a217a85a67a13a85a67,a207a13 a110a9a62a124a216a145a110
a83a62a214a169a217a85a67a13a85a67(a2a110a83a62a124a135a67),a65a79a47,a8a9a76a161a180a165
a158,a110a9a62a124a61a180a152a135a100a160a117a165a165a37a27a62a214 q + Qa23a41a27a62a124a34
114/384
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115/384
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a141a152a53a189a110a57a65a65a94a94,a107a19a78a127a51a158a27a141a152a53a189a110
a17a225a19a78a23a41a27a27a62a62a179a169a217a20a39a117a167a167a164a164a145a27a111a62a254a34
a23a19a78τ,a164a145a111a62a254Q,a19a78a9a62a179a169a217φ(vectorr).
·(epsilon1?φ) = 0
contintegraldisplay
τ
dvectorS·epsilon1?φ = Q a195a161a15a103a44a62a94a135
a121,a8Qa134a143λQa0,a62a179a169a217a217a67a67a143λφ(vectorr).
a144a167a62a94a135a247a118a141a152a53a189a110,a144a73a121a8Qa134a143λQa0,λφ(vectorr)a69a180a41.
·(epsilon1?λφ) = 0
contintegraldisplay
τ
dvectorS·epsilon1λ?φ = λQ a195a161a15a103a44a62a94a135
λφa180a41.
116/384
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a141a152a53a189a110a57a65a65a94a94,a107a19a78a127a51a158a27a141a152a53a189a110
a19a78a124a164a27a78a88,a49ia135a19a78a254a27a111a62a214qi,a19a78a9a140a177a107a48a159,a2
a195a103a100a62a214,a166a121a152a109a63a191a191a152a152a58vectorra63a62a179a134qia27a39a88a180a130a53a224a103a27.
φ(vectorr,q1,q2,...,qn) =
nsummationdisplay
i=1
pi(vectorr)qi
pi(vectorr)a134qia195a39.
a23a49ia135a19a78a254a27a27a62a62a179a143φi,a75φi =
nsummationdisplay
j=1
pijqj,a135a41a26qi =
nsummationdisplay
j=1
cijφj,
pij:a62a179a88a234,a180a19a78ja145a145a252a252a160a62a254,a217a167a19a78a122a135a19a78a254a111a62a214a1430a158,
a49ia135a19a78a27a27a62a62a179a138.cij:a62a78a88a234,a189a194 cj(vectorr) =
nsummationdisplay
i=1
pi(vectorr)cij a75,
φ(vectorr) =
nsummationdisplay
i=1
pi(vectorr)qi =
nsummationdisplay
i,j=1
pi(vectorr)cijφj =
nsummationdisplay
j=1
cj(vectorr)φj
cj(vectorr)a180a19a78ja143a252a160a62a179,a217a167a19a78a1430a62a179a179a158a158,a152a109a27a27a62a62a179a169a217.
117/384
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a141a152a53a189a110a57a65a65a94a94,a107a19a78a127a51a158a27a141a152a53a189a110
a233a254a33a130a53a48a159,a62a179a169a217a247a118?2φ =?ρf/epsilon1 =?ρf/(epsilon10epsilon1r)
a233a39a253a152a165a27a27a62a62a179a169a217a144a167?2φ =?ρ/epsilon10
ρ = ρf +ρprime = ρfepsilon1
r
a61 ρprime = (1epsilon1
r
1)ρf
a51a254a33a130a53a48a159a159a165a165,a52a122a62a214a209a121a51a103a100a62a214a63,ρprime = (1epsilon1
r
1)ρf,a51a195
a103a100a62a214a63a143a195a52a122a62a214,a52a122a166a103a100a62a214a63a27a111a62a214a254a67a143a6a53
a271epsilon1
r
(a182a45a8a65),a101a180a195a161a140a254a33a130a53a48a159,a62a179a169a217a247a118
2φ =?ρfepsilon1 =? ρfepsilon1
0epsilon1r
φvextendsinglevextendsingler→∞→ 1r
a233a39a253a152a165a27a27a62a62a179a169a217a144a167
2φf =?ρfepsilon1
0
φfvextendsinglevextendsingler→∞→1r
φ = φfepsilon1
r
vectorE = vectorEf
epsilon1r a180a41
118/384
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a46a202a46a100a144a167,a169a108a67a254a123
a254a33a48a159,a51a152a109a171a141τa165,a118a107a103a100a62a214a158a27a27a62a62a179a169a217a247a118a144a167:
2φ = 0
a107a103a100a62a214a156a47a101,a62a179a169a217a53a103a252a220a169,a78a62a214ρ = ρf/epsilon1r a27a0
a122; τa76a161a161a161a161a62a214a57τa171a141a9a27a27a62a62a214a27a0a122 (a80a143φprime).
a78a62a214a169a217ρf/epsilon1ra152a132a180a174a127a27,a167a233a233a62a62a179a169a217a27a0a122a143,14piepsilon1
0
integraltext dτprime
τ
ρf
epsilon1rR,
a242a217a169a209,φ = 14piepsilon1 integraltext dτprimeρfR +φprime,φprimea51τa165a247a118a46a202a46a100a144a167.
2φprime =?2[φ? 14piepsilon1
integraldisplay
dτprimeρfR] =?ρfepsilon1? 14piepsilon1
integraldisplay
dτprimeρf?2 1R= 0
a161a62a214a233a233a62a62a179a169a217a27a0a122a247a118a224a103a46a202a46a100a144a167.
a252a171a156a185,a175a75a209a122a143a166a41a224a103a46a202a46a100a144a167?2φ = 0
119/384
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a46a202a46a100a144a167,a169a108a67a254a123,a165a139a73
a0
a0a0a9
a45
a54
a0
a0a0a18
a64
a64a64a82
z
θ
ψ
vectorr
x
ya165a139a73a27a110a135a139a73a73a67a67a254a143r,θ,ψ,
a169a108a67a234a123a137a209a224a103a46a202a46a100a144a167a207a41a27a152a132a47a170a143:
φ(r,θ,ψ) =
summationdisplay
n,m
(an,mrn + bn,mrn+1)Pmn (cosθ)cosmψ
+
summationdisplay
n,m
(cn,mrn + dn,mrn+1)Pmn (cosθ)sinmψ
a101a164a63a216a27a175a75a39a117za182a233a161,a75φa134ψa195a39:
φ(r,θ) =
∞summationdisplay
n=0
(anrn + bnrn+1)Pn(cosθ)
Pmn (cosθ),Pn(cosθ)a143a53a220a86a52a25a188a234a218a86a52a25a188a234.
a88a234a100a62a94a135a251a189.
120/384
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a46a202a46a100a144a167,a169a108a67a254a123,a165a139a73
1
|vectorr?vectorrprime| = (r
2 + rprime2?2rrprimecosθ)?12 =
∞summationdisplay
n=0
rn<
rn+1> Pn(cosθ)
integraldisplay 1
1
dxPm(x)Pn(x) = 2δmn2n + 1
∞summationdisplay
n=0
Pn(x)Pn(xprime)2n + 12 = δ(x?xprime)
Pn(x) = 12nn!( ddx)n(x2?1)n
P0(x) = 1 P1(x) = x P2(x) = 12(3x2?1)
121/384
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a46a202a46a100a144a167,a169a108a67a254a123,a206a139a73
a0
a0a0a9
a45
a54
a64
a64a64a82
z
θ
vectorrx
ya139a73a27a110a135a139a73a73a67a67a254a143r,θ,z,a169a108a67a234a123a137a209
a224a103a46a202a46a100a144a167a134za195a39a207a41a27a152a132a47a170a143:
φ(r,θ) = A0 lnr + B0θ+ C0 +
∞summationdisplay
n=1
(Anrn + Bnr?n)cos(nθ+αn)
ln|vectorr?vectorrprime|= ln(r2 + rprime2?2rrprimecosθ)12 =?
∞summationdisplay
n=1
parenleftbigr<
r>
parenrightbign cosnθ+ lnr
>
integraldisplay 2pi
0
dθcosnθcosmθ = piδmn
∞summationdisplay
n=0
cosnθcosnθprime = piδ(θ?θprime)
122/384
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a46a202a46a100a144a167,a169a108a67a254a123,a165a139a73a126
a26a25
a27a24a45a45
a45
epsilon1
vectorE0?2φ
1 = 0 φ1
vextendsinglevextendsingle
r=a = φ2
vextendsinglevextendsingle
r=a φ1
vextendsinglevextendsingle
r=0a107a129
2φ2 = 0 epsilon1r?φ1?r vextendsinglevextendsingler=a =?φ2?r vextendsinglevextendsingler=a φ2vextendsinglevextendsingler→∞→?E0rP1(cosθ) + C
φ1 =
∞summationdisplay
n=0
AnrnPn φ2 =
∞summationdisplay
n=0
Bn
rn+1Pn + B
primeP0 + BprimeprimerP1
A0 = B0a + Bprime; epsilon1rA1 =?2B1a3 + Bprimeprime; A1a = B1a2 + Bprimeprimea;?B0a = 0
Anan = Bnan+1; epsilon1rnAnan?1 =?(n + 1) Bnan+2 n>1; Bprimeprime =?E0; Bprime = C
A0 = C; A1 =?3E0epsilon1
r + 2; B1 = epsilon1r?1epsilon1
r + 2
E0a3; An = Bn = 0 n>0
φ1 = C?3
vectorE0·vectorr
epsilon1r + 2 φ2 = C +
epsilon1r?1
epsilon1r + 2a
3vectorE0·vectorr
r3?
vectorE0·vectorr
a8epsilon1r = 1a158,φ1 = φ2,a8epsilon1r→∞a158,a48a159a67a164a19a78.
123/384
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a46a202a46a100a144a167,a169a108a67a254a123,a206a139a73a126
φ = φ0 +φ1
φ0=?2k1ηln|vectorr?vectorr0|+C=2k1η
∞summationdisplay
n=1
1
n
parenleftbigr<
r>
parenrightbigncosnθ?2k
1ηlnr>+C
2φ1 = 0 →φ1(r,θ) = A0 lnr + B0 +
∞summationdisplay
n=1
Bnr?n cos(nθ+αn)
φ(r,θ) = φ(r,?θ)→αn = 0; φ(a,θ) = a126a234;?a
integraldisplay 2pi
0
dθ?φ?rvextendsinglevextendsingler=a= η
prime
epsilon10
C+2k1η
∞summationdisplay
n=1
1
n(
a
r0)
ncosnθ?2k1ηlnr0+A0lna+B0+
∞summationdisplay
n=1
Bna?ncosnθ=a126a234
integraldisplay 2pi
0
dθ[2k1η
∞summationdisplay
n=1
( ar
0
)n?1 1r
0
cosnθ+ A0a +
∞summationdisplay
n=1
nBna?n?1 cosnθ] =?η
prime
aepsilon10
2k1η( ar
0
)n1n + Bna?n = 0; Bn =?2k1η( ar
0
)na
n
n ; 2pi
A0
a =?
ηprime
aepsilon10
124/384
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a46a202a46a100a144a167,a169a108a67a254a123,a206a139a73a126
φ(r,θ) = φ0(r,θ) +φ1(r,θ)
φ0(r,θ) = 2k1η
∞summationdisplay
n=1
1
n
parenleftbigr<
r>
parenrightbigcosnθ+ lnr
> + C
=?2k1ηln|vectorr?vectorr0|+ C
φ1(r,θ) =?2k1ηprimelnr?2k1η
∞summationdisplay
n=1
( a
2
r0r)
n1
n cosnθ
= 2k1ηln|a
2
r20vectorr0?vectorr|?2k1(η
prime +η)lnr
a11a206a19a130a254a27a62a214a233a206a9a62a179a27a0a122a140a94a160a117a186a150a58a151a221?η,a218
a160a117a206a37,a151a221ηprime +ηa27a134a206a206a182a182a178a49a27a130a62a214a23a41a27a179a147a79.
a38a37
a39a36
a114vectorr0
η
a114 a114
a2
r20 vectorr0
η
η + ηprime
125/384
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a46a202a46a100a144a167,a169a108a67a254a123,a165a139a73a126
za182a254,a6a58a58a20a20a9a130a44a152a58a58a27a27a165a254vectorc,c =√a2 + b2
a167a20za182a254a124a58a229a108:R =|zvectorez?vectorc| 1R=
∞summationdisplay
n=0
rn<
rn+1> Pn(cosγ)
γ,a9a130a233za182a27a220a14 r< = min(z,c) r> = max(z,c)
φ(z,0) =
contintegraldisplay
a130
dlk1
q
2pia
R =
k1q
R = k1q
∞summationdisplay
n=0
rn<
rn+1> Pn(cosγ)
a152a132a154a9a130a63,?2φ = 0 a6a58a58a62a62a179a107a129; a195a161a15a103a44a62a94a135
φ(r,θ) =
∞summationdisplay
n=0
Bn
rn+1Pn(cosθ) r>c
∞summationdisplay
n=0
AnrnPn(cosθ) r<c
za182a254a62a179a140a189a209a88a234,
Bn = k1qcnPn(cosγ)
An = k1q 1cn+1Pn(cosγ)
b = 0? cosγ = 0
φ(r,θ) = k1q
∞summationdisplay
n=0
rn<
rn+1> Pn(cosγ)Pn(cosθ) P2n+1(0) = 0,P2n(0) = (?1)
n
2 1·3·5···(2n?1)2·4·6···2n
126/384
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a46a202a46a100a144a167,a169a108a67a254a123,a165a139a73a126
a152a132a154a9a130a63,φ(r,θ) =
contintegraldisplay
a130
dlk1
q
2pia
R a134a26a200a169a79a142
vectorr = rcosθvectorez + rsinθcosφvectorex + rsinθsinφvectorey
vectora = acosφ0vectorex + asinφ0vectorey dl = adφ0
R2 = r2 + a2?2vectorr·vectora = r2 + a2?2rasinθcos(φ0?φ)
φ(r,θ) = k1qa2pia
integraldisplay φ+2pi
φ
dφ0 1radicalbigr2 + a2?2rasinθcos(φ
0?φ)
φ0?φ = 2α+pi cos(φ0?φ) =?cos(2α) = 2sin2α?1 dφ0 = 2dα
φ(r,θ) = k1qpi
integraldisplay pi/2
pi/2
dα 1radicalbigr
2 + a2?2rasinθ(2sin2α?1)
φ(r,θ) = 2k1qpi√r2 + a2 + 2rasinθ
integraldisplay pi/2
0
dα 1radicalBig
1? 4rasinθr2+a2+2rasinθ sin2α
= 2k1qpi√r2+a2+2rasinθK(
radicalbigg 4rasinθ
r2+a2+2rasinθ) a49a152a97a17a28a253a11a200a169
127/384
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a186a148a123,a196a29a103a142
a253a152a165,?2φ =?ρepsilon1
0
a83a92a152a135a100a160a117vectorr0a63a62a254a143qa27a27a58a58a58a62a62a214a62a179φprime = φ+ q4piepsilon1
0|vectorr?vectorr0|
2φprime =?2φ? qepsilon1
0
δ(vectorr?vectorr0)=?ρepsilon1
0
qepsilon1
0
δ(vectorr?vectorr0)
a58a58a62a62a214a160a117a166a41a171,φprimea134φa247a118a216a211a27a144a167
a58a58a62a62a214a216a51a166a41a171,φprimea134φa247a118a131a211a27a144a167
a154a166a41a171a92a58a58a62a62a214a216a75a143a124a144a167 !
a51a154a166a41a171a92a62a214a53a78a33a41a27a49a143,a166a131a216a61a247a118a124a144a167,a132
a247a118a62a94a135
a51a154a41a171a92a27a62a214a23a150a62a214a34a152a132a216a180a162a83a127a51a27a62a214,a2
a167a51a166a41a171a23a41a27 a62a179a134a162a83a51a62a46a254a218a154a166a41a171a127a51a27
a62a214a51a166a41a171a23a41a27a27a62a62a179a131a211!
a186a150a123a180a124a94a186a150a62a214a27a62a179a53a8a69a224a103a46a202a46a100a144a167a41
a207a41,a44a0a50a124a94a62a94a135a189a209a207a41a165a27a63a191a126a234
128/384
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a186a148a123,a1261,2,3
a1261,a9a76a161a143a165a27a27a19a19a78a138,a110a83a62a254Q,a138a254a62a254 q,
φvextendsinglevextendsinglea165a161a31a179,
contintegraldisplay
a165a76a161
dvectorS·?φ =?1epsilon1
0
(q + Q)
2φ = 0 → φ = q + Q4piepsilon1
0r
a150a62a214a160a117a165a37a27a27a62a62a254a143q + Q !
a1262,a195a161a140a26a47a47a19a19a78a134a9a107a152a58a58a62a62a214q,a114x0
q
a114
q
x0
a150a62a214a160a117a19a78a134a83a186a150a233a161a58,a62a254a143?q !a85a216a85a216a26a47a186
a1263,a140a187a143r0a27a26a47a47a19a19a78a165a9a229a165a37da63a107a152a58a58a62a62a214q,
φ(vectorr) = 14piepsilon1
0
bracketleftbig q
|vectorr?vectord|?
qr0
d
|vectorr? r20d2vectord|
bracketrightbig
a150a62a214a51qa134a165a37a233a130a254,a229a165a37r20da63,a62a254a143qprime =?qr0d !
a38a37
a39a36
a114
vectord
qqprime
a114
r20
d2
vectord
129/384
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a186a148a123,a1263
a51a165a161a254 a181
φ(vectorr)vextendsinglevextendsingler=r
0
= 14piepsilon1
0
bracketleftbig q
|vectorr?vectord|?
qr0
d
|vectorr? r20d2vectord|
bracketrightbig
r=r0
= 14piepsilon1
0
bracketleftbig qradicalbig
r20 + d2?2r0dcosθ?
qr0
dradicalBig
r20 + r40d2?2r0r20d cosθ
bracketrightbig
= 14piepsilon1
0
bracketleftbig qradicalbig
r20 + d2?2r0dcosθ?
qradicalbig
d2 + r20?2r0dcosθ
bracketrightbig
= 0
a216a26a47a78a111a141a186
a137a189a189a19a19a78a111a62a214Q?a51a165a37a92a148a62a214Q+qr0d
a137a189a189a19a19a78a62a179φ0?a51a165a37a92a148a62a2144piepsilon10r0φ0 +qr0d
130/384
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a186a148a123,a1264,5
a1264,a114q
φ0
a114?q
φ1
φ1 + φ0
a114?q
φ2
φ0 + φ2
a114q
φ3
φ3 + φ2
a114q
φ4
φ1 + φ4
a114?q
φ5
φ5 + φ4
a114?q
φ6
φ3 + φ6··· ···
φ = φ0 +φ1 +φ2 +φ3 +φ4 +φ5 +φ6 +···
a1265,?2φ1 =? q0epsilon1
r1epsilon10
δ(x?x0)δ(y)δ(z)?2φ2 = 0
φ1vextendsinglevextendsinglex=0 = φ2vextendsinglevextendsinglex=0 epsilon1r1?φ1?xvextendsinglevextendsinglex=0 = epsilon1r2?φ2?xvextendsinglevextendsinglex=0
a114 a114x0?x0
epsilon11epsilon12
q2 q0epsilon1r1,q3
φ1 = k1q0epsilon1
r1r1
+ k1q2r
2
φ2 = k1q0epsilon1
r1r1
+ k1q3r
3
vectorr2 =vectorr + x0vectorex vectorr3 =vectorr1 =vectorr?x0vectorex q2 = q3 = epsilon1r1?epsilon1r2epsilon1
r1(epsilon1r1 +epsilon1r2)
q0
131/384
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a186a148a123,a152a132a156a185
φ = φa83 +φa9 +φa76a161= φa83 +φa150
φa150 = φa9 +φa76a161
a51a107a48a159a27a171a141,a103a100a62a214a177a140a135a78a88a52a122a62a214,a166a111a62a254a67
a143a6a53a62a254a27 1/epsilon1ra21a34
a186a150a62a214a27a23a152a73a135a179a178a8,a152a135a216a49,a210a23a245a135,a134a20a85a247
a118a62a94a135a143a142a34
132/384
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a130a21a188a234,a152a132a156a185
a137a189Va83a62a214a169a217ρa218
Va62a46Sa254a136a58a58a27a27a27a62a62a179φS
a189a62a124a123a149a169a254?φ?n
vextendsinglevextendsingle
S
a189a195a46a152a109a158a27a195a161a15a103a44a62a94a135
a166Va83a136a58a58a27a27a27a62a62a179a138?
2φ(vectorr) =?1epsilon1
0
ρ(vectorr)
a62a94a135
a233a65a27a27a58a58a58a62a62a214a62a138a175a75=======?
2G(vectorr,vectorrprime) =?1epsilon1
0
δ(vectorr?vectorrprime)
a62a94a135
a242a154a219a103a62a214a169a217a27a27a62a62a179a166a41a122a143a58a58a62a62a214a27a27a62a62a179a166a41a175a75
a58a58a62a62a214a27a27a62a62a179a140a94a186a148a123a31a144a144a123a123a166a41
133/384
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a130a21a188a234,a152a132a156a185
φ(vectorr)
=
integraldisplay
V
dτprimeφ(vectorrprime)δ(vectorrprime?vectorr) =?epsilon10
integraldisplay
V
dτprimeφ(vectorrprime)?prime2G(vectorrprime,vectorr)
=?epsilon10
integraldisplay
V
dτprimebraceleftbig?prime·[φ(vectorrprime)?primeG(vectorrprime,vectorr)]primeφ(vectorrprime)·?primeG(vectorrprime,vectorr)bracerightbig
= epsilon10
integraldisplay
V
dτprimebraceleftbig?prime·bracketleftbig?φ(vectorrprime)?primeG(vectorrprime,vectorr) + [?primeφ(vectorrprime)]G(vectorrprime,vectorr)bracketrightbig
prime2φ(vectorrprime)G(vectorrprime,vectorr)bracerightbig
= epsilon10
integraldisplay
V
dτprimebraceleftbig?prime·bracketleftbig?φ(vectorrprime)?primeG(vectorrprime,vectorr) + [?primeφ(vectorrprime)]G(vectorrprime,vectorr)bracketrightbig
+1epsilon1
0
ρ(vectorrprime)G(vectorrprime,vectorr)bracerightbig
=
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr) +epsilon10
contintegraldisplay
S
dvectorSprime·[G(vectorrprime,vectorr)?primeφ(vectorrprime)?φ(vectorrprime)?primeG(vectorrprime,vectorr)]
=
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr) +epsilon10
contintegraldisplay
S
dSprime[G(vectorrprime,vectorr)?φ(vectorr
prime)
nprime?φ(vectorr
prime)?G(vectorr
prime,vectorr)
nprime ]
134/384
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a130a21a188a234,a62a138a175a75
φ(vectorr) =
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr) +epsilon10
contintegraldisplay
S
dSprime[G(vectorrprime,vectorr)?φ(vectorr
prime)
nprime?φ(vectorr
prime)?G(vectorr
prime,vectorr)
nprime ]
a49a152a97a62a138a175a75,φvextendsinglevextendsinglevectorrprime?Sa174a127,a18,G(vectorrprime,vectorr)vextendsinglevextendsinglevectorrprime?S = 0
φ(vectorr) =
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr)?epsilon10
contintegraldisplay
S
dSprimeφ(vectorrprime)?G(vectorr
prime,vectorr)
nprime
a49a19a97a62a138a175a75,?φ?nprimevextendsinglevextendsingleSa174a127,a18,?G(vectorr
prime,vectorr)
nprime
vextendsinglevextendsingle
vectorrprime?S = c 1 =?epsilon10cSV
φ(vectorr) =
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr) +epsilon10
contintegraldisplay
S
dSprimeG(vectorrprime,vectorr)?φ(vectorr
prime)
nprime
a49a152,a19a97a183a218a62a138a175a75,S = S1 + S2,φvextendsinglevextendsingleS
1
,?φ?nprimevextendsinglevextendsingleS
2
a174a127,
G(vectorrprime,vectorr)vextendsinglevextendsinglevectorrprime?S
1
= 0,?G(vectorr
prime,vectorr)
nprime
vextendsinglevextendsingle
vectorrprime?S2 = c
a195a46a152a109(Va143a143a195a195a161a140),φvextendsinglevextendsingle∞a247a118a195a161a15a103a44a62a94a135
G(vectorrprime,vectorr)vextendsinglevextendsinglevectorrprime?∞a247a118a195a161a15a103a44a62a94a135 φ(vectorr) =
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr)
135/384
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a130a21a188a234,a49a19a97a62a138a175a75
φ(vectorr) =
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr) +epsilon10
contintegraldisplay
S
dSprime[G(vectorrprime,vectorr)?φ(vectorr
prime)
nprime?φ(vectorr
prime)?G(vectorr
prime,vectorr)
nprime ]
a49a19a97a62a138a175a75:?φ?nprimevextendsinglevextendsingleSa174a127,a18,?G(vectorr
prime,vectorr)
nprime
vextendsinglevextendsingle
vectorrprime?S = 0
1 =
integraldisplay
V
dτprimeδ(vectorrprime?vectorr) =?epsilon10
integraldisplay
V
dτprime?prime2G(vectorrprime,vectorr)
=?epsilon10
integraldisplay
V
dτprime?prime·[?primeG(vectorrprime,vectorr)] =?epsilon10
integraldisplay
V
dvectorSprime·[?primeG(vectorrprime,vectorr)]
=?epsilon10
integraldisplay
V
dSprimenprimeG(vectorrprime,vectorr)
G(vectorrprime,vectorr)
nprime
vextendsinglevextendsingle
vectorrprime?S = c 1 =?epsilon10cSV
φ(vectorr) =
integraldisplay
V
dτprimeρ(vectorrprime)G(vectorrprime,vectorr) +epsilon10
contintegraldisplay
S
dSprimeG(vectorrprime,vectorr)?φ(vectorr
prime)
nprime
136/384
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a130a21a188a234,a126a195a46a152a109a27a130a21a188a234:
G(vectorr,vectorrprime) = 14piepsilon1
0
1
|vectorr?vectorrprime| φ(vectorr) =
1
4piepsilon10
integraldisplay
∞
dτprime ρ(vectorr
prime)
|vectorr?vectorrprime|
a254a140a152a109(z > 0)a49a152a97a62a138a175a75a27a130a21a188a234:
G(vectorr,vectorrprime) = 14piepsilon1
0
parenleftbig 1
|vectorr?vectorrprime|?
1
|vectorr +vectorrprime|
parenrightbig
φ(vectorr) = 14piepsilon1
0
integraldisplay
∞
dτprimeρ(vectorrprime)parenleftbig 1|vectorr?vectorrprime|? 1|vectorr +vectorrprime|parenrightbig
14pi
integraldisplay
z=0
dSprimeφ(vectorrprime)zprimeparenleftbig 1|vectorr?vectorrprime|? 1|vectorr +vectorrprime|parenrightbig
a165a9a152a109(r>r0)a49a152a97a62a138a175a75a27a130a21a188a234:
G(vectorr,vectorrprime) = 14piepsilon1
0
parenleftbig 1
|vectorr?vectorrprime|?
r0
rprime
|vectorr? r20rprime2vectorrprime|
parenrightbig
φ(vectorr) = 14piepsilon1
0
integraldisplay
∞
dτprimeρ(vectorrprime)parenleftbig 1|vectorr?vectorrprime|?
r0
r
|vectorrprime?r20r2vectorr|
parenrightbig
14pi
integraldisplay
r0
dSprimeφ(vectorrprime)rprimeparenleftbig 1|vectorr?vectorrprime|?
r0
r
|vectorrprime?r20r2vectorr|
parenrightbig
137/384
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a183a62a124a85a254
W = 12
integraldisplay
∞
dτvectorE·vectorD = 12
integraldisplay
∞
dτ(φ)·vectorD
= 12
integraldisplay
∞
dτ[·(φvectorD) +φ?·vectorD]
=?12
contintegraldisplay
∞
dvectorS·φvectorD + 12
integraldisplay
∞
dτρfφ
a233a183a62a124,φvextendsinglevextendsingler→∞～1r,Dvextendsinglevextendsingler→∞～ 1r2 →
contintegraldisplay
∞
dvectorS·φvectorD→0
W = 12
integraldisplay
dτvectorE·vectorD = 12
integraldisplay
dτρfφ = 18piepsilon1
0
integraldisplay
dτdτprimeρf(vectorr)ρ(vectorr
prime)
R
a233a253a152a165a27a27a62a62a214a169a217,ρ(vectorr) = ρf(vectorr)
W = 18piepsilon1
0
integraldisplay
dτdτprimeρf(vectorr)ρf(vectorr
prime)
R
138/384
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a183a62a124a85a254,a131a112a138a94a85
W = 18piepsilon1
0
integraldisplay
dτdτprimeρf(vectorr)ρf(vectorr
prime)
R
a253a152a165a107a252a124a62a214a169a217a124a164a27a78a88,ρ = ρ1 +ρ2
W = W1 + W2 + Wa131a112
W = 18piepsilon1
0
integraldisplay
dτdτprime1R[ρ1(vectorr) +ρ2(vectorr)][ρ1(vectorrprime) +ρ2(vectorr)prime]
= 18piepsilon1
0
integraldisplay
dτdτprime1R[ρ1(vectorr)ρ1(vectorrprime) +ρ2(vectorr)ρ2(vectorrprime) +ρ1(vectorr)ρ2(vectorrprime) +ρ2(vectorr)ρ1(vectorrprime)]
Wa131a112 = 14piepsilon1
0
integraldisplay
dτdτprime1Rρ1(vectorr)ρ2(vectorrprime) =
integraldisplay
dτρ1φ2 =
integraldisplay
dτρ2φ1
a62a214ρa51a9a124φa165a27a85a254,Wa131a112 =
integraldisplay
dτρφ
a58a58a62a62a214a51a9a124a165a27a85a254,qφ a183a62a160a85a180a62a124a85a27a141a150a76a227a156
139/384
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a183a62a124a85a254,a131a112a138a94a85
a70a126a41a185a165a107 m = m1 + m2 Frank.Wilczek,a218a238a238a49a49a34a189a198
a85 E = mc2 a110a41 W = W1 + W2 + Wa131a112
a218a238a238a49a49a34a189a198a216a50a164a225a156 Wa131a112a178a126< 0
a22a20a135a42 Wa131a112 a229a27a138a94a22a140a156
a131a112a138a94a51a135a42a43a141a229a216a37a138a94
a51a212a159a216a211a27a0a103a107a216a211a27a198a137a59a128a239a196Wa131a112a156
140/384
triangleleftsldtriangleleftsld
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a183a62a124a85a254,a41a48a154a189a110
a62a214a51a216a26a47a47a27a27a27a19a19a78a254a27a183a21a169a217a240a166a183a62a85a18a52a2a138 !
a127a196a19a78a254a27a62a214a27a67a122(a253a14a48a159a165a27a62a214a195a123a67,a122a135a19a78a254
a27a111a62a254a143a195a123a67)a19a151a27a27a62a62a124a67a122δvectorE
δW = 12
integraldisplay
dτ epsilon1(vectorE +δvectorE)·(vectorE +δvectorE)?12
integraldisplay
dτ epsilon1E2
=
integraldisplay
dτ epsilon1[vectorE·δvectorE + 12δvectorE·δvectorE] a217a165:
integraldisplay
dτ epsilon1vectorE·δvectorE =?
integraldisplay
dτ?φ·δvectorD =?
integraldisplay
dτ[?·(φδvectorD)?φ?·δvectorD]
=?
contintegraldisplay
dvectorS·φδvectorD +
integraldisplay
dτ φδ?·vectorD =
integraldisplay
dτ φδρ =
summationdisplay
a19a78i
φiδqi = 0
δW = 12
integraldisplay
dτ epsilon1δvectorE·δvectorE≥0
141/384
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trianglerightsld
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a183a62a124a85a254,a41a48a154a189a110
a62a214a51a216a26a47a47a27a27a27a19a19a78a254a27a183a21a169a217a240a166a183a62a85a18a52a2a138 !
a127a196a19a78a254a27a62a214a27a67a122(a253a14a48a159a165a27a62a214a195a123a67,a122a135a19a78a254
a27a111a62a254a143a195a123a67)a19a151a27a27a62a62a124a67a122δvectorE
δW = 12
integraldisplay
dτ epsilon1(vectorE+δvectorE)·(vectorE+δvectorE)?12
integraldisplay
dτ epsilon1E2 = 12
integraldisplay
dτ epsilon1δvectorE·δvectorE≥0
a135a131a101a62a214a169a217a166a183a62a85a136a20a52a2.a167a131a8a117a88a101a21a229a67a169a175a75:
δqi =
integraldisplay
a19a78i
dτδρ = 0
summationdisplay
a19a78i
integraldisplay
a19a78i
dτ φδρ = 0
a218a92a46a188a166a102φi:
summationdisplay
a19a78i
integraldisplay
a19a78i
dτ (φ?φi)δρ=0 φ(vectorr)=φi a19a78a143a31a179a78!
142/384
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a240a189a189a62a62a54a27a27a62a62a124
·vectorD = ρf?×vectorE = 0 vectorD = epsilon1vectorE vectorjc = γvectorE
a240a189a189a62a62a54a88a218,?
vectorjc
t = 0 →
vectorE
t = 0
a173a21a88a218,a108a62a124a85a61a122a53a27a62a102a27a196a85,a207a76a62a102a134a172a130a45a69,a242
a196a85a68a52a137a172a130,a47a164a57a85a181
a0a15a189a198,a57a245a199a151a221 W = Q?φ?t?τ = jc?t?SδlE?t?S?l =vectorjc·vectorE = j
2
c
γ
a73a9a46a74a248a85a254a156
a218a92a154a183a62a229a57a131a65a27a27a31a31a8a62a124vectorEprime = vectorFa154/q + +
a63a85a27a238a238a48a48a189a198a181vectorjc = γ(vectorE + vectorEprime)
a62a62a196a196a179a181 ε =
contintegraldisplay
dvectorl·vectorFa154/q =
contintegraldisplay
dvectorl·(vectorE + vectorEprime)
=
contintegraldisplay
dvectorl·vectorjc/γ =
contintegraldisplay
dlJλS = JR
a9a53a85a254a207a76a154a183a62a229a27a62a196a179a216a228a61a122a164a62a124a85a167a129a0a158a209a180a27
a62a123a254a61a67a143a57a85a156
143/384
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a240a189a189a62a62a54a27a27a62a62a124,a124a144a167a124
vectorj,vectorEprimea216a145a158a109a85a67→vectorEa216a145a158a109a85a67.
·vectorD = ρf?×vectorE = 0
ρf
t +?·
vectorjc = 0 vectorjc = γ(vectorE + vectorEprime)
ρf
t = 0 →?·
vectorjc = 0 →?·(γvectorE) =·(γvectorEprime)
×vectorE = 0?·(epsilon1vectorE) = ρf
γa233a65epsilon1,·(γvectorEprime)a233a65a65a117a117ρf.
144/384
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a240a189a189a62a62a54a27a27a62a62a124,a1261
a62a52A,B(γ =∞)a152a117a19a62a120a159γ1a165,γ1a9 a157a140a44a152a171a19a62a120a159γ2:
×vectorE1 = 0?×vectorE2 = 0 γ1E1n = γ2E2na62a54a235a89
·(γ1vectorE1) = 0?·(γ1vectorE2) = 0 E1t = E2t
a154a62a52a63,vectorEprime = 0,γ1,γ2a229epsilon1a27a138a94,a124a94vectorE =φ
2φ1 = 0?2φ2 = 0 γ1?φ1?n = γ2?φ2?n φ1 = φ2
a92a254a62a52A,Ba27a53a159a57a195a161a15a27a103a44a62a94a135,a62a52a180a110a142a19a78a181
γ =∞ vectorjc = γvectorE(vectorEprime = 0) → vectorE→0 a62a52a143a31a179a78!
a62a94a135a181
braceleftbiggφ
A?φB = V
QA =?QB
braceleftbiggφ
B = 0
φA = V
a92a254a195a161a15a103a44a62a94a135φvextendsinglevextendsingler→∞→ 1r,a144a167a124a27a41a180a141a152a27.
ρf =?·vectorD =?·(epsilon1vectorE) σf = D2n?D1n σ = epsilon10(E2n?E1n)
a101a120a159a254a33a27,γ,epsilon1a143a126a234,a75ρf = 0.
145/384
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a240a189a189a62a62a54a27a27a62a62a124,a1262
a140a187a143aa27a140a165a47a62a52a238a51a62a19a199a143γa27a47a101,a62a52a178a161a134 a47a161a178
a224,a108a62a52a83a54a209a27a27a62a62a54a114a221I,a166a47a101a27a27a62a62a54a169a217?
a47a101a154a62a52a63a27a27a62a62a179φa247a118?2φ = 0
a51a47a161a62a54a144a85a247a131a149a54a196,a107a62a94a135En = 0
a51a62a52a27a165a76a161I = integraltext dvectorS·γvectorE
a192a193a38a41φ = cr,a167a247a118a51a47a161a27a62a94a135En = 0
I =
integraldisplay
a140a165
dvectorS·cγvectorrr3 =
integraldisplay
a140a165
dS cγ 1a2 =?2picγ
→ c = I2piγ → φ = I2piγr
a62a52a229a152a135a145a152a189a189a62a62a214a27a27a19a19a78a27a138a94.
146/384
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a173a240a62a54a27a94a124,a165a165a179a179
braceleftbigg?×vectorH =vectorj
c
·vectorB = 0
braceleftbigg vectorB = μvectorH
vectorf =vectorjc×vectorB a62a94a135,
braceleftbiggvectorn·(vectorB
2?vectorB1) = 0
vectorn×(vectorH2?vectorH1) =vectoric
a195a209a27a165a254a124a152a189a140a177a76a143a44a135a165a254a124a27a94a221 a156
a189a194a181vectorA(vectorr)≡
integraldisplay 1
0
dλ vectorB(λvectorr)×(λvectorr)
×(vectorf×vectorg) =?vectorg?·vectorf?vectorf·?vectorg +vectorg·?vectorf +vectorf?·vectorg
×vectorA(vectorr) =
integraldisplay 1
0
dλ λ?×[vectorB(λvectorr)×vectorr]
=
integraldisplay 1
0
dλ λ[?vectorr?·vectorB(λvectorr)?vectorB(λvectorr)·?vectorr +vectorr·?vectorB(λvectorr) + vectorB(λvectorr)?·vectorr]
=
integraldisplay 1
0
dλ λ[?vectorB(λvectorr)+λ ddλvectorB(λvectorr)+3vectorB(λvectorr)]=
integraldisplay 1
0
dλ λ[2vectorB(λvectorr)+λ ddλvectorB(λvectorr)]
=
integraldisplay 1
0
dλ ddλ[λ2vectorB(λvectorr)] = [λ2vectorB(λvectorr)]vextendsinglevextendsingleλ=1λ=0 = vectorB(vectorr)
147/384
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a173a240a62a54a27a94a124,a165a165a179a179
a189a194 vectorB =?×vectorAa40a189a189a27a27vectorAa216a141a152.
vectorB =?×vectorA =?×(vectorA +?χ) vectorAprime = vectorA +?χ ←a53a137a67a134
a53a137a67a134a166vectorAa144a85a79a40a20a131a11a152a135a63a191a70a221a188a234.
a143a166vectorA a40a189,a76a78a92a152a135a53a137a27a189a94a135a22a75a53a137a103a100a221:
a165a212a53a137a27a189a94a135,?·vectorA = 0
a126a181 a233a126a234a124vectorB,a192vectorA·vectorr = 0,a75vectorA = 12vectorB×vectorr
×vectorA = 12(?·vectorr)vectorB?12(vectorB·?)vectorr = 32vectorB?12vectorB·arrowrighttophalfarrowrighttophalfI = vectorB
a53a137a27a189a94a135a233a245,a212a110a134a53a137a27a189a94a135a27a192a74a195a39.
vectorB =?×vectorAa137a209?·vectorB =?·(?×vectorA) = 0,a191a155a155a195a195a94a94a252a252a52.
148/384
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a173a240a62a54a27a94a124,a165a165a179a179
a165a212a53a137a181vectorAvextendsinglevextendsinglea165a212a53a137 =
integraldisplay
dvectorlprimeμ0J4piR =
integraldisplay
dτprime μ0
vectorj
4piR
a51a165a212a53a137a101a27a165a254a179a247a118a181
·vectorAvextendsinglevextendsinglea165a212a53a137 = μ04pi
integraldisplay
dτprimevectorj·?1R =?μ04pi
integraldisplay
dτprimevectorj·?prime1R
=?μ04pi
integraldisplay
dτprime?prime·
vectorj
R =?
μ0
4pi
integraldisplay
∞
dvectorSprime·
vectorj
R = 0
a165a165a179a179a57a217a247a118a27a144a167,(a254a33a48a159)
2vectorA(?·vectorA) =×(?×vectorA) =×vectorB =?μ?×vectorH=?μvectorjc
a233a165a212a53a137,?2vectorA =?μvectorjc.
149/384
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a173a240a62a54a27a94a124,a165a165a179a179,a173a240a62a54a94a124a27a85a254
W = 12
integraldisplay
∞
dτvectorB·vectorH = 12
integraldisplay
∞
dτ(?×vectorA)·vectorH
= 12
integraldisplay
∞
dτ[?·(vectorA×vectorH) + vectorA·(?×vectorH)]
= 12
integraldisplay
∞
dvectorS·(vectorA×vectorH) + 12
integraldisplay
∞
dτvectorA·vectorjc
a233a173a240a62a54a27a94a124,vectorAvextendsinglevextendsingler→∞～1r,vectorHvextendsinglevextendsingler→∞～ 1r2
→
integraldisplay
∞
dvectorS·(vectorA×vectorH)→ 0
W = 12
integraldisplay
dτvectorB·vectorH = 12
integraldisplay
dτvectorA·vectorjc = μ08pi
integraldisplay
dτdτprime
vectorjc(vectorr)·vectorj(vectorrprime)
R
a233a253a152a165a27a27a62a62a54a169a217:vectorj(vectorr) =vectorjc(vectorr) W = μ08pi
integraldisplay
dτdτprime
vectorjc(vectorr)·vectorjc(vectorrprime)
R
150/384
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a173a240a62a54a27a94a124,a165a165a179a179,a173a240a62a54a94a124a27a85a254
a233a253a152a165a27a27a62a62a54a169a217:vectorj(vectorr) =vectorjc(vectorr) W = μ08pi
integraldisplay
dτdτprime
vectorj(vectorr)·vectorj(vectorrprime)
R
a252a124a62a54a169a217a124a164a27a78a88:vectorj =vectorj1 +vectorj2,W = W1 + W2 + Wa131a112
W = μ08pi
integraldisplay
dτdτprime1R[vectorj1(vectorr) +vectorj2(vectorr)]·[vectorj1(vectorrprime) +vectorj2(vectorr)prime]
= μ08pi
integraldisplay
dτdτprime1R[vectorj1(vectorr)·vectorj1(vectorrprime) +vectorj2(vectorr)·vectorj2(vectorrprime) +vectorj1(vectorr)·vectorj2(vectorrprime)
+vectorj2(vectorr)·vectorj1(vectorrprime)]
Wa131a112 = μ04pi
integraldisplay
dτdτprime1Rvectorj1(vectorr)·vectorj2(vectorr)prime =
integraldisplay
dτvectorA1·vectorj2 =
integraldisplay
dτvectorA2·vectorj1
a62a54vectorja51a9a124vectorAa165a27a85a254,Wa131a112 =
integraldisplay
dτvectorj·vectorA
151/384
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a94a124a175a75a27a152a132a41a123,a195a68a19a19a62a62a54a195a144a94
a152a135a131a233a94a19a199μra27a140a187a143aa27a254a33a165,a152a51a254a33a94a124a114a221vectorH0 a27
a9a94a124a165,a166a166a165a165a83a9a27a94a124
a26a25
a27a24a45a45
a45
μ
vectorH0
braceleftbigg?×vectorH
1 = 0
·vectorB1 = 0
braceleftbigg?×vectorH
2 = 0
·vectorB2 = 0braceleftbigg
H1tvextendsinglevextendsingler=a = H2tvextendsinglevextendsingler=a
B1nvextendsinglevextendsingler=a = B2nvextendsinglevextendsingler=a
braceleftbigg vectorB
2 = μ0vectorH2vector
B1 = μvectorH1
vectorH2vextendsinglevextendsingle
r→∞→
vectorH0
a131a211a140a2a27a27a62a62a48a159a165a152a51a254a33a62a124vectorE0a165:
a26a25
a27a24a45a45
a45
epsilon1
vectorE0
braceleftbigg?×vectorE
1 = 0
·vectorD1 = 0
braceleftbigg?×vectorE
2 = 0
·vectorD2 = 0braceleftbigg
E1tvextendsinglevextendsingler=a = E2tvextendsinglevextendsingler=a
D1nvextendsinglevextendsingler=a = D2nvextendsinglevextendsingler=a
braceleftbigg vectorD
2 = epsilon10vectorE2vector
D1 = epsilon1vectorE1
vectorE2vextendsinglevextendsingle
r→∞→
vectorE0
vectorH?vectorE vectorB?vectorD μ?epsilon1
152/384
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a46a202a46a100a144a167,a169a108a67a254a123,a165a139a73a126
2φ1 = 0 φ1vextendsinglevextendsingler=a = φ2vextendsinglevextendsingler=a φ1vextendsinglevextendsingler=0a107a129
2φ2 = 0 epsilon1r?φ1?r vextendsinglevextendsingler=a =?φ2?r vextendsinglevextendsingler=a φ2vextendsinglevextendsingler→∞→?E0rP1(cosθ) + C
a26a25
a27a24a45a45
a45
epsilon1
vectorE0
φ1 =
∞summationdisplay
n=0
AnrnPn φ2 =
∞summationdisplay
n=0
Bn
rn+1Pn + B
primeP0 + BprimeprimerP1
A0 = B0a + Bprime; epsilon1rA1 =?2B1a3 + Bprimeprime; A1a = B1a2 + Bprimeprimea;?B0a = 0
Anan = Bnan+1; epsilon1rnAnan?1 =?(n + 1) Bnan+2 n>1; Bprimeprime =?E0; Bprime = C
A0 = C; A1 =?3E0epsilon1
r + 2; B1 = epsilon1r?1epsilon1
r + 2
E0a3; An = Bn = 0 n>0
φ1 = C?3
vectorE0·vectorr
epsilon1r + 2 φ2 = C +
epsilon1r?1
epsilon1r + 2a
3vectorE0·vectorr
r3?
vectorE0·vectorr
a8epsilon1r = 1a158,φ1 = φ2,a8epsilon1r→∞a158,a48a159a67a164a19a78.
153/384
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a94a124a175a75a27a152a132a41a123,a195a68a19a19a62a62a54a195a144a94
vectorH1 =φm1 vectorH2 =φm2
2φm1 = 0 φm1vextendsinglevextendsingler=a = φm2vextendsinglevextendsingler=a φm1vextendsinglevextendsingler=0a107a129
2φm2 = 0 μr?φm1?r vextendsinglevextendsingler=a =?φm2?r vextendsinglevextendsingler=a φm2vextendsinglevextendsingler→∞?H0rP1(cosθ) + C
φm1 =? 3H0μ
r + 2
rP1(cosθ)
φm2 =?H0rP1(cosθ) + μr?1μ
r + 2
a3H0r2P1(cosθ)
vectorH1 = 3
μr + 2
vectorH0 vectorH2 = vectorH0 + 1
4pi[?
vectorm
r3 +
3vectorr(vectorm·vectorr)
r5 ]
vectorB1 = 3μr
μr + 2
vectorB0 vectorB2 = μ0vectorH2 vectorm = 4pia3μr?1
μr + 2
vectorH0
a195a68a19a19a62a62a54?×vectorH = 0→vectorH =φm,vectorjc =?×(φm) = 0.
154/384
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a94a124a175a75a27a152a132a41a123,a195a68a19a19a62a62a54a107a144a94
a152a172a144a94a225a14,a48a159a83
braceleftbigg?×vectorH
1 = 0
·vectorB1 = 0,a48a159a9
braceleftbigg?×vectorH
2 = 0
·vectorB2 = 0
a62a46a254
braceleftbigg H
1t=H2t
B1n=B2n a62a94a53a159a144a167
braceleftbigg vectorB
2 = μ0vectorH2vector
B1 = μ1vectorH1+μ0 vectorM0 a94
vectorHa76a136
braceleftbigg?·(μvectorH
1) =·(μ0 vectorM0)
×vectorH1 = 0
braceleftbigg?·(μ
0vectorH2) = 0
×vectorH2 = 0
braceleftbiggμ
0H2n?μH1n = μ0M0nvector
H2t = vectorH1t
braceleftbigg?·(epsilon1vectorE
1) = ρf
×vectorE1 = 0
braceleftbigg?·(epsilon1
0vectorE2) = 0
×vectorE2 = 0
braceleftbiggepsilon1
0E2n?epsilon1E1n = σfvector
E2t = vectorE1t
μ～epsilon1 μ0～epsilon10 μr～epsilon1r vectorH～vectorE·(μ0 vectorM0)～ρf μ0M0n～σf
155/384
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a94a124a175a75a27a152a132a41a123,a195a68a19a19a62a62a54a107a144a94
μ～epsilon1 vectorH～vectorE·(μ0 vectorM0)～ρf μ0M0n～σf
a91a200a94a99,μ = μ0
a233a65epsilon1 = epsilon10,a195a48a159,a207a13a195a52a122a62a214,a62a124a28a220a100ρf,σfa23a41
vectorE =
integraldisplay
dτprime ρf
vectorR
4piepsilon10R3 +
integraldisplay
dSprime σf
vectorR
4piepsilon10R3
vectorH =?
integraldisplay
dτprime?
prime·(μ0 vectorM0)vectorR
4piμ0R3 +
integraldisplay
dSprimeμ0M0n
vectorR
4piμ0R3
a101a154a91a200a94a99,a75a65a92a92a52a122a62a214(a94a214)a27a0a122.
156/384
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a94a124a175a75a27a152a132a41a123,a107a68a19a19a62a62a54
braceleftbigg?×vectorH =vectorj
c
·vectorB = 0
vectorB = μvectorH
braceleftbiggvectorn×(vectorH
2?vectorH1) =vectoric
vectorn·(vectorB2?vectorB1) = 0
a23a123a242a100a175a75a122a143a143a195a195a68a19a19a62a62a54a27a175a75:
vectorB0 =
integraldisplay
dτprimeμ0
vectorjc×vectorR
4piR3 +
integraldisplay
dSprimeμ0
vectoric×vectorR
4piR3
·vectorB0 = 0?×vectorB0 = μ0vectorjc
vectorn×(vectorB02?vectorB01) = μ0vectoric vectorn·(vectorB02?vectorB01) = 0
157/384
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a94a124a175a75a27a152a132a41a123,a107a68a19a19a62a62a54
·vectorB0=0?×vectorB0=μ0vectorjc vectorn×(vectorB02?vectorB01)=μ0vectoric vectorn·(vectorB02?vectorB01)=0
vectorH = 1
μ0
vectorB0 + vectorHprime vectorB = μvectorH = μ
μ0
vectorB0 +μvectorHprime
×vectorHprime =?×(vectorH? 1μ
0
vectorB0) =vectorjc? 1
μ0?×
vectorB0 =vectorjc?vectorjc = 0
·(μvectorHprime) =?·(vectorB? μμ
0
vectorB0) =?·vectorBμ
μ0 ·
vectorB0? μ
μ0?·
vectorB0 =μ·vectorB0
μ0
vectorn×(vectorHprime2?vectorHprime1) = vectorn×bracketleftbig(?1μ
0
vectorB02 + vectorH2)?(?1
μ0
vectorB01 + vectorH1)bracketrightbig
=?1μ
0
vectorn×(vectorB02?vectorB01) +vectorn×(vectorH2?vectorH1) =vectoric?vectoric = 0
vectorn·(μ2vectorHprime2?μ1vectorHprime1) = vectorn·bracketleftbigvectorB2?μ2μ
0
vectorB02?vectorB1 + μ1
μ0
vectorB01bracketrightbig = μ1?μ2
μ0 vectorn·
vectorB0
vectorHprime～vectorE μ～epsilon1?(?μ)vectorB0
μ0 ～ρf
μ1?μ2
μ0 vectorn·
vectorB0～σf
a233a245a183a94a175a75a134a183a62a175a75a107a233a65a39a88,a131a65a27a212a110a121a121a150a150a143a107a233a65.
158/384
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a94a124a175a75a27a152a132a41a123,a94a186a148a123braceleftbigg
×vectorB1 = μ1vectorjc
·vectorB1 = 0
braceleftbigg?×vectorB
2 = 0
·vectorB2 = 0
braceleftBigg
1
μ1
vectorB1tvextendsinglevextendsingle
x=0 =
1
μ2
vectorB2tvextendsinglevextendsingle
x=0
B1nvextendsinglevextendsinglex=0 = B2nvextendsinglevextendsinglex=0
a94a122a62a54a152a220a169a78a88a51vectorjca254,a166a167a67a143μrvectorjc,a44a152a220a169a51a48a159 a2a46a161
a254vectoriprime,a23μrvectorjca23a41a27a94a97a65a114a221vectorB0,vectoriprimea511,2a48a159a165a23a41a27 a94a97a65a114
a221a169a79a143vectorB10,vectorB20,vectorB1 = vectorB0 + vectorB10 vectorB2 = vectorB0 + vectorB20
a186a150a123a242vectoriprimea23a41a27a94a97a65a114a221a94a160a117a154a166a41a171a27a186a150a62a54vectorjprime,vectorjprimeprimea23a41
a27a94a97a65a114a221a147a76.
a0a18a64a64a73 a65
a65a75
a8a8
a8a8a8a42 x0?x0
vectorR2 vectorR1 μ1μ2
vectorj” μ1rvectorj,vectorjprimea100a100a117a117vectoriprimea51a46a161a252a62a23a41a27a124a233a161,a23a41a249a135a124
a27a186a150a62a54a27a160a152a144a149a143a233a161.vectorjprimea134vectorjc a211a160a152,
vectorjprimeprimea160a117a48a1592a165a27vectorjca27a186a150a233a161a160a152.
vectorjprimen =?vectorjprimeprimen vectorjprimet =vectorjprimeprimet
a23vectorjca20a124a58a58a27a27a165a254vectorR1,vectorjprimeprimea20a124a58a58a27a27a165a254vectorR2,
vectorB0 = μ0μ1rvectorjc×vectorR1
4piR31 dτ
prime vectorB10 = μ0vectorj
primeprime×vectorR2
4piR32 dτ
prime vectorB20 = μ0vectorj
prime×vectorR1
4piR31 dτ
prime
159/384
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a94a124a175a75a27a152a132a41a123,a94a186a148a123a51a48a159a2a2a46a46a161a254,vector
R1 =?x0vectore1 + yvectore2 + zvectore3 vectorR2 = x0vectore1 + yvectore2 + zvectore3 R1 = R2≡R
a0a18a64a64a73 a64
a64a64a73
a0
a0a0a18 x0?x0
vectorR2 vectorR1 μ1μ2
vectorj” μ1rvectorj,vectorjprimevectorB1 = μ0
4pidτ
prime(vectorj
primeprime×vectorR2
R32 +
μ1rvectorjc×vectorR1
R31 )
= μ04piR3dτprime[(vectorjprimeprime?μ1rvectorjc)×(x0vectore1) + (vectorjprimeprime +μ1rvectorjc)×(yvectore2 + zvectore3)]
= μ04piR3dτprime[(vectorjprimeprimet?μ1rvectorjct)×(x0vectore1) + (vectorjprimeprimet +vectorjprimeprimen +μ1rvectorjct +μ1rvectorjcn)
×(yvectore2 + zvectore3)]
vectorB2 = μ0
4pidτ
prime(vectorj
prime×vectorR1
R31 +
μ1rvectorjc×vectorR1
R31 )
= μ04piR3dτprime[(vectorjprime +μ1rvectorjc)×(?x0vectore1) + (vectorjprime +μ1rvectorjc)×(yvectore2 + zvectore3)]
= μ04piR3dτprime[(vectorjprimet +μ1rvectorjct)×(?x0vectore1) + (vectorjprimet +vectorjprimen +μ1rvectorjct +μ1rvectorjcn)
×(yvectore2 + zvectore3)]
= μ04piR3dτprime[(vectorjprimeprimet +μ1rvectorjct)×(?x0vectore1) + (vectorjprimeprimet?vectorjprimeprimen +μ1rvectorjct +μ1rvectorjcn)
×(yvectore2 + zvectore3)]
160/384
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a94a124a175a75a27a152a132a41a123,a94a186a148a123
(vectorjprimeprimet +μ1rvectorjt)×(yvectore2 + zvectore3) = (vectorjprimeprimet +μ1rvectorjt)×(yvectore2 + zvectore3)
1
μ1[(
vectorjprimeprimet?μ1rvectorjt)×(x0vectore1) + (vectorjprimeprimen +μ1rvectorjn)×(yvectore2 + zvectore3)]
= 1μ
2
[(vectorjprimeprimet +μ1rvectorjt)×(?x0vectore1) + (?vectorjprimeprimen +μ1rvectorjn)×(yvectore2 + zvectore3)]
1
μ1(
vectorjprimeprimet?μ1rvectorjct) =?1
μ2(
vectorjprimeprimet +μ1rvectorjct) 1
μ1(
vectorjprimeprimen +μ1rvectorjcn) = 1
μ2(?
vectorjprimeprimen +μ1rvectorjcn)
vectorjprimet =vectorjprimeprimet = μ2?μ1
μ2 +μ1μ1r
vectorjct vectorjprimen =?vectorjprimeprimen = μ2?μ1
μ2 +μ1μ1r
vectorjcn
Jprime = Jprimeprime = μ2?μ1μ
2 +μ1
μ1rJc
161/384
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a94a124a175a75a27a152a132a41a123,a126a181a195a161a127a134a19a130
a206a139a73a181 vectorB(r,θ,z) = B(r)vectoreθ
μ0J =
contintegraldisplay
dvectorl·vectorB = 2pirB(r)
vectorB(r,θ,z) = μ0J
2pirvectoreθ
a134a19a130a94a124a27a28a28a91a91
162/384
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a94a124a175a75a27a152a132a41a123,a126a181a130a47a19a130
a206a139a73,vectorA(r,θ,z) =
contintegraldisplay
a130
dvectorl μ0J4piR a134a26a200a169a79a142
vectorr = zvectorez + rvectorer vectora = acosθprimevectorer + asinθprimevectoreθ
R2 = z2+ r2+ a2?2vectorr·vectora = z2+ r2+ a2?2racosθprime
dvectorl=adθprimevectoreθprime=adθprime(vectoreθprime·vectoreθvectoreθ+vectoreθprime·vectorervectorer)=adθprime(cosθprimevectoreθ?sinθprimevectorer)
vectorA(r,θ,z) = aμ0J
4pi
integraldisplay 2pi
0
dθprime cosθ
primevectoreθ?sinθprimevectorer
√z2 + r2 + a2?2racosθprime
θprime = 2α+pi cosθprime =?cos(2α) = 2sin2α?1 dθprime = 2dα
vectorA(r,θ,z) = aμ0Jvectoreθ
4pi
integraldisplay pi/2
pi/2
dα 2sin
2α?1
radicalbigz
2 + r2 + a2?2ra(2sin2α?1)
= 2μ0Javectoreθpi√z2 + r2 + a2 + 2ra
integraldisplay pi/2
0
dα sin
2α?1/2
radicalBig
1? 4raz2+r2+a2+2ra sin2α
= 2μ0Javectoreθpi√z2+r2+a2+2ra
integraldisplay pi/2
0
dαsin
2α?1/2
√1?k
2sin2α k
2≡ 4ra
z2+r2+a2+2ra
163/384
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a94a124a175a75a27a152a132a41a123,a126a181a130a47a19a130
a206a139a73,vectorA(r,θ,z) =
contintegraldisplay
a130
dvectorl μ0J4piR a134a26a200a169a79a142
vectorr = zvectorez + rvectorer vectora = acosθprimevectorer + asinθprimevectoreθ
R2 = z2+ r2+ a2?2vectorr·vectora = z2+ r2+ a2?2racosθprime
dvectorl = adθprimevectoreθprime = adθprime(cosθprimevectoreθ?sinθprimevectorer) k2≡ 4raz2+r2+a2+2ra
vectorA(r,θ,z) = 2μ0Jvectoreθ
pi√z2 + r2 + a2 + 2ra
integraldisplay pi/2
0
dα sin
2α?1/2
√1?k
2 sin2α
sin2α?1/2 =?1k2(1?k2 sin2α) + 1k2?12
vectorA(r,θ,z)= 2μ0Javectoreθ
pi√z2+r2+a2+2ra
integraldisplay pi/2
0
dα[ 1/k
2?1/2
√1?k
2sin2α?
1
k2
radicalbig
1?k2sin2α]
= 2μ0Javectoreθpi√z2 + r2 + a2 + 2ra[( 1k2?12)K(k)? 1k2E(k)]
= μ0Jvectoreθ
√z2 + r2 + a2 + 2ra
2rpi [(1?
k2
2 )K(k)?E(k)]
164/384
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a94a124a175a75a27a152a132a41a123,a126a181a130a47a19a130
a206a139a73,vectorA(r,θ,z) =
contintegraldisplay
a130
dvectorl μ0J4piR = Aθ(r,z)vectoreθ
Aθ(r,z) = μ0Jvectoreθ
√z2+r2+a2+2ra
2rpi [(1?
k2
2 )K(k)?E(k)]
k2≡ 4raz2+r2+a2+2ra?k?z =?zk
3
4ar
k
r =
k
2r
a2+z2?r2
(a+r)2+z2
vectorB(r,θ,z) =?×vectorA(r,θ,z) =Aθ(r,z)
z vectorer + [
Aθ(r,z)
r +
Aθ(r,z)
r ]vectorez
K(k) =
integraldisplay pi/2
0
dα 1√1?k2 sin2α E(k) =
integraldisplay pi/2
0
dα
radicalbig
1?k2 sin2α
K(k)
k =
E(k)
k(1?k2)?
K(k)
k
E(k)
k =
E(k)?K(k)
k
vectorB(r,θ,z) = μ0J
2piradicalbig(a + r)2 + z2{
vectorer
r [
a2 + r2 + z2
(a?r)2 + z2E(k)?K(k)]
+vectorez[a
2?r2?z2
(a?r)2 + z2E(k) + K(k)]} a130a47a19a130a94a124a27a28a28a91a91
165/384
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a245a52a208a109
a152a236a62a214a51a51a15a15a63a27a27a62a62a198a198a8a8a65
≡a152a135a58a58a62a62a214+a152a135a62a243a52a102+a152a135a62a111a52a102+···
a51a51a15a15a63a27a27a62a62a198a198a8a8a65
a152a236a62a54a51a51a15a15a63a27a94a198a198a8a8a65
≡a152a135a94a243a52a102+a152a135a94a111a52a102+···a51a51a15a15a63a27a94a198a198a8a8a65
166/384
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a245a52a208a109g(xprime) a51xprime = 0a63a208a109:
g(xprime) = g(0) +
∞summationdisplay
n=1
1
n!
ng(xprime)
xprimen
vextendsinglevextendsingle
xprime=0 x
primen
f(vectorrprime) a51vectorrprime = 0a63a208a109:
f(vectorrprime) = f(0) +
∞summationdisplay
n=1
3summationdisplay
i1i2...in=1
1
n!
bracketleftbig?
xprimei1
xprimei2···
xprimeinf(vectorr
prime)bracketrightbig
vectorrprime=0x
prime
i1x
prime
i2···x
prime
in
= f(0) +
∞summationdisplay
n=1
1
n!
bracketleftbig[vectorrprime·?”]n f(vectorr”)bracketrightbig
vectorr”=0
1
|vectorr?vectorrprime| =
1
r +
∞summationdisplay
n=1
3summationdisplay
i1i2...in=1
1
n!
bracketleftbig?
xprimei1
xprimei2···
xprimein
1
|vectorr?vectorrprime|
bracketrightbig
vectorrprime=0x
prime
i1x
prime
i2···x
prime
in
= 1r +
∞summationdisplay
n=1
3summationdisplay
i1i2...in=1
(?1)n
n!
bracketleftbig?
xi1
xi2···
xin
1
|vectorr?vectorrprime|
bracketrightbig
vectorrprime=0x
prime
i1x
prime
i2···x
prime
in
= 1r +
∞summationdisplay
n=1
3summationdisplay
i1i2...in=1
(?1)n
n!
bracketleftbig?
xi1
xi2···
xin
1
r
bracketrightbigxprime
i1x
prime
i2···x
prime
in
167/384
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a245a52a208a109,a62a179a27a27a245a245a52a208a109,a62a62a245a245a52a221
φ(vectorr) = 14piepsilon1
0
integraldisplay
dτprime ρ(vectorr
prime)
|vectorr?vectorrprime|
= 14piepsilon1
0
braceleftbig1
r
integraldisplay
dτprimeρ(vectorrprime) +
∞summationdisplay
n=1
3summationdisplay
i1i2...in=1
(?1)n
n!
bracketleftbig?
xi1
xi2···
xin
1
r
bracketrightbig
×
integraldisplay
dτprimexprimei1xprimei2···xprimeinρ(vectorrprime)bracerightbig =
∞summationdisplay
n=0
φn(vectorr)
φ0(vectorr) = 14piepsilon1
0r
integraldisplay
dτprimeρ(vectorrprime)
n>0,
φn(vectorr) = 14piepsilon1
0
3summationdisplay
i1i2...in=1
(?1)n
n!
bracketleftbig?
xi1
xi2···
xin
1
r
bracketrightbigq
i1i2...in
qi1i2...in =
integraldisplay
dτprimexprimei1xprimei2···xprimeinρ(vectorrprime) a79a142a165(a253a165)a47a169a217a62a214a27qi1i2...in,a96a178a217a112a30a145a233a233a62a62a179a195a0a122!
vectorE = vectorE0 + vectorE1 +··· vectorEn =φn φn+1/φn～rprime/r
168/384
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a245a52a208a109,a62a179a27a27a245a245a52a208a109,a62a62a245a245a52a221
a62a214:
φ0(vectorr) = 14piepsilon1
0
Q
r
Q =
integraldisplay
dτprimeρ(vectorrprime)
vectorE0(vectorr) = Qvectorr
4piepsilon10r3
a78a88a26a160a164a152a135a62a254a143Qa27a27a58a58a58a62a62a214,a78a88a88a130a130a221a221a27a27a8a65a26a209a22a10.
169/384
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a245a52a208a109,a62a179a27a27a245a245a52a208a109,a62a62a245a245a52a221
a62a243a52a102:
φ1(vectorr) =? 14piepsilon1
0
3summationdisplay
i1=1
bracketleftbig?
xi1
1
r
bracketrightbigq
i1 =
1
4piepsilon10
3summationdisplay
i1=1
xi1
r3 qi1 =
vectorr·vectorp
4piepsilon10r3
vectorp =
3summationdisplay
i=1
qivectorei =
integraldisplay
dτprimevectorrprimeρ(vectorrprime) vectorE1 = 14piepsilon1
0
[?vectorpr3 + 3vectorr(vectorp·vectorr)r5 ]
a78a88a26a160a164a152a135a62a243a52a221a143vectorPa27a62a243a52a102,a152a135a100a252a135a62a254±q,a131
a229vectorl(a108a75a141a149a20),a133l→0,q→∞,a2a2a177qla143a107a129a27a58a62a214a8
a164a27a78a88,a167a140a140a119a119a164a180a134a160a27a20a75a62a214a78a88a34
a1
a1
a1
a1a1a21
a16a16a16
a16a16a16
a16a16a16a49
a80a80a80
a80a80a80a113a109
a109
vectorr2
vectorr1
vectorl
+
a233a100a78a88,a111a62a254Q = q?q = 0,a129a36a30a27a210a180a243a52a221a27a0a122,
vectorp =
integraldisplay
dτprimevectorrprimeρ = q
integraldisplay
dτprimevectorrprime[δ(vectorrprime?vectorrq)?δ(vectorrprime?vectorr?q)] = q(vectorrq?vectorr?q) = qvectorl
a112a30a145a20a39a117 ql(lr)n→0(n>1)a8l→0 a158a170a1170.
170/384
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a245a52a208a109,a62a179a27a27a245a245a52a208a109,a62a62a245a245a52a221a62a111a52a102:
φ2(vectorr) =? 14piepsilon1
0
3summationdisplay
i1,i2=1
1
2
bracketleftbig?
xi1
xi2
1
r
bracketrightbigq
i1i2 =
1
8piepsilon10
3summationdisplay
i1,i2=1
[3xi1xi2r5?δi1i2r3 ]qi1i2
= 18piepsilon1
0r5
3summationdisplay
i1,i2=1
integraldisplay
dτprimeρ(vectorrprime)[3xi1xprimei1xprimei2xi2?rprime2r2]
= 18piepsilon1
0r5
3summationdisplay
i1,i2=1
xi1xi2
integraldisplay
dτprimeρ(vectorrprime)[3xprimei1xprimei2?rprime2δi1i2] =
3summationdisplay
i1,i2=1
xi1Qi1i2xi2
8piepsilon10r5
= vectorr·
arrowrighttophalfarrowrighttophalfQ ·vectorr
8piepsilon10r5 Qij =
integraldisplay
dτprime[3xprimeixprimej?rprime2δij]ρ(vectorrprime)
summationdisplay
i
Qii = 0
arrowrighttophalfarrowrighttophalfQ= 3summationdisplay
i,j=1
Qijvectoreivectorej =
integraldisplay
dτprime[3vectorrprimevectorrprime?rprime2 arrowrighttophalfarrowrighttophalfI ]ρ(vectorrprime)
vectorE2 = 1
8piepsilon10[
(arrowrighttophalfarrowrighttophalfQ ·vectorr +vectorr·arrowrighttophalfarrowrighttophalfQ)
r5?
5vectorr(vectorr·arrowrighttophalfarrowrighttophalfQ ·vectorr)
r7 ]
171/384
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a245a52a208a109,a62a179a27a27a245a245a52a208a109,a62a62a245a245a52a221
a78a88a26a160a164a152a135a62a111a52a221a143
arrowrighttophalfarrowrighttophalfQ
a27 a62a111a52a102.
a152a135a100a111a135a152a152a51a152a135a178a49a111a62a47a111a135a186a14A B C D(a62a127vectorl1 Aa141a149D,vectorl2 Aa141a149B)a254
a27a169a79a143q,?q,q,?q,a133l1→0,l2→0,q→∞,a2a2a177ql1l2a143a107a129a27a58a62a214a8a164a27a78
a88,a167a140a140a119a119a164a180a134a160a27a20a135a243a52a102a88a218a34
a45a1
a1a1a21
a1
a1a1a21
a45
vectorl1
vectorl2
A D
B C
a233a100a78a88,a111a62a254Q = q?q + q?q = 0,a111a62a243a52a221
vectorp = qvectorrA?qvectorrB+ qvectorrC?qvectorrD =?qvectorl2+ qvectorl2 = 0,a129a36a30a180a62a111a52a221a0a122,
arrowrighttophalfarrowrighttophalfQ= q(3vectorr
AvectorrA?r2A
arrowrighttophalfarrowrighttophalfI?3vectorr
BvectorrB + r2B
arrowrighttophalfarrowrighttophalfI +3vectorr
CvectorrC?r2C
arrowrighttophalfarrowrighttophalfI
3vectorrDvectorrD + r2D arrowrighttophalfarrowrighttophalfI )
= q[3vectorrAvectorrA?r2A arrowrighttophalfarrowrighttophalfI?3(vectorrA +vectorl2)(vectorrA +vectorl2) + (vectorrA +vectorl2)·(vectorrA +vectorl2) arrowrighttophalfarrowrighttophalfI
+3(vectorrA +vectorl1 +vectorl2)(vectorrA +vectorl1 +vectorl2)?(vectorrA +vectorl1 +vectorl2)·(vectorrA +vectorl1 +vectorl2) arrowrighttophalfarrowrighttophalfI
3(vectorrA +vectorl1)(vectorrA +vectorl1) + (vectorrA +vectorl1)·(vectorrA +vectorl1) arrowrighttophalfarrowrighttophalfI ]
= q(3vectorl1vectorl2 + 3vectorl2vectorl1?2vectorl1·vectorl2 arrowrighttophalfarrowrighttophalfI )
a112a30a145a20a39a117 ql2(lr)n→0(n>1)a8l→0 a158a170a1170.
172/384
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φ(vectorr) = φ(0) +
∞summationdisplay
n=1
3summationdisplay
i1i2...in=1
1
n!
bracketleftbig?
xi1
xi2···
xinφ(vectorr)
bracketrightbig
vectorr=0xi1xi2···xin
= φ(0)?
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
xi1xi2···xinbracketleftbigx
i2
xi3···
xinEi1(vectorr)
bracketrightbig
vectorr=0
vectorE(vectorr) = vectorE(0) +
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
xi1xi2···xinbracketleftbigx
i1
xi2···
xin
vectorE(vectorr)bracketrightbig
vectorr=0
W =
integraldisplay
dτρ(vectorr)φ(vectorr) ρ,a102a53a169a217
=
integraldisplay
dτρ(vectorr)braceleftbigφ(0)?
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
xi1xi2···xinbracketleftbigx
i2
···x
in
Ei1(vectorr)bracketrightbigvectorr=0bracerightbig
= Qφ(0)?
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
qi1i2...inbracketleftbigx
i2
xi3···
xinEi1(vectorr)
bracketrightbig
vectorr=0 =
∞summationdisplay
n=0
Wn
W0 = Qφ(0); Wn =?1n!
3summationdisplay
i1i2...in=1
qi1i2...inbracketleftbigx
i2
xi3···
xinEi1(vectorr)
bracketrightbig
vectorr=0
173/384
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a245a52a208a109,a145a62a78a51a9a124a165a27a85a254,a201a229a218a229a221
W =
∞summationdisplay
n=0
Wn
W0 = Qφ(0)
Wn =?1n!
3summationdisplay
i1i2...in=1
qi1i2...inbracketleftbigx
i2
xi3···
xinEi1(vectorr)
bracketrightbig
vectorr=0
W1 =?
3summationdisplay
i=1
qiEi(0) =?vectorp·vectorE(0)
W2 =?12
3summationdisplay
i,j=1
qijx
j
Ei =?16
3summationdisplay
i,j=1
(3qij?δij
3summationdisplay
k=1
qkk)x
j
Ei
=?16
3summationdisplay
i,j=1
Qijx
j
Ei?·vectorEvextendsinglevextendsinglevectorr=0 = 0
174/384
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a245a52a208a109,a145a62a78a51a9a124a165a27a85a254,a201a229a218a229a221vectorF =
integraldisplay
dτρ(vectorr)vectorE(vectorr)
=
integraldisplay
dτρ(vectorr)braceleftbigvectorE(0) +
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
xi1xi2···xinbracketleftbigx
i1
···x
in
vectorE(vectorr)bracketrightbig
vectorr=0
bracerightbig
= QvectorE(0) +
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
qi1i2...inbracketleftbigx
i1
xi2···
xin
vectorE(vectorr)bracketrightbig
vectorr=0 =
∞summationdisplay
n=0
vectorFn
vectorF0 = QvectorE(0) vectorFn = 1
n!
3summationdisplay
i1i2...in=1
qi1i2...inbracketleftbigx
i1
xi2···
xin
vectorE(vectorr)bracketrightbig
vectorr=0
vectorF1 =
3summationdisplay
i=1
qix
i
vectorE(vectorr)vextendsinglevextendsingle
vectorr=0 = vectorp·?
vectorE(vectorr)vextendsinglevextendsingle
vectorr=0
vectorF2 =?1
2
3summationdisplay
i,j=1
qij[x
i
xj
vectorE(vectorr)]vextendsinglevextendsingle
vectorr=0 =
1
6
3summationdisplay
i,j=1
(3qij?δij
3summationdisplay
k=1
qkk)?
2
xi?xj
vectorEvextendsinglevextendsingle
vectorr=0
= 16
3summationdisplay
i,j=1
Qij[?
2
xi?xj
vectorE(vectorr)]vextendsinglevextendsingle
vectorr=0?
2vectorEvextendsinglevextendsingle
vectorr=0 =[?
2φ]vextendsinglevextendsingle
vectorr=0 = 0
175/384
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a245a52a208a109,a145a62a78a51a9a124a165a27a85a254,a201a229a218a229a221vectorL = integraldisplay dτρ(vectorr)vectorr×vectorE(vectorr)
=
integraldisplay
dτρ(vectorr)vectorr×
∞summationdisplay
n=0
1
n!
3summationdisplay
i1i2...in=1
xi1···xinbracketleftbigx
i1
···x
in
vectorE(vectorr)bracketrightbig
vectorr=0
=
3summationdisplay
i,j,k=1
epsilon1ijk
integraldisplay
dτρ(vectorr)xivectorek
∞summationdisplay
n=0
1
n!
3summationdisplay
i1i2...in=1
xi1···xinbracketleftbigx
i1
···x
in
Ej(vectorr)bracketrightbigvectorr=0
=
∞summationdisplay
n=1
1
n!
3summationdisplay
i,j,k,i1i2...in=1
epsilon1ijkqii1...invectorekbracketleftbigx
i1
···x
in
Ej(vectorr)bracketrightbigvectorr=0 =
∞summationdisplay
n=0
vectorLn
vectorLn = 1
n!
3summationdisplay
i,j,k,i1i2...in=1
epsilon1ijkqii1...invectorekbracketleftbigx
i1
···x
in
Ej(vectorr)bracketrightbigvectorr=0
vectorL0 = summationdisplay
i,j,k
epsilon1ijkqivectorekEj(0) = vectorp×vectorE(0)
summationdisplay
ijk
epsilon1ijkvectorekx
i
Ej =?×vectorE = 0
vectorL1 = summationdisplay
i,j,k,i1
epsilon1ijkqii1vectorekx
i1
Ejvextendsinglevextendsinglevectorr=0 =
3summationdisplay
i,j,k,i1=1
epsilon1ijk[qii1?δii13
3summationdisplay
k=1
qkk]vectorek?Ej?x
i1
vextendsinglevextendsingle
vectorr=0
176/384
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a245a52a208a109,a165a254a179a27a27a245a245a52a208a109,a94a245a52a221
vectorA(vectorr) = μ0
4pi
integraldisplay
dτprime
vectorj(vectorrprime)
|vectorr?vectorrprime|
= μ04pi
∞summationdisplay
n=0
3summationdisplay
i1i2...in=1
(?1)n
n!
bracketleftbig?
xi1
xi2···
xin
1
r
bracketrightbigintegraldisplay dτprimexprime
i1x
prime
i2···x
prime
invectorj(vectorr
prime)
=
∞summationdisplay
n=0
vectorAn(vectorr)
vectorAn(vectorr) = μ0
4pi
3summationdisplay
i1i2...in=1
(?1)n
n!
bracketleftbig?
xi1
xi2···
xin
1
r
bracketrightbigvectorJ
i1i2...in
vectorJi
1i2...in =
integraldisplay
dτprimexprimei1xprimei2···xprimein vectorj(vectorrprime)
vectorB = vectorB0 + vectorB1 +··· vectorBn =?×vectorAn
177/384
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a245a52a208a109,a165a254a179a27a27a245a245a52a208a109,a94a245a52a221
a94a214:
vectorA0(vectorr) = μ0
4pi
vectorJ
r
vectorJ =
integraldisplay
dτprimevectorj(vectorrprime) =
integraldisplay
dτprime?prime·[vectorj(vectorrprime)vectorrprime] =
integraldisplay
∞
dvectorSprime·[vectorj(vectorrprime)vectorrprime] = 0
vectorA0(vectorr) = 0 vectorB0(vectorr) = 0
a78a88a88a130a130a221a221a27a27a8a65a26a209a22,a26a160a164a152a135a58a94a214,a100a100a117a117a195a94a214,a94a124a1430.
178/384
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a245a52a208a109,a165a254a179a27a27a245a245a52a208a109,a94a245a52a221a94a243a52a221:
vectorA1(vectorr) =?μ0
4pi
3summationdisplay
i1=1
bracketleftbig?
xi1
1
r
bracketrightbigvectorJ
i1 =
μ0
4pi
3summationdisplay
i1=1
xi1
r3
vectorJi
1 =
μ0
4pivectorm×
vectorr
r3
vectorJi =
3summationdisplay
k=1
vectorek
integraldisplay
dτprimexprimeijk(vectorrprime) = 12
3summationdisplay
k=1
vectorek
integraldisplay
dτprime[(xprimeijk?xprimekji) + (xprimeijk + xprimekji)]
= 12
3summationdisplay
k=1
vectorek
integraldisplay
dτprime(xprimeijk?xprimekji)
integraldisplay
dτprime(xprimeijk + xprimekji) =
3summationdisplay
l=1
integraldisplay
dτprimexprime
l
(jlxprimeixprimek) =
integraldisplay
∞
dvectorSprime·(vectorjxprimeixprimek) = 0
3summationdisplay
i=1
xivectorJi = 12
3summationdisplay
i,k=1
integraldisplay
dτprime(xixprimeijk?xprimekjixi)vectorek = 12
integraldisplay
dτprime(vectorr·vectorrprimevectorj?vectorrprimevectorr·vectorj)
= 12
integraldisplay
dτprime(vectorrprime×vectorj)×vectorr = vectorm×vectorr vectorm = 12
integraldisplay
dτprimevectorrprime×vectorj
179/384
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a245a52a208a109,a165a254a179a27a27a245a245a52a208a109,a94a245a52a221a94a243a52a221:
vectorA1(vectorr) = μ0
4pivectorm×
vectorr
r3
vectorB1 = μ0
4pi[?
vectorm
r3 +
3vectorr(vectorm·vectorr)
r5 ] vectorm =
1
2
integraldisplay
dτprimevectorrprime×vectorj
a78a88a26a160a164a152a135a94a243a52a221a143vectorma27a94a243a52a102,a152a135a100 a62a54Ja254a33a169a217
a51a161vectorSa254,a133S → 0,J →∞,a2a2a177JSa143a107a129a27a62a54a23a8a164a27a78
a88,a233a100a78a88,a129a36a30a27a210a180a94a243a52a221a27a0a122,
vectorm = 12
integraldisplay
dτprimevectorrprime×vectorj =?12
integraldisplay
dτprimevectorj×vectorrprime =?12J
contintegraldisplay
dvectorlprime×vectorrprime
=?12J
integraldisplay
(dvectorSprime×?prime)×vectorrprime =?12J
3summationdisplay
i,j=1
integraldisplay
(dvectorSprime×vectorei)×vectorej?primeixprimej
=?12J
3summationdisplay
i=1
integraldisplay
(dvectorSprime×vectorei)×vectorei =?12J
3summationdisplay
i=1
integraldisplay
[(dvectorSprime·vectorei)vectorei?dvectorSprime(vectorei·vectorei)]
= J
integraldisplay
dvectorSprime = JvectorS
a112a30a145a20a39a117Jlprimen(n > 3,lprime2～S)～JSn/2 = JSSn?22 →0a167a180a100a94
a111a52a221,a94a108a52a221...a31a31a31a31a0a122a27a8a65.
180/384
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a62a94a197a27a68a194
181/384
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a110a142a253a14a48a159a159a165a165a27a197a196a144a167a57a178a161a62a94a197a41,a197a196a144a167
vectorB = μvectorH vectorD = epsilon1vectorE
a110a142a253a14a48a159a159a165a165γ = 0,a127a196a103a100a62a214a214a218a218a68a19a19a62a62a54a143a34a27a156a185,
×vectorE=
vectorB
t
1
μ?×
vectorB+?1
μ×
vectorB=epsilon1?vectorE
t epsilon1?·
vectorE=epsilon1·vectorE?·vectorB=0
×vectorB?(?lnμ)×vectorB = μepsilon1?
vectorE
t?·
vectorE =?(?lnepsilon1)·vectorE
2vectorE =[?·vectorE + (?lnepsilon1)·vectorE]+?2vectorE =×(?×vectorE)[(?lnepsilon1)·vectorE]
=?×?
vectorB
t[(?lnepsilon1)·
vectorE] =μepsilon1?2
t2
vectorE+(?lnμ)×?vectorB
t[(?lnepsilon1)·
vectorE]
=μepsilon1?
2
t2
vectorE?(?lnμ)×(?×vectorE)[(?lnepsilon1)·vectorE]
=μepsilon1?
2
t2
vectorE+(?lnμ)·?vectorE?(?vectorE)·(?lnμ)[(?lnepsilon1)·vectorE]
2vectorE?1v2?
2vectorE
t2 =(?lnμ)·?
vectorE?(?vectorE)·(?lnμ)[(?lnepsilon1)·vectorE] v2= 1
μepsilon1
182/384
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a110a142a253a14a48a159a159a165a165a27a197a196a144a167a57a178a161a62a94a197a41,a197a196a144a167
vectorB = μvectorH vectorD = epsilon1vectorE
a110a142a253a14a48a159a159a165a165γ = 0,a127a196a103a100a62a214a214a218a218a68a19a19a62a62a54a143a34a27a156a185,
×vectorE=
vectorB
t?×
vectorB?(?lnμ)×vectorB=μepsilon1?vectorE
t?·
vectorE=?(?lnepsilon1)·vectorE?·vectorB=0
2vectorE?1v2?
2vectorE
t2 =(?lnμ)·?
vectorE?(?vectorE)·(?lnμ)[(?lnepsilon1)·vectorE] v2= 1
μepsilon1
2vectorB=(?·vectorB)+?2vectorB=×(?×vectorB)=×[(μepsilon1?
vectorE
t )+?(lnμ)×
vectorB]
= μepsilon1?
2
t2
vectorB(μepsilon1)×?vectorE
t×[?(lnμ)×
vectorB]
= μepsilon1?
2
t2
vectorBln(μepsilon1)×[?×vectorB?(?lnμ)×vectorB]×[?(lnμ)×vectorB]
2vectorB? 1v2?
2vectorB
t2 =ln(μepsilon1)×[?×
vectorB?(?lnμ)×vectorB]×[?(lnμ)×vectorB]
183/384
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a110a142a253a14a48a159a159a165a165a27a197a196a144a167a57a178a161a62a94a197a41,a253a152a165a197a196a144a167
(?2? 1c2?
2
t2)
vectorB = 0 (?2? 1
c2
2
t2)
vectorE = 0
a108a254a102a27a14a221a119a181
iplanckover2pi1vectorp iplanckover2pi1t?E
21planckover2pi12vectorp·vectorp?1c2?2?t2? 1c2planckover2pi12E2
(?vectorp·vectorp + E2/c2)vectorB = 0 (?vectorp·vectorp + E2/c2)vectorE = 0
E2/c2?vectorp·vectorp = m2c2? m2a49a102 = 0
a88a74a74a49a49a102a107a154a34a183a142a142a159a159a254m2a49a102:
(?vectorp·vectorp + E2/c2?m2a49a102c2)vectorB = 0 (?vectorp·vectorp + E2/c2?m2a49a102c2)vectorE = 0
(?2? 1c2?
2
t2?
m2a49a102c2
planckover2pi12 )
vectorB = 0 (?2? 1
c2
2
t2?
m2a49a102c2
planckover2pi12 )
vectorE = 0
184/384
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a110a142a253a14a48a159a159a165a165a27a197a196a144a167a57a178a161a62a94a197a41,a178a161a197a41
a254a33a48a159a181 vectorE(vectorr,t) = vectorE(t?vectorn·vectorrv ) vectorB(vectorr,t) = vectorB(t?vectorn·vectorrv ) vectorn·vectorn=1
vectorE(t?vectorn·vectorrv ) =?(t?vectorn·vectorrv )?
vectorE
t =?
vectorn
v
vectorE
t
·?vectorE =?vectornv·
vectorE
t =
vectorn
v·?(t?
vectorn·vectorr
v )
2vectorE
t2 =
1
v2
2vectorE
t2
a64
a64
a64a64
a64
a64
a64a64
a0
a0
a0
a0a18
a2
a2a2a14
a8a8
a8a42
a0a18
t1
t2vectorn
vectorr2
vectorr1
vectorr2?vectorr1
a233a137a189a158a143,a31a138a161a143a134vectorna82a134a27a178a161,vectorr⊥·vectorn = 0 vectorn·vectorr =
vectorn·(vectorr +vectorr⊥)
t?vectorn·vectorrv = t?vectorn·(vectorr +vectorr⊥)v
a31a138a161a177a132a221vectorva247vectorna144a149a36a196,a252a135a158a143t1 < t2,a31a138a161
a100vectorr1a163a20vectorr2,
t1?vectorn·vectorr1v = t2?vectorn·vectorr2v → v = vectorn·parenleftbigvectorr2?vectorr1t
2?t1
parenrightbig
185/384
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a110a142a253a14a48a159a159a165a165a27a197a196a144a167a57a178a161a62a94a197a41,a178a161a197a41
×vectorE =?(t?vectorr·vectornv )×?
vectorE
t =?
vectorn
v×
vectorE
t =?
vectorB
t
×vectorB =?(t?vectorr·vectornv )×?
vectorB
t =?
vectorn
v×
vectorB
t = μ?×
vectorH = μ?vectorD
t
= μepsilon1?
vectorE
t =
1
v2
vectorE
t
t(v
vectorB?vectorn×vectorE) = 0 → vvectorB = vectorn×vectorE +a158a109a195a39a145
t(vectorn×v
vectorB + vectorE) = 0 → vectorE =?vectorn×vvectorB +a158a109a195a39a145
=?vectorv×vectorB +a158a109a195a39a145
186/384
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a110a142a253a14a48a159a159a165a165a27a197a196a144a167a57a178a161a62a94a197a41,a178a161a197a41
vvectorB = vectorn×vectorE +a158a109a195a39a145 vectorE =?vectorv×vectorB +a158a109a195a39a145
0 =?·vectorE =?(t?vectorr·vectornv )·?
vectorE
t =?
t
parenleftbigvectorn
v·
vectorEparenrightbig → vectorn·vectorE = a158a109a195a39a145
0 =?·vectorB =?(t?vectorr·vectornv )·?
vectorB
t =?
t
parenleftbigvectorn
v·
vectorBparenrightbig → vectorn·vectorB = a158a109a195a39a145
a209a22a158a109a195a39a145(a183a21a62a94a124),vectorE,vectorB,vectorn a209a108a109a195a39a88! vB = E
We = 12epsilon1E2 = 12epsilon1v2B2 = 12μB2 = Wm
vectorS = vectorE×vectorH = vectorE×vectorB
μ =?(vectorv×
vectorB)×vectorB
μ = vectorv
B2
μ = vectorv(We + Wm)
187/384
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a189a21a197a196a144a167a57a178a161a197a41,a19a78
a19a78a134a253a14a48a159a27a11a79a51a117a233a233a19a19a78a127a51a238a238a48a48a189a198vectorj = γvectorE,γ = 0a75
a163a20a110a142a253a14a48a159.
ρ
t =·
vectorj =·(γvectorE) =?γ
epsilon1?·
vectorD =?γ
epsilon1ρ
ρ(t) = ρ(0)e?γepsilon1t
a51a19a78a165,a62a214a151a221a180a233a175a80a126a27!
(a51a183a62a158,a19a78a83a220a220a216a216a145a62,a62a214a144a169a217a51a76a161a254).
a63a216a88a101a252a171a156a185:
γ～0 ρ(0) = 0
γgreatermuchepsilon1ω ω,a62a94a197a27a14a170a199
a67a113a47a18 ρ(t) = 0
188/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197
vectorD(ω) = epsilon1(ω)vectorE(ω) vectorB(ω) = μ(ω)vectorH(ω) vectorj(ω) = γ(ω)vectorE(ω)
a189a21a62a94a197,vectorE(vectorr,t) = vectorE(vectorr)e?iωt vectorB(vectorr,t) = vectorB(vectorr)e?iωt?
×vectorE = iωvectorB
×vectorH = vectorjc?iωvectorD
·vectorD = 0
·vectorB = 0
a147a92a212a159a62a94a144a167
×vectorE = iωvectorB
×vectorB = μ(γ?iωepsilon1)vectorE
·vectorE = 0
·vectorB = 0
2vectorE =?2vectorE(?·vectorE) =×(?×vectorE) =?iω?×vectorB
=?iωμ(γ?iωepsilon1)vectorE
2vectorB =?2vectorB(?·vectorB) =×(?×vectorB) =?μ(γ?iωepsilon1)?×vectorE
=?iωμ(γ?iωepsilon1)vectorB
(?2 + k2)vectorE = 0
·vectorE = 0
vectorB =?i
ω?×
vectorE
vectorE,vectorBa131a112a216a213a225
←→
k2≡ω2μepsilon1+ iωμγ
(?2 + k2)vectorB = 0
·vectorB = 0
vectorE = 1
μ(γ?iωepsilon1)?×
vectorB
189/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
vectorE(vectorr) = vectorE0eivectork·vectorr vectorE(vectorr,t) = vectorE0ei(vectork·vectorr?ωt) vectork·vectork≡k2 = ω2μepsilon1+ iωμγ
vectorB(vectorr) = vectorB0eivectork·vectorr vectorB(vectorr,t) = vectorB0ei(vectork·vectorr?ωt)
vectorka181 a197a165 vectork = vectorkR + ivectorkI.
vectorE0,vectorB0,a69a8a204 E0n =bardblE0nbardbleiδEn B0n =bardblB0nbardbleiδBn.
vectorE(vectorr,t) =
3summationdisplay
n=1
vectorenbardblE0nbardble?vectorkI·vectorrei(vectorkR·vectorr?ωt+δEn)
vectorB(vectorr,t) =
3summationdisplay
n=1
vectorenbardblB0nbardble?vectorkI·vectorrei(vectorkR·vectorr?ωt+δBn)
bardblE0nbardble?vectorkI·vectorr a180a62a124a114a221a51na144a149a169a254a27a8a204.
bardblB0nbardble?vectorkI·vectorr a180a94a97a65a114a221a51na144a149a169a254a27a8a204.
vectorkR·vectorr?ωt +δEn a180a62a124a114a221a51na144a149a169a254a27a160a131.
vectorkR·vectorr?ωt +δBn a180a94a97a65a114a221a51na144a149a169a254a27a160a131.
190/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
vectorE(vectorr,t) =
3summationdisplay
n=1
vectorenbardblE0nbardble?vectorkI·vectorrei(vectorkR·vectorr?ωt+δEn)
a51a137a189a158a143t,a31a131a161a134vectorkRa82a134.
vectorkR·vectorr = vectorkR·(vectorr +vectorr⊥) vectorr⊥·vectorkR = 0
a31a131a161a177a132a199vφ = ω/kRa247vectorkRa144a149a36a196.
ωt1+vectorkR·vectorr1 =?ωt2+vectorkR·vectorr2 → 1 =
vectorkR
ω ·
(vectorr2?vectorr1)
t2?t1 =
vectorkR
ω ·vectorvφ
a8vectorkI = 0a158,a31a131a161a210a180a31a138a161.
a8vectorkInegationslash= 0a158,a31a138a171a141a100 vectorkR·vectorr =a126a234a218 vectorkI·vectorr =a126a234a233a220a166
a41a251a189,a152a132a180a152a94a130,a247vectorkIa144a149a167a8a204a80a126a129a175a167a135vectorkIa144
a149a167a8a204a79a114a129a175,
191/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
a197a165:
vectorkRa27a144a149a135a78a27a180a160a131a27a68a194a144a149,vectorkIa27a144a149a135a78a27a180a8
a204a27a129a140a80a126a144a149.
k2 = ω2μepsilon1+ iωμγ = vectork·vectork = k2R?k2I + 2ivectorkR·vectorkI
k2R?k2I = μepsilon1ω2 vectorkR·vectorkI = 12ωμγ
vectorkR·vectorkI >0? a247a160a131a68a194a144a149a197a111a180a80a126a27a156
kIa147a76a62a94a197a8a204a27a80a126.a143a135a78a10a212a159a233a62a94a197a27a225a194.
a233a253a14a48a159,γ = 0,kI = 0,a233a253a14a48a159,a62a94a197a195a225a194,a62a94
a85a254a216a209a209.
a197a204a252a150a6a1381/ea27 a68a194a229a108a143a66a223a29a221δ = 1k
I cosθ
.
kIa134ωa107a39,a135a78a249a171a212a159 a233a44a10a170a199a27a27a62a62a94a197a225a194a27a114a102.
192/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
a8a204,a69a8a204vectorE0,vectorB0a247a118 vectork·vectorE0 = 0 vectorB0 = vectorkω×vectorE0
kI = 0,vectork,vectorE0,vectorB0 a164a109a195a39a88,B0 = kωE0,vectorBa134vectorEa211a160a131.
kInegationslash= 0,a109a195a39a88a216a164a225,vectorBa134vectorEa216a211a160a131.
vectorE0 = vectorE0R + ivectorE0I vectorB0 = vectorB0R + ivectorB0I
vectorkR·vectorE0R?vectorkI·vectorE0I = 0 vectorkR·vectorE0I +vectorkI·vectorE0R = 0
vectorB0R = vectorkR
ω ×
vectorE0R?vectorkI
ω ×
vectorE0I vectorB0I = vectorkR
ω ×
vectorE0I + vectorkI
ω ×
vectorE0R
vectorE0R =
3summationdisplay
n=1
bardblE0nbardblvectoren cosδEn vectorE0I =
3summationdisplay
n=1
bardblE0nbardblvectoren sinδEn
vectorB0R =
3summationdisplay
n=1
bardblB0nbardblvectoren cosδBn vectorB0I =
3summationdisplay
n=1
bardblB0nbardblvectoren sinδBn
193/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
a160a8,vectorka143a162,za182a143vectorka144a149(a62a94a197a247za182a182a20a20a144a149a68a194)
E0z = 0
vectorE(vectorr,t) =bardblE0xbardblei(kz?ωt+δEx)vectore1 +bardblE0ybardblei(kz?ωt+δEy)vectore2
a51a44a152za18a189a189a27a27xya178a161a254,a62a124a114a221a165a254a145a158a109a216a228a67a122
δEx?δEy = 0,pi a130a160a8
bardblE0xbardbl=bardblE0ybardbl
δEx?δEy = pi2 a134a94
δEx?δEy = 3pi2 a109a94
δEx?δEya216a27a189 a103a44a49
a152a132,a253a11
194/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
a85a254a151a221,a85a54a151a221:
We = 12epsilon1vectorER·vectorER Wm = 12μvectorBR·vectorBR vectorS = 1μvectorER×vectorBR
vectorER·vectorER =
3summationdisplay
n=1
ERnERn =
3summationdisplay
n=1
bardblE0nbardbl2e?2vectorkI·vectorr cos2(vectorkR·vectorr?ωt +δEn)
vectorBR·vectorBR =
3summationdisplay
n=1
BRnBRn =
3summationdisplay
n=1
bardblB0nbardbl2e?2vectorkI·vectorr cos2(vectorkR·vectorr?ωt +δBn)
vectorER×vectorBR =
3summationdisplay
n,m,p=1
epsilon1nmpERnBRmvectorep
=
3summationdisplay
n,m,p=1
bardblE0nbardblbardblB0mbardble?2vectorkI·vectorrvectorep cos(vectorkR·vectorr?ωt +δEn)
×cos(vectorkR·vectorr?ωt +δBn)
195/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
a85a254a151a221,a85a54a151a221:
cos2(vectorkR·vectorr?ωt +δEn) = 12[1 + cos2(vectorkR·vectorr?ωt +δEn)] = 12
cos2(vectorkR·vectorr?ωt +δBn) = 12[1 + cos2(vectorkR·vectorr?ωt +δBn)] = 12
cos(vectorkR·vectorr?ωt +δEn)cos(vectorkR·vectorr?ωt +δBn)
= 12[cos(2vectorkR·vectorr?2ωt +δEn +δBn) + cos(δEn?δBn)] = 12 cos(δEn?δBn)
vectorER·vectorER = 1
2
3summationdisplay
n=1
bardblE20nbardble?2vectorkI·vectorr = 12Re(vectorE·vectorE?)
vectorBR·vectorBR = 1
2
3summationdisplay
n=1
bardblB20nbardble?2vectorkI·vectorr = 12Re(vectorB·vectorB?)
vectorER×vectorBR = 1
2
3summationdisplay
n,m,p=1
epsilon1nmpbardblE0nbardblbardblB0mbardblvectorepe?2vectorkI·vectorr cos(δEn?δBn)
= 12Re(vectorE×vectorB?)
196/384
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a189a21a197a196a144a167a57a178a161a197a41,a189a21a62a94a197a27a178a161a197a41
a85a254a151a221,a85a54a151a221:
We = 12epsilon1vectorER·vectorER = 14epsilon1Re(vectorE·vectorE?)
Wm = 12μvectorBR·vectorBR = 14μRe(vectorB·vectorB?) = 14μRebracketleftbig(
vectork
ω×
vectorE)·(vectork?
ω ×
vectorE?)bracketrightbig
= 14μω2Rebracketleftbig(vectork·vectork?)(vectorE·vectorE?)?(vectork·vectorE?)(vectork?·vectorE)bracketrightbig
a233a253a14a78,vectork = vectork?,vectork·vectorE = 0→ We = Wm a233a233a19a19a78,Wenegationslash= Wm
vectorS = vectorER×vectorBR
μ =
1
2μRe(
vectorE×vectorB?)
= 12μRe(vectorE?×vectorB) = 12μRe[vectorE?×(
vectork
ω×
vectorE)]
= 12μωRe[vectork(vectorE·vectorE?)?vectorE(vectork·vectorE?)]
197/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a62a46a94a135
a92a19a197,vectorA1 = vectorA10ei(vectork1·vectorr?ωt) a223a19a197,vectorA2 = vectorA20ei(vectork2·vectorr?ω2t)
a135a19a197,vectorA3 = vectorA30ei(vectork3·vectorr?ω3t) vectorA?(vectorE,vectorH)
a0
a0a0a18
a64
a64a64a73
a45zθ3
θ1
epsilon11,μ1,γ1 = 0 epsilon12,μ2,γ2
vectork1 = k1(sinθ1vectore1 + cosθ1vectore3) vectork3 = k3xvectore1+ k3yvectore2+ k3zvectore3 vectork2 = vectork2R+ivectork2I
k1 = ωv =√μ1epsilon11ω k3 = ω3v =√μ1epsilon11ω3
braceleftbigg k2
2R?k
2
2I = μ2epsilon12ω
2
2vector
k2R·vectork2I = 12ω2μ2γ2
a174a127,vectork1,ω,vectorA10,μ1,epsilon11,γ1,μ2,epsilon12,γ2
a73a166,vectork2,vectork3,ω2,ω3,vectorA20,vectorA30
a62a94a135,vectorn×(vectorE2?vectorE1) = 0,vectorn×(vectorH2?vectorH1) =vectoric
vectorn·(vectorD2?vectorD1) = σf,vectorn·(vectorB2?vectorB1) = 0
vectoric = 0,σf = 0 vectorn×(vectorA1 + vectorA3)vextendsinglevextendsingle
z=0 = vectorn×
vectorA2vextendsinglevextendsingle
z=0
vectorn×[vectorA10ei(k1xx+k1yy?ωt) + vectorA30ei(k3xx+k3yy?ω3t)] = vectorn×vectorA20ei(k2xx+k2yy?ω2t)?
ω = ω2 = ω3
k1x = k2x = k3x = k1 sinθ1 =√μ1epsilon11 sinθ1
k1y = k2y = k3y = 0
vectorn×(vectorA10 + vectorA30) = vectorn×vectorA20
198/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a197a165
a0
a0a0a18
a64
a64a64a73
a45zθ3
θ1
epsilon11,μ1,γ1 = 0 epsilon12,μ2,γ2
a135a19a197:
ω3 = ω,k3 = k1
vectork3 = k3 sinθ3vectore1?k3 cosθ3vectore3
k3x = k1x → k3 sinθ3 = k1 sinθ1 → θ3 = θ1 a135a19a189a198
a223a19a197:
ω2 = ω,k2Rx =√μ1epsilon11ωsinθ1,k2Ix = 0,k2Ry = k2Iy = 0
k22R?k22I = μ2epsilon12ω22,vectork2R·vectork2I = 12ω2μ2γ2
k22Rz?k22Iz = k22R?k22Rx?k22Ry?k22I + k22Ix + k22Iy
= μ2epsilon12ω2?μ1epsilon11ω2 sin2θ1
k2Rzk2Iz = vectork2R·vectork2I?k2Rxk2Ix?k2Ryk2Iy = 12ωμ2γ2
199/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a197a165
a223a19a197:
ω2 = ω,k2Rx =√μ1epsilon11ωsinθ1,k2Ix = 0,k2Ry = k2Iy = 0
k22Rz?k22Iz = μ2epsilon12ω2?μ1epsilon11ω2 sin2θ1 k2Rzk2Iz = 12ωμ2γ2
k2Rz
= 1√2
radicalbigg
(μ2epsilon12?μ1epsilon11 sin2θ1)ω2 +
radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1)2ω4 +μ22γ22ω2
k2Iz
= ωμ2γ2√
2
radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1)ω2 +radicalbig(μ2epsilon12?μ1epsilon11 sin2θ1)2ω4 +μ22γ22ω2
200/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a197a165～a253a14a223a19a48a159
a0
a0a0a18
a64
a64a64a73
a45zθ3
θ1
epsilon11,μ1,γ1 = 0 epsilon12,μ2,γ2 = 0
γ2 = 0 a242a19a199,n≡√μrepsilon1r = c√μepsilon1
k2 =√μ2epsilon12ω = n2c ω k2k
1
= n2n
1
k2Rz = ω√2
radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1) +bardblμ2epsilon12?μ1epsilon11 sin2θ1bardbl
k2Iz = ωμ2γ2√2radicalbig(μ
2epsilon12?μ1epsilon11 sin2θ1) +bardblμ2epsilon12?μ1epsilon11 sin2θ1bardbl
μ2epsilon12≥μ1epsilon11→ n2≥n1,a49a213→a49a151
k2Rz = ω
radicalBig
μ2epsilon12?μ1epsilon11 sin2θ1 k2Iz = 0 a195a80a126
vectork2 = k2 sinθ2vectore1 + k2 cosθ2vectore3 = ω√μ2epsilon12(sinθ2vectore1 + cosθ2vectore3)
→ ω√μ2epsilon12 sinθ2 = ω√μ1epsilon11 sinθ1→n2 sinθ2 = n1 sinθ1 a242a19a189a198
sinθ2 = n1n
2
sinθ1≤1
201/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a197a165～a253a14a223a19a48a159
a0
a0a0a18
a64
a64a64a73
a45zθ3
θ1
epsilon11,μ1,γ1 = 0 epsilon12,μ2,γ2 = 0
γ2 = 0 a242a19a199,n≡√μrepsilon1r = c√μepsilon1
k2 =√μ2epsilon12ω = n2c ω k2k
1
= n2n
1
k2Rz = ω√2
radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1) +bardblμ2epsilon12?μ1epsilon11 sin2θ1bardbl
k2Iz = ωμ2γ2√2radicalbig(μ
2epsilon12?μ1epsilon11 sin2θ1) +bardblμ2epsilon12?μ1epsilon11 sin2θ1bardbl
μ2epsilon12 <μ1epsilon11 sinθ1≤
radicalbiggμ
2epsilon12
μ1epsilon11 =
n2
n1,a49a151→a49a213
k2Rz = ω
radicalBig
μ2epsilon12?μ1epsilon11 sin2θ1 k2Iz = 0 a195a80a126
vectork2 = k2 sinθ2vectore1 + k2 cosθ2vectore3 = ω√μ2epsilon12(sinθ2vectore1 + cosθ2vectore3)
→ ω√μ2epsilon12 sinθ2 = ω√μ1epsilon11 sinθ1→n2 sinθ2 = n1 sinθ1 a242a19a189a198
sinθ2 = n1n
2
sinθ1≤1
202/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a197a165～a253a14a223a19a48a159
a0
a0a0a18
a64
a64a64a73
a45zθ3
θ1
epsilon11,μ1,γ1 = 0 epsilon12,μ2
γ2 = 0
μ2epsilon12 <μ1epsilon11 sinθ1 >
radicalbiggμ
2epsilon12
μ1epsilon11 =
n2
n1,a49a151→a49a213
k2Rz = ω√2
radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1) +bardblμ2epsilon12?μ1epsilon11 sin2θ1bardbl= 0
√2k
2Iz =
ωμ2γ2radicalbig
(μ2epsilon12?μ1epsilon11 sin2θ1) +bardblμ2epsilon12?μ1epsilon11 sin2θ1bardbl =
0
0
= ωμ2γ2radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1)ω2 +radicalbig(μ2epsilon12?μ1epsilon11 sin2θ1)2ω4 +μ22γ22ω2
= ωμ2γ2radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1)ω2+(μ1epsilon11 sin2θ1?μ2epsilon12)ω2[1+ μ22γ22ω22(μ
1epsilon11 sin2 θ1?μ2epsilon12)2ω4
]
vextendsinglevextendsingle
γ2→0
=√2ω
radicalBig
μ1epsilon11 sin2θ1?μ2epsilon12 vectork2I = ω
radicalBig
μ1epsilon11 sin2θ1?μ2epsilon12vectore3
vectork2R =√μ1epsilon11ωsinθ1vectore1 v2φ = v2φ,x = ω
kR =
1√
μ1epsilon11 sinθ1 =
v1
sinθ1
vectorA2=vectorA20e?ωz√μ1epsilon11sinθ1?μ2epsilon12ei(√μ1epsilon11ωsinθ1x?ωt) a28a135a19a27a28a28a91a91a252a171 a162a126a252a171
203/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a197a165～a19a19a62a62a223a19a48a159
a0
a0a0a18
a64
a64a64a73
a45zθ3
θ1
epsilon11,μ1,γ1 = 0 epsilon12,μ2,γ2
γnegationslash= 0 vectorA2 = vectorA20e?k2Izzei(k2Rxx+k2Rzz?ωt)
k2Rx =√μ1epsilon11ωsinθ1,k2Ix = 0,k2Ry = k2Iy = 0
k2Rz
= 1√2
radicalbigg
(μ2epsilon12?μ1epsilon11 sin2θ1)ω2 +
radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1)2ω4 +μ22γ22ω2
k2Iz
= ωμ2γ2√
2
radicalBig
(μ2epsilon12?μ1epsilon11 sin2θ1)ω2 +radicalbig(μ2epsilon12?μ1epsilon11 sin2θ1)2ω4 +μ22γ22ω2
a223a19a197a31a131a161a247vectork2R = k2Rxvectore1 + k2Rzvectore3a144a149a68a194,a170a199ω,a132a199v =
ω
k2R =
ω√
k22Rx+k22RZ,a223a19a197a107a80a126,a144a149a51za144a149.
204/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a8a204～a92a19a197a62a124a160a117a92a19a161
a0
a0a0a18
a64
a64a64a73
a45
a54
z
x
y
θ
vectorE10 θ
epsilon11,μ1,γ1 = 0 epsilon12,μ2,γ2
a102a114
a102a114a64a64a73
braceleftbiggvectork
3·vectorE30 = 0vector
k2·vectorE20 = 0 →
braceleftbigg k
3xE30x + k3zE30z = 0
k2xE20x + k2zE20z = 0
→
braceleftbigg E
30z =?k3xk3zE30x
E20z =?k2xk
2z
E20x
braceleftBigg vector
B30 = vectork3ω ×vectorE30
vectorB20 = vectork2
ω ×
vectorE20
vectorE10 = E10(vectore1 cosθ1?vectore3 sinθ1) vectorB10 = vectork1
ω ×
vectorE10 = k1
ωE10vectorey
vectorE10t + vectorE30t = vectorE20t vectorH10t + vectorH30t = vectorH20t
braceleftbigg E
30y = E20y?
1
μ1B30x =
1
μ2B20x?
braceleftbigg E
10 cosθ1 + E30x = E20x?
1
μ1
k1
ω E10 +
1
μ1B30y =
1
μ2B20y?
205/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a8a204～a92a19a197a62a124a160a117a92a19a161
braceleftbigg E
30y = E20y?
1
μ1B30x =
1
μ2B20x?
braceleftbigg E
10 cosθ1 + E30x = E20x?
1
μ1
k1
ω E10 +
1
μ1B30y =
1
μ2B20y?
→ 1μ
1
(
vectork3
ω ×
vectorE30)x = 1
μ2(
vectork2
ω ×
vectorE20)x
→ 1μ
1
(k3yE30z?k3zE30y) = 1μ
2
(k2yE20z?k2zE20y)
→ 1μ
1
k3zE30y = 1μ
2
k2zE20y?→E30y = E20y = 0?→B30x = B20x = 0
vectore2bardbl=k2zk
2
vectore1?k2xk
2
vectore3⊥vectork2 vectore3bardbl=k3zk
3
vectore1?k3xk
3
vectore3=?cosθ1vectore1?sinθ1vectore3⊥vectork3
vectorE20 = E20xvectore1 + E20zvectore3 = E20xvectore1?k2x
k2zvectore3E20x =vectore2bardbl
k2
k2zE20x
vectorE30=E30xvectore1+E30zvectore3=E30xvectore1?k3x
k3zvectore3E30x=vectore3bardbl
k3
k3zE30x=?
vectore3bardbl
cosθ1E30x
206/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a8a204～a92a19a197a62a124a160a117a92a19a161
braceleftbigg E
30y = E20y?
1
μ1B30x =
1
μ2B20x?
braceleftbigg E
10 cosθ1 + E30x = E20x?
1
μ1
k1
ω E10 +
1
μ1B30y =
1
μ2B20y?
→ 1μ
2
(
vectork2
ω ×
vectorE20)y? 1
μ1(
vectork3
ω ×
vectorE30)y = 1
μ1
k1
ωE10
→ 1μ
2
(k2zE20x?k2xE20z)? 1μ
1
(k3zE30x?k3xE30z) = k1μ
1
E10
→ 1μ
2
(k2z + k
2
2x
k2z)E20x?
1
μ1(k3z +
k23x
k3z)E30x =
1
μ1k1E10
→ 1μ
2
k22
k2zE20x?
1
μ1
k23
k3zE30x =
1
μ1k1E10
+?→ μ1μ
2
k22
k1k2zE20x +
1
cosθ1E30x = E10 E20x?E30x = E10 cosθ1
→E20x = 2cosθ11 + μ
1k22 cosθ1
μ2k1k2z
E10 E30x = 1?
μ1k22 cosθ1
μ2k1k2z
1 + μ1k22 cosθ1μ
2k1k2z
cosθ1E10
207/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a8a204～a92a19a197a62a124a160a117a92a19a161
→E20x = 2cosθ11 + μ
1k22 cosθ1
μ2k1k2z
E10 E30x = 1?
μ1k22 cosθ1
μ2k1k2z
1 + μ1k22 cosθ1μ
2k1k2z
cosθ1E10
vectorE20bardbl = 2cosθ1
k2z
k2 +
k2μ1
k1μ2 cosθ1
E10bardblvectore2bardbl vectorE30bardbl =?
k2z
k2 +
k2μ1
k1μ2 cosθ1
k2z
k2 +
k2μ1
k1μ2 cosθ1
E10bardblvectore3bardbl
γ2 = 0,μ2 = μ1→k2z = k2 cosθ2; k2x = k2 sinθ2
→vectore2bardbl = cosθ2vectore1?sinθ2vectore3; k2/k1 = n2/n1 = sinθ1/sinθ2
vectorE20bardbl = 2cosθ1
cosθ2 + sinθ1sinθ
2
cosθ1E10bardblvectore2bardbl =
2cosθ1 sinθ2
1
2 sin2θ2 +
1
2 sin2θ1
E10bardblvectore2bardbl
= 2cosθ1 sinθ2sin(θ
1 +θ2)cos(θ1?θ2)
E10bardblvectore2bardbl
vectorE30bardbl =?sinθ2 cosθ2 + sinθ1 cosθ1
sinθ2 cosθ2 + sinθ1 cosθ1 E10bardblvectore3bardbl =
sin2θ2 + sin2θ1
sin2θ2 + sin2θ1 E10bardblvectore3bardbl
= tan(θ1?θ2)tan(θ
1 +θ2)
E10bardblvectore3bardbl
208/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a8a204～a92a19a197a62a124a160a117a92a19a161
a0
a0a0a18
a64
a64a64a73
a17a17
a17a17a17a51
a45
a54
z
x
y
θ1 θ2
vectorE10
vectore3bardbl
vectore2bardbl
θ1
epsilon11,μ1=μ2,γ1= 0 epsilon12,γ2= 0
a102a114
a102a114
a102a114
a64a64a73
a0a9
a102a114a74a74a93γ
2 = 0,μ2 = μ1 k2/k1 = n2/n1 = sinθ1/sinθ2
vectore2bardbl = cosθ2vectore1?sinθ2vectore3 vectore3bardbl =?cosθ1vectore1?sinθ1vectore3
vectorE30bardbl = tan(θ1?θ2)
tan(θ1 +θ2)E10bardblvectore3bardbl = E10bardblvectore3bardbl×
n2?n1
n2+n1 θ1→0
1 θ1→ pi2
n1 = n2 sinθ2 ≤ n2
a49a213→a49a151(n2 >n1)? vectorE30bardbl
a178a49a117 θ1 → 0
a135a178a49a117 θ1 → pi2
vectore3bardbl? a140a197a155a148
vectorE20bardbl = 2cosθ1 sinθ2
sin(θ1 +θ2)cos(θ1?θ2)E10bardblvectore2bardbl = E10bardblvectore2bardbl×
n1
n2+n1 θ1→0
0 θ1→ pi2
a49a213→a49a151 (n2 >n1)? vectorE20bardbla178a49a117vectore2bardbl? a195a140a197a155a148
209/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a8a204～a92a19a197a62a124a82a134a92a19a161
a0
a0a0a18
a64
a64a64a73
a45
a54
z
x
y
θ
vectorE10
vectorB10
θ
epsilon11,μ1,γ1 = 0 epsilon12,μ2,γ2
a102a114
a102a114
a64a64a82
braceleftbiggvectork
3·vectorE30 = 0vector
k2·vectorE20 = 0 →
braceleftbigg k
3xE30x + k3zE30z = 0
k2xE20x + k2zE20z = 0
→
braceleftbigg E
30z =?k3xk3zE30x
E20z =?k2xk
2z
E20x
braceleftBigg vector
B30 = vectork3ω ×vectorE30
vectorB20 = vectork2
ω ×
vectorE20
vectorE10 = E10vectore2 vectorB10 = vectork1
ω ×
vectorE10 = k1
ωE10(?vectore1 cosθ1 +vectore3 sinθ1)
vectorE10t + vectorE30t = vectorE20t vectorH10t + vectorH30t = vectorH20t
braceleftbigg E
30x = E20x?
1
μ1B30y =
1
μ2B20y?
braceleftbigg E
10y + E30y = E20y?
1
μ1(B10x + B30x) =
1
μ2B20x?
210/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a8a204～a92a19a197a62a124a82a134a92a19a161
vectorE20⊥ = 2
1 + μ1k2zμ
2k1 cosθ1
E10⊥vectore2 vectorE30⊥ = 1?
μ1k2z
μ2k1 cosθ1
1 + μ1k2zμ
2k1 cosθ1
E10⊥vectore2
γ2 = 0,μ2 = μ1→k2z = k2 cosθ2 = k1 sinθ1 cosθ2/sinθ2
vectorE30⊥ = 1?
sinθ1 cosθ2
sinθ2 cosθ1
1 + sinθ1 cosθ2sinθ
2 cosθ1
E10⊥vectore2 = E10⊥vectore2×
n1?n2
n1+n2 θ1→0
1 θ1→ pi2
a49a213→a49a151? n2 >n1? vectorE30bardbl a135a178a49a117vectore2 a140a197a155a148
vectorE20⊥ = 2
1 + sinθ1 cosθ2sinθ
2 cosθ1
E10⊥vectore2 = E10⊥vectore2×
2n2
n1+n2 θ1→0
0 θ1→ pi2
a49a213→a49a151? n2 >n1? vectorE20bardbl a178a49a117vectore2 a195a140a197a155a148
211/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a197a27a8a204～a152a132a156a185
vectork1 = k1(sinθ1vectore1 + cosθ1vectore3) vectork3 = k3(sinθ1vectore1?cosθ1vectore3) k1 = k3
vectorE10 = vectorE10bardbl + vectorE10⊥ = E10bardbl(vectore1 cosθ1?vectore3 sinθ1) + E10⊥vectore2
= E10bardbl(vectore2×
vectork1
k1) + E10⊥vectore2 vectore2=
vectore3×vectork1
|vectore3×vectork1|=
vectore3×vectork3
|vectore3×vectork3|
vectorE30 = vectorE30bardbl + vectorE30⊥
= E10bardbl(vectore1 cosθ1 +vectore3 sinθ1)
k2z
k2?
μ1k2
μ2k1 cosθ1
k2z
k2 +
μ1k2
μ2k1 cosθ1
+ E10⊥vectore21?
μ1k2z
μ2k1 cosθ1
1 + μ1k2zμ
2k1 cosθ1
= E10bardbl(vectore2×
vectork3
k3)
k2zk
2
+ μ1k2μ
2k1
cosθ1
k2z
k2 +
μ1k2
μ2k1 cosθ1
+ E10⊥vectore21?
μ1k2z
μ2k1 cosθ1
1 + μ1k2zμ
2k1 cosθ1
γ2=0,μ2=μ1======= E
10bardbl(vectore2×
vectork3
k3)
tan(θ1?θ2)
tan(θ1 +θ2) + E10⊥vectore2
sin(θ2?θ1)
sin(θ1 +θ2)
θ1=0==== n2?n1
n2 + n1[E10bardbl(vectore2×
vectork3
k3)?E10⊥vectore2] a109a94?a134a94
212/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a223a19a197a27a8a204～a152a132a156a185
vectork1 = k1(sinθ1vectore1 + cosθ1vectore3) vectork3 = k3(sinθ1vectore1?cosθ1vectore3) k1 = k3
vectorE10 = vectorE10bardbl + vectorE10⊥ = E10bardbl(vectore1 cosθ1?vectore3 sinθ1) + E10⊥vectore2
= E10bardbl(vectore2×
vectork1
k1) + E10⊥vectore2
vectorE20 = vectorE20bardbl + vectorE20⊥
= E10bardbl(vectore1k2zk
2
vectore3k2xk
2
) 2cosθ1k
2z
k2 +
μ1k2
μ2k1 cosθ1
+ E10⊥vectore2 21 + μ
1k2z
μ2k1 cosθ1
= E10bardbl(vectore2×
vectork2
k2)
2cosθ1
k2z
k2 +
μ1k2
μ2k1 cosθ1
+ E10⊥vectore2 21 + μ
1k2z
μ2k1 cosθ1
γ2=0,μ2=μ1======= E
10bardbl(vectore2×
vectork2
k2)
2cosθ1 sinθ2
sin(θ1+θ2)cos(θ1?θ2) + E10⊥vectore2
2sinθ2 cosθ1
sin(θ1+θ2)
θ1=0==== n1
n2 + n1[E10bardbl(vectore2×
vectork2
k2) + E10⊥vectore2] a134a94?a134a94a182 a109a94?a109a94
213/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a85a54
vectorS = 1
2μωRe[
vectork(vectorE·vectorE?)?vectorE(vectork·vectorE?)]
vectorS1 = 1
2μ1ωRe[
vectork1(vectorE10·vectorE?10)] = 1
2μ1ω
vectork1(vectorE10bardbl·vectorE?10bardbl + vectorE10⊥·vectorE?10⊥)
= vectorS1bardbl +vectorS1⊥
vectork1 = vectork?1 vectork1·vectorE10 = 0
vectorS1bardbl = vectork1
2μ1ω(
vectorE10bardbl·vectorE?10bardbl) = vectork1
2μ1ωE10bardblE
10bardbl
vectorS1⊥ = vectork1
2μ1ω(
vectorE10⊥·vectorE?10⊥) = vectork1
2μ1ωE10⊥E
10⊥
214/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a85a54
vectorS = 1
2μωRe[
vectork(vectorE·vectorE?)?vectorE(vectork·vectorE?)]
vectorS1 = vectorS1bardbl +vectorS1⊥
vectorS3 = 1
2μ1ωRe[
vectork3(vectorE30·vectorE?30)] = 1
2μ1ω
vectork3(vectorE30bardbl·vectorE?30bardbl + vectorE30⊥·vectorE?30⊥)
= vectorS3bardbl +vectorS3⊥
vectork3 = vectork?3 vectork3·vectorE30 = 0
vectorS3bardbl = vectork3
2μ1ω(
vectorE30bardbl·vectorE?30bardbl) = vectork3
2μ1ωE30bardblE
30bardbl
vectorS3⊥ = vectork3
2μ1ω(
vectorE30⊥·vectorE?30⊥) = vectork3
2μ1ωE30⊥E
30⊥
215/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a85a54
vectorS = 1
2μωRe[
vectork(vectorE·vectorE?)?vectorE(vectork·vectorE?)]
vectorS1 = vectorS1bardbl +vectorS1⊥ vectorS3 = vectorS3bardbl +vectorS3⊥
vectorS2 = 1
2μ2ωRe[
vectork2(vectorE2·vectorE?2)?vectorE2(vectork2·vectorE?2)]
= 12μ
2ω
Re[vectork2(vectorE2bardbl·vectorE?2bardbl + vectorE2⊥·vectorE?2⊥)?(vectorE2bardbl + vectorE2⊥)(vectork2·vectorE?2bardbl)]
= vectorS2bardbl +vectorS2⊥
vectorS2bardbl = 1
2μ2ω{
vectork2R(vectorE2bardbl·vectorE?2bardbl)?Re(vectorE2bardbl + vectorE2⊥)(vectork2·vectorE2bardbl)} a195a101a730
vectorS2⊥ = vectork2R
2μ2ω(
vectorE2⊥·vectorE?2⊥) a195a101a730
216/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a85a54
a18a46a161(z = 0)a254a152a135a2a161a3?vectorS =?Svectore3
D⊥≡
vectorS2⊥·?vectorS
vectorS1⊥·?vectorS
vextendsinglevextendsingle
z=0 =
S2⊥z
S1⊥z
vextendsinglevextendsingle
z=0 =
k2zR
2μ2ω(
vectorE2⊥·vectorE?2⊥)
k1z
2μ1ω(
vectorE10⊥·vectorE?10⊥)
= μ1k2zRμ
2k1 cosθ1
vectorE20⊥·vectorE?20⊥
E10⊥E?10⊥ e
2vectork2I·vectorrvextendsinglevextendsingle
z=0 =
μ1k2zR
μ2k1 cosθ1
vextenddoublevextenddouble 2
1 + μ1k2zμ
2k1 cosθ1
vextenddoublevextenddouble2
R⊥≡?
vectorS3⊥·?vectorS
vectorS1⊥·?vectorS
vextendsinglevextendsingle
z=0 =?
S3⊥z
S1⊥z
vextendsinglevextendsingle
z=0 =?
k3z
2μ1ω(
vectorE30⊥·vectorE?30⊥)
k1z
2μ1ω(
vectorE10⊥·vectorE?10⊥)
=
vectorE30⊥·vectorE?30⊥
E10⊥E?10⊥ =
vextenddoublevextenddouble1? μ1k2zμ
2k1 cosθ1
1 + μ1k2zμ
2k1 cosθ1
vextenddoublevextenddouble2
217/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a85a54
R⊥ + D⊥ =
4μ1k2zR
μ2k1 cosθ1 + (1?
μ1k2z
μ2k1 cosθ1)(1?
μ1k?2z
μ2k1 cosθ1)
bardbl1 + μ1k2zμ
2k1 cosθ1
bardbl2
=
2μ1
μ2k1 cosθ1(k2z + k
2z) + 1?
μ1
μ2k1 cosθ1(k2z + k
2z) +
μ21k2zk?2z
μ22k21 cos2 θ1
bardbl1 + μ1k2zμ
2k1 cosθ1
bardbl2
= 1
(vectorS2⊥?vectorS3⊥)·?vectorS = (D⊥ + R⊥)vectorS1⊥·?vectorS = vectorS1⊥·?vectorS
a108a92a19a197a54a92?Sa27⊥a62a94a85a254a31a117a108?Sa54a209a27 ⊥a27a135a19a197a218a223
a19a197a62a94a85a254a27a27a83a83a92.
218/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a135a19a223a19a197a27a85a54
D⊥ = μ1k2zRμ
2k1 cosθ1
vextenddoublevextenddouble 2
1 + μ1k2zμ
2k1 cosθ1
vextenddoublevextenddouble2 R
⊥ =
vextenddoublevextenddouble1? μ1k2zμ
2k1 cosθ1
1 + μ1k2zμ
2k1 cosθ1
vextenddoublevextenddouble2
Dbardbl =
4μ1 cosθ1Re(k2k?2zk?
2
)
μ2k1vextenddoublevextenddoublek2zk
2
+ k2μ1k
1μ2
cosθ1vextenddoublevextenddouble2
Rbardbl = vextenddoublevextenddouble
k2z
k2?
k2μ1
k1μ2 cosθ1
k2z
k2 +
k2μ1
k1μ2 cosθ1
vextenddoublevextenddouble
a126,γ2 = 0,μ1=μ2,k2=k1 sinθ1/sinθ2,k2z=k2 cosθ2=k1 sinθ1 cothθ2
Rωbardbl = vextenddoublevextenddouble?cosθ2 +
sinθ1
sinθ2 cosθ1
cosθ2 + sinθ1sinθ
2
cosθ1
vextenddoublevextenddouble2 = vextenddoublevextenddoublesin2θ1?sin2θ2
sin2θ1 + sin2θ2
vextenddoublevextenddouble2
= vextenddoublevextenddoublesin(θ1?θ2)cos(θ1 +θ2)sin(θ
1 +θ2)cos(θ1?θ2)
vextenddoublevextenddouble2 = tan2(θ1?θ2)
tan2(θ1 +θ2)
θ1+θ2=pi2===?0
a217a86a100a65a189a198
Rω⊥ = vextenddoublevextenddouble?sinθ1 cothθ2 + cosθ1sinθ
1 cothθ2 + cosθ1
vextenddoublevextenddouble2 = bracketleftbig?sinθ1 cosθ2 + cosθ1 sinθ2
sinθ1 cosθ2 + cosθ1 sinθ2
bracketrightbig2
= sin
2(θ1?θ2)
sin2(θ1 +θ2)
θ1+θ2=pi2===?cos22θ
1
Dbardbl = 1?Rbardbl D⊥ = 1?R⊥
219/384
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220/384
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a62a94a197a51a46a161a254a27a135a19a218a242a19,a49a216
a62a94a197a108a253a152a19a20a212a159a76a161,a252a160a76a161a201a27a229a143
vectorf =?vectorn· arrowrighttophalfarrowrighttophalfJ =vectorfE +vectorfB
vectorfE = 1
2epsilon10E
2(vectorncos2θE +vectoreE sin2θE)
vectorfB = 1
2μ0B
2(vectorncos2θB +vectoreB sin2θB)
a92a19a197a62a94a124a27a144a149a216a40a189,a51a164a107a144a149a254a209a209a31a31a65a199a209a121:
cos2θ =
integraltext d?cos2θ
integraltext d? =
integraltext2pi
0 dφ
integraltextpi
0 dθsinθcos2θintegraltext
2pi
0 dφ
integraltextpi
0 dθsinθ
=?13
sin2θ =
integraltext d?sin2θ
integraltext d? =
integraltext2pi
0 dφ
integraltextpi
0 dθsinθsin2θintegraltext
2pi
0 dφ
integraltextpi
0 dθsinθ
= 0
vectorf =?vectorn
3(
1
2epsilon10E
2 + 1
2μ0H
2) =?vectorn
3W
221/384
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a20a8a110,a144a167a57a62a94a135
a251a19a78a8a164a27a152a189a62a46a83a27a27a62a62a94a197,a189a21a62a94a197a247a118a181
(?2 + k2)vectorE = 0?·vectorE = 0 vectorB =?iω?×vectorE
k2 = ω2μepsilon1
a62a46a254,γ >>1,a62a94a197a144a85a66a63a19a78a233a102,a51a108a76a161a65a135a66a223a29a221
a63,a162a83a254a174 a118a107a62a94a124
vectorn×vectorE = 0 vectorn×vectorH =vectoric vectorn·vectorD = σf vectorn·vectorB = 0
a51a19a78a76a161a254a62a124a114a221a82a134a117a76a161,a94a97a65a114a221a134a76a161a131a131.
222/384
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a20a8a110,a221a47a47a20a20a8a110
a139a73a6a58a192a51a221a47a20a8a110a27a152a135a14,a177a131a25a27a110a135a62a143a139a73a182.
u(x,y,z)a143Ex,Ey,Eza165a63a152a135
a0
a0a0a9
a45
a54
a0a0
a0a0
a0a0
z
x
y
L1 L2
L3
2u + k2u = 0 u(x,y,z) = X(x)Y(y)Z(z)
2u + k2u=YZ?
2X
x2 +XZ
2Y
y2 +XY
2Z
z2 +(k
2
x + k
2
y + k
2
z)XYZ= 0
d2X
dx2 +k
2
xX=0
d2Y
dy2 +k
2
yY=0
d2Z
dz2 +k
2
zZ=0 k
2=k2
x+k
2
y+k
2
z
u(x,y,z) = (C1 coskxx + D1 sinkxx)(C2 coskyy + D2 sinkyy)
×(C3 coskzz + D3 sinkzz)
x = 0,y = 0,z = 0 a63a62a46a94a135,
Ex y=0;z=0==== 0 Ex=(A1 coskxx+Aprime1 sinkxx)sinkyysinkzz
Ey x=0;z=0==== 0 Ey=sinkxx(A2 coskyy+Aprime2 sinkyy)sinkzz
Ez x=0;y=0==== 0 Ez=sinkxxsinkyy(A3 coskzz+Aprime3 sinkzz)
·vectorE = 0? Aprime1 = Aprime2 = Aprime3 = 0 kxA1 + kyA2 + kzA3 = 0
223/384
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a20a8a110,a221a47a47a20a20a8a110
a0
a0a0a9
a45
a54
a0a0
a0a0
a0a0
z
x
y
L1 L2
L3
Ex = A1 coskxxsinkyysinkzz Ey = A2 sinkxxcoskyysinkzz
Ez = A3 sinkxxsinkyycoskzz x = L1,y = L2,z = L3a63a27a62a94a135→
kx = mpiL
1
ky = npiL
2
kz = ppiL
3
m,n,p = 0,1,2,···
ω2μepsilon1 = k2 = k2x + k2y + k2z = pi2[(mL
1
)2 + ( nL
2
)2 + ( pL
3
)2]
·vectorE = 0 → kxA1 + kyA2 + kzA3 = 0
a137a189(m,n,p)a143a152a171a29a14a14a8a8a11,a252a135a213a225a160a8a197a47,a20a8a170a199a143
ωnmp = pi√μepsilon1
radicalbigg
( nL
1
)2 + (mL
2
)2 + ( pL
3
)2
λnmp = 2piω
mnp
√μepsilon1 = 2radicalBig
( nL
1
)2 + (mL
2
)2 + ( pL
3
)2
a62a94a124a160a117a110a83,a129a36a170a1540a62a94a197a143(110),(101),(011),a197a127a134a20a8a110a130a221a211a254a63.
a135a197a137a140,a230a94a228a107a55a225a57a161a27a20a8a110a23a41a112a170a8a11.
a49a198a137a140,a135a19a186a124a164a49a198a198a20a20a8a110a23a41a67a252a218a27a45a49a229.
224/384
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a62a94a197a27a27a189a189a149a68a194,a144a167a57a62a94a135braceleftbigg
×vectorE = iωvectorB
×vectorB =?iμepsilon1ωvectorE
braceleftbigg?·vectorE = 0
·vectorB = 0 vectorn×
vectorEvextendsinglevextendsingle
a62a46 = 0
(2)vectorE = [vectorE
z(x,y) + vectorEt(x,y)]eikzz?=?xy +
zvectorez
vectorB = [vectorBz(x,y) + vectorBt(x,y)]eikzz?xy =?
xvectorex +
yvectorey
braceleftbigg?iμepsilon1ωvectorE
z =?xy×vectorBt
iωvectorBz =?xy×vectorEt
braceleftbigg?iμepsilon1ωvectorE
t =?xy×vectorBz + ikzvectorez×vectorBt
iωvectorBt =?xy×vectorEz + ikzvectorez×vectorEtbraceleftbigg
xy·vectorEt + ikzEz = 0
xy·vectorBt + ikzBz = 0
a62a46(a43a57)a254a27a27a62a62a54:vectoric = vectorn×vectorH = 1μvectorn×vectorB = 1μvectorn×(vectorBt + vectorBz)ei(kzz?ωt)
a62a46(a43a57)a254a27a27a62a62a214,σf = vectorn·vectorD = epsilon1vectorn·vectorE = epsilon1vectorn·vectorEtei(kzz?ωt)
a41,Ez = Bz = 0 Ez = 0 Bz = 0
225/384
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a62a94a197a27a27a189a189a149a68a194,TEMa197braceleftbigg
xy·vectorEt = 0
xy×vectorEt = 0
braceleftbigg?
xy·vectorBt = 0
xy×vectorBt = 0
braceleftBigg vector
Et =?kzμepsilon1ωvectorez×vectorBt
ω
kz
vectorBt =vectorez×vectorEt
vectorn×vectorEtvextendsinglevextendsinglea62a46 = 0
vectorEt =xyφt?2xyφt = 0 φtvextendsinglevextendsingle
a62a46 = a126a234
a101a165a152,φt = a126a234a240a180a41→vectorEt = vectorBt = 0
a101a165a216a152,a101a83a9a62a46a62a179a131a211φt = a126a234a69a240a180a41.
a101a83a9a62a46a62a179a216a211,a19a145a183a62a175a75,za152a189 vectorE:a252a145a179a124,φta51a62a46a31a179
vectoric = 1
μvectorn×(
vectorBt + vectorBz)ei(kzz?ωt) = 1
μvectorn×(
kz
ωvectorez×
vectorEt)ei(kzz?ωt)
= kzμepsilon1vectorezvectorn·vectorEtei(kzz?ωt) =?kzμωvectorez?φt?nei(kzz?ωt)
σf = epsilon1vectorn·vectorEtei(kzz?ωt) = epsilon1?φt?nei(kzz?ωt) vectoric =? kzμepsilon1ωσfvectorez
226/384
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a62a94a197a27a27a189a189a149a68a194,TEa197
braceleftbigg?
xy·vectorEt = 0
xy×vectorEt = iωvectorBz
braceleftbigg?
xy·vectorBt =?ikzBz
xy×vectorBt = 0braceleftBigg
ω
kz
vectorBt =vectorez×vectorEt
vectorEt =?kz
μepsilon1ωvectorez×
vectorBt + i
μepsilon1ω?xy×
vectorBz vectorn×vectorEt
vextendsinglevextendsingle
a62a46 = 0
ω
kz
vectorBt =vectorez×(? kz
μepsilon1ωvectorez×
vectorBt + i
μepsilon1ω?xy×
vectorBz) = kz
μepsilon1ω
vectorBt + i
μepsilon1ω?xyBz
vectorBt = i
μepsilon1ω
1
ω
kz?
kz
μepsilon1ω
xyBz = ikzμepsilon1ω2?k2
z
xyBz
vectorEt =? kz
μepsilon1ω
ikz
μepsilon1ω2?k2zvectorez×?xyBz +
i
μepsilon1ω?xy×
vectorBz
=?iμepsilon1ω( k
2
z
μepsilon1ω2?k2z + 1)vectorez×?xyBz =
iω
μepsilon1ω2?k2zvectorez×?xyBz
ikz
μepsilon1ω2?k2z?
2
xyBz =?ikzBz → (?
2
xy +μepsilon1ω
2?k2
z)Bz = 0
227/384
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a62a94a197a27a27a189a189a149a68a194,TEa197braceleftbigg
xy·vectorEt = 0
xy×vectorEt = iωvectorBz
braceleftbigg?
xy·vectorBt =?ikzBz
xy×vectorBt = 0braceleftBigg
ω
kz
vectorBt =vectorez×vectorEt
vectorEt =?kz
μepsilon1ωvectorez×
vectorBt + i
μepsilon1ω?xy×
vectorBz vectorn×vectorEt
vextendsinglevextendsingle
a62a46 = 0
vectorBt = ikz
μepsilon1ω2?k2z?xyBz
vectorEt =?iω
μepsilon1ω2?k2zvectorez×?xyBz
(?2xy +μepsilon1ω2?k2z)Bz = 0
vectorn×vectorEtvextendsinglevextendsinglea62a46 =?iωμepsilon1ω2?k2
z
vectorn·?xyBzvextendsinglevextendsinglea62a46 = 0 →?Bz?n vextendsinglevextendsinglea62a46 = 0
kz =
radicalBig
μepsilon1ω2?k2x?k2y≥0 → ω≥k
2
x + k
2
y
μepsilon1 a31a142a170a199
vectoric = 1
μvectorn×(
vectorBt + vectorBz)ei(kzz?ωt) = 1
μ(
ikz
μepsilon1ω2?k2zvectorn×?xy +vectorez)Eze
i(kzz?ωt)
σf = epsilon1vectorn·vectorEtei(kzz?ωt) =?iωepsilon1μepsilon1ω2?k2
z
(vectorn×vectorez)·?xyBzei(kzz?ωt)
228/384
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a62a94a197a27a27a189a189a149a68a194,a221a47a197a19
a221a47a62a127a169a79a143a,b,a139a73a6a58a18a51a221a47a27a152a135a14,a252a131a25a27a62a169
a79a143xya182,a207a143a165a152,a195TEMa197.
a233TEa197:
(?2xy +μepsilon1ω2?k2z)Bz = 0
→ Bz = (a1 sinkxx + b1 coskxx)(a2 sinkyy + b2 coskyy)
Bz
n
vextendsinglevextendsingle
a62a46 = 0 →
Bz
x
vextendsinglevextendsingle
x=0,x=a =
Bz
y
vextendsinglevextendsingle
y=0,y=b = 0
Bz = Acoskxxcoskyy kxa = mpi kyb = npi m,n = 0,1,2,···
(mpia )2 + (npib )2 = ω2μepsilon1?k2z
ωc = 1√μepsilon1
radicalbigg
(mpia )2 + (npib )2 ω≥ωc λ≤λc = 2pi√μepsilon1ω
c
229/384
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a62a94a197a27a27a189a189a149a68a194,a221a47a197a19
a221a47a62a127a169a79a143a,b,a139a73a6a58a18a51a221a47a27a152a135a14,a252a131a25a27a62a169
a79a143xya182,a207a143a165a152,a195TEMa197.
a233TMa197:
(?2xy +μepsilon1ω2?k2z)Ez = 0
→ Ez = (a1 sinkxx + b1 coskxx)(a2 sinkyy + b2 coskyy)
Ezvextendsinglevextendsinglea62a46 = 0 → Ezvextendsinglevextendsinglex=0,x=a = Ezvextendsinglevextendsingley=0,y=b = 0
Ez = Asinkxxsinkyy kxa = mpi kyb = npi m,n = 0,1,2,···
(mpia )2 + (npib )2 = ω2μepsilon1?k2z
ωc = 1√μepsilon1
radicalbigg
(mpia )2 + (npib )2
a211a182a62a67a165a209a121a27a180a183a21a124; a197a19a43a165a209a121a27a180a55a197a124,a107a252a171a197a47TE,TMa197.
230/384
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a62a94a197a27a65a219a219a49a49a198a52a129,a197a196a144a167a27a65a219a219a49a49a198a67a113
a233a154a254a33a48a159a167a63a216epsilon1 = epsilon1(vectorr),a2μa69a143a126a234a27a156a185a181
2vectorE? 1v2?
2vectorE
t2 = (?lnμ)·?
vectorE?(?vectorE)·(?lnμ)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
=0
[(?lnepsilon1)·vectorE]
×vectorE =
vectorB
t?·
vectorE =?(?lnepsilon1)·vectorE a154a254a33a253a14a48a159,v2≡ 1
μepsilon1 =
c2
n2(vectorr)
a152a132a27a27a189a189a21a197,vectorE(vectorr,t) = vectorC(vectorr)eiΦ(vectorr)?iωt vectorB(vectorr,t) = vectorCprime(vectorr)eiΦ(vectorr)?iωt
vectorC(vectorr),vectorCprime(vectorr),Φ(vectorr)a143a162a188a234,vectorC(vectorr),vectorCprime(vectorr)a180a8a204,Φ(vectorr)a180a160a131
iωvectorB(vectorr,t) =?×vectorE = [?×vectorC + i(?Φ)×vectorC]bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
iωvectorCprime
eiΦ(vectorr)?iωt
·vectorC + i(?Φ)·vectorC =?(?lnepsilon1)·vectorC
·vectorC =?(?lnepsilon1)·vectorC (?Φ)·vectorC = 0
231/384
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a62a94a197a27a65a219a219a49a49a198a52a129,a197a196a144a167a27a65a219a219a49a49a198a67a113
2vectorE? 1v2?
2vectorE
t2 =[(?lnepsilon1)·
vectorE] v2≡ 1
μepsilon1 =
c2
n2(vectorr)
vectorE(vectorr,t) = vectorC(vectorr)eiΦ(vectorr)?iωt vectorB(vectorr,t) = 1
ω[?i?×
vectorC + (?Φ)×vectorC]eiΦ(vectorr)?iωt
·vectorC =?(?lnepsilon1)·vectorC (?Φ)·vectorC = 0
(?2?1v2?
2
t2)
vectorE(vectorr,t) = [(?2vectorC)eiΦ+2?eiΦ·(?vectorC)+vectorC?2eiΦ+vectorCω2
v2e
iΦ]e?iωt
= eiΦ?iωt[?2vectorC?vectorC(?Φ)·?Φ + vectorCω
2n2
c2 + i(2?Φ·?
vectorC + vectorC?2Φ)]
[(?lnepsilon1)·vectorE] = [(vectorC·?lnepsilon1)?i(?Φ)(?lnepsilon1)·vectorC]eiΦ?iωt
2vectorC?vectorC(?Φ)·?Φ + vectorCω
2n2
c2 =(
vectorC·?lnepsilon1) =?(?·vectorC)
2?Φ·?vectorC + vectorC?2Φ =?(?Φ)(?lnepsilon1)·vectorC = (?Φ)?·vectorC
a144a167a252a62a58a166vectorC,a191a124a94vectorC ·?Φ = 0·(C2?Φ) =?Φ·?C2 + C2?2Φ = 0
232/384
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a62a94a197a27a65a219a219a49a49a198a52a129,a197a196a144a167a27a65a219a219a49a49a198a67a113
2vectorE? 1v2?
2vectorE
t2 =[(?lnepsilon1)·
vectorE] v2≡ 1
μepsilon1 =
c2
n2(vectorr)
vectorE(vectorr,t) = vectorC(vectorr)eiΦ(vectorr)?iωt vectorB(vectorr,t) = 1
ω[?i?×
vectorC + (?Φ)×vectorC]eiΦ(vectorr)?iωt
·vectorC =?(?lnepsilon1)·vectorC (?Φ)·vectorC = 0
2vectorC?vectorC(?Φ)·?Φ + vectorCω
2n2
c2 ==?(?·
vectorC)
2?Φ·?vectorC + vectorC?2Φ = (?Φ)?·vectorC·(C2?Φ) = 0
vectorS = 1
2μωRe(
vectorE×vectorB?) = 1
2μωRe{
vectorC×[i?×vectorC + (?Φ)×vectorC]}= C2?Φ
2μω
·(vectorS) = 0 a173a240a62a94a85a54
a65a219a219a49a49a198a67a113,vectorCa145a152a109a27a67a122a122a133a133a250?
Φ·?Φ = ω2n2c2
vectorC?2Φ = 0 a173a240a62a94a85a54
vectorC·?Φ = 0 vectorC⊥?Φ
vectorC·?epsilon1 = 0 vectorC⊥?epsilon1
233/384
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a62a94a197a27a65a219a219a49a49a198a52a129,a49a198a144a167
vectorE(vectorr,t) = vectorC(vectorr)eiΦ(vectorr)?iωt vectorB(vectorr,t) = 1
ω[?i?×
vectorC + (?Φ)×vectorC]eiΦ(vectorr)?iωt
a65a219a219a49a49a198a67a113,vectorCa145a152a109a27a67a122a122a133a133a250?
Φ·?Φ = ω2n2c2
vectorC?2Φ = 0 a173a240a62a94a85a54
vectorC·?Φ = 0 vectorC⊥?Φ
vectorC·?epsilon1 = 0 vectorC⊥?epsilon1
a49a130a27a59a44a189a194a143a100a31a131a161Φ =a126a234a27a123a130a27a59a44,a94a235a234sa157a54a27
a59a44vectorr(s)a163a227,a51a49a130a59a44a254,ds =√dx2 + dy2 + dz2 =√dr2.
Φ∝a31a131a161Φ =a126a234a27a123a130,dvectorr/ds∝a31a131a161Φ =a126a234a27a123a130.
Φ = advectorrds? a2 = ω2n2c2? a = ωncΦ = ωnc dvectorrds
vectorC⊥dvectorr
ds
vectorCprime= 1
ω?Φ×
vectorC = n
c
dvectorr
ds×
vectorC?a238a197:vectorE,vectorB,dvectorr
dsa164a109a195a39a88
a233a178a161a197,Φ = vectork·vectorr,k2 = ω2n2c2? dvectorrds = vectorkk
234/384
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a62a94a197a27a65a219a219a49a49a198a52a129,a49a198a144a167
vectorE(vectorr,t) = vectorC(vectorr)eiΦ(vectorr)?iωt vectorB(vectorr,t) = 1
ω[?i?×
vectorC + (?Φ)×vectorC]eiΦ(vectorr)?iωt
a65a219a219a49a49a198a67a113,vectorCa145a152a109a27a67a122a122a133a133a250?
Φ·?Φ = ω2n2c2
vectorC?2Φ = 0 a173a240a62a94a85a54
vectorC·?Φ = 0 vectorC⊥?Φ
vectorC·?epsilon1 = 0 vectorC⊥?epsilon1
a49a130a27a59a44a189a194a143a100a31a131a161Φ =a126a234a27a123a130a27a59a44,a94vectorr(s)a163a227.
Φ = ωnc dvectorrds ds =
radicalbig
dx2 + dy2 + dz2 =
√
dr2
a51a49a130a254sa63,dds(ndvectorrds) = cω dds?Φ = cωdvectorrds·Φ = cω
3summationdisplay
i=1
dxi
ds?
Φ
xi
=
3summationdisplay
i=1
dxi
ds?(n
dxi
ds ) = (?n)
dvectorr
ds·
dvectorr
ds+
1
2n?(
dvectorr
ds·
dvectorr
ds) =?n
a254a33a48a159==?dvectorr
ds=a126a234
a254a33a48a159a159a165a165a27a49a114a134a130! Φ(s)≡Φ(vectorr(s)) arrowtripleright dΦds =dvectorrds·?Φ=ωnc dvectorrds·dvectorrds=ωnc
235/384
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a62a94a197a27a65a219a219a49a49a198a52a129,a49a167a141a188a134a164a234a6a110
vectorE(vectorr,t) = vectorC(vectorr)eiΦ(vectorr)?iωt vectorB(vectorr,t) = 1
ω[?i?×
vectorC + (?Φ)×vectorC]eiΦ(vectorr)?iωt
a65a219a219a49a49a198a67a113,vectorCa145a152a109a27a67a122a122a133a133a250?
Φ·?Φ = ω2n2c2
vectorC?2Φ = 0 a173a240a62a94a85a54
vectorC·?Φ = 0 vectorC⊥?Φ
vectorC·?epsilon1 = 0 vectorC⊥?epsilon1
a49a130a27a59a44a189a194a143a100a31a131a161Φ =a126a234a27a123a130a27a59a44,a94vectorr(s)a163a227.
Φ = ωnc dvectorrds ds =
radicalbig
dx2 + dy2 + dz2 =
√
dr2 dds(ndvectorrds) =?n
Φ(s)≡Φ(vectorr(s)) arrowtripleright dΦds = ωnc arrowtripleright Φ(s1)?Φ(s2) = ωc
integraldisplay s2
s1
ds n(vectorr(s))
a49a167a141a188,S(vectorr1,vectorr2)≡
integraldisplay s2
s1
ds n(vectorr(s)) =
integraldisplay vectorr2
vectorr1
√
dr2n(vectorr) vectorr1=vectorr(s1),vectorr2=vectorr(s2)
a164a234a6a110,a162a83a27a49a130a59a44a166a49a167a141a188a18a52a138 !
236/384
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a62a94a197a27a65a219a219a49a49a198a52a129,a49a167a141a188a134a164a234a6a110
vectorE(vectorr,t) = vectorC(vectorr)eiΦ(vectorr)?iωt vectorB(vectorr,t) = 1
ω[?i?×
vectorC + (?Φ)×vectorC]eiΦ(vectorr)?iωt
a65a219a219a49a49a198a67a113,vectorCa145a152a109a27a67a122a122a133a133a250?
vectorC?2Φ = 0 a173a240a62a94a85a54
vectorC·?Φ = 0 vectorC⊥?Φ
vectorC·?epsilon1 = 0 vectorC⊥?epsilon1
Φ = ωnc dvectorrds ds =
radicalbig
dx2 + dy2 + dz2 =
√
dr2 dds(ndvectorrds) =?n
a49a167a141a188,S(vectorr1,vectorr2)≡
integraldisplay s2
s1
ds n(vectorr(s)) =
integraldisplay vectorr2
vectorr1
√
dr2n(vectorr) vectorr1=vectorr(s1),vectorr2=vectorr(s2)
a164a234a6a110,a162a83a27a49a130a59a44a166a49a167a141a188a18a52a138 !
0 = δS =
integraldisplay
[(δn)
√
dr2 + ndvectorr·δdvectorr√dr2 ] =
integraldisplay
[(?n)·δvectorr
√
dr2 + ndvectorr·dδvectorr√dr2 ]
=
integraldisplay
[(?n)·δvectorr
√
dr2?d( ndvectorr√dr2)·δvectorr] =
integraldisplay
[?n? d√dr2( ndvectorr√dr2)]·δvectorr
√
dr2
=
integraldisplay
[?n? dds(ndvectorrds )]·δvectorr
√
dr2 arrowtripleright dds(ndvectorrds ) =?n
237/384
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a207a165a127a193
238/384
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a207a165a127a193
a158a109,07a9912a271a70(a40a207a56) a101a2042:00-5:00
a47a47a58a58,a152a19205
a137a166,11a2730a70(a40a207a202)a20a2547:00-9:00 a47a47a58a58a110a137a1623408
12a271a70(a40a207a56)a254a2049:00-11:30 a47a47a58a58a110a137a1623408
a193a75,5a75a1924a75a182a122a7525a169a182a183a62a94a1243a75a33a68a1942a75
a127a123,a109a242(a140a145a214a33a249a194a33a41a80a33a138a146a29); a247a169100a169 a163a177a249a194a254a27a83a78a143a79a164
239/384
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a83a75a49a19a217a492a33a214a191a75(a):
a100a19a78a124a164a27a78a88,a174a127a49ia135a19a78a254a27a111a62a214qi,a133a19a78a9a140a177a107a48
a159,a2a195a103a100a62a214,a166a121a152a109a63a191a152a58vectorra63a27a62a179a134qia27a39a88a180a130a53
a224a103a27,a61,φ(vectorr,q1,q2,...,qn) =
nsummationdisplay
i=1
pi(vectorr)qi a217a165pi(vectorr)a134qia195a39.
a121a178:
a49a152a218,φ = φ1 +φ2 +···+φn,φia180a19a78ia145qi,a217a167a216a145a62a158a27a169
a217,a175a175a162a162a254,a122a152a135φka247a118
·[epsilon1?φk] = 0 φkvextendsinglevextendsingleS
k
= a126a234 φkvextendsinglevextendsingleS
i
= a126a234contintegraldisplay
Sk
dvectorS·epsilon1?φk =?qk
contintegraldisplay
Si
dvectorS·epsilon1?φk = 0
·[epsilon1?φ] =
nsummationdisplay
i=1
·[epsilon1?φi] = 0 φvextendsinglevextendsingleS
i
=
nsummationdisplay
j=1
φjvextendsinglevextendsingleS
i
= a126a234
contintegraldisplay
Si
dvectorS·epsilon1?φ =
nsummationdisplay
j=1
contintegraldisplay
Si
dvectorS·epsilon1?φj =?qi φa180a41
240/384
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a83a75a49a19a217a492a33a214a191a75(a):
a100a19a78a124a164a27a78a88,a174a127a49ia135a19a78a254a27a111a62a214qi,a133a19a78a9a140a177a107a48
a159,a2a195a103a100a62a214,a166a121a152a109a63a191a152a58vectorra63a27a62a179a134qia27a39a88a180a130a53
a224a103a27,a61,φ(vectorr,q1,q2,...,qn) =
nsummationdisplay
i=1
pi(vectorr)qi a217a165pi(vectorr)a134qia195a39.
a49a152a218,φ = φ1 +φ2 +···+φn,φia180a19a78ia145qi,a217a167a216a145a62a158a27a169
a217.
a49a19a218,a121a178φi = pi(vectorr)qia189a31a100a47a121 φi(vectorr,λqi) = λφi(vectorr,qi).
·[epsilon1?φi] = 0
φivextendsinglevextendsingleS
j
= a126a234
contintegraldisplay
Sj
dvectorS·epsilon1?φi =?qiδij
·[epsilon1?λφi] = 0
λφivextendsinglevextendsingleS
j
= a126a234
contintegraldisplay
Sj
dvectorS·epsilon1?λφi =?λqiδij
a61φi(vectorr,λqi) = λφi(vectorr,qi),a121a46.
241/384
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a83a752.19:
a140a187r0a27a165a161a167a51a165a139a730 < θ < pi/2a27a140a165a161a254a27a62a179a143φ0,a51
pi/2 <θ<pia27a140a165a161a254a27a62a179a143?φ0,a94a130a21a188a234a144a123a166a152a109a136a58
a27a27a62a62a179a169a217a186
φ(vectorr) = 14piepsilon1
0
integraldisplay
∞
dτprimeρ(vectorrprime)parenleftbig 1|vectorr?vectorrprime|?
r0
r
|vectorrprime?r20r2vectorr|
parenrightbig
14pi
integraldisplay
r0
dvectorSprimeφ(vectorrprime)·?primeparenleftbig 1|vectorr?vectorrprime|?
r0
r
|vectorrprime?r20r2vectorr|
parenrightbig
dvectorSprime =?vectorerprime r20d?prime,a166a41a171a27a9a123a130!
=
1
4pi
integraltext
r0 dS
primeφ(vectorrprime)·?
rprime
parenleftbig 1
|vectorr?vectorrprime|?
r0
r
|vectorrprime?r
20
r2vectorr|
parenrightbig r>r
0 (?dvectorSprime//vectorerprime)
14pi integraltextr
0
dSprimeφ(vectorrprime)·rprimeparenleftbig 1|vectorr?vectorrprime|? r0r
|vectorrprime?r
20
r2vectorr|
parenrightbig r<r
0 (dvectorSprime//vectorerprime)
242/384
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a83a752.19:
a140a187r0a27a165a161a167a51a165a139a730 < θ < pi/2a27a140a165a161a254a27a62a179a143φ0,a51
pi/2 <θ<pia27a140a165a161a254a27a62a179a143?φ0,a94a130a21a188a234a144a123a166a152a109a136a58
a27a27a62a62a179a169a217a186
Pn(cosΘ) =
nsummationdisplay
m=?n
(n?m)!
(n + m)!P
m
n (cosθ)P
m
n (cosθ
prime)eim(φ?φprime)
cosΘ = cosθcosθprime + sinθsinθprimecos(φ?φprime)
a1
a1
a1
a1
a1
a1
a1
a1
a1
a1a1a21
a0
a0
a0
a0
a0
a0
a0
a0a0a18
Θintegraldisplay
d?prime f(θ
prime)
|vectorr?vectorrprime| =
integraldisplay
d?primef(θprime)
∞summationdisplay
n=0
rn<
rn+1> Pn(cosΘ)
=
integraldisplay 2pi
0
dφprime
integraldisplay pi
0
dθprimesinθprimef(θprime)
∞summationdisplay
n=0
nsummationdisplay
m=?n
rn<
rn+1>
(n?m)!
(n + m)!P
m
n (cosθ)P
m
n (cosθ
prime)eim(φ?φprime)
a83a75a41a137a27a63a216a216a233a156 = 2pi
∞summationdisplay
n=0
rn<
rn+1> Pn(cosθ)
integraldisplay pi
0
dθprimesinθprimef(θprime)Pn(cosθprime)
243/384
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a83a75
14pi
integraldisplay
r0
dSprimeφ(vectorrprime) 1|vectorr?vectorrprime| Pn(1) = 1 Pn(0) n=a243==== (?1)n2 1·3·5···(n?1)2·4·6···n Pn(0) n=a219==== 0
=?r
2
0
4pi
integraldisplay
d?prime[θ(pi2?θprime)φ0?θ(θprime?pi2)θ(pi?θprime)φ0] 1|vectorr?vectorrprime|
=?r
2
0
2
∞summationdisplay
n=0
rn<
rn+1> Pn(cosθ)
integraldisplay pi
0
dθprimesinθprimePn(cosθprime)[θ(pi2?θprime)φ0?θ(θprime?pi2)θ(pi?θprime)φ0]
= r
2
0φ0
2
∞summationdisplay
n=0
rn<
rn+1> Pn(cosθ)[
integraldisplay 0
1
dx Pn(x)?
integraldisplay?1
0
dx Pn(x)] Pn(?x)=(?)nPn(x)
=?r20φ0
∞summationdisplay
n=1,3,···
rn<
rn+1> Pn(cosθ)
Pn+1(x)?Pn?1(x)
2n + 1
vextendsinglevextendsingle1
0
R 1
0 dxPn(x)=
Pn+1(x)?Pn?1(x)
2n+1
1
0
= r20φ0
∞summationdisplay
n=1,3,···
rn<Pn(cosθ)
(2n + 1)rn+1> [(?1)
n+1
2
1·3···n
2·4···(n+1)?(?1)
n?1
2
1·3···(n?2)
2·4···(n?1)]
= r20φ0
∞summationdisplay
n=1,3,···
rn<Pn(cosθ)
(2n + 1)rn+1> (?1)
n?1
2
1·3···(n?2)
2·4···(n + 1)[?n?n?1]
244/384
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a83a75
14pi
integraldisplay
r0
dSprimeφ(vectorrprime)[θ(r0?r)?θ(r?r0)]rprime 1|vectorr?vectorrprime|
=?r20φ0
∞summationdisplay
n=1,3,···
Pn(cosθ)[θ(r0?r)?θ(r?r0)]rprime r
n
<
rn+1> (?1)
n?1
2
1·3···(n?2)
2·4···(n + 1)
= r20φ0
∞summationdisplay
n=1,3,···
Pn(cosθ)(?1)n?12 1·3···(n?2)2·4···(n + 1)×
braceleftBigg (n+ 1) rn
rn+20 r<r0
nrn?10rn+1 r>r0
14pi
integraldisplay
r0
dSprimeφ(vectorrprime)[θ(r0?r)?θ(r?r0)]rprime
r0
r
|vectorrprime?r20r2vectorr|
=?r20φ0
∞summationdisplay
n=1,3,···
Pn(cosθ)(?1)n?12 1·3···(n?2)2·4···(n+1)[θ(r0?r)?θ(r?r0)]rprime
r0(r
20
r )
n
rrprime(n+1)
r20
r<r0
r0rprimen
r(r
20
r )n+1
r20
r>r0
=?r20φ0
∞summationdisplay
n=1,3,···
Pn(cosθ)(?1)n?12 1·3···(n?2)2·4···(n + 1)
braceleftBigg
(n+ 1)rn?10rn+1 r>r0
n rnrn+2
0
r<r0
245/384
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a83a75
14pi
integraldisplay
r0
dSprimeφ(vectorrprime)[θ(r0?r)?θ(r?r0)]rprime 1|vectorr?vectorrprime|
= r20φ0
∞summationdisplay
n=1,3,···
Pn(cosθ)(?1)n?12 1·3···(n?2)2·4···(n + 1)×
braceleftBigg (n+ 1) rn
rn+20 r<r0
nrn?10rn+1 r>r0
14pi
integraldisplay
r0
dSprimeφ(vectorrprime)[θ(r0?r)?θ(r?r0)]rprime
r0
r
|vectorrprime?r20r2vectorr|
=?r20φ0
∞summationdisplay
n=1,3,···
Pn(cosθ)(?1)n?12 1·3···(n?2)2·4···(n + 1)
braceleftBigg
(n+ 1)rn?10rn+1 r>r0
n rnrn+2
0
r<r0
φ(vectorr) =?14pi
integraldisplay
r0
dSprimeφ(vectorrprime)[θ(r0?r)?θ(r?r0)]rprimeparenleftbig 1|vectorr?vectorrprime|?
r0
r
|vectorrprime?r20r2vectorr|
parenrightbig
= φ0
∞summationdisplay
n=1,3,···
Pn(cosθ)(?1)n?12 1·3···(n?2)2·4···(n + 1)(2n + 1)×
rn
rn0 r<r0
rn+10
rn+1 r>r0
246/384
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a83a75a240a189a189a62a62a54a27a27a62a62a124a214a191a83a75:
a252a135a229a108a233a15a140a187a143aa27a55a225a140a165a238a92a62a19a199a143γa27a47a161,a166a140a165
a178a161a220 a176a134a47a161a178a224.a51a100a252a140a165a62a52a254a92a62a216V,a166a47a161a101a52
a161a78a67a62a54a169a217,a191a166a252a140a165a109 a27a62a123a218a122a135a140a165a27a26a47a62a123?
φ = cr
1
cr
2
V = (ca?cd)?(cd?ca) = 2ca?2cd →c = V2
a?
2
d
≈Va2
vectorj =?γ?( c
r1?
c
r2) =
γVa
2 (
vectorr1
r31?
vectorr2
r32)
I =
integraldisplay
dvectorS·vectorj≈γVa2a2 2pia2 = aVγpi a252a140a165a109a62a123 = VI = 1aγpi
φ = φ1 +φ2 φ1 = cr
1
a62a521a62a54a54a54a54a149a195a161a15 φ2 =?cr
1
a195a161a15a62a54a54a54a54a149a62a522
a122a135a140a165a27a26a47a47a62a62a123 = c/aI = V/2aVpi = 12aγpi
247/384
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a83a75a240a189a189a62a62a54a27a27a62a62a124a214a191a83a75:
a51a152a172a195a129a140a27a19a62a178a134a254a44a58pa63a207a92a62a54a191a149a195a161a15a54a209,a51a134a254a63a191a63a109 a152
a11a154,a216a157a185pa58.a193a121a51a100a11a154a62a14a254a63a191a252a58a27a62a179a11a131a138a143a216a109a11a154a158a84a252a58
a27a27a62a62a179a11a131a131a138a138a27a152a21
a38a37
a39a36
a114
vectord
p
a114
r20
d2
vectord
φ=cln|vectorr?vectord|+cprimeln|vectorr?a
2
d2
vectord|?c”lnr vectorE=?c(vectorr?vectord)
|vectorr?vectord|2?
cprime(vectorr?a2d2vectord)
|vectorr?a2d2vectord|2 +
c”vectorr
r2
vectorer·vectorEvextendsinglevextendsingler=a =vectorer·bracketleftbig c(vectorr?
vectord)
a2 + d2?2adcosθ +
cprime(vectorr?a2d2vectord)
a2 + a4d2?2aa2d cosθ?
c”vectorr
a2
bracketrightbig
= c(a?dcosθ) + c
primed2
a2(a?
a2
d cosθ)?
c”
a (d
2 + a2?2adcosθ)
d2 + a2?2adcosθ
= ca+c
primed2
a?
c”
a (d
2+a2)?(cd+cprimed?2c”d)cosθ
d2 + a2?2adcosθ
cprime=c=c”→0
cln|vectorr?vectord|= cln|vectorr1?
vectord|
|vectorr2?vectord| =
c
2 ln
a2 + d2?2adcosθ1
a2 + d2?2adcosθ2
cprimeln|vectorr?a
2
d2
vectord|= cln|vectorr1?a
2
d2
vectord|
|vectorr2?a2d2vectord| =
c
2 ln
a2 + a4d2?2aa2d cosθ1
a2 + a4d2?2aa2d cosθ2
248/384
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a83a753.2:
a254a33a195a161a127a134a11a206a47a218a130a43a167a122a252a160a127a221a130a23a42a234a143n,a62a54a114a221
a143I,a193a94a141a152a53a189a110a166a43a83a9a94a97a65a114a221vectorBa186
a43a83a9:?·vectorB = 0?×vectorB = 0
a43a57a254,vectorn·(vectorBa9?vectorBa83) = 0 μ0vectorer×(vectorBa9?vectorBa83) =vectoreθnI
a195a161a15,vectorB→0
a223a41a181 vectorB0a9 = 0 vectorB0a83 = nIμ
0
vectorez vectorer×vectorez =?vectoreθ
vectorB = vectorB0 + vectorBprime
a43a83a9:?·vectorBprime = 0?×vectorBprime = 0
a43a57a254,vectorn·(vectorBprimea9?vectorBprimea83) = 0 μ0vectorer×(vectorBprimea9?vectorBprimea83) = 0
a195a161a15,vectorBprime→ 1r2
a183a62a141a152a53a189a110→vectorBprime = 0
249/384
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a83a753.13:
a107a152a135a254a33a145a62a27a0a19a78a138,a217a140a187a143R0,a111a62a214a143Q,a56a166a165a138a55a103
a28a44a152a134a187a177a14a132a221ωa61a196,a166a166a165a165a83a9a27a94a124vectorB?
vectorBa83 =?μ0?φa83 vectorBa9 =?μ0?φa9
a165a83a9:?2φa83 = 0?2φa9 = 0 a195a161a15,vectorB→ 1r2
a165a138a254,?φa83?rφa9?r = 0
μ0vectorer×(?φa83φa9) = Q4piR2
0
ωvectorez×R0vectorer =? Q4piR
0
vectoreφ?cosθ?θ
vectorer×?φ =vectorer×(vectorer?φ?r +vectoreθ1r?φ?θ +vectoreφ 1rsinθ?φ?φ) =vectoreφ 1R
0
φ
θ
μ0(?φa83?θφa9?θ )vextendsinglevextendsingler=R
0
=?Q4pi?cosθ?θ
μ0(φa83?φa9)vextendsinglevextendsingler=R
0,θa157a54
=?Q4piP1(cosθ)
250/384
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a83a75a62a54a78a88a245a63a208a109a214a191a83a75:
W =
∞summationdisplay
n=0
Wn Wn = 1n
3summationdisplay
i1,···in
vectorJi
1···in·[
n
xi1···?xin
vectorA(vectorr)]
W0 = 0 W1 = vectorm·vectorB
vectorF =
∞summationdisplay
n=0
vectorFn vectorFn = 1
n
3summationdisplay
i1,···in
vectorJi
1···in×[
n
xi1···?xin
vectorB(vectorr)]
vectorF0 = 0 vectorF1 = (?vectorB)·vectorm
vectorL =
∞summationdisplay
n=0
vectorLn vectorJi
1···in =
3summationdisplay
i
vectoreiJii1···in
vectorLn = 1
n
3summationdisplay
i,i1,···in
braceleftbigvectorJ
ii1···in[
n
xi1···?xinBi(vectorr)]?J
i
ii1···in[
n
xi1···?xin
vectorB(vectorr)]bracerightbig
vectorL0 = vectorm×vectorB
251/384
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a83a75a62a54a78a88a245a63a208a109a214a191a83a75:
W =
integraldisplay
dτvectorj(vectorr)·vectorA(vectorr)
=
integraldisplay
dτvectorj(vectorr)·braceleftbigvectorA(0) +
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
xi1xi2···xinbracketleftbigx
i1
···x
in
vectorA(vectorr)bracketrightbig
vectorr=0
bracerightbig
=
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
vectorJi
1i2...in·
bracketleftbig?
xi1
xi2···
xin
vectorA(vectorr)bracketrightbig
vectorr=0 =
∞summationdisplay
n=0
Wn
W0 = 0 Wn = 1n!
3summationdisplay
i1i2...in=1
vectorJi
1i2...in·
bracketleftbig?
xi1
xi2···
xin
vectorA(vectorr)bracketrightbig
vectorr=0 nnegationslash= 0
W1 =
3summationdisplay
i=1
vectorJi·[?
xi
vectorA(vectorr)]bracketrightbig
vectorr=0 =
1
2
3summationdisplay
i,k=1
integraldisplay
dτprime(xprimeijk?xprimekji)[?Ak?x
i
]bracketrightbigvectorr=0
= 12
3summationdisplay
i,j,k,l,m=1
epsilon1jlmepsilon1jik
integraldisplay
dτprimexprimeljm[?Ak?x
i
]bracketrightbigvectorr=0 =
3summationdisplay
j=1
mjBj(0) = vectorm·vectorB(0)
252/384
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a83a75vectorF =
integraldisplay
dτvectorj(vectorr)×vectorB(vectorr)
=
integraldisplay
dτvectorj(vectorr)×
∞summationdisplay
n=0
1
n!
3summationdisplay
i1i2...in=1
xi1xi2···xinbracketleftbigx
i1
···x
in
vectorB(vectorr)bracketrightbig
vectorr=0
=
∞summationdisplay
n=1
1
n!
3summationdisplay
i1i2...in=1
vectorJi
1i2...in×
bracketleftbig?
xi1
xi2···
xin
vectorB(vectorr)bracketrightbig
vectorr=0 =
∞summationdisplay
n=0
vectorFn
vectorFn = 1
n!
3summationdisplay
i1i2...in=1
vectorJi
1i2...in×
bracketleftbig?
xi1
xi2···
xin
vectorB(vectorr)bracketrightbig
vectorr=0
vectorF0 = 0
vectorF1 =
3summationdisplay
i=1
vectorJi×?
xi
vectorB(vectorr)vextendsinglevextendsingle
vectorr=0 =
1
2
3summationdisplay
i,k=1
integraldisplay
dτprime(xprimeijk?xprimekji)vectorek×?
vectorB(vectorr)
xi
vextendsinglevextendsingle
vectorr=0
= 12
3summationdisplay
i,k,l,n=1
epsilon1kln
integraldisplay
dτprime(xprimeijk?xprimekji)vectoren?Bl(vectorr)?x
i
vextendsinglevextendsingle
vectorr=0 =
3summationdisplay
i,j,k,l,n=1
mjepsilon1jikepsilon1klnvectoren?Bl(vectorr)?x
i
vextendsinglevextendsingle
vectorr=0
=
3summationdisplay
i,j,k,l,n=1
mj(δjlδin?δjnδil)vectoren?Bl?x
i
vextendsinglevextendsingle
vectorr=0=
3summationdisplay
i,j=1
mjvectorei?Bj?x
i
vextendsinglevextendsingle
vectorr=0=
3summationdisplay
i,j=1
mjvectorei?Bi?x
j
vextendsinglevextendsingle
vectorr=0=[vectorm·?
vectorB]vextendsinglevextendsingle
vectorr=0
253/384
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a83a75
vectorL =
integraldisplay
dτvectorr×[vectorj(vectorr)×vectorB(vectorr)] =
integraldisplay
dτbracketleftbig[vectorr·vectorB(vectorr)]vectorj(vectorr)?[vectorr·vectorj(vectorr)]vectorB(vectorr)]bracketrightbig
=
integraldisplay
dτ
bracketleftBig
xiBi(vectorr)vectorj(vectorr)?xiji(vectorr)vectorB(vectorr)
bracketrightBig
=
∞summationdisplay
n=0
vectorLn
vectorLn = 1
n!
3summationdisplay
i,i1i2...in=1
braceleftbigg
vectorJii
1...in
xi1···
xinBi(vectorr)?J
i
ii1...in
xi1···
xin
vectorB(vectorr)
bracerightbigg
vectorr=0
vectorL0 =
3summationdisplay
i=1
[vectorJiBi(0)?JiivectorB(0)] =
3summationdisplay
i,k=1
integraldisplay
dτprimexprimeijkvectorekBi(0)
Jii≡
integraldisplay
dτprimexprimeiji = 12
integraldisplay
dτprime?primel[xprimeixiprimejl] = 0 xprimeijk = 12(xprimeijk+xprimekji+xprimeijk?xprimekji)
= 12
3summationdisplay
i,k=1
integraldisplay
dτprime[(xprimeijk?xprimekji)vectorekBi(0)] =
3summationdisplay
i,k,n=1
mnepsilon1nikvectorekBi(0) = vectorm×vectorB(0)
254/384
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a126a75
a62a254a143qa27a27a58a58a58a62a62a214a26a152a51a229a195a161a140a26a47a47a19a19a78a134aa63a167a166a217a201a229a186
a177a58a58a62a62a214a134a217a186a148a62a214a27a235a130a165a58a143a139a73a6a58a181 φ= q4piepsilon1
0|vectorr?avectorex|
q4piepsilon1
0|vectorr+avectorex|
a114a
q
a114
q
a
vectorf = qvectorEa9vextendsinglevextendsingle
vectorr=avectorex =?q?[?
q
4piepsilon10|vectorr + avectorex|]
vextendsinglevextendsingle
vectorr=avectorex
=? q
2(vectorr + avectorex)
4piepsilon10|vectorr + avectorex|3
vextendsinglevextendsingle
vectorr=avectorex =?
q2
4piepsilon10(2a)2vectorex
fx =?qa?q4piepsilon1
0(2a)
=? q
2
4piepsilon10(2a2) a131a11a207a1022 a186
a177a186a148a62a214a27a160a152a143a139a73a6a58a181 φ = q4piepsilon1
0|vectorr?2avectorex|
q4piepsilon1
0r
vectorf=qvectorEa9vextendsinglevextendsingle
vectorr=2avectorex=?q?[?
q
4piepsilon10r]
vextendsinglevextendsingle
vectorr=2avectorex=?
q2vectorr
4piepsilon10r3
vextendsinglevextendsingle
vectorr=2avectorex=?
q2
4piepsilon10(2a)2vectorex
255/384
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a83a75a62a94a197a51a46a161a254a27a135a19a218a242a19a214a191a83a75(c):
Dbardbl≡
vectorS2bardbl·?vectorS
vectorS1bardbl·?vectorS
vextendsinglevextendsingle
z=0 =
S2bardblz
S1bardblz
vextendsinglevextendsingle
z=0
vectorE20bardbl=E10bardbl(vectore1k2z
k2?vectore3
k2x
k2 )
2cosθ1
k2z
k2 +
μ1k2
μ2k1cosθ1
=
1
2μ2ω[k2zR(
vectorE2bardbl·vectorE?2bardbl)?Re(E2bardblz + E2⊥z)(vectork2·vectorE?2bardbl)]
k1z
2μ1ω(E10bardblE
10bardbl)
vextendsinglevextendsingle
z=0
= μ1μ
2k1cosθ1
braceleftBigg
k2zR
4cos2θ1(k2zk
2
k?2z
k?2 +
k2x
k2
k2x
k?2)
bardblk2zk
2
+μ1k2μ
2k1
cosθ1bardbl2 +
Re[(k2xk?2zk?
2
k2xk2zk?
2
)k2xk
2
]4cos2θ1
bardblk2zk
2
+μ1k2μ
2k1
cosθ1bardbl2
bracerightBigg
= 4μ1 cosθ1μ
2k1
k2zR(k2zk?2z+k2xk?2xk
2k?2
) + Re[k2xk2xk
2k?2
(k?2z?k2z)]
bardblk2zk
2
+ μ1k2μ
2k1
cosθ1bardbl2
a129a0a249a145=0
= 4μ1 cosθ1μ
2k1
Re(k2zk2zk?2zk
2k?2
+ k?2zk2xk2xk
2k?2
)
bardblk2zk
2
+ μ1k2μ
2k1
cosθ1bardbl2 =
4μ1 cosθ1Re(k2k?2zk?
2
)
μ2k1bardblk2zk
2
+ μ1k2μ
2k1
cosθ1bardbl2
Rbardbl≡?
vectorS3bardbl·?vectorS
vectorS1bardbl·?vectorS
vextendsinglevextendsingle
z=0 =?
S3bardblz
S1bardblz
vextendsinglevextendsingle
z=0 =
vectorE30bardbl·vectorE?30bardbl
vectorE10bardbl·vectorE?10bardbl =
vextenddoublevextenddouble?k2zk
2
+ μ1k2μ
2k1
cosθ1
k2z
k2 +
μ1k2
μ2k1 cosθ1
vextenddoublevextenddouble2
256/384
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a83a75
a62a94a197a51a48a161a254a27a135a19a218a242a19a214a191a83a75(c):
Dbardbl =
4μ1 cosθ1Re(k2k?2zk?
2
)
μ2k1bardblk2zk
2
+ μ1k2μ
2k1
cosθ1bardbl2 Rbardbl =
vextenddoublevextenddouble?k2zk
2
+ μ1k2μ
2k1
cosθ1
k2z
k2 +
μ1k2
μ2k1 cosθ1
vextenddoublevextenddouble2
Rbardbl + Dbardbl
=
k2zk?2z
k2k?2?
μ1 cosθ1
μ2k1 (
k2k?2z
k?2 +
k?2k2z
k2 ) +
μ21 cos2 θ1
μ22k21 k2k
2 +
2μ1 cosθ1
μ2k1 (
k2k?2z
k?2 +
k?2k2z
k2 )
bardblk2zk
2
+ μ1k2μ
2k1
cosθ1bardbl2
= 1
(vectorS2bardbl?vectorS3bardbl)·?vectorS = (Dbardbl + Rbardbl)vectorS1bardbl·?vectorS = vectorS1bardbl·?vectorS
257/384
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a207a165a127a193
258/384
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a207a165a127a193
a138a146,a174a101a136a65a17a45 a156
a209a75a103a180,a67a67a88a88a204a135a212a110a83a78a182 a8a127a160a234a198a144a161a27a83a78a156
a202a192a111a27a83a78:
– a183a142a27a27a62a62a214a169a217a23a41a27a27a62a62a124
– a240a189a189a27a27a27a62a62a54a169a217a23a41a27a27a62a62a124
– a240a189a189a27a27a27a62a62a54a169a217a23a41a27a94a124
– a189a21a62a94a197a51a195a161a140a178a161a46a161a254a27a135a223a19
– a62a94a197a27a27a189a189a149a68a194
a141a152a53a189a110a216a252a213a127
a130a21a188a234a216a252a213a127
a245a52a208a109a216a252a213a127
259/384
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a62a94a197a27a203a19
260/384
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a62a94a124a27a165a165a179a179a218a73a179,a94a179a163a227a62a94a124; a53a137a67a134a218a218a53a53a137a216a216a67a67a53
·vectorE = ρepsilon1
0
·vectorB = 0?×vectorE =
vectorB
t?×
vectorB=μ0vectorj+μ0epsilon10?vectorE
t
×vectorE =t?×vectorA =×?
vectorA
t →?×(
vectorE +?vectorA
t ) = 0
vectorE +?vectorA
t =φ →
vectorE =φvectorA
t
braceleftbigg vectorE =φvectorA
tvector
B =?×vectorA
vectorA,φa216a141a152
braceleftbigg vectorA→vectorAprime = vectorA +?χ
φ→φprime = φχ?t
a122a152a124(vectorA,φ)a23a152a171a53a137,a216a211a53a137a233a65a211a152a124vectorE,vectorB,vectorE,vectorBa180a212a110a42a9a254.
a164a107a212a110a110a254a254a218a212a110a53a198a134a65a207a27a53a137a192a74a195a39—a53a137a216a216a67a67a53!
a51a234a198a254a152a171a216a67a53a210a233a65a152a171a233a161a53,a134a254a161a53a137a216a67a53a233a65a27a233a161a53a23U(1)a53
a137a233a161a53,a207a100,n a62a94a131a112a138a94a228a107U(1)a53a137a233a161a53.
a51a162a83a79a142a165,a192a74a152a189a27a94a135a53a114vectorA,φa164a228a107a27a216a40a189a27a53a137a103a100a221a129a155a52,a192
a74a152a171a53a137a27a189a94a135a23a192a152a171a53a137.
a165a212a53a137,?·vectorA = 0 a226a226a212a212a91a53a137,?·vectorA + 1c2?φ?t = 0
261/384
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a62a94a124a27a165a165a179a179a218a73a179,a136a75a203(d’Alembert)a144a167
×vectorB = μ0vectorj +μ0epsilon10?
vectorE
t →?×(?×
vectorA)=μ0vectorj+μ0epsilon10[φ
t?
2vectorA
t2 ]
·vectorE = ρepsilon1
0
→2φt?·vectorA = ρepsilon1
0
a165a212a53a137,?·vectorA = 0
2vectorA? 1c2?
2vectorA
t2 =?μ0
vectorj + 1
c2
t?φ =?μ0
vectorj2φ =?ρ
epsilon10
a226a226a212a212a91a53a137,?·vectorA + 1c2?φ?t = 0
2vectorA? 1c2?
2vectorA
t2 =?μ0
vectorj?2φ? 1
c2
2φ
t2 =?
ρ
epsilon10
c→∞a158,a163a20a183a62a94a156a47
262/384
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a237a180a179,a154a224a103(a107a13)a197a196a144a167a27a165a161a197a41
2φ(vectorr,t)? 1c2?
2
t2φ(vectorr,t) =?
ρ(vectorr,t)
epsilon10
2vectorrU(vectorr,vectorrprime,t)? 1c2?
2
t2U(vectorr,vectorr
prime,t) =?1
epsilon10δ(vectorr?vectorr
prime)ρ(vectorrprime,t)
φ(vectorr,t) =
integraldisplay
dτprime U(vectorr,vectorrprime,t)
U(vectorr,vectorrprime,t) = U(R,vectorrprime,t) vectorR =vectorr?vectorrprime =
3summationdisplay
i=1
Xivectorei
2vectorrU(R,vectorrprime,t) =
3summationdisplay
i=1
i?iU(R,vectorrprime,t) =
3summationdisplay
i=1
i[?U?R?iR] =
3summationdisplay
i=1
i[?U?RXiR]
=
3summationdisplay
i=1
parenleftbigXi
R
2U
R2?iR+
1
R
U
R+
U
RXi?i
1
R
parenrightbig=3summationdisplay
i=1
parenleftbigXiXi
R2
2U
R2+
1
R
U
R?
XiXi
R3
U
R
parenrightbig
=?
2U
R2 +
2
R
U
R =
1
R[
2U
R2R + 2
U
R] =
1
R
2(RU)
R2
263/384
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a237a180a179,a154a224a103(a107a13)a197a196a144a167a27a165a161a197a41
2φ(vectorr,t)? 1c2?
2
t2φ(vectorr,t) =?
ρ(vectorr,t)
epsilon10
2vectorrU(vectorr,vectorrprime,t)? 1c2?
2
t2U(vectorr,vectorr
prime,t) =?1
epsilon10δ(vectorr?vectorr
prime)ρ(vectorrprime,t)
φ(vectorr,t) =
integraldisplay
dτprime U(vectorr,vectorrprime,t) U(vectorr,vectorrprime,t) = U(R,vectorrprime,t)
vectorR =vectorr?vectorrprime =
3summationdisplay
i=1
Xivectorei?2vectorrU(R,vectorrprime,t) = 1R?
2(RU)
R2
1
R
2(UR)
R2?
1
c2
2U
t2 =?
1
epsilon10δ(
vectorR)ρ(vectorrprime,t)
2(UR)
R2?
1
c2
2(UR)
t2 = 0 RU(R,vectorr
prime,t) = f(t?R
c,vectorr
prime) + g(t + R
c,vectorr
prime)
U1(R,vectorrprime,t)≡1Rf(t?Rc,vectorrprime) a237a180a41 U2(R,vectorrprime,t)≡1Rg(t+Rc,vectorrprime) a135a99a41
264/384
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a237a180a179,a154a224a103(a107a13)a197a196a144a167a27a165a161a197a41
2vectorrU(vectorr,vectorrprime,t)? 1c2?
2
t2U(vectorr,vectorr
prime,t) =?1
epsilon10δ(vectorr?vectorr
prime)ρ(vectorrprime,t)
U1(R,vectorrprime,t)≡1Rf(t?Rc,vectorrprime) a237a180a41 U2(R,vectorrprime,t)≡1Rg(t+Rc,vectorrprime) a135a99a41
1epsilon1
0
δ(vectorR)ρ(vectorrprime,t) = (?2vectorr? 1c2?
2
t2)[
1
Rf(t?
R
c,vectorr
prime)]
= (?2vectorr 1R)f(t?Rc,vectorrprime) + 2?vectorrprime 1R·?vectorrf(t?Rc,vectorrprime)
+1R(?2vectorr? 1c2?
2
t2)f(t?
R
c,vectorr
prime)
=?4piδ(vectorR)f(t?Rc,vectorrprime) + 2c
vectorR
R3·
vectorR
R
tf(t?
R
c,vectorr
prime)
+1R(?
2
R2 +
2
R
R?
1
c2
2
t2)f(t?
R
c,vectorr
prime) =?4piδ(vectorR)f(t?R
c,vectorr
prime)
=?4piδ(vectorR)f(t) f(t,vectorrprime) = ρ(vectorr
prime,t)
4piepsilon10
U1(R,vectorrprime,t) = ρ(vectorr
prime,t?R
c )
4piepsilon10R U2(R,vectorr
prime,t) = ρ(vectorr
prime,t + R
c )
4piepsilon10R
265/384
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a237a180a179,a154a224a103(a107a13)a197a196a144a167a27a165a161a197a41
2φ(vectorr,t)? 1c2?
2
t2φ(vectorr,t) =?
ρ(vectorr,t)
epsilon10
2vectorrU(vectorr,vectorrprime,t)? 1c2?
2
t2U(vectorr,vectorr
prime,t) =?1
epsilon10δ(vectorr?vectorr
prime)ρ(vectorrprime,t)
φ(vectorr,t) =
integraldisplay
dτprime U(vectorr,vectorrprime,t) U(vectorr,vectorrprime,t) = U(R,vectorrprime,t)
U1(R,vectorrprime,t)=ρ(vectorr
prime,t?R
c )
4piepsilon10R U2(R,vectorr
prime,t)=ρ(vectorr
prime,t+R
c )
4piepsilon10R
vectorR=vectorr?vectorrprime=
3summationdisplay
i=1
Xivectorei
a233a137a189a158a143t,a31a138a161a143a177vectorrprimea143a165a37a27a165a161,t?Rc =a126a234→R =|vectorr?vectorrprime|=a126a234.
a31a138a161a177a132a199ca36a196,a31a138a161a27a138a27a140a2a851Ra67a122,a233a237a180a41,a31a138a161a42a220,a31a138a161a27a138
a160a2,a233a135a99a41,a31a138a161a194a160,a31a138a161a27a138a79a140,t2 >t1,a31a138a161a100vectorr1a163a20vectorr2
a237a180a41,R2=|vectorr2?vectorrprime|>R1=|vectorr1?vectorrprime| t1?R1c =t2?R2c R2?R1t
2?t1
=c
a135a99a41,R2=|vectorr2?vectorrprime|<R1=|vectorr1?vectorrprime| t1+R1c =t2+R2c R2?R1t
2?t1
=?c
266/384
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a237a180a179,a226a226a212a212a91a53a137a237a180a8a65,R/c a145 vectorE,vectorB a51a195a161a15a63a216a28a851/r2a80a126.
φ(vectorr,t) =
integraldisplay
dτprimeρ(vectorr
prime,t?R
c )
4piepsilon10R
vectorA(vectorr,t) =
integraldisplay
dτprimeμ0
vectorj(vectorrprime,t?R
c )
4piR
·vectorA = μ04pi?·
integraldisplay
dτprime
vectorj(vectorrprime,t?)
R =
μ0
4pi
integraldisplay
dτprime?·bracketleftbig
vectorj(vectorrprime,t?)
R
bracketrightbig
= μ04pi
integraldisplay
dτprime bracketleftbig(?1R)·vectorj(vectorrprime,t?) + 1R?·vectorj(vectorrprime,t?)bracketrightbig
= μ04pi
integraldisplay
dτprime bracketleftbig?(?prime1R)·vectorj(vectorrprime,t?) + 1R?
vectorj(vectorrprime,t?)
t? ·?t
bracketrightbig
= μ04pi
integraldisplay
dτprime bracketleftbigprime·[
vectorj(vectorrprime,t?)
R ] +
1
R?
prime·vectorj(vectorrprime,t?) + 1
R
vectorj(vectorrprime,t?)
t? ·?t
bracketrightbig
= μ04pi
integraldisplay
dτprime bracketleftbig1R?prime·vectorj(vectorrprime,t?)vextendsinglevextendsinglet? + 1R?
vectorj(vectorrprime,t?)
t? ·(?t
+?primet?)bracketrightbig
= μ04pi
integraldisplay
dτprime (?1R)t?ρ(vectorrprime,t?) = μ04pi
integraldisplay
dτprime (?1R)tρ(vectorrprime,t?)
=tμ04pi
integraldisplay
dτprime ρ(vectorr
prime,t?)
R =?μ0epsilon10
φ(vectorr,t)
t
267/384
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a237a180a179,a226a226a212a212a91a53a137;a165a212a53a137
a226a226a212a212a91a53a137:
φ(vectorr,t) =
integraldisplay
dτprimeρ(vectorr
prime,t?R
c )
4piepsilon10R
vectorA(vectorr,t) =
integraldisplay
dτprimeμ0
vectorj(vectorrprime,t?R
c )
4piR
a165a212a53a137:
φ(vectorr,t) =
integraldisplay
dτprimeρ(vectorr
prime,t)
4piepsilon10R
vectorA(vectorr,t) =
integraldisplay
dτprimeμ0
vectorj?(vectorrprime,t?R
c )
4piR
vectorj?(vectorr,t) =vectorj(vectorr,t)?epsilon10
tφ(vectorr,t)
a233a152a135a88a62a54a124,ρ(vectorr,t) = 0→φ(vectorr,t) = 0 a40a74a134a226a226a212a212a91a53a137a131a211.
268/384
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a237a180a179,a49a102a159a254a233a178a144a144a135a135a39a199a27a63a20
a88a74a74a49a49a102a107a154a34a183a142a142a159a159a254m2a49a102:
(?2? 1c2?
2
t2?
m2a49a102c2
planckover2pi12 )
φ(vectorr,t) =?ρ(vectorr,t)
epsilon10
(?2vectorr? 1c2?
2
t2?
m2a49a102c2
planckover2pi12 )
U(vectorr,vectorrprime,t) =?1
epsilon10δ(
vectorR)ρ(vectorrprime,t) vectorR=vectorr?vectorrprime
φ(vectorr,t) =
integraldisplay
dτprime?U(vectorr,vectorrprime,t)?U(vectorr,vectorrprime,t) =?U(R,vectorrprime,t)
U(R,vectorrprime,t) m2a49a102=0===== ρ(vectorrprime,t?Rc )
4piepsilon10R a195a159a254a254a226a226a102a152a189a189a19a19a151a178a144a144a135a135a39a199a156
a49a102a159a254a134a233a178a144a144a135a135a39a199a27a63a20a53a103a117?U(R,vectorrprime,t) a134
ρ(vectorrprime,t?Rc )
4piepsilon10R a27a11a79a156
a193a38,?U(R,vectorrprime,t) = ρ(vectorr
prime,t?R
c )
4piepsilon10R h(R)
269/384
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a237a180a179,a49a102a159a254a233a178a144a144a135a135a39a199a27a63a20
a88a74a74a49a49a102a107a154a34a183a142a142a159a159a254m2a49a102,?U(R,vectorrprime,t) = ρ(vectorrprime,t?Rc )4piepsilon1
0R
h(R)
1epsilon1
0
δ(vectorR)ρ(vectorrprime,t) = (?2vectorr? 1c2?
2
t2?
m2a49a102c2
planckover2pi12 )
U(vectorr,vectorrprime,t)
= h(R)(?2vectorr? 1c2?
2
t2?
m2a49a102c2
planckover2pi12 )
ρ(vectorrprime,t?Rc )
4piepsilon10R
+ρ(vectorr
prime,t?R
c )
4piepsilon10R?
2
vectorrh(R) + 2[?vectorr
ρ(vectorrprime,t?Rc )
4piepsilon10R ]·?vectorrh(R)
=?1epsilon1
0
δ(vectorR)ρ(vectorrprime,t)h(R) + ρ(vectorr
prime,t?R
c )
4piepsilon10R (
2
R2 +
2
R
R
m
2
a49a102c
2
planckover2pi12 )h(R) + 2[
R
ρ(vectorrprime,t?Rc )
4piepsilon10R ]
Rh(R)
=?h(0)epsilon1
0
δ(vectorR)ρ(vectorrprime,t) + ρ(vectorr
prime,t?R
c )
4piepsilon10R (
2
R2?
m2a49a102c2
planckover2pi12 )h(R)
+2R[Rρ(vectorr
prime,t?R
c )
4piepsilon10 ]
Rh(R)
270/384
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a237a180a179,a49a102a159a254a233a178a144a144a135a135a39a199a27a63a20
a88a74a74a49a49a102a107a154a34a183a142a142a159a159a254m2a49a102,?U(R,vectorrprime,t) = ρ(vectorrprime,t?Rc )4piepsilon1
0R
h(R)
1epsilon1
0
δ(vectorR)ρ(vectorrprime,t) =?h(0)epsilon1
0
δ(vectorR)ρ(vectorrprime,t) + ρ(vectorr
prime,t?R
c )
4piepsilon10R (
2
R2?
m2a49a102c2
planckover2pi12 )h(R)
+2R[Rρ(vectorr
prime,t?R
c )
4piepsilon10 ]
Rh(R)
h(0) = 1 {?
2
R2?
m2a49a102c2
planckover2pi12 + 2[
R lnρ(vectorr
prime,t?R
c )]
R}h(R) = 0
a144a107a62a214a151a221a216a145a158a109a67a122a226a103a84a181 h(R) = e±cplanckover2pi1 ma49a102R
a127a196a195a161a15a27a62a94a135,?U(R,vectorrprime,t) = ρ(vectorr
prime)
4piepsilon10Re
cplanckover2pi1 ma49a102R
a49a102a159a254a233a178a144a144a135a135a39a199a27a63a20,1R2? 1R2 e?cplanckover2pi1 ma49a102R
a77a77a94a94 1R2+epsilon1a28a28a91a91,epsilon1 = cplanckover2pi1 ma49a102 RlnR a15a167a131a112a138a94a67a164a67a167a131a112a138a94!
271/384
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a203a19a62a94a124,a152a132a53a159
φ(vectorr,t) =
integraldisplay
dτprimeρ(vectorr
prime,t?)
4piepsilon10R
vectorA(vectorr,t) =
integraldisplay
dτprimeμ0
vectorj(vectorrprime,t?)
4piR t
= t?R
c
vectorB =?×vectorA vectorE =φvectorA
tcontintegraldisplay
∞
dvectorσ·vectorS =
contintegraldisplay
∞
dvectorσ·(vectorE×vectorH) =? 0
vectorn≡
vectorR
R =
vectorr
r + O(
1
r)
t?
t = 1?t
=R
c =?
vectorn
c
ψ(t?) =?tψ?t =?vectornc?ψ?t
·vectorf(t?) =?t?·?
vectorf
t =?
vectorn
c·
vectorf
t
×vectorf(t?) =?t?×?
vectorf
t =?
vectorn
c×
vectorf
t
272/384
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a203a19a62a94a124,a152a132a53a159
vectorBa203 = (?×vectorA)vextendsinglevextendsingle1
r
=
integraldisplay
dτprime μ04piR?×vectorj(vectorrprime,t?) =
integraldisplay
dτprime μ04picR?
vectorj(vectorrprime,t?)
t ×vectorn
vectorEa203 = (φvectorA
t )
vextendsinglevextendsingle
1
r
=
integraldisplay
dτprimebracketleftbig? 14piRepsilon1
0
ρ(vectorrprime,t?)? μ04piR?
vectorj(vectorrprime,t?)
t
bracketrightbig
=
integraldisplay
dτprimebracketleftbig 14piepsilon1
0cR
ρ(vectorrprime,t?)
t vectorn?
μ0
4piR
vectorj(vectorrprime,t?)
t
bracketrightbig
=?φ?tvectornc?
integraldisplay
dτprime μ04piR?
vectorj(vectorrprime,t?)
t =?cvectorn?·
vectorA?
integraldisplay
dτprime μ04piR?
vectorj(vectorrprime,t?)
t
=
integraldisplay
dτprime μ04piRbracketleftbig?cvectorn?·vectorj(vectorrprime,t?)
vectorj(vectorrprime,t?)
t
bracketrightbig
=
integraldisplay
dτprime μ04piRbracketleftbigvectorn?
vectorj(vectorrprime,t?)
t ·vectorn?
vectorj(vectorrprime,t?)
t
bracketrightbig
vectorn×vectorEa203 = cvectorBa203 cvectorBa203×vectorn = vectorEa203
273/384
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a203a19a62a94a124,a152a132a53a159
vectorBa203 =
integraldisplay
dτprime μ04picR?
vectorj(vectorrprime,t?)
t ×vectorn
vectorn×vectorEa203 = cvectorBa203 cvectorBa203×vectorn = vectorEa203
We = 12epsilon10E2 Wm = 12μ0H2 = B
2
2μ0 =
E2
2c2μ0 =
1
2epsilon10E
2 = We
vectorS = vectorE×vectorH = cB2
μ0vectorn = 2Wmcvectorn = (We + Wm)cvectorn
a203a19a245a199:
I =
integraldisplay
a165
dvectorσ·vectorS =
integraldisplay
a165
dσ Svectorn·vectorn =
integraldisplay
a165
dσ S =
integraldisplay
d? Sr2 =
integraldisplay
d? dId?
dI
d? = Sr
2
274/384
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a203a19a62a94a124,a245a52a208a109
R =|vectorr?vectorrprime|= (r2+rprime2?2vectorr·vectorrprime)12 = r(1?vectorn·vectorr
prime
r +···) = r?vectorn·vectorr
prime+···
vectorA(vectorr,t)vextendsinglevextendsinglea203 =
integraldisplay
dτprime μ04piRvectorj(vectorrprime,t?Rc ) =
integraldisplay
dτprime μ04piRvectorj(vectorrprime,t?rc+vectorn·vectorr
prime
c?···)
= μ04pir
integraldisplay
dτprime bracketleftbig
∞summationdisplay
k=0
1
k!
kvectorj(vectorrprime,t?rc)
tk (
vectorn·vectorrprime
c?···)
kbracketrightbig
= μ04pir
integraldisplay
dτprime
∞summationdisplay
k=0
1
k!
kvectorj(vectorrprime,t?rc)
tk (
vectorn·vectorrprime
c )
k
= μ04pir
∞summationdisplay
k=0
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck
k
tk
integraldisplay
dτprimevectorj(vectorrprime,t?rc)xprimei1···xprimeik
= μ04pir
∞summationdisplay
k=0
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck
k
tk
vectorJi
1···ik(t?
r
c)
vectorJi
1···ik(t) =
integraldisplay
dτprimevectorj(vectorrprime,t)xprimei1···xprimeik
275/384
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a203a19a62a94a124,a245a52a208a109
vectorJi
1···ik(t) =
integraldisplay
dτprimevectorj(vectorrprime,t)xprimei1···xprimeik
vectorA(vectorr,t)vextendsinglevextendsinglea203 = μ0
4pir
∞summationdisplay
k=0
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck
k
tk
vectorJi
1···ik(t?
r
c)
vectorB(vectorr,t)vextendsinglevextendsinglea203 = (?×vectorA)vextendsinglevextendsingle1
r
= μ04pir
∞summationdisplay
k=0
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck?×
k
tk
vectorJi
1···ik(t?
r
c)
= μ04pir
∞summationdisplay
k=0
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck (?
vectorn
c×
t)
k
tk
vectorJi
1···ik(t?
r
c)
= μ04pir
∞summationdisplay
k=0
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck+1
k+1
tk+1
vectorJi
1···ik(t?
r
c)×vectorn =
∞summationdisplay
k=0
vectorBk
vectorBk = μ0
4pir
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck+1
k+1
tk+1
vectorJi
1···ik(t?
r
c)×vectorn
vectorBk+1a39vectorBka245 1cxprimet ～ lprimeωc ～ lprimeλ,a135a208a109a208a55a76lprimelessmuchλ
276/384
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a203a19a62a94a124,a245a52a208a109
vectorB(vectorr,t)vextendsinglevextendsinglea203=
∞summationdisplay
k=0
vectorBk vectorBk= μ0
4pir
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck+1
k+1
tk+1
vectorJi
1···ik(t?
r
c)×vectorn
vectorB0 = μ0
4picr
t
vectorJ(t?r
c)×vectorn =
μ0
4picr
t
integraldisplay
dτprimevectorj(vectorrprime,t?rc)×vectorn
= μ04picr
3summationdisplay
i,j,k=1
tepsilon1ijk
integraldisplay
dτprime ji(vectorrprime,t?rc)njvectorek
= μ04picr
3summationdisplay
i,j,k=1
epsilon1ijkt
integraldisplay
dτprime bracketleftbig?primel[jl(vectorrprime,t?rc)xprimei]?xprimei?primeljl(vectorrprime,t?rc)bracketrightbignjvectorek
= μ04picr
3summationdisplay
i,j,k=1
epsilon1ijkt
integraldisplay
dτprime xprimei?ρ(vectorr
prime,t?r
c)
t njvectorek
= μ04picr
3summationdisplay
i,j,k=1
epsilon1ijk?
2
t2qi(t?
r
c)njvectorek =
μ0
4picr
¨vectorP(t?r
c)×vectorn
vectorP(t) = summationdisplayqi(t)vectorei qi(t) =
integraldisplay
dτprime xprimeiρ(vectorrprime,t)
277/384
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a203a19a62a94a124,a245a52a208a109
vectorB(vectorr,t)vextendsinglevextendsinglea203=
∞summationdisplay
k=0
vectorBk vectorBk= μ0
4pir
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck+1
k+1
tk+1
vectorJi
1···ik(t?
r
c)×vectorn
vectorB1 = μ0
4pic2r
3summationdisplay
i=1
ni?
2
t2
vectorJi(t?r
c)×vectorn
= μ04pic2r
3summationdisplay
i=1
2
t2
integraldisplay
dτprime xprimeivectorj(vectorrprime,t?rc)×vectorn
= μ04pic2r
3summationdisplay
i,j,k,l=1
ni?
2
t2epsilon1jkl
integraldisplay
dτprime xprimeijj(vectorrprime,t?rc)nkvectorel
= μ08pic2r
3summationdisplay
i,j,k,l=1
ni?
2
t2epsilon1jkl
integraldisplay
dτprime [xprimeijj + xprimejji + xprimeijj?xprimejji]nkvectorel
= μ08pic2r
3summationdisplay
i,j,k,l=1
ni?
2
t2epsilon1jkl
integraldisplay
dτprimebracketleftbig
3summationdisplay
m=1
[?primem(jmxprimeixprimej)?xprimeixprimej(?primemjm)]+
3summationdisplay
m,n,p=1
epsilon1ijpepsilon1nmpxprimenjmbracketrightbignkvectorel
278/384
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a203a19a62a94a124,a245a52a208a109
vectorB(vectorr,t)vextendsinglevextendsinglea203=
∞summationdisplay
k=0
vectorBk vectorBk= μ0
4pir
1
k!
3summationdisplay
i1,···,ik=1
ni1···nik
ck+1
k+1
tk+1
vectorJi
1···ik(t?
r
c)×vectorn
vectorB1 = μ0
4pic2r
3summationdisplay
i=1
ni?
2
t2
vectorJi(t?r
c)×vectorn
= μ08pic2r
3summationdisplay
i,j,k,l=1
ni?
2
t2epsilon1jkl
integraldisplay
dτprime bracketleftbig
3summationdisplay
m=1
[?primem(jmxprimeixprimej)?xprimeixprimej(?primemjm)]
+
3summationdisplay
m,n,p=1
epsilon1ijpepsilon1nmpxprimenjmbracketrightbignkvectorel
= μ08pic2r
3summationdisplay
i,j,k,l=1
ni?
2
t2
integraldisplay
dτprime bracketleftbigepsilon1jklxprimeixprimejtρ(vectorrprime,t?rc)
+
3summationdisplay
m,n,p=1
epsilon1jklepsilon1ijpepsilon1nmpxprimenjm(vectorrprime,t?rc)bracketrightbignkvectorel
279/384
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a203a19a62a94a124,a245a52a208a109
vectorB1 = μ0
8pic2r
3summationdisplay
i,j,k,l=1
ni?
2
t2
integraldisplay
dτprime bracketleftbigepsilon1jklxprimeixprimejtρ(vectorrprime,t?rc)
+
3summationdisplay
m,n,p=1
epsilon1jklepsilon1ijpepsilon1nmpxprimenjm(vectorrprime,t?rc)bracketrightbignkvectorel
= μ08pic2r
3summationdisplay
i,j,k,l=1
nibracketleftbig?
3
t3epsilon1jklqij(t?
r
c) + 2
2
t2
3summationdisplay
p=1
epsilon1jklepsilon1ijpmp(t?rc)bracketrightbignkvectorel
= μ08pic2r
3summationdisplay
i,j,k,l=1
nibracketleftbig?
3
t3epsilon1jkl
1
3Qij(t?
r
c) + 2
2
t2
3summationdisplay
p=1
epsilon1jklepsilon1ijpmp(t?rc)bracketrightbignkvectorel
= μ04pic2rbracketleftbig16vectorn·
···arrowrighttophalfarrowrighttophalf
Q ×vectorn + (¨vectorm×vectorn)×vectornbracketrightbig
arrowrighttophalfarrowrighttophalfQ(t?r
c)=
integraldisplay
dτprime(3vectorrprimevectorrprime?arrowrighttophalfarrowrighttophalfIrprime2)ρ(vectorrprime,t?rc) vectorm(t?rc)=12
integraldisplay
dτprimevectorrprime×vectorj(vectorrprime,t?rc)
280/384
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a203a19a62a94a124,a245a52a208a109,a62a243a52a203a19
a152a225a85a130a100A,Ba252a135a2a165a124a164,(vectorla233a225,a100Ba141a149A),A,Ba254a136a145a131a135a62a214,a191a62a0a51A
Ba109a27a27a19a19a130a254a23a41a112a170a62a54
a0
a0
a0
a0a0a18
a115
a115 r
vectornA
B
J = dQAdt = J0 cosωt QA = J0ω sinωt
l→0,J0ω →0,lJ0ωa27a189 vectorP = QAvectorl = J0ωvectorlsinωt
vectorB(vectorr,t) = μ0
4picr
¨vectorP(t?r
c)×vectorn =?
μ0J0ω
4picr
vectorlsinω(t?r
c)×vectorn
= μ0
vectorl
4picr
˙J(t?r
c)×vectorn
vectorS(vectorr,t) = c
μ0B
2vectorn = μ0l
2
(4pi)2cr2
˙J2(t?r
c)sin
2θvectorn
dI
d? =
μ0l2
(4pi)2c
˙J2(t?r
c)sin
2θ
281/384
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a203a19a62a94a124,a245a52a208a109,a62a243a52a203a19
a152a225a85a130a100A,Ba252a135a2a165a124a164,(vectorla233a225,a100Ba141a149A),A,Ba254a136a145a131a135a62a214,a191a62a0a51A
Ba109a27a27a19a19a130a254a23a41a112a170a62a54
a0
a0
a0
a0a0a18
a115
a115 r
vectornA
B
dI
d? =
μ0l2
(4pi)2c
˙J2(t?r
c)sin
2θ
I(t) =
integraldisplay
d? dId? =
integraldisplay 2pi
0
dφ
integraldisplay pi
0
dθsinθ dId?
= 2pi
integraldisplay pi
0
dθ μ0l
2
(4pi)2c
˙J2(t?r
c)sin
3θ = μ0l
2
6pic
˙J2(t?r
c) =
μ0l2ω2J20
6pic sin
2ω(t?r
c)
I = lim
T→∞
1
T
integraldisplay T
0
dt I(t) = lim
T→∞
1
T
integraldisplay T
0
dt μ0l
2ω2J2
0
6pic
1
2[1?cos2ω(t?
r
c)]
= μ0l
2ω2J2
0
12pic limT→∞
1
T[t?
1
2ω sin2ω(t?
r
c)]
vextendsinglevextendsingleT
0 =
μ0l2ω2J20
12pic
= μ0l
2ω2
6picR(
1
2J
2
0R) =
Ra203
R (
1
2J
2
0R)
Ra203 = μ0l
2ω2
6pic
1
2J
2
0R,a57a245a199 R,a85a130a62a123
282/384
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a203a19a62a94a124,a245a52a208a109,a62a111a52a203a19
a62a254a143q,q,?2qa27a110a135a2a165a169a79a160a117a127a1432la27a225a85a130a27a252a224a218a165a109,vectorla143 a108a85a130a165a58
a141a149a152a224a224a27a27a165a254,q = q0 cosωt,q0→∞,l→0a2q0l2a27a189.
a0
a0
a0
a0a0a18
a115
a118
a115
r
vectorn
q
2q
q J = ˙q =?q0ωsinωt = J0 sinωt
vectorP =
integraldisplay
dτprimevectorrprimeρ = qvectorl + q(?vectorl) = 0
vectorm = 12
integraldisplay
dτprimevectorrprime×vectorj = 12
integraldisplay
vectorrprime×Jdvectorlprime = 0
arrowrighttophalfarrowrighttophalfQ= integraldisplay dτprime3vectorrprimevectorrprimeρ = 3vectorlvectorlq + 3(?vectorl)(?vectorl)q = 6qvectorlvectorl
vectorB(vectorr,t) = μ0
6c2r4pivectorn·
···arrowrighttophalfarrowrighttophalf
Q ×vectorn = μ06c2r4pi6 ···qvectorn·vectorlvectorl×vectorn
= l
2μ0q0ω3
4pic2r sinω(t?
r
c)cosθsinθ
vectorS(vectorr,t) = c
μ0B
2vectorn = μ0l
4q2
0ω
6
c3r2(4pi)2 sin
2ω(t?r
c)sin
2θcos2θvectorn
283/384
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a203a19a62a94a124,a245a52a208a109,a62a111a52a203a19
a62a254a143q,q,?2qa27a110a135a2a165a169a79a160a117a127a1432la27a225a85a130a27a252a224a218a165a109,vectorla143 a108a85a130a165a58
a141a149a152a224a224a27a27a165a254,q = q0 cosωt,q0→∞,l→0a2q0l2a27a189.
a0
a0
a0
a0a0a18
a115
a118
a115
r
vectorn
q
2q
q vectorS(vectorr,t) = c
μ0B
2vectorn = μ0l
4q2
0ω
6
c3r2(4pi)2 sin
2ω(t?r
c)sin
2θcos2θvectorn
dI
d? =
μ0l4q20ω6
c3(4pi)2 sin
2θcos2θsin2ω(t?r
c)
I = μ0l
4q2
0ω
6
c3(4pi)2 sin
2ω(t?r
c)2pi
integraldisplay pi
0
dθ sin3θcos2θ
= μ0l
4q2
0ω
6
30c3pi sin
2ω(t?r
c)
I = μ0l
4q2
0ω
6
60c3pi =
μ0l4J20ω2
60c3pi =
Ra203
R (
1
2J
2
0R)
Ra203 = μ0l
4ω4
30pic
284/384
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a203a19a62a94a124,a245a52a208a109,a94a243a52a203a19
a62a54a23a140a187a143a,I = I0 cosωt,vectorm = Ipia2vectorez
vectorj(vectorr,t) = jθ(r,z,t)vectoreθ jr = jz = 0
˙ρ(vectorr,t) =·vectorj =?1r?jθ(r,z,t)?θ = 0 ˙arrowrighttophalfarrowrighttophalfQ = 0
vectorB(vectorr,t) = μ0
4pic2r
¨vectorm×vectorn = ω2μ0pia2
4pic2r I0 cosωt
vectorez×vectorn
vectorS(vectorr,t) = c
μ0B
2vectorn = ω
4μ0a4I2
0
16c3r2 cos
2ωt?(vectorez×vectorn)·(vectorez×vectorn)vectorn
= ω
4μ0a4I2
0
16c3r2 cos
2ωt?sin2θvectorn
dI
d? =
ω4μ0a4I20
16c3 cos
2ωt?sin2θ I(t) = pi
3ω4μ0a4I2
0
6c3 cos
2ωt?
I = pi
3ω4μ0a4I2
0
12c3 =
Ra203
R (
1
2J
2
0R) Ra203 =
μ0a4ω4pi3
6c3
285/384
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a203a19a62a94a124,a62a94a197a27a251a19
a168a141a100a6a110,a51a197a68a194a27a197a10a161a254a27a122a152a58a209a180a103a63a197a13,a122a152a135
a197a13a117a209a2a27a165a161a102a197,a165a161a102a197a27a157a161a47a164a35a27a197a10a161.
u?Ex,Ey,Ez,Bx,By,Bz
2u(vectorr,t)? 1c2?
2
t2u(vectorr,t) = 0
vectorR =vectorrprime?vectorr vectorr?τ
u(vectorr,t) =
integraldisplay
dτprimeδ(vectorrprime?vectorr)u(vectorrprime,t?Rc ) =
integraldisplay
dτprime(?14pi)(?prime2 1R)u(vectorrprime,t?Rc )
=
integraldisplay
dτprime(?14pi)braceleftbig?prime·[(?prime1R)u(vectorrprime,t?Rc )]?(?prime1R)·?primeu(vectorrprime,t?Rc )bracerightbig
=
integraldisplay
dτprime(?14pi)braceleftbig?prime·[(?prime1R)u(vectorrprime,t?Rc )?1R?primeu(vectorrprime,t?Rc )]+ 1R?prime2u(vectorrprime,t?Rc )bracerightbig
primet? =?1c?primeR =?
vectorR
cR
286/384
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a203a19a62a94a124,a62a94a197a27a251a19
1
R?
prime2u(vectorrprime,t?R
c ) =
1
R?
prime·[?primeu(vectorrprime,t?)vextendsinglevextendsingle
t
u(vectorrprime,t?)
t?
vectorR
cR]
= 1R?prime·[?primeu(vectorrprime,t?)vextendsinglevextendsinglet?]?
vectorR
cR2·?
prime?u(vectorr
prime,t?)
t
1
cR
u(vectorrprime,t?)
t
prime·vectorR
R
= 1R?prime2u(vectorrprime,t?)vextendsinglevextendsinglet
vectorR
cR2·
t
primeu(vectorrprime,t?)vextendsinglevextendsingle
t
vectorR
cR2·?
prime?
t?u(vectorr
prime,t?)vextendsinglevextendsingle
t?
+ 1c2R?
2
t?2u(vectorr
prime,t)u(vectorr
prime,t?)
t?
1
cR[
3
R?
1
R]
= 1R[?prime2u(vectorrprime,t?)vextendsinglevextendsinglet? + 1c2?
2
t?2u(vectorr
prime,t?)]? 2vectorR
cR2·?
prime?
t?u(vectorr
prime,t?)vextendsinglevextendsingle
t?
2cR2?u(vectorr
prime,t)
t?
= 2Rc2?
2
t?2u(vectorr
prime,t?)? 2vectorR
cR2·?
prime?
t?u(vectorr
prime,t?)vextendsinglevextendsingle
t
2
cR2
u(vectorrprime,t)
t?
= 2cbraceleftbig
vectorR
cR·
vectorR
R2
2
t?2u(vectorr
prime,t?)? vectorR
R2·?
prime?
t?u(vectorr
prime,t?)vextendsinglevextendsingle
t(?
prime· vectorR
R2)
u(vectorrprime,t?)
t?
bracerightbig
287/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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Close
a203a19a62a94a124,a62a94a197a27a251a19
1
R?
prime2u(vectorrprime,t?R
c )
= 2cbraceleftbig
vectorR
cR·
vectorR
R2
2
t?2u(vectorr
prime,t?)? vectorR
R2·?
prime?
t?u(vectorr
prime,t?)vextendsinglevextendsingle
t(?
prime· vectorR
R2)
u(vectorrprime,t?)
t?
bracerightbig
=?2cbraceleftbig
vectorR
R2·[?
vectorR
cR
t? +?
primevextendsinglevextendsingle
t?]
t?u(vectorr
prime,t?) + (?prime· vectorR
R2)
u(vectorrprime,t?)
t?
bracerightbig
=?2cbraceleftbig
vectorR
R2·?
prime?
t?u(vectorr
prime,t?)+(?prime· vectorR
R2)
u(vectorrprime,t?)
t?
bracerightbig =?2
c?
prime·braceleftbig vectorR
R2
t?u(vectorr
prime,t?)bracerightbig
u(vectorr,t) = (?14pi)
integraldisplay
τ
dτprime?prime·braceleftbig(?prime1R)u(vectorrprime,t?Rc )? 1R?primeu(vectorrprime,t?Rc )
2c
vectorR
R2
t?u(vectorr
prime,t?)bracerightbig
=(?14pi)
integraldisplay
τ
dτprime?prime·braceleftbig?
vectorR
R3u(vectorr
prime,t?R
c )?
1
R?
primeu(vectorrprime,t?)vextendsinglevextendsingle
t
vectorR
cR2
t?u(vectorr
prime,t?)bracerightbig
= 14pi
integraldisplay
τ
dvectorSprime ·braceleftbig1R?primeu(vectorrprime,t?) +
vectorR
R3u(vectorr
prime,t?) + vectorR
cR2
t?u(vectorr
prime,t?)bracerightbigvextendsinglevextendsingle
t?=t?Rc
288/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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Close
a203a19a62a94a124,a62a94a197a27a251a19～a196a16a191a197a98a23
a127a196a178a161a197u(vectorr,t) a19a149a109a107a2a154a27a195a161a140a178a161Sa254.a138a88a101a196a16a191a197a98a23:
a216a154a83a9,u,?ua51Sa161a254a63a63a63a63a1430.
a154a83a27u,?ua27a138a31a117a118a107a63a219a182a189a230a78a212a158,a92a19a197a27a138.
u(vectorr,t) =
braceleftbigg 0
a154a9
Aei(ωt?ωzc ) a154a83
u(vectorr,t) = 14pi
integraldisplay
τ
dvectorSprime·braceleftbig1R?primeu(vectorrprime,t?) +
vectorR
R3u(vectorr
prime,t?) + vectorR
cR2
t?u(vectorr
prime,t?)bracerightbigvextendsinglevextendsingle
t?=t?Rc
primeu(vectorrprime,t?) =?iωcvectorezu(vectorrprime,t?)t?u(vectorrprime,t?) = iωu(vectorrprime,t?)
u(vectorr,t) = A4pi
integraldisplay
a154
dvectorSprime·braceleftbig1R(?iωcvectorez) +
vectorR
R3 +
vectorR
cR2iω
bracerightbigu(vectorrprime,t?)vextendsinglevextendsingle
t?=t?Rc
= A4pi
integraldisplay
a154
dvectorSprime·braceleftbig1R(?iωcvectorez) +
vectorR
R3 +
vectorR
cR2iω
bracerightbigeiωt?vextendsinglevextendsingle
t?=t?Rc
289/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a203a19a62a94a124,a62a94a197a27a251a19～a196a16a191a197a98a23
a127a196a178a161a197u(vectorr,t) a19a149a109a107a2a154a27a195a161a140a178a161Sa254.a138a88a101a196a16a191a197a98a23:
a216a154a83a9,u,?ua51Sa161a254a63a63a63a63a1430.
a154a83a27u,?ua27a138a31a117a118a107a63a219a182a189a230a78a212a158,a92a19a197a27a138.
u(vectorr,t) =
braceleftbigg 0
a154a9
Aei(ωt?ωzc ) a154a83
u(vectorr,t) = A4pi
integraldisplay
a154
dvectorSprime·braceleftbig1R(?iωcvectorez) +
vectorR
R3 +
vectorR
cR2iω
bracerightbigeiωt?vextendsinglevextendsingle
t?=t?Rc
dvectorSprime=?dSprimevectorez=====?A
4pie
iωt
integraldisplay
a154
dSprimebraceleftbig?iωcR +?cosθR2? iωcR cosθbracerightbige?iωRc
= A4pieiωt
integraldisplay
a154
dSprimebraceleftbigiωcR(1 + cosθ) + cosθR2 bracerightbige?iωRc
a8a124a58a108a182a22a15a158,a204a135a49a152a145a0a122,a23a197a70a218a164a251a19
a8a124a58a108a182a22a22a67a67a158,a204a135a49a19a145a0a122,a23a153a114a16a251a19
290/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a203a19a62a94a124,a62a94a197a27a251a19～a197a70a218a164a251a19
a233a152a135a127a176a169a79a143a,ba27a221a47a47a2a2a154:
a1
a1
a1
a1
a1a1a1
a1
a1
a1
a1a1
a45
a54
a1
a1
a1
a1a1a21
a8a8
a8a8
a8a8
a8a8a8a42
a3
a3a3a23
a24a24a24a24
a24a24a24a24a58
vectorez
vectorey vectorex
vectorr
vectorRvectorρ
b a
vectorρ = xvectorex + yvectorey ρ2 = x2 + y2
vectorR =vectorr?vectorρ vectorr·vectorez = rcosθ3
vectorr·vectorex = rcosθ1 vectorr·vectorey = rcosθ2
R = (r2 +ρ2?2vectorr·vectorρ)12 = [r2 +ρ2?2r(xcosθ1 + ycosθ2)]12
cosθ =?
vectorR·vectorez
R =
rcosθ3
R =
rcosθ3
[r2 +ρ2?2r(xcosθ1 + ycosθ2)]12
ρlessmuchr
R = r[1 + ρ
2
r2?
2
r(xcosθ1 + ycosθ2)]
1
2 = r?xcosθ1?ycosθ2 +···
cosθ = cosθ3[1 + xr cosθ1 + yr cosθ2 +···]
u(vectorr,t) = A4pieiωt
integraldisplay
a154
dSprimebraceleftbigiωcR(1 + cosθ) + cosθR2 bracerightbige?iωRc
= A4pieiωt
integraldisplay
a154
dSprimeiωcr(1 + cosθ3)e?iωc[r?xcosθ1?ycosθ2]
291/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a203a19a62a94a124,a62a94a197a27a251a19～a197a70a218a164a251a19
a233a152a135a127a176a169a79a143a,ba27a221a47a47a2a2a154:
a1
a1
a1
a1
a1a1a1
a1
a1
a1
a1a1
a45
a54
a1
a1
a1
a1a1a21
a8a8
a8a8
a8a8
a8a8a8a42
a3
a3a3a23
a24a24a24a24
a24a24a24a24a58
vectorez
vectorey vectorex
vectorr
vectorRvectorρ
b a
u(vectorr,t) = A4pieiωt
integraldisplay
a154
dSprimebraceleftbigiωcR(1+cosθ)+cosθR2 bracerightbige?iωRc
= A4pieiωt
integraldisplay
a154
dSprimeiωcr(1 + cosθ3)e?iωc[r?xcosθ1?ycosθ2]
= A4pieiωtiωcr(1 + cosθ3)
integraldisplay a
a
dx
integraldisplay b
b
dyeiωc cosθ1x+iωc cosθ2ye?iωcr
= A4pieiωt?iωcriωcrc
2
ω2(1 + cosθ3)
2isin(ωac cosθ1)
cosθ1
2isin(ωbc cosθ2)
cosθ2
= A4piei(ωt?ωcr)4icωr(1 + cosθ3)sin(
ω
cacosθ1)
cosθ1
sin(ωcbcosθ2)
cosθ2
a251a19a197a143a165a161a197,a8a204a51ωcacosθ1 = mpi,ωcbcosθ2 = npi,(m,n,1,2,3)a63a1430(a86a58).
a252a191a251a19a27a28a28a91a91
292/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a100a194a131a233a216
293/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a100a194a131a233a216,a196a58～a6a207
a143a159a111a51a62a62a196a196a229a198a145a165a63a216a100a194a131a233a216,a167a180a62a62a196a196a229a198a117a208a27a23a212
a240a142a100a137a27a27a62a62a94a110a216a1631865a164a18a26a10a227a140a27a164a245
a167a143a107a101a90a134a162a8a216a20a218a27a27a47a47a144
–a220a169a175a175a162a162a163a231a78a203a19a167a6a102a117a49a164?a254a102a216
–a220a169a175a175a162a162a163a132a0a164?a131a233a216
a254a102a216a218a131a233a216a8a164a19a155a173a86a212a110a198a129a149a140a27a252a145a117a121a182
a143a180a19a155a155a173a173a86a212a110a198a26a177a37a199a117a208a27a196a58
a100a194a131a233a216a180a51a233a62a94a110a216a63a49a10a28a161a48a11(a183a62a94a33a68
a194a33a203a19)a0a167a233a101a90a216a218a20a144a161a27a29a92a239a196
294/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
Back
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a100a194a131a233a216,a196a58～a131a233a53a6a110
a216a37a181a152a131a46a53a235a127a88a51a212a110a198a254a180a17a28a31a100a27
a162a8a181a51a216a211a46a53a88a137a211a24a27a162a8a55a107a211a24a27a40a74a156
a110a216a181a63a219a196a29a212a110a53a198a51a216a211a46a53a88a165a55a107a131a211a27a144a167a47a170!
a131a233a53a181a132a221a180a131a233a233a27a27a88a51a196a186a195a253a233a235a127a88a182a92a132a221a216a180a131a233a233a27a27!
a179a124a209a158a147a1631564-1642a164a27a131a233a53a6a110a181
a70a37a96a163a120a120a901473-1563a164a134a47a37a96a27a216a212(a19a172a167a70a126a178a8)
a47a165a131a233a20a19a27a36a196a132a221,30a250a112/a166?a154a74,a8a158a118a107a110a216a167a144a85a157a227a121a121a150a150
a233a148a31a218a156a106a64a24a85a248a108a47a161a13a51a152a165a202a51a152a227a27a192a220a167a167a130a119a53a216a140
a85a139a254a47a165a27a132a221a34a183a130a65a84a250a26a167a130a180a156a175a47a149a220a144a36a196a34
a106a14a78a111a139a26a254a249a135a63a167a81a186a183a130a119a20a27a180a167a106a14a149a192a189a246a149a220a167a189a246
a149a79a27a144a149a156a49a167a118a107a63a219a11a201a34
a88a74a183a130a228a234a27a182a158a250a26a78a51a242a254a27a186a131a8a136a17a167a51a47a165a249a24a152a135a95a152
a237a13a49a27a156a132a63a167a165a167a183a130a84a97a250a152a24a245a114a167a27a192a186a66a156a44a13a37a250a9a216
a20a249a24a27a75a143a34
295/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a100a194a131a233a216,a196a58～a131a233a53a6a110
a216a37a181a152a131a46a53a235a127a88a51a212a110a198a254a180a17a28a31a100a27
a162a8a181a51a216a211a46a53a88a137a211a24a27a162a8a55a107a211a24a27a40a74a156
a110a216a181a63a219a196a29a212a110a53a198a51a216a211a46a53a88a165a55a107a131a211a27a144a167a47a170!
a131a233a53a181a132a221a180a131a233a233a27a27a88a51a196a186a195a253a233a235a127a88a182a92a132a221a216a180a131a233a233a27a27!
a179a124a209a158a147a1631564-1642a164a27a131a233a53a6a110a181
a70a37a96a163a120a120a901473-1563a164a134a47a37a96a27a216a212(a19a172a167a70a126a178a8)
a47a165a131a233a20a19a27a36a196a132a221,30a250a112/a166?a154a74,a140a237,a156a106
a39a117a247a86a151a218a120a120a90a252a140a173a46a78a88a27a27a233a233a123(1632):
a233a154a74a27a216a121a180a131a233a53a6a110a129a64a27a76a227:
a114a92a218a152a10a42a108a39a51a152a94a140a69a96a134a101a27a204a242a112a167a50a52a92a130a145a65a144a241a71a33
a8a82a218a217a167a2a156a193a34a242a83a152a152a144a140a89a18a167a217a165a152a65a94a126a34a44a0a33a254a152a135
a89a180a167a52a89a152a37a152a37a37a47a47a47a37a37a37a20a20a101a161a27a152a135a176a176a157a157a45a112a34
a69a202a88a216a196a158a181a2a193a209a177a31a132a149a242a83a136a144a149a156a49a182a126a149a136a135a144a149a145a66a105
a196a182a89a37a37a63a101a161a27a45a102a165a182a92a114a63a219a192a220a68a137a92a27a42a108a158a167a144a135a229a108
a131a31a167a149a249a152a144a149a216a55a39a44a152a144a149a94a141a245a27a229a182a92a86a12a224a97a167a195a216a149a61
a135a144a149a97a76a27a229a108a209a131a31a34
a50a166a69a177a63a219a132a221a99a63a167a144a135a36a196a180a33a132a27a167a143a216a3a134a3a109a47a123a196a34
296/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a100a194a131a233a216,a196a58～a131a233a53a6a110
a50a166a69a177a63a219a132a221a99a63a167a144a135a36a196a180a33a132a27a167a143a216a3a134a3a109a47a123a196a34
a92a242a117a121a167a164a107a254a227a121a150a106a206a118a107a67a122a167a92a195a123a108a217a165a63a219a152a135a121a150a53
a40a189a167a69a180a51a51a36a36a196a132a180a202a88a216a196a196a27a27a167a61a166a69a36a196a196a27a27a131a8a175a34
a51a97a23a158a167a92a242a218a177a99a152a24a51a69a46a134a254a97a76a131a211a27a229a108a167a92a97a149a69a151a143
a216a172a39a97a149a69a222a53a27a15a167a143a44a92a97a20a152a165a158a167a12a101a27a69a46a134a149a88a92a97a27
a131a135a135a144a144a149a163a196a34
a92a114a216a216a159a111a192a220a68a137a92a27a211a138a158a167a216a216a166a180a51a69a222a132a180a51a69a151a167a144a135
a92a103a67a213a213a51a51a233a161a167a92a143a216a73a135a94a141a245a245a27a27a229a34
a89a37a242a148a107a99a152a24a167a37a63a101a161a27a45a102a167a152a37a143a216a172a37a149a69a151a167a143a44a89a37
a51a152a165a158a167a69a174a49a168a10a78a245a221a34
a126a51a89a165a105a149a89a18a99a220a164a94a27a229a167a216a39a105a149a89a18a0a220a53a27a140a34a166a130a152a24
a97a115a47a105a149a152a51a89a18a62a14a63a219a47a144a27a160a17a34
a8a82a218a241a71a242a85a89a145a66a47a20a63a156a49a167a166a130a251a216a172a149a69a151a56a165a34a191a216a207a143
a166a130a127a158a109a51a51a152a165a167a248a108a10a69a27a36a196a167a143a96a254a69a27 a36a196a119a209a92a27a24
a102a34
a88a74a58a45a134a235a167a92a242a119a20a235a148a152a249a31a152a24a149a254a44a229a167a216a149a63a219a152a62a163
a196a34
a212a78a228a107a46a53a34a143a44a64a158a132a118a107a218a238a238a27a27a46a53a189a198
297/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a100a194a131a233a216,a196a58～a131a233a53a6a110
a216a37a181a152a131a46a53a235a127a88a51a212a110a198a254a180a17a28a31a100a27
a162a8a181a51a216a211a46a53a88a137a211a24a27a162a8a55a107a211a24a27a40a74a156
a110a216a181a63a219a196a29a212a110a53a198a51a216a211a46a53a88a165a55a107a131a211a27a144a167a47a170!
a131a233a53a181a132a221a180a131a233a233a27a27a88a51a196a186a195a253a233a235a127a88a182a92a132a221a216a180a131a233a233a27a27!
a179a124a209a158a147a1631564-1642a164a27a131a233a53a6a110a181
a70a37a96a163a120a120a901473-1563a164a134a47a37a96a27a216a212(a19a172a167a70a126a178a8)
a47a165a131a233a20a19a27a36a196a132a221,30a250a112/a166?a154a74,a140a237,a156a106
a39a117a247a86a151a218a120a120a90a252a140a173a46a78a88a27a27a233a233a123(1632)
a233a154a74a27a216a121a180a131a233a53a6a110a129a64a27a76a227 a137a63a219a162a8a195a123a40a189a69a180a196a180a51a51a36a36a196a196a27a27a156
a218a238a158a147a1631642-1727a164a27a131a233a53a6a110a181 a218a238a239a225a10a110a216
a103a44a243a198a27a234a198a6a110(1685),a218a238a110a189a198,a25a107a218a229a189a198,a19a209a109a202a86a189a198,...
a163a137vectorF = mvectoraa177a159a111a143a235a236a212?
– a127a51a253a233a152a109a134a253a233a158a109,a167a130a166 vectorF = mvectora a164a225
– a131a233a167a130a137a33a132a36a196a196a27a27a152a109a134a158a109a143a166 vectorF = mvectora a164a225
a131a233a53a6a110a144a129a117a229a198a27a137a198!
298/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a100a194a131a233a216,a196a58～a131a233a53a6a110
a178a59a229a198,a229a198a196a29a36a196a196a189a189a198a233a164a107a46a53a88a209a164a225,a46a53a88a131a109a207a76a179a124a209a67a134a233a88
a45
a54
a8a8
a8a25
a54
a8a8
a8a25
a45 vectorV
z zprime
x xprime
y yprime
xprime = x?Vt
yprime = y
zprime = z
tprime = t
a178a8a27a132a221a220a164a39a88:?
vprimex = vx?V
vprimey = vy
vprimez = vz
vectoraprime =vectora
a218a238a238a189a189a198a51a179a124a209a67a134a118a107a237a19a101a180a216a216a67a67a27a156
a62a94a121a121a150a150a27a196a29a53a198a183a94a94a117a117a159a111a235a127a88? a240a142a100a137a118a107a163a137!
a179a124a209a27a158a152a42a103,a101a212a159a36a196a132a221a131a233a44a235a127a88a143c,a51a44a152a235a127a88a217a132a221a216a140a85
a247a136a136a135a135a144a149a209a180c.
a6a107a27a179a124a209a27a131a233a53a6a110a233a233a62a62a62a196a196a229a198a216a164a225!
a62a94a121a121a150a150a40a189a152a135a65a207a235a127a88.a131a233a84a65a207a235a127a88a27a36a196a143a253a233a36a196.
a8a158a27a110a41,a64a143a62a94a197a51”a177a20”a48a159a159a165a165a68a194,a131a233a53a6a110a216a164a225a180a103a44a27! a40a197,a89a197
299/384
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a100a194a131a233a216,a196a58～a162a8a196a58
a36a196a48a159a27a49a27a68a194a175a75—a220a162a162a8,a55a76a98a23a177a20a26a89a220a169a86a218
a207a233a20a216a211a235a127a88a165a49a132a27a11a201:
a114 a0a0 l1
l2
a240a142a16a154-a35a88a162a8,a23a164a236a131a233a253a233a183a142a88a177a89a178a132a221va36a196
T1= l1c?v+ l1c+v = 2cl1c2?v2 T2=
2
radicalBig
l22+(vT22 )2
c =
2l2√
c2?v2
a1
a1
a1
a1a1a21a65a65
a65
a65a65a85
T2
2 cl2
T2v?T
(1) = T1?T2 = 2cl1
c2?v2?
2l2√
c2?v2
a242a164a236a61 90?,
T(2) = 2l1√c2?v2? 2cl2c2?v2 → v = 0T
(1) =?T(2)
vnegationslash= 0T(1)negationslash=?T(2)
a101vnegationslash= 0,a90a21a94a171a51a61a196a76a167a165a152a189a135a117a41a67a122(v > 5Km/s?a163a196a152a138a94a171).a162a8
a40a216a180 v = 0,a51a152a99a111a71a209a118a255a209a47a165a254a49a132a51a216a211a144a149a254a27a11a201,a40a74a134a20a19a132a221a195a39
a86a40a162a8,a196a189a10a49a13a134a49a132a107a39.
a164a107a162a8a209a141a209,a49a132a216a157a54a117a42a9a246a164a51a27a235a127a88,a133a134a49a13a13a36a36a196a132a221a195a39.
300/384
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a131a233a216a196a29a6a110,a226a226a212a212a91a67a134,a196a29a6a110
a226a226a213a213a91a74a209a194a160a98a23,a247a177a20a36a196a144a149a107a127a221a194a160radicalbig1?v2/c2
a79a207a100a34a34a74a74a209:
a131a233a53a6a110,a164a107a46a53a235a127a88a209a180a31a100a27
a49a132a216a216a67a67a6a110,a253a152a165a49a132a51a63a219a219a46a46a53a88a247a63a152a144a149a240a143c,a134a49a13a195a39.
a131a233a53a6a110a119a138a183a130,a51a216a211a46a53a88a137a211a24a27a162a8a55a107a211a24a27a40a74a156
a63a219a196a29a212a110a53a198a51a216a211a46a53a88a165a55a107a131a211a27a144a167a47a170!
a132a221a131a233a53-a195a253a233a183a142:a207a76a229a198,a62a94a189a217a167a121a121a150a150a195a123a255a209a253a233a36a196.
a131a233a53a6a110a180a26a201,a167a177a110a53a27a229a229a254a254a93a212a10a177a99200a245a99a27a110a216a218a162a8a156
a49a132a216a216a67a67a6a110a119a138a183a130,a49a197a134a217a167a197a196a216a211a167a217a68a194a132a221a134a235a127a88a195a39.
a49a132a216a216a67a67a6a110a45a60a164a41a186 a118a107a177a20! a62a94a197a197a216a216a180a48a159a27a8a11!
301/384
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DavidMattingly,UniversityofCaliforniaatDavis,gr-qc/0502097(2005.2.22)?1a102a146/a166
302/384
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a131a233a216a196a29a6a110,a226a226a212a212a91a67a134,a179a124a209a67a134
a98a23,a152a109a180a254a33a134a136a149a211a53a27a167a158a109a180a254a33a27!
(t,x,y,z)a218(tprime,xprime,yprime,zprime)a131a109a27a39a88a55a76a180a130a53a27,a143a159a111?
a140a177a192a74:a252a135a131a233a36a196a132a221a143vectorVa27a46a53a88Sa218S’,a252a88a27a139a73a88a139a73a182
a144a149a131a211,xa182a18a51vectorVa144a149,a191a23t = 0,tprime = 0a158a252a139a88a173a220.
a45
a54
a8a8
a8a25
a54
a8a8
a8a25
a45 vectorVS S
prime
z zprime
x xprime
y yprime
xprime = αx +βt yprime = y zprime = z tprime = γx +δt
a98a23,a36a196a196a27a27a131a233a53:
a108Sa119,S’a107a36a196a132a221V? dxprime = 0 dxdt = V?β =?αV
S’a119,Sa107a36a196a132a221-V? dx = 0 dxprimedtprime =?V?β =?δV
xprime = α(x?Vt) yprime = y zprime = z tprime = γx +αt
a179a124a209a67a134,tprime = t? γ = 0,α = 1 a253a233a158a109! a196a189a179a124a209a67a134a55a76a196a189a253a233a158a109
303/384
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a131a233a216a196a29a6a110,a226a226a212a212a91a67a134,a196a29a226a226a212a212a91a67a134
a252a135a131a233a36a196a132a221a143vectorVa27a46a53a88Sa218S’,a252a88a27a139a73a88a139a73a182
a144a149a131a211,xa182a18a51vectorVa144a149,a191a23t = 0,tprime = 0a158a252a139a88a173a220.
a45
a54
a8a8
a8a25
a54
a8a8
a8a25
a45 vectorVS S
prime
z zprime
x xprime
y yprime
a152a135a175a135a51Sa218S’a88a165a27a158a152a139a73a169a79a143(t,x,y,z)a218 (tprime,xprime,yprime,zprime)
xprime = αx +βt yprime = y zprime = z tprime = γx +δt
a51t = 0,tprime = 0a158a6a58a117a209a152a135a165a161a62a94a197,a51ta158a143a62a94a197a136a20a177a6a58Oa143a165a37a27a44
a135a165a197a10a161a254,a131a65a51tprimea158a143a62a94a197a136a20a177a6a58O’a143a165a37a27a44a135a165a197a10a161a254,
braceleftbigg x2 + y2 + z2 = c2t2
xprime2 + yprime2 + zprime2 = c2tprime2 →
braceleftbiggα2(x?Vt)2 + y2 + z2 = c2(γx +αt)2
xprime2 + yprime2 + zprime2 = c2tprime2
→
α2?c2γ2 = 1
2Vα2?2c2γα = 0
c2α2?V2α2 = c2
→ α = 1radicalBig
1?V2c2
γ =?
V
c2radicalBig
1?V2c2
xprime = x?VtradicalBig
1?V2c2
yprime = y zprime = z tprime = t?
V
c2xradicalBig
1?V2c2
V ≤c Vlessmuchc==?a179a124a209a67a134
304/384
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a131a233a216a196a29a6a110,a226a226a212a212a91a67a134,a196a29a226a226a212a212a91a67a134～a63a191a36a196a144a149a139a73a132a221a67a134?
vectorrprime·vectorV
V =
vectorr·vectorV
V?VtradicalBig
1?V2c2
vectorrprime?vectorrprime·vectorVV2 vectorV =vectorr?vectorr·vectorVV2 vectorV
tprime = t?vectorr·
vectorV
c2radicalBig
1?V2c2
→
vectorrprime =vectorr?vectorr·vectorVV2 vectorV + vectorr·vectorVV?VtradicalBig
1?V2c2
vectorV
V
tprime = t?vectorr·
vectorV
c2radicalBig
1?V2c2
vectoru = dvectorrdt vectoruprime≡dvectorr
prime
dtprime =
dvectorrprime
dt
dtprime
dt
=
vectoru?vectoru·vectorVV2 vectorV + vectoru·vectorVV?VradicalBig
1?V2c2
vectorV
V
1?vectorV·vectoruc2radicalBig
1?V2c2
vectoruprime =
radicalBig
1?V2c2 (vectoru?vectoru·vectorVV2 vectorV) + (vectoru·vectorVV2?1)vectorV
1? vectorV·vectoruc2 a131a233a216a165a27a132a221a220a164a39a88a156
Uα≡dx
α
dτ =
1radicalBig
1?V2c2
dxα
dt →
Ui= viradicalBig
1?v2c2
x1=x,x2=y,x3=z
U4= icradicalBig
1?v2c2
x4 = ict
305/384
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a131a233a216a196a29a6a110,a226a226a212a212a91a67a134,a196a29a226a226a212a212a91a67a134～a63a191a36a196a144a149a139a73a132a221a67a134
vectorrprime =vectorr?vectorr·
vectorV
V2
vectorV + vectorr·
vectorV
V?VtradicalBig
1?V2c2
vectorV
V
vectoruprime =
radicalBig
1?V2c2 (vectoru?vectoru·vectorVV2 vectorV) + (vectoru·vectorVV2?1)vectorV
1? vectorV·vectoruc2 |
vectoruprime| |vectoru|=c==?c |vectoru|<c?|vectoruprime|<c
vectora≡dvectorudt = d
2vectorr
dt2 vectora
prime≡dvectoru
prime
dtprime =
d2vectorrprime
dtprime2 =
dvectoruprime
dt
dtprime
dt
vectoraprime = 1
(1? vectorV·vectoruc2 )2
braceleftbig(1?V2
c2 )(vectora?
vectora·vectorV
V2
vectorV) +
radicalbigg
1?V
2
c2
vectora·vectorV
V2
vectorV
+vectora·vectorV
(1?V2c2 )(vectoru?vectoru·vectorVV2 vectorV) + (vectoru·vectorVV2?1)vectorV
radicalBig
1?V2c2
(1? vectorV·vectoruc2 )c2
bracerightbig
306/384
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a131a233a216a27a158a152a110a216,a211a158a53a167a36a196a158a168a134a186a102
a252a135a175a135P1a218P2a51a252a235a127a88a165a136a107a139a73(x1,t1),(xprime1,tprime1)a218 (x2,t2),(xprime2,tprime2)
a211a158a27a131a233a53:
tprime =?t?
V
c?xradicalBig
1?V2c2
t=0,?xnegationslash=0→?tprimenegationslash= 0
a51a152a135a46a53a88a165a216a211a47a47a58a58a211a158a117a41a27a252a175a135a167
a51a44a9a27a46a53a88a119a191a191a216a216a211a158a117a41a156
a211a158a131a233a53a27a28a28a91a91
a51a44a9a27a46a53a88a167a38a210a114a27a180a167a216a152a152a24a24a181
– a88a74a74a49a49a132a216a216a67a67a210a242a19a151a216a211a158a156
– a88a74a74a49a49a132a85a179a124a209a67a134a67a167a242a45a158a180a167a27a11a79a167a69a206a2a177a211a158a156
307/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a131a233a216a27a158a152a110a216,a211a158a53a167a36a196a158a168a134a186a102
a252a135a175a135P1a218P2a51a252a235a127a88a165a136a107a139a73(x1,t1),(xprime1,tprime1)a218 (x2,t2),(xprime2,tprime2)
a36a196a158a168a67a250:
xprime = 0→?t =?t
prime + V
c2?x
prime
radicalBig
1?V2c2
=?t
prime
radicalBig
1?V2c2
tprimea180a36a196a168a103a67a27a27a214a214a234a163S’a88a252a175a135a135a109a109a27a158a109a11a164
ta180a183a142a142a168a168a27a27a214a214a234a163Sa88a252a175a135a135a109a109a27a158a109a11a164?t≥?tprime
a158a109a109a109a109a133a131a233a53a27a28a28a91a91
a216a211a47a47a58a58a58a27a27a183a142a142a168a168a55a76a31a158a233a79a217a167a88a119a118a233a79
Sa119S’a168a67a250! S’a119Sa168a143a67a250a156 (a86a41a102a15a184)
308/384
triangleleftsldtriangleleftsld
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trianglerightsld
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a131a233a216a27a158a152a110a216,a211a158a53a167a36a196a158a168a134a186a102
a252a135a175a135P1a218P2a51a252a235a127a88a165a136a107a139a73(x1,t1),(xprime1,tprime1)a218 (x2,t2),(xprime2,tprime2)
a36a196a186a102a67a225:
t = 0→?xprime =?x?V?tradicalBig
1?V2c2
=?xradicalBig
1?V2c2
a36a196a186a102a127a221a221a27a27a255a254a162a83a254a180a252a135a186a102a252a224a224a27a27a31a158a39a22a34
a252a135a88a27a27a31a31a158a158a180a180a216a152a151a27a34 a196a127a221?x≤?xprime
a127a221a131a233a53a27a28a28a91a91
Sa119S’a186a67a225,S’a119Sa186a143a67a225a156(a187a144a76a236a201; a186a102a75a150a112)
a164a107a249a10a209a180a36a196a198a167a13a216a180a196a229a198a198a8a8a65a156
309/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a131a233a216a27a158a152a110a216,a39a117a158a109a27a181a53
a64a10a168a254a141a171a27a158a109a180a216a180a47a212a110a27a158a109a48?
a218a238a238a27a27a42a58a181 a253a233a233a27a27a158a109a103a28a254a33a47a54a178
a226a226a212a212a91a27a42a58a181a218a92a131a233a253a233a158a109a47a88a47a170a76a136a48a27a35a27a219a141a158a109
a117a178a226a226a212a212a91a67a134a167a2a118a107a152a239a253a233a158a109! a226a226a213a213a91a195a10a234a198a167a2a118a195a212a110 a13a8a119
a36a92a52a27a42a58a181 a242a219a141a158a109a41a186a143a131a233a177a20a27a36a196a88a165a183a142a27
a207a76a76a49a49a38a210a4a233a233a27a27a158a168a27a27a214a214a234a208a148a49a132a51a136a46a53a88a209a152a152a24a24
a171a64a53a198a131a233a53a167a2a118a107a152a239a253a233a158a109! a36a92a52a195a10a243a198? a2a118a195a212a110 a13a8a119
a79a207a100a34a27a42a58a181a118a107a253a233a158a109a156a219a141a158a109a210a180a46a53a88a27a212a110a158a109a156
310/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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a131a233a216a27a158a152a110a216,a109a133a27a216a216a67a67a53
ca107a129,a168a218a186a102a255a254a27a40a74a134a235a127 a88a27a192a74a107a39,a167a130a216a85a50a138a143a253a233a73a79a53a239a254
a175a135a131a109a27a158a152a39a88,a73a135a207a233a35a27a134 a235a127a88a192a74a195a39a27a254,a61a51a226a212a91a67a134a101a216
a67a27a254a254a53a53a138a143a253a233a73a79.
a233a152a135a235a127a88a165a27a252a135a175a135(x1,y1,z1,t1),(x2,y2,z2,t2),a189a194a167a130a131a109a27a109a133a143:
s2≡c2(t1?t2)2?(x1?x2)2?(y1?y2)2?(z1?z2)2
= c2(?t)2?(?x)2?(?y)2?(?z)2
sprime2≡c2(tprime1?tprime2)2?(xprime1?xprime2)2?(yprime1?yprime2)2?(zprime1?zprime2)2
=c2[t1?t2?
V
c2(x1?x2)radicalBig
1?V2c2
]2?[x1?x2?V(t1?t2)radicalBig
1?V2c2
]2?(y1?y2)2?(z1?z2)2
=(t1?t2)2[ c
2
1?V2c2?
V2
1?V2c2 ]+(x1?x2)
2[
V2
c2
1?V2c2?
1
1?V2c2 ]?(y1?y2)
2?(z1?z2)2
=?s2
a152a132a27a226a226a212a212a91a67a134a51a234a198a254a189a194a143a2a177a109a133a216a216a67a67a27 (x,y,z,t)a27a224a103a130a53a67a134.
311/384
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a131a233a216a27a158a152a110a216,a109a133a27a216a216a67a67a53
a233a195a161a2a109a133:
ds2 = c2dt2?dvectorr·dvectorr = c2dt2(1?v
2
c2) = c
2dτ2 =?dl2
0 vectorv =
dvectorv
dt
a233ds2 > 0,dτ ≡ 1c√ds2a180a168a27a27a107a158(a6a158),a167a180a100a109a133
a218a209a27a228a107a158a109a254a106a27a216a216a67a67a254.
a233ds2 < 0,dl0≡√?ds2a180a186a27a27a107a127a221,a167a180a100a109a133a218a209
a27a228a107a127a221a254a106a27a216a216a67a67a254.
a252a175a135a39a88a27a169a97:
s2 >0,a97a158a158a175a175a135,a97a158a109a133
s2 <0,a97a152a175a135,a97a152a109a133
s2 = 0,a97a49a175a135,a97a49a109a133
312/384
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trianglerightsldtrianglerightsld
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a131a233a216a27a158a152a110a216,a97a158a109a133?s2 >0
a38a210a132a221,ˉv =
radicalbigg
x2
t2 +
y2
t2 +
z2
t2
0<?s2 = c2?t2x2y2z2 =?t2(c2?ˉv2)→ˉv <c
a252a175a135a131a109a140a177a94a49a38a210a239a225a225a233a233a88,a207a143a239a225a225a233a233a88a164a73a27a38a210a132a221a2a117c.
a158a109a103a83,?tprime =?t?
V
c2?xradicalBig
1?V2c2
=?tradicalBig
1?V2c2
[1?Vc2?x?t]
x
t ≤ˉv<cv <c
bracerightbiggv?x
t <c
2→ v?x
c2?t<1→1?
V
c2
x
t>0→?t
primea134?ta211a210
0<?s2=x2y2z2+c2?t2=xprime2yprime2zprime2+c2?tprime2
→?tnegationslash= 0?tprimenegationslash= 0
a252a175a135a216a211a158a117a41a167a167a130a131a109a27a158a109a103a83a216a172a54a54a16a16,a207a74a53a2a177!
a152a109a103a83,?y =?z = 0?xprime =?x?V?tradicalBig
1?V2c2
=?x 1?
V
ˉVradicalBig
1?V2c2
313/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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trianglerightsld
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a131a233a216a27a158a152a110a216,a97a158a109a133?s2 >0
a38a210a132a221,a252a175a135a131a109a140a177a94a49a38a210a239a225a225a233a233a88.
a158a109a103a83,a252a175a135a131a109a27a158a109a103a83a216a172a54a54a16a16,a207a74a53a2a177!
a152a109a103a83,?y =?z = 0?xprime =?x?V?tradicalBig
1?V2c2
=?x 1?
V
ˉVradicalBig
1?V2c2
x a134?xprime a216a152a189a211a210,a192 V = ˉV =||?x?t||
xprime = 0?yprime = 0?zprime = 0?tprime =?t?
V
c2?xradicalBig
1?V2c2
=?t?
x2
c2?t2radicalBig
1x2c2?t2
a175a135a131a109a27a152a109a103a83a145a235a127a88a192a74a13a67,a127a51a152a135a235a127a88,a252a175a135a51a152a109a211a152a47a47a58a58a117a41!
314/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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trianglerightsld
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a131a233a216a27a158a152a110a216,a97a152a109a133?s2 <0
a38a210a132a221,0>?s2=c2?t2x2y2z2=?t2(c2?ˉv2)→ˉv >c
a252a175a135a131a109a195a123a94a49a38a210a239a225a225a233a233a88,a207a143a239a225a225a233a233a88a164a73a27a38a210a132a221a140a117c.
a158a109a103a83,?tprime =?t?
V
c2?xradicalBig
1?V2c2
=?tradicalBig
1?V2c2
[1?Vc2?x?t]
V
c <1?x
c?ta61a140a140a1171a143a140a2a1171
bracerightbiggV?x
c2?ta140a140a1171a189a2a1171→1?
V?x
c2?ta140a140a1170a189a2a1170
tprimea216a152a189a134?ta211a210.
a192?y = 0,?z = 0
V = c
2?t
x = c
c
x
t
= ccˉv <c?tprime =?t?
V
c2?xradicalBig
1?V2c2
= 0
a175a135a131a109a27a158a109a103a83a180a216a40a189a189a27a27,a145a235a127a88a192a74a13a67.
a127a51a152a135a235a127a88,a51a217a165a252a175a135a211a158a117a41,a207a74a53a216a216a2a2a177.
315/384
triangleleftsldtriangleleftsld
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a131a233a216a27a158a152a110a216,a97a152a109a133?s2 <0
a38a210a132a221,a252a175a135a131a109a195a123a94a49a38a210a239a225a225a233a233a88.
a158a109a103a83,a175a135a131a109a27a158a109a103a83a180a216a40a189a189a27a27,a145a235a127a88a192a74a13a67.
a127a51a152a135a235a127a88,a51a217a165a252a175a135a211a158a117a41,a207a74a53a216a216a2a2a177.
a152a109a103a83,?y =?z = 0
xprime =?x?V?tradicalBig
1?V2c2
=?x 1?
V
ˉVradicalBig
1?V2c2
V <c
ˉV >c
bracerightbigg
V < ˉV→1?VˉV >0
xa134?xprimea211a210,a65a79a47,
0>?s2=x2y2z2+c2?t2=xprime2yprime2zprime2+c2?tprime2
→a216a127a51?xprime =?yprime =?zprime = 0
a51a63a219a235a127a88a119,a252a175a135a131a109a27a152a109a160a152a216a211,a133a160a152a131a233a103a83a216a172a54a54a16a16
a216a127a51a51a249a249a24a24a152a152a135a235a127a88,a51a217a165a252a175a135a51a211a152a47a47a58a58a117a41.
316/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a131a233a216a27a158a152a110a216,a97a49a109a133?s2 = 0
a38a210a132a221,0=?s2=c2?t2x2y2z2=?t2(c2?ˉv2)→ˉv=c
a252a175a135a131a109a140a177a94a49a38a210a239a225a225a233a233a88,a207a143a239a225a225a233a233a88a164a73a27a38a210a132a221a221a31a31a117c.
a158a109a103a83,?tprime =?t?
V
c2?xradicalBig
1?V2c2
=?tradicalBig
1?V2c2
[1?Vc2?x?t]
x
t ≤ˉv = cv <c
bracerightbiggv?x
t <c
2→ v?x
c2?t <1→1?
V
c2
x
t >0
tprimea134?ta211a210.
y = 0,?z = 0 →?x = c?t
tprime =?t?
V
c2?xradicalBig
1?V2c2
=?t 1?
V
cradicalBig
1?V2c2
=?t
radicalbiggc?v
c + v
v→c→0
a252a175a135a131a109a27a158a109a103a83a69a216a172a54a54a16a16.a207a74a53a69a69a44a44a2a177.
317/384
triangleleftsldtriangleleftsld
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a131a233a216a27a158a152a110a216,a97a49a109a133?s2 = 0
a38a210a132a221,a252a175a135a131a109a140a177a94a49a38a210a239a225a225a233a233a88.
a158a109a103a83,a252a175a135a131a109a27a158a109a103a83a69a216a172a54a54a16a16.a207a74a53a69a69a44a44a2a177.
a127a51a51a249a249a24a24a152a152a135a235a127a88,a51a217a165a252a175a135a211a158a117a41,a249a135a235a127a88a177a49a132a131a233a6a235a127a88a36a196.
a152a109a103a83,?y =?z = 0→?x = c?t
xprime =?x?V?tradicalBig
1?V2c2
=?x 1?
V
cradicalBig
1?V2c2
=?x
radicalbiggc?v
c + v
v→c→0
a252a175a135a131a109a27a152a109a103a83a216a172a54a54a16a16
a127a51a152a135a235a127a88,a51a217a165a252a175a135a51a211a152a47a47a58a58a117a41,a249a135a235a127a88a177a49a132a131a233a6a235a127a88a36a196.
a178a59a229a198a131a8a117a18c→∞,a249a158a144a107a97a158a158a175a175a135.
a177(x,y,z,t)a143a139a73a228a107?s2a216a216a67a67a53a27a152a109a161a143Minkowskia152a109.
a111a145a158a152a51a110a145a102a152a109(a152a145a158a109,a252a145a152a109)a27a221a75a171a165,a242a121a51(t = 0,
x = y = z = 0)a18a143a152a175a135,a75a75a249a249a158a109a133a169a97a177’a49a73’a143a46,a49a73a144a167a143
c2t2 = x2 +y2,a49a73a83a143a97a158a109a133,a49a73a254a143a97a49a109a133,a49a73a9a9a143a143a97a152a109
a133,a97a158a171a26a121a51(t = 0,x = y = z = 0)a169a143a253a233a76a22a218a253a233a153a53.
318/384
triangleleftsldtriangleleftsld
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a131a233a216a27a158a152a110a216,a207a74a53a134a129a140a38a210a68a52a132a221
a107a207a74a53a27a175a135(a97a158a189a97a49)a131a109a27a38a210a68a52a132a221≤c
a195a207a74a53a27a175a135a163a97a152a164a131a109a27a38a210a68a52a132a221>c
a129a140a27a212a110a38a210a68a52a132a221a143c a135a49a132a226a102(tachyon)a19a151a187a128a207a74a53
a164a107a27a131a112a138a94a143a129a245a144a85a177a132a221ca68a52a182a195a135a229a138a94
a110a41a49a132a216a216a67a67a6a110a181
– a49a132a81a180a129a140a27a212a110a38a210a68a52a132a221
– a131a233a53a6a110a135a166a129a140a212a110a38a210a68a52a132a221a134a235a127a88a192a74a195a39
a39a117a38a210a181
– a159a58a138a143a38a210a132a221a78a180a40a189
– a197a138a143a38a210a132a221a216a78a180a40a189
319/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a131a233a216a110a216a27a14a67a47a170,a111a145a158a152a139a73a73a67a67a134
a152a132a226a226a212a212a91a67a134a107a56a135a213a225a103a100a221,a110a135a131a233a36a196,a110a135a139a73a182a131a233a61a196.
c2t2?(x2+y2+z2)=c2tprime2?(xprime2+yprime2+zprime2) x1=x,x2=y,x3=z,x4=ict→
x21+x22+x23+x24=xprime12+xprime22+xprime32+xprime42 a189
4summationdisplay
μ=1
xμxμ =
4summationdisplay
μ=1
xprimeμxprimeμ
a247a118xμ = 0a134xprimeμ = 0a233a65a27a152a132a130a53a67a134a143,xprimeμ =
4summationdisplay
μ=1
aμνxν
4summationdisplay
μ=1
xprimeμxprimeμ =
4summationdisplay
μ,ν,λ=1
aμνxνaμλxλ =
4summationdisplay
μ,ν,λ=1
aμνaμλxνxλ =
4summationdisplay
μ=1
xμxμ
4summationdisplay
μ=1
aμνaμλ = δνλ a137a20910a135a21a229a144a167 xμ=
4summationdisplay
ν=1
δμνxν=
4summationdisplay
ν,λ=1
aλμaλνxν=
4summationdisplay
λ=1
aλμxprimeλ
a44a152a171a76a136a47a170,
4summationdisplay
μ=1
aνμaλμ = δνλ
320/384
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a131a233a216a110a216a27a14a67a47a170,a111a145a158a152a139a73a73a67a67a134
a152a132a226a226a212a212a91a67a134a107a56a135a213a225a103a100a221,a110a135a131a233a36a196,a110a135a139a73a182a131a233a61a196.
xprimeμ =
4summationdisplay
μ=1
aμνxν
4summationdisplay
μ=1
aμνaμλ = δνλ xμ =
4summationdisplay
λ=1
aλμxprimeλ
4summationdisplay
μ=1
aνμaλμ = δνλ
X =
x1
x2
x3
x4
A =
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
Xprime = AX X = ATXprime ATA = I AAT = I
A =
1radicalBig
1?V2c2
0 0 iVcradicalBig
1?V2c2
0 1 0 0
0 0 1 0
iVcradicalBig
1?V2c2
0 0 1radicalBig
1?V2c2
1 = detI = det(ATA) = (detAT)(detA) = (detA)2→(detA) =±1
detA = 1a100a88a61a196a8a164 detA =?1a100a88a61a196a196a83a83a92a254a152a109a135a19a189a158a109a135a252a8a164
321/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a131a233a216a110a216a27a14a67a47a170,a212a110a110a254a254a85a158a152a67a134a53a159a169a97
a131a233a53a6a110a27a234a198a76a136a181 3+1a145a158a152a220a254a144a167
0a30a220a254:(a73a254,a226a226a212a212a91a216a216a67a67a254) φprime = φ
1a30a220a254:(a165a254) Aprimeμ =
4summationdisplay
ν=1
aμνAν
2a30a220a254:Bprimeμν =
4summationdisplay
μprime,νprime=1
aμμprimeaννprimeBμprimeνprime
···
na30a220a254,Tprimeμ1···μn =
4summationdisplay
ν1,···,νn=1
aμ1ν1···aμnνnTν1···νn
na30a220a254a27a65a58a180a1074na135a169a254,a51a226a226a212a212a91a67a134a101,a85na135a139a73a166a200a27a67a134a144a170a67a134.
322/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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trianglerightsld
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a131a233a216a110a216a27a14a67a47a170,a212a110a110a254a254a85a158a152a67a134a53a159a169a97
Tprimeμ1···μn =
4summationdisplay
ν1,···,νn=1
aμ1ν1···aμnνnTν1···νn
Tμ1···μn = Fμ1···μn
4summationdisplay
ν1,·,νn=1
aν1μ1···aνnμnTμ1···μn =
4summationdisplay
ν1,·,νn=1
aν1μ1···aνnμnFμ1···μn →Tprimeμ1···μn =Fprimeμ1···μn
a51a35a139a73a88a165a27a144a167a170a47a170a254a134a206a139a73a88a165a27a144a167a170a152a152a24a24—a131a233a53a6a110
a124a94a220a254,a140a242a163a227a212a110a53a198a27a144a167a21a164a247a118a131a233a53a6a110a27a47a170—a212a110a53a198a27a14a67a67a76a76a136.
a242a164a107a212a110a53a198a209a94a220a254a76a136,a73a127a23:
a220a254a131a109a27a36a142a123a75.
a17a23a27a220a254a56a107a245a8a171a220a254.
a228a78a27a212a110a53a198a88a219a94a220a254a76a136.
323/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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trianglerightsld
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a131a233a216a110a216a27a14a67a47a170,a220a254a36a142
a92a92a126a126a123,a211a30a220a254a92a92a126a126a0a69a180a249a30a220a254
Fprimeμ1···μn =
4summationdisplay
ν1,···,νn=1
aμ1ν1···aμnνnFν1···νn Gprimeμ1···μn =
4summationdisplay
ν1,···,νn=1
aμ1ν1···aμnνnGν1···νn
Fprimeμ1···μn±Gprimeμ1···μn =
4summationdisplay
ν1,···,νn=1
aμ1ν1···aμnνn[Fν1···νn±Gμ1···μn]
a166a123,ma30a220a254a134na30a220a254a27a166a200a143m + na30a220a254
Aprimeμ1···μm =
4summationdisplay
ν1,···,νm=1
aμ1ν1···aμmνmAν1···νm Bprimeμ1···μn =
4summationdisplay
ν1,···,νn=1
aμ1ν1···aμnνnBν1···νn
Tμ1···μm+n ≡Aμ1···μmBμm+1···μm+n
Tprimeμ1···μm+n = Aprimeμ1···μmBprimeμm+1···μm+n
=
4summationdisplay
ν1,···,νm=1
aμ1ν1···aμmνmAν1···νm
4summationdisplay
νm+1,···,νm+n=1
aμm+1νm+1···aμm+nνm+nBνm+1···νm+n
324/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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trianglerightsld
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a131a233a216a110a216a27a14a67a47a170,a220a254a36a142
a92a92a126a126a123,a211a30a220a254a92a92a126a126a0a69a180a249a30a220a254
a166a123,ma30a220a254a134na30a220a254a27a166a200a143m + na30a220a254
Aprimeμ1···μm =
4summationdisplay
ν1,···,νm=1
aμ1ν1···aμmνmAν1···νm Bprimeμ1···μn =
4summationdisplay
ν1,···,νn=1
aμ1ν1···aμnνnBν1···νn
Tμ1···μm+n ≡Aμ1···μmBμm+1···μm+n
Tprimeμ1···μm+n = Aprimeμ1···μmBprimeμm+1···μm+n
=
4summationdisplay
ν1,···,νm=1
aμ1ν1···aμmνmAν1···νm
4summationdisplay
νm+1,···,νm+n=1
aμm+1νm+1···aμm+nνm+nBνm+1···νm+n
=
4summationdisplay
ν1,···,νm+n=1
aμ1ν1···aμm+nνm+nAν1···νmBνm+1···νm+n =
4summationdisplay
ν1,···,νm+n=1
aμ1ν1···aμm+nνm+nTν1···νm+n
a160a191,na30a220a254a194a160a152a103a67a164n-2a30a220a254
4summationdisplay
μi,μj=1
δμiμjTμ1···μi···μj···μna161a143Tμ1···μi···μj···μna233a141a73μi,μja27a160a191.
325/384
triangleleftsldtriangleleftsld
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a131a233a216a110a216a27a14a67a47a170,a220a254a27a126a102
c,h,a62a214,a183a142a142a159a159a254···a143a226a226a212a212a91a73a254.
a111a145a78a200a3d4x≡dx1dx2dx3dx4 = icdVdta27a67a134a39a88a143:
d4xprime =||detA||d4x = d4x→a226a226a212a212a91a73a254dVprimedtprime = dVdt
a111a145a132a221uμ = dxμdτ a143a111a165a254,a207dxμa143a111a165a254,dτa180a216a216a67a67a254.
a14a67a135a251x
μ
a180a152a30a220a254
xprimeμ =
4summationdisplay
ν=1
xν
xprimeμ
xν =
4summationdisplay
ν=1
aμνx
ν
4summationdisplay
μ=1
xμ
xμ =
x
x +
y
y +
z
z?
1
c2
t
t =?
2? 1
c2
2
t2
δμνa180a19a30a252a160a220a254,δprimeμν = δμν =
4summationdisplay
λ=1
aμλaνλ =
4summationdisplay
λ,λprime=1
aμλaνλprimeδλλprime
326/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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trianglerightsld
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a131a233a216a53a212a110a198
a174a127a127a51a152a64a220a254a254a110a110a216a140a177a108a234a198a254a76a136a131a233a53a6a110
a144a135a114a212a110a144a167a21a164a180a111a145a158a152a165a27a220a254a144a167
a183a130a78a111a127a23a198a76a27a212a110a144a167a85a196a21a164a220a254a144a167?
a216a139a73a9a167a183a130a216a127a23a217a167a167a164a164a107a212a110a110a254a254a218a111a145a220a254a107a159a111a39a88a186
a173a35a34a34a192a192a183a130a27a229a198a218a62a62a196a196a229a198a110a216
a135a166a110a216a51a152a109a169a210a247a118a131a233a53a6a110a19a209a212a110a110a254a254a218a111a145a220a254a27a39a88a156
a173a35a237a19a209a229a198a218a62a62a196a196a229a198a110a216
a191a239a225a94a111a145a220a254a76a136a27a212a110a144a167
a66a23a172a181a216a108a162a8a167a13a108a110a216a6a6a75a75a19a209a10a229a198a218a62a62a196a196a229a198a27a164a107a144a167
327/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
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trianglerightsld
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a131a233a216a229a198,a129a2a138a94a254a6a110
a233a117a122a152a135a229a198a78a88,a127a51a152a135a23a138a94a254a27a200a169S,a167a233a117a162a83a36a196a107a129a2a138.
a233a152a135a103a100a221a221a27a27a78a88,S =
integraldisplay β
α
dt L(q.˙q)
0 = δS =
integraldisplay β
α
dt[?L?˙qδ˙q +?L?qδq] =
integraldisplay β
α
dt[?L?˙q ddtδq +?L?qδq]
=
integraldisplay β
α
dt [ ddt(?L?˙qδq) + (?L?q? ddt?L?˙q)δq] →L?q + ddt?L?˙q = 0
p =?L?˙q H = p˙q?L
dp
dt =
L
q
dH
dt =
dp
dt ˙q + p
d˙q
dt?
L
q
dq
dt?
L
˙q
d˙q
dt = 0
328/384
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a131a233a216a229a198,a58a226a102a229a198
S =?m0c
integraldisplay b
a
ds ds = c
radicalbigg
1?v
2
c2dt→S =?m0c
2
integraldisplay b
a
dt
radicalbigg
1?v
2
c2
L =?m0c2
radicalbigg
1?v
2
c2
c→∞→?m
0c2 +
1
2m0v
2
L?x
i
+ ddt?L?v
i
= 0 → ddt?L?v
i
= 0
pi=?L?v
i
= m0viradicalBig
1?v2c2
vectorp= m0vectorvradicalBig
1?v2c2
=m0vectoru=mvectorv m= m0radicalBig
1?v2c2
H = vectorp·vectorv?L = m0v
2
radicalBig
1?v2c2
+ m0c2
radicalbigg
1?v
2
c2 =
m0c2radicalBig
1?v2c2
= mc2
dpi
dt = 0
dvectorp
dt = 0
dH
dt = 0
329/384
triangleleftsldtriangleleftsld
trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a131a233a216a229a198,a58a226a102a229a198
S =?m0c2
integraldisplay b
a
dt
radicalbigg
1?v
2
c2 m =
m0radicalBig
1?v2c2
vectorp = m0vectorvradicalBig
1?v2c2
= mvectorv dvectorpdt = 0 H = mc2 dHdt = 0
uμ = dx
μ
dτ =
1radicalBig
1?v2c2
dxμ
dt = (
vectorvradicalBig
1?v2c2
,icradicalBig
1?v2c2
)
pμ = m0uμ = ( m0vectorvradicalBig
1?v2c2
,im0cradicalBig
1?v2c2
) = (vectorp,iHc ) dp
μ
dτ = 0
summationdisplay
μ
pμpμ = m
2
0v
2?m2
0c
2
radicalBig
1?v2c2
=?m20c2 → H
2
c2?p
2 = m2
0c
2
330/384
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a131a233a216a229a198,a145a62a62a58a58a226a102a57a62a214a169a217a51a9a62a94a124a165
a62a94a124a88a219a163a227a186
a212a110a110a254a254a140a85a217a158a152a67a134a53a159a169a97,a216a211a220a254a147a76a76a216a216a211a212a110
a62a94a124a65a84a225a117a44a152a220a254,a98a23a217a143a111a165a254Aμ = (vectorA,icφ)
a233a39a158a152a139a73a165a254xμ = (x,y,z,ict)a57a217a226a226a212a212a91a67a134a39a88
xprime = x?vtradicalBig
1?v2c2
yprime = y zprime = z tprime = t?
v
c2xradicalBig
1?v2c2
a140a26(vectorA,φ)a27a226a226a212a212a91a67a134a39a88
Aprimex = Ax?
v
c2φradicalBig
1?v2c2
Aprimey = Ay Aprimez = Az φprime = φ?vAxradicalBig
1?v2c2
a143a159a111a135a94a111a221a165a254a124a53a163a227a62a94a124a186
331/384
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a131a233a216a229a198,a145a62a62a58a58a226a102a57a62a214a169a217a51a9a62a94a124a165
a98a23a23a165a165a254a124Aμ = (vectorA,icφ)a163a227a62a94a124,a53a189a10 (vectorA,φ)a27a226a226a212a212a91a67a134a39a88,
Aprimex = Ax?
v
c2φradicalBig
1?v2c2
Aprimey = Ay Aprimez = Az φprime = φ?vAxradicalBig
1?v2c2
a143a163a227a145a62a58a226a102a51a9a62a94a124a27a49a143,a55a76a51a103a100a58a226a102a138a94a254a92a254a152a135a135a65a62a94
a124a134a145a62a226a102a117a41a131a112a138a94a27a145a34 a249a152a145a150a8a65a157a185a58a226a102a27a53a159a218a62a94a124a27a53
a159a167a191a133a180a226a226a212a212a91a67a134a73a254a181
S =?m0c
integraldisplay b
a
ds + e
4summationdisplay
μ=1
integraldisplay b
a
dxμ Aμ a129a2a205a220
a249a24a27a27a62a62a94a131a112a138a94a228a107a53a137a216a216a67a67a53a181Aμ→Aprimeμ = Aμ +?μχ
S=
integraldisplay b
a
(?m0cds+evectorA·dvectorr?eφdt)=
integraldisplay b
a
dt (?m0c2
radicalbigg
1?v
2
c2 +e
vectorA·vectorv?eφ)
332/384
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a131a233a216a229a198,a145a62a62a58a58a226a102a57a62a214a169a217a51a9a62a94a124a165
S =
integraldisplay b
a
dt(?m0c2
radicalbigg
1?v
2
c2 +e
vectorA·vectorv?eφ) L=?m0c2
radicalbigg
1?v
2
c2 +e
vectorA·vectorv?eφ
vectorp = m0vectorvradicalBig
1?v2c2
+ evectorA a20a75a196a254negationslash= a197a22a196a254 H = m0c
2
radicalBig
1?v2c2
+ eφ
(vectora·vectorb) =vectora·?vectorb +vectorb·?vectora +vectorb×(?×vectora) +vectora×(?×vectorb)
3summationdisplay
i=1
L
xivectorei=?L=e?(
vectorA·vectorv)?e?φ=evectorv·?vectorA + evectorv×(?×vectorA)?e?φ
d
dt
m0vectorvradicalBig
1?v2c2
= dvectorpdt?ed
vectorA
dt
L?x
i
+ ddt?L?v
i
= 0=========?L?edvectorA
dt
= evectorv·?vectorA + evectorv×(?×vectorA)?e?φ?e(?
vectorA
t +
vectorr
t·?
vectorA)
= e(
vectorA
tφ) + evectorv×(?×
vectorA)
333/384
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a131a233a216a229a198,a145a62a62a58a58a226a102a57a62a214a169a217a51a9a62a94a124a165
S =
integraldisplay b
a
dt(?m0c2
radicalbigg
1?v
2
c2 +e
vectorA·vectorv?eφ) L=?m0c2
radicalbigg
1?v
2
c2 +e
vectorA·vectorv?eφ
vectorp = m0vectorvradicalBig
1?v2c2
+ evectorA a20a75a196a254negationslash= a197a22a196a254 H = m0c
2
radicalBig
1?v2c2
+ eφ
d
dt
m0vectorvradicalBig
1?v2c2
= e(
vectorA
tφ) + evectorv×(?×
vectorA)≡vectorF = evectorE + evectorv×vectorB
vectorE =vectorA
tφ
vectorB =?×vectorA
Eprimex = Ex Eprimey = Ey?vBzradicalBig
1?v2c2
Eprimez = Ez + vByradicalBig
1?v2c2
Bprimex = Bx Bprimey = By +
v
c2EzradicalBig
1?v2c2
Bprimez = Bz?
v
c2EyradicalBig
1?v2c2
334/384
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a131a233a216a229a198,a62a214a169a217a51a9a62a94a124a165
a58a58a62a62a214a57a217a54a196a180a235a89a62a214a169a217a57a217a54a196a196a27a27a65a126
ρ(vectorr,vectorrprime) = eδ(vectorr?vectorrprime) vectorj(vectorr,vectorrprime) = ρ(vectorr,vectorrprime)vectorv
edxμ = eδ(vectorr?vectorrprime)dτdxμ = ρdτdxμ = ρdxμdt dtdτ = jμdtdτ
jμ≡ρdxμdt = (vectorj,icρ)
jprimex = jx?vρradicalBig
1?v2c2
jprimey = jy jprimez = jz ρprime = ρ?
v
c2jxradicalBig
1?v2c2
4summationdisplay
μ=1
integraldisplay b
a
dxμ eAμ =
4summationdisplay
μ=1
integraldisplay
dtdτAμjμ =
integraldisplay
dtdτ(vectorA·vectorj?φρ)
0 =
4summationdisplay
μ=1
integraldisplay
dtdτjμ?μχ =?
4summationdisplay
μ=1
integraldisplay
dtdτ(?μjμ)χ
4summationdisplay
μ=1
μjμ = 0 a189?ρ?t +?·vectorj = 0
335/384
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a131a233a216a229a198,a14a67a67a76a76a136P
μ = pμ + eAμ pμ = (
m0vectorvradicalBig
1?v2c2
,im0cradicalBig
1?v2c2
)
Fμν≡?μAννAμ=
0 B3?B2?icE1
B3 0 B1?icE2
B2?B1 0?icE3
i
cE1
i
cE2
i
cE3 0
Fprimeμν =
4summationdisplay
λ,λprime=1
aμλFλλprimeaνλprime
Fprime = AFAT?
Fij =
3summationdisplay
k=1
epsilon1ijkBk
F4i =?Fi4 = icEi
Bi =
3summationdisplay
j,k=1
1
2epsilon1ijkFjk
Ei =?icF4i
fμ≡
4summationdisplay
ν=1
Fμνjν = (ρvectorE +vectorj×vectorB,icvectorj·vectorE)
fi = Fi4j4 +
3summationdisplay
k=1
Fikjk =?icEiicρ+
3summationdisplay
k,l=1
epsilon1ikljkBl = (ρvectorE +vectorj×vectorB)i
f4 =
3summationdisplay
k=1
F4kjk = icEkjk = icvectorE·vectorj
336/384
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a131a233a216a229a198,a14a67a67a76a76a136
fμ≡
4summationdisplay
ν=1
Fμνjν = (ρvectorE +vectorj×vectorB,icvectorj·vectorE)
a233a233a58a58a58a62a62a214,jμ = eδ(vectorr?vectorrprime)dxμdt =
integraldisplay
dtprime eδ(vectorr?vectorrprime)δ(t?tprime)dxμdtprime
=
integraldisplay
dτprimeeδ(4)(x?xprime)dxμdτprime =
integraldisplay
dτprimeeδ(4)(x?xprime)uμ
dpμ
dτ = Kμ
a138a146===?K
μ = (vectorK,
i
c
vectorK·vectorv) =
4summationdisplay
ν=1
eFμνuν fμ =
integraldisplay
dτδ(4)(x?xprime)Kμ
= parenleftbig eradicalBig
1?v2c2
(vectorE +vectorv×vectorB),
ie
c
vectorE·vectorv
radicalBig
1?v2c2
parenrightbig
a189,vectorF = vectorK
radicalbigg
1?v
2
c2
vectorF = dvectorP
dt
vectorF·vectorv = dH
dt
337/384
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a131a233a216a62a62a196a196a229a198,a138a94a254
a143a163a227a62a94a124a36a196,a65a51a138a94a254a165a92a152a145a88a135a65a62a94a124a27a145,a247a118a53a137a216a216a67a67a6a110.
4summationdisplay
μ,ν=1
FμνFμν = 2(B2? 1c2E2)
4summationdisplay
μ,ν,σ,ρ=1
epsilon1μνσρFμνFσρ = icvectorE·vectorB =
4summationdisplay
μ,ν,σ,ρ=1
4epsilon1μνσρ?μ(Aν?σAρ)
S =
integraldisplay
dtdτbracketleftbig(? 14μ
0
)
4summationdisplay
μ,nu=1
FμνFμν +
4summationdisplay
μ=1
jμAμbracketrightbig
0 = δS =
integraldisplay
dtdτbracketleftbig(? 14μ
0
)
4summationdisplay
μ,nu=1
δ(FμνFμν) +
4summationdisplay
μ=1
jμδAμbracketrightbig
=
integraldisplay
dtdτbracketleftbig(? 12μ
0
)
4summationdisplay
μ,ν=1
FμνδFμν +
4summationdisplay
μ=1
jμδAμbracketrightbig
338/384
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a131a233a216a62a62a196a196a229a198,a240a142a100a137a144a167a124
0 = δS =
integraldisplay
dtdτbracketleftbig(? 14μ
0
)
4summationdisplay
μ,nu=1
δ(FμνFμν) +
4summationdisplay
μ=1
jμδAμbracketrightbig
=
integraldisplay
dtdτbracketleftbig(? 12μ
0
)
4summationdisplay
μ,ν=1
FμνδFμν +
4summationdisplay
μ=1
jμδAμbracketrightbig
=
integraldisplay
dtdτbracketleftbig(?1μ
0
)
4summationdisplay
μ,ν=1
Fμνδ(?μAν) +
4summationdisplay
μ=1
jμδAμbracketrightbig
=
integraldisplay
dtdτbracketleftbig(?1μ
0
)
4summationdisplay
μ,ν=1
Fμν(?μδAν) +
4summationdisplay
μ=1
jμδAμbracketrightbig
=
integraldisplay
dtdτbracketleftbig(?1μ
0
)
4summationdisplay
μ,ν=1
[?μ(FμνδAν)?(?μFμν)δAν] +
4summationdisplay
μ=1
jμδAμbracketrightbig
4summationdisplay
μ=1
μFμν =?μ0jν?μFνλ +?νFλμ +?λFμν = 0
339/384
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a131a233a216a62a62a196a196a229a198,a240a142a100a137a144a167a1244summationdisplay
μ=1
μFμν =?μ0jν?μFνλ +?νFλμ +?λFμν = 0
3summationdisplay
j=1
jFji =?μ0ji4F4i→
3summationdisplay
j,k=1
jepsilon1jikBk =?μ0ji? 1c2?Ei?t
→?×vectorB=μ0vectorj+μ0epsilon10?
vectorE
t
3summationdisplay
i=1
iFi4 =?μ0j4 →
3summationdisplay
i=1
iEi(?ic) =?μ0icρ →?·vectorE = ρepsilon1
0
1F23+?2F31+?3F12=0→
3summationdisplay
i,j,k=1
iepsilon1jkiFjk=0→
3summationdisplay
i=1
iBi=0→?·vectorB=0
jFk4 +?kF4j +?4Fjk = 0→
3summationdisplay
j,k=1
epsilon1ijk?j(icFk4) =?ic?4(12
3summationdisplay
j,k=1
epsilon1jkiFjk)
→
3summationdisplay
j,k=1
epsilon1ijk?jEk =tBi→?×vectorE =
vectorB
t
340/384
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a131a233a216a62a62a196a196a229a198,a165a254a179a134a240a142a100a137a144a167a124
a88a219a110a41a240a142a100a137a144a167a124a27a53a13a186 a162a8a189a198a182a46a130a75a70a254a182
4summationdisplay
μ=1
μFμν =?μ0jν?μFνλ +?νFλμ +?λFμν = 0
a144a135a189a194a10 a165a254a179Aμ
a210a140a177a189a194a124a114 Fμν ≡?μAννAμ
a124a114a27a40a8a166a144a167?μFνλ +?νFλμ +?λFμν = 0 a103a44a164a225
a124a114a27a40a8a132a166a144a167 summationtextμ,ν?μ?νFμν = 0 a103a44a164a225
a167a191a155a88a140a177a189a194a54 jν ≡?1μ
0
summationtext
μ?μFμν
a54a54a247a247a118a54a197a240a144a167 summationtextν?νjν = 0
341/384
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a131a233a216a62a62a196a196a229a198,a85a196a254a197a240
4summationdisplay
μ=1
μFμν =?μ0jν?μFνλ+?νFλμ+?λFμν=0 fμ≡
4summationdisplay
ν=1
Fμνjν
fμ=
4summationdisplay
ν=1
Fμνjν =?1μ
0
4summationdisplay
ν,νprime=1
Fμν?νprimeFνprimeν =?1μ
0
4summationdisplay
ν,νprime=1
[?νprime(FμνFνprimeν)?(?νprimeFμν)Fνprimeν]
=?1μ
0
4summationdisplay
ν,νprime=1
[?νprime(FμνFνprimeν)?12(?νprimeFμν)Fνprimeν?12(?νFμνprime)Fννprime]
=?1μ
0
4summationdisplay
ν,νprime=1
[?νprime(FμνFνprimeν) + 12(?νprimeFμν +?νFνprimeμ)Fννprime]
=?1μ
0
4summationdisplay
ν,νprime=1
[?νprime(FμνFνprimeν)?12(?μFννprime)Fννprime]
=?1μ
0
4summationdisplay
ν,νprime=1
νprime[FμνFνprimeν?14δμνprime
4summationdisplay
ν”=1
Fνν”Fνν”]≡?
4summationdisplay
νprime=1
νprimeTμνprime
342/384
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a131a233a216a62a62a196a196a229a198,a85a196a254a197a240
fμ =?
4summationdisplay
ν=1
νTμν Tμν ≡ 1μ
0
[
4summationdisplay
νprime=1
FμνprimeFννprime?14δμν
4summationdisplay
μprime,νprime=1
FμprimeνprimeFμprimeνprime] = Tνμ
4summationdisplay
μprime,νprime=1
FμprimeνprimeFμprimeνprime = 2B2? 2c2E2 Fij =
3summationdisplay
k=1
epsilon1ijkBk F4i =?Fi4 = icEi
Tij = 1μ
0
[
4summationdisplay
k=1
FikFjk + Fi4Fj4?12δij(B2? 1c2E2)
= 1μ
0
[
4summationdisplay
k,l,lprime=1
epsilon1iklepsilon1jklprimeBlBlprime? 1c2EiEj?12δij(B2? 1c2E2)]
= 1μ
0
[δijB2?BiBj? 1c2EiEj?12δij(B2? 1c2E2)]
= 1μ
0
[?BiBj? 1c2EiEj + 12δij(B2 + 1c2E2)]
= 1μ
0
(?BiBj + 12δijB2) +epsilon10(?EiEj + 12δijE2) =Jij
343/384
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a131a233a216a62a62a196a196a229a198,a85a196a254a197a240
(ρvectorE +vectorj×vectorB,icvectorj·vectorE) = fμ =?
4summationdisplay
ν=1
νTμν
Tμν ≡ 1μ
0
[
4summationdisplay
νprime=1
FμνprimeFννprime?14δμν
4summationdisplay
μprime,νprime=1
FμprimeνprimeFμprimeνprime] = Tνμ
4summationdisplay
μprime,νprime=1
FμprimeνprimeFμprimeνprime = 2B2? 2c2E2 Fij =
3summationdisplay
k=1
epsilon1ijkBk F4i =?Fi4 = icEi
Tij= 1μ
0
(?BiBj + 12δijB2) +epsilon10(?EiEj + 12δijE2) =Jij
T44= 1μ
0
[
4summationdisplay
i=1
F24i?12(B2?1c2E2)]=?12μ
0
(B2+ 1c2E2) =?12μ
0
B2?epsilon102E2=?W
Ti4=T4i = 1μ
0
4summationdisplay
j=1
FijF4j = iμ
0c
4summationdisplay
j,k=1
epsilon1ijkBkEj = iμ
0c
(vectorE×vectorB)i = icSi
344/384
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a131a233a216a62a62a196a196a229a198,a85a196a254a197a240
(ρvectorE +vectorj×vectorB,icvectorj·vectorE) = fμ =?
4summationdisplay
ν=1
νTμν
Tμν ≡ 1μ
0
[
4summationdisplay
νprime=1
FμνprimeFννprime?14δμν
4summationdisplay
μprime,νprime=1
FμprimeνprimeFμprimeνprime] = Tνμ
Tij= 1μ
0
(?BiBj + 12δijB2) +epsilon10(?EiEj + 12δijE2) =Jij
T44=? 12μ
0
B2?epsilon102E2=?W Ti4=T4i = iμ
0c
(vectorE×vectorB)i = icSi = icgi
fi =?
4summationdisplay
ν=1
νTiν =?
4summationdisplay
j=1
jTijTi4ic?t arrowtripleright vectorf =· arrowrighttophalfarrowrighttophalfJvectorg?t
f4 =?
4summationdisplay
ν=1
νT4ν =?
4summationdisplay
i=1
iT4iT44ic?t arrowtripleright vectorf·vectorv =·vectorSW?t
vectorf·vectorv = ρvectorv·vectorE =vectorj·vectorE
345/384
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a145a62a226a102
a218a62a94a124a27a131a112a138a94
346/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a36a196a145a62a226a102a27a163a227,a237a180a8a65
ρ(vectorr,t) = qδ(vectorr?vectorr0) vectorj(vectorr,t) = ρ(vectorr,t)vectorv0(t) = qvectorv0(t)δ(vectorr?vectorr0)
φ(vectorr,t) =
integraldisplay
d3xprimeρ(vectorr
prime,t?R
c )
4piepsilon10R =
integraldisplay
d3xprimeqδ(vectorr
prime?vectorr0(t?|vectorr?vectorrprime|
c ))
4piepsilon10|vectorr?vectorrprime|
vectorA(vectorr,t)=
integraldisplay
d3xprimeμ0
vectorj(vectorrprime,t?R
c )
4piR =
integraldisplay
d3xprimeμ0qvectorv0(t?
|vectorr?vectorrprime|
c )δ(vectorr
prime?vectorr0(t?|vectorr?vectorrprime|
c ))
4pi|vectorr?vectorrprime|
δa188a234a144a51vectorrprime?vectorr0(t?|vectorr?vectorr
prime|
c ) = 0a63a216a1430,a23a249a135a144a167a27a41a143:vectorr
braceleftbiggvectorrvectorr
0(t?
|vectorr?vectorr?|
c ) = 0
t?≡t?|vectorr?vectorr?|c →
braceleftbiggvectorr? =vectorr
0(t?) vectorv?≡vectorv0(t?)
|vectorr?vectorr?|= c(t?t?)
φ(vectorr,t) = q4piepsilon1
0|vectorr?vectorr?|
integraldisplay
d3xprimeδ(vectorrprime?vectorr0(t?|vectorr?vectorr
prime|
c ))
vectorA(vectorr,t) = μ0qvectorv?
4pi|vectorr?vectorr?|
integraldisplay
d3xprimeδ(vectorrprime?vectorr0(t?|vectorr?vectorr
prime|
c ))
347/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a111a66-a145a28a16a179 (Li′enard-Wiechert)
φ(vectorr,t) = q4piepsilon1
0|vectorr?vectorr?|
integraldisplay
d3xprimeδ(vectorrprime?vectorr0(t?|vectorr?vectorr
prime|
c ))
vectorA(vectorr,t) = μ0qvectorv?
4pi|vectorr?vectorr?|
integraldisplay
d3xprimeδ(vectorrprime?vectorr0(t?|vectorr?vectorr
prime|
c ))integraldisplay
d3xprimeδ(vectorrprime?vectorr0(t?|vectorr?vectorr
prime|
c )) =
integraldisplay
d3x?prime 1J(vectorr,vectorrprime)δ(vectorr?prime) = 1J(vectorr,vectorr?)
vectorr?prime =vectorrprime?vectorr0(t?|vectorr?vectorr
prime|
c )
J=||?x
i
prime
xprimej||
vextendsinglevextendsingle
vectorrprime=vectorr? =||
xprimej[x
prime
i?x0i(t?
|vectorr?vectorrprime|
c )]||
vextendsinglevextendsingle
vectorrprime=vectorr? =1?
(vectorr?vectorr?)·vectorv?
|vectorr?vectorr?|c
φ(vectorr,t) = q4piepsilon1
0|vectorr?vectorr?|
1
1?(vectorr?vectorr?)·vectorv?|vectorr?vectorr?|c
vectorA(vectorr,t) = μ0qvectorv?
4pi|vectorr?vectorr?|
1
1?(vectorr?vectorr?)·vectorv?|vectorr?vectorr?|c
348/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a62a94a124
braceleftbigg vectorE =φvectorA
tvector
B =?×vectorA
braceleftBiggφ(vectorr,t) = q
4piepsilon10R?
1
1?vectorR?·vectorv?R?c
vectorA(vectorr,t) = μ0qvectorv?
4piR?
1
1?vectorR?·vectorv?R?c
braceleftbigg vector
R?=vectorr?vectorr?
vectorr?=vectorr?(vectorr,t)=vectorr0(t?)
vectorR? =vectorr?vectorr? vectorr? =vectorr0(t?),
R?
t =
1
2R?
R?2
t =
1
2R?
vectorR?·vectorR?
t =
vectorR?
R?·(?
vectorr?
t?)
t?
t =?vectorv
·vectorR
R?
t?
t
R? = 12RR?2 = 12R(vectorR?·vectorR?) = 1R?(?vectorR?)·vectorR?
= 1R?(arrowrighttophalfarrowrighttophalfItvectorr
t?)·
vectorR? = 1
R?(
vectorRt?vectorv?·vectorR?)
R? = c(t?t?)→?R
t = c?
t?
t?R
=?c?t?
t?
t =
1
1?vectorR?·vectorv?R?c
R?
t =
vectorR?R? ·vectorv?
1?vectorR?·vectorv?R?c?t
=?
vectorR?
cR?
1?vectorR?·vectorv?R?c?R
=
vectorR?
R?
1?vectorR?·vectorv?R?c
vectorR?
t =?
vectorr?
t?
t?
t =
vectorv?
1? vectorR?·vectorv?R?c?
vectorR? =arrowrighttophalfarrowrighttophalfItvectorr?
t? =
arrowrighttophalfarrowrighttophalfI? vectorR?vectorv?R?c
1? vectorR?·vectorv?R?c
349/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a62a94a124
braceleftbigg vectorE =φvectorA
tvector
B =?×vectorA
braceleftBiggφ(vectorr,t) = q
4piepsilon10R?
1
1?vectorR?·vectorv?R?c
vectorA(vectorr,t) = μ0qvectorv?
4piR?
1
1?vectorR?·vectorv?R?c
braceleftbigg vector
R?=vectorr?vectorr?
vectorr?=vectorr?(vectorr,t)=vectorr0(t?)
t?
t =
1
1?vectorR?·vectorv?R?c
R?
t =
vectorR?R? ·vectorv?
1?vectorR?·vectorv?R?c?t
=?
vectorR?
cR?
1?vectorR?·vectorv?R?c?R
=
vectorR?
R?
1?vectorR?·vectorv?R?c
vectorR?
t =
vectorv?
1? vectorR?·vectorv?R?c?
vectorR? =arrowrighttophalfarrowrighttophalfI?
vectorR?vectorv?
R?c
1? vectorR?·vectorv?R?c
vectorE(vectorr,t)= q
4piepsilon10S?3(
vectorRR?vectorv?
c )(1?
v?2
c2)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
vectorEa154a154a203a203
+ q4piepsilon1
0c2S?3
bracketleftbigvectorR?×[(vectorRR?vectorv?
c )×vectora
]bracketrightbig
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
vectorEa203
vectorB(vectorr,t) = vectorR?
cR?×
vectorE(vectorR,t)
vectorR? =vectorr?vectorr? vectora?≡d2vectorr0(t?)
dt?2 S
≡R?(1?vectorR
·vectorv?
R?c )
350/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a62a94a124
vectorE = vectorEa154a154a203a203 + vectorEa203 vectorB = vectorBa154a154a203a203 + vectorBa203
vectorEa154a154a203a203 = q
4piepsilon10S?3(
vectorRR?vectorv?
c )(1?
v?2
c2 )
vectorBa154a154a203a203 = vectorR?
cR?×
vectorEa154a154a203a203
vectorEa203 = q
4piepsilon10c2S?3
bracketleftbigvectorR?×[(vectorRR?vectorv?
c )×vectora
]bracketrightbig vectorB
a203 =
vectorR?
cR?×
vectorEa203
vectorR = vectorRR?vectorv?
c
vectorR·vectorv? = vectorR?·vectorvR?v?2
c
R2 = R?2(1 + v
2
c2 )?2R
vectorR
·vectorv?
c = R
2(1?v
2
c2 )?R
2vectorR·vectorv
c
→R?2(1?v
2
c2 ) = R
2 + R?2vectorR·vectorv
c
S? = RvectorR?·vectorv
c = R
1
c(
vectorR·vectorv? + R?v?2
c ) = R
(1?v
2
c2 )?
1
c
vectorR·vectorv?
S?2 = R?2(1?v
2
c2 )
2?(1?v
2
c2 )R
2vectorR·vectorv
c +
1
c2(
vectorR·vectorv?)2
351/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a62a94a124
vectorEa154a154a203a203 = q
4piepsilon10S?3(
vectorRR?vectorv?
c )(1?
v?2
c2 )
vectorBa154a154a203a203 = vectorR?
cR?×
vectorEa154a154a203a203
R?2(1?v
2
c2 ) = R
2 + R?2vectorR·vectorv
c S
= R?(1?v
2
c2 )?
1
c
vectorR·vectorv?
S?2 = R?2(1?v
2
c2 )
2?(1?v
2
c2 )R
2vectorR·vectorv
c +
1
c2(
vectorR·vectorv?)2
= (1?v
2
c2 )[R
2+R?2vectorR·vectorv
c ]?(1?
v?2
c2 )R
2vectorR·vectorv
c +
1
c2(
vectorR·vectorv?)2
= (1?v
2
c2 )R
2 + (vectorR·vectorv
)2
c2 ≡S
2 vectorR = vectorRR?vectorv
c
vectorEa154a154a203a203 = qvectorR
4piepsilon10S3(1?
v?2
c2 )
vectorBa154a154a203a203 = vectorv?
c2 ×
vectorEa154a154a203a203
352/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a62a94a124
vectorEa154a154a203a203 = qvectorR
4piepsilon10S3(1?
v?2
c2 )
vectorBa154a154a203a203 = vectorv?
c2 ×
vectorEa154a154a203a203
S2 = (1?v
2
c2 )R
2 + (vectorR·vectorv
)2
c2
vectorR = vectorRR?vectorv?
c
vectorr0(t) a33a132=== vectorVt
vectorv? = vectorV
vectorr? = vectorVt?
c(t?t?)=R?=|vectorr?vectorr?|
vectorR=vectorr?vectorr(t?t?)vectorv?=vectorr?vectorVt
a233vectorR//vectorv?
braceleftBigg vector
Ea154a154a203a203//= qvectorR4piepsilon1
0R3
(1?v?2c2 )
vectorBa154a154a203a203//=0
a233vectorR⊥vectorv?
vectorEa154a154a203a203⊥= qvectorR
4piepsilon10R3
1radicalBig
1?v?2c2
vectorBa154a154a203a203⊥= μ0qvectorv?×vectorR
4piR3
1radicalBig
1?v?2c2
a8v?lessmuchca158,Ea154a154a203a203a136a144a149a211a234a138; a8v?→ca158,Ea154a154a203a203//→0,Ea154a154a203a203⊥→∞.
353/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a62a94a124
vectorE = vectorEa154a154a203a203 + vectorEa203 vectorB = vectorBa154a154a203a203 + vectorBa203 S? = RvectorR?·vectorv?
c
vectorEa203 = q
4piepsilon10c2S?3
bracketleftbigvectorR?×[(vectorRR?vectorv?
c )×vectora
]bracketrightbig vectorB
a203 =
vectorR?
cR?×
vectorEa203
vectorR?·vectorEa203 = 0 vectorn? = vectorR?R? vectorn?·vectorEa203 = 0
a92a132a36a196a196a27a27a226a102a172a23a41a203a19a156
a33a132a36a196a196a27a27a226a102a152a189a216a172a23a41a203a19a237a186a144a8v? <ca158
a233a33a132a36a196a196a27a27a226a102a167a8vectorn?·vectorv?c = 1a158S? = 0a167vectorEa203a165a209a1210/0?Cerenkova203a19
vectorS= 1
μ0
vectorEa203×vectorBa203= 1
μ0c
vectorEa203×(vectorn?×vectorEa203)= 1
μ0c[
vectorE2a203vectorn(vectorn?·vectorEa203)vectorEa203]
= q
2
16pi2epsilon120c5μ0S?6|
vectorR?×[(vectorRR?vectorv?
c )×vectora
]|2vectorn?
= q
2vectorn?
16pi2epsilon10c3R?2
|vectorn?×[(vectornvectorv?c )×vectora?]|2
(1?vectorv?·vectorn?c )6
354/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a203a19a245a199a57a14a169a217
vectorS= 1
μ0
vectorEa203×vectorBa203 = q2vectorn?
16pi2epsilon10c3R?2
|vectorn?×[(vectornvectorv?c )×vectora?]|2
(1?vectorv?·vectorn?c )6
I?≡
integraldisplay
S?
d dI
d =
integraldisplay
S?
dvectorσ?·(vectorSdtdt?) =
integraldisplay
S?
d R?2vectorn?·(vectorSdtdt?)
dI?
d=R
2vectorn?·(vectorSdt
dt?)=
q2vectorn?
16pi2epsilon10c3R?2·vectorn
R?2|vectorn
×[(vectornvectorv?
c )×vectora
]|2
(1?vectorv?·vectorn?c )6 (1?
vectorv?·vectorn?
c )
= q
2
16pi2epsilon10c3
|vectorn?×[(vectornvectorv?c )×vectora?]|2
(1?vectorv?·vectorn?c )5
|vectorn?×[(vectornvectorv
c )×vectora
]|2 =|vectorn?·vectora?(vectornvectorv
c )?(1?
vectorn?·vectorv?
c )vectora
|2
=(vectorn?·vectora?)2(1+vectorv
2
c2?2
vectorn?·vectorv?
c )?2vectorn
·vectora?(vectorn?·vectoravectorv
·vectora?
c )(1?
vectorn?·vectorv?
c )+(1?
vectorn?·vectorv?
c )
2vectora?2
= 2(vectorn
·vectora?)(vectorv?·vectora?)
c (1?
vectorn?·vectorv?
c ) + a
2(1?vectorn
·vectorv?
c )
2?(1?v
2
c2 )(vectorn
·vectora?)2
I? =
integraldisplay
d ( dI
d)
a177vectorv?a143za182=====? q2
6piepsilon10c3
[a?2?(vectora?×vectorv?)2c2 ]
(1?v?2c2 )3
355/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a203a19a245a199a57a14a169a217
|vectorn?×[(vectornvectorv
c )×vectora
]|2
= 2(vectorn
·vectora?)(vectorv?·vectora?)
c (1?
vectorn?·vectorv?
c ) + a
2(1?vectorn
·vectorv?
c )
2?(1?v
2
c2 )(vectorn
·vectora?)2
vectora?bardblvectorv?,a18vectorv?a134vectorn?a27a89a14a143θ?
|vectorn?×[(vectornvectorv
c )×vectora
]|2
= 2ca?2v?cosθ?(1?v
c cosθ
)+a?2(1?v
c cosθ
)2?(1?v
2
c2 )a
2 cos2θ?
= a?2(1?cos2θ?) = a?2 sin2θ?
dI?
d =
q2a?2 sin2θ?
16pi2epsilon10c3(1?v?c cosθ?)5 I
= q
2
6piepsilon10c3
a?2
(1?v?2c2 )3
a111a245a199a39a154a131a233a216a156a47a245a10a207a102(1?v?2c2 )?3
a203a19a129a114a27a144a149θ?≤pi/2a167a133a8a22a112a132a167
a22a26a26a67a67θ? = 0
356/384
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a36a196a145a62a226a102a27a27a62a62a94a124,a203a19a245a199a57a14a169a217
|vectorn?×[(vectornvectorv
c )×vectora
]|2
= 2(vectorn
·vectora?)(vectorv?·vectora?)
c (1?
vectorn?·vectorv?
c ) + a
2(1?vectorn
·vectorv?
c )
2?(1?v
2
c2 )(vectorn
·vectora?)2
vectora?⊥vectorv?,a177vectorv?a143za182,vectora?a143xa182,vectorn?·vectorv? = v? cosθ?,vectorn?·vectora? = a? sinθ? cosφ?
|vectorn?×[(vectornvectorv
c )×vectora
]|2 = a?2[(1?v
c cosθ
)2?(1?v
2
c2 )sin
2θ?cos2φ?]
dI?
d =
q2a?2
16pi2epsilon10c3
1
(1?v?c cosθ?)3
bracketleftbig1?(1?v?2c2 )sin2θ?cos2φ?
(1?v?c cosθ?)2
bracketrightbig
I?= q
2
6piepsilon10c3(1?v?2c2 )2 a233xza178a161,φ
=0 dI
d=
q2a?2
16pi2epsilon10c3
(v?c?cosθ?)2
(1?v?c cosθ?)5
a111a245a199a39a154a131a233a216a156a47a245a10a207a102(1?v?2c2 )?2,
a2a39//a156a47a8a10(1?v?2c2 )?1
a203a19a169a217a154a182a233a161a167a134φ? = 0,pia129a114a167φ? = pi/2a129a102
a112a132a36a196a158a203a19a129a114a27a144a149θ? = 0
357/384
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a203a19a170a170a204a204a169a219
vectorEa203(vectorr,t) =
integraldisplay ∞
∞
dω vectorEω(vectorr)e?iωt vectorEω(vectorr) = 12pi
integraldisplay ∞
∞
dt vectorEa203(vectorr,t)eiωt
vectorEω(vectorr) = 1
2pi
integraldisplay ∞
∞
dt eiωt q4piepsilon1
0c2R?
vectorn?×[(vectornvectorv?c )×vectora?]
(1?vectorn?·vectorv?c )3
= 12pi
integraldisplay ∞
∞
dt? dtdt?eiω(t?+R?c ) q4piepsilon1
0c2R?
vectorn?×[(vectornvectorv?c )×vectora?]
(1?vectorn?·vectorv?c )3
= 12pi
integraldisplay ∞
∞
dt? eiωt? qe
iωR?c
4piepsilon10c2R?
vectorn?×[(vectornvectorv?c )×vectora?]
(1?vectorn?·vectorv?c )2?
ω→∞→ 0
ω→0,v?lessmuchc,R?,vectorn?a134t?a195a39→ qeiωR?c
8piepsilon10c2R?vectorn
×(vectorn?×?vectorv?)
a220a151a203a19a181a145a62a226a102a92a19a20a212a159a113a254a167a51a45a69a76a167a165a126a132a23a41a27a203a19a34
358/384
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a203a19a170a170a204a204a169a219
(dWd? )?≡
integraldisplay ∞
∞
dt? ( dI
d) =
integraldisplay ∞
∞
dt?epsilon10cvectorEa203(vectorr,t)·vectorEa203(vectorr,t)R?2( dtdt?)
=
integraldisplay ∞
∞
dtepsilon10cR?2
integraldisplay ∞
∞
dωdωprimee?iωte?iωprimetvectorEω(vectorr)·vectorEωprime(vectorr)
= 2piepsilon10c
integraldisplay ∞
∞
dωR?2vectorEω(vectorr)·vectorE?ω(vectorr)
= 4piepsilon10c
integraldisplay ∞
0
dωR?2vectorEω(vectorr)·vectorE?ω(vectorr)≡
integraldisplay ∞
0
dω (dW
ω
d? )
dW?ω
d? = 4piepsilon10cR
2vectorEω(vectorr)·vectorE?ω(vectorr)
ω→∞→ 0?vectorv?·vectorn?≡?v?cosΘ
ω→0,v?lessmuchc,R?,vectorn?a134t?a195a39→ q2
16pi3epsilon10c3[(vectorn
·?vectorv?)vectornvectorv?]2 = q2?v?2 sin2 Θ
16pi3epsilon10c3
W?ω =
integraldisplay
d? 4piepsilon10cR?2vectorEω(vectorr)·vectorE?ω(vectorr)
ω→∞→ 0
ω→0,v?lessmuchc,R?,vectorn?a134t?a195a39→ q2?v?2
6pi2epsilon10c3?a36a170a170a117a126a234?a112a170a170a117a34
359/384
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a131a212a133a197(Cerenkov)a203a19
a195a161a140a254a33a48a159(μ,epsilon1)a165a27a203a19a124:
vectorj(vectorr,t) =
integraldisplay ∞
∞
dωvectorjω(vectorr)e?iωt vectorjω(vectorr) = 12pi
integraldisplay ∞
∞
dtvectorj(vectorr,t)eiωt
vectorA(vectorr,t) =
integraldisplay ∞
∞
dω vectorAω(vectorr)e?iωt vectorAω(vectorr) = 12pi
integraldisplay ∞
∞
dt vectorA(vectorr,t)eiωt
vectorA(vectorr,t) = μ
4pi
integraldisplay
dτprime
vectorj(vectorrprime,t?nR
c )
r =
μ
4pir
integraldisplay
dτprime
integraldisplay ∞
∞
dωvectorjω(vectorrprime)e?iω(t?nRc )
vectorAω(vectorr) = μ
4pir
integraldisplay
dτprimevectorjω(vectorrprime)eiωnRc = μ8pi2r
integraldisplay
dτprime
integraldisplay ∞
∞
dtvectorj(vectorrprime,t)eiω(t+nRc )
= μ8pi2r
integraldisplay
dτprime
integraldisplay ∞
∞
dt evectorV0(t)δ(vectorrprime?vectorr0(t))eiω(t+nRc )
≈ eμ8pi2reiωrnc
integraldisplay
dτprime
integraldisplay ∞
∞
dtvectorV0(t)δ(vectorrprime?vectorr0(t))eiω(t?ncvectorn·vectorrprime)
= eμ8pi2reiωrnc
integraldisplay ∞
∞
dtvectorV0(t)eiω(t?ncvectorn·vectorr0(t))
360/384
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a131a212a133a197(Cerenkov)a203a19
vectorAω(vectorr) = eμ
8pi2re
iωrnc
integraldisplay ∞
∞
dtvectorV0(t)eiω(t?ncvectorn·vectorr0(t))
a63a216a33a132a36a196a196a27a27a156a47,vectorV0(t) = vectorv0 vectorr0(t) =vectorr0 +vectorv0t nc = 1v
vectorAω(vectorr)=eμvectorv0
8pi2re
iωv(r?vectorn·vectorr0)
integraldisplay ∞
∞
dt e?iωt(1?vectorn·vectorv0v )=eμvectorv04pir eiωv(r?vectorn·vectorr0)δ[ω(1?vectorn·vectorv0v )]
a233a253a152,v = c,v0 <c,vectorn·vectorv0v <1,δa188a234a240a1430.
a233a48a159,v <c,v0 <c,a107a140a85a127a51vectorn·vectorv0v = 1.
a38a37
a39a36
a108a114a114a45v0 < v
a38a37
a39a36
a22a21
a23a20
a114a114a45v0 = v
a38a37
a39a36
θa114 a45v0 > v a80a80a80
a80a80a80
a2
a2
cosθ = v/v0 a38a37
a39a36
a30a29
a31a28
a114a114 a45a80a80a80
a80a80a80
a2
a2
a2a2
a100a158a27a203a19a124a23Cerenkova203a19,a167a180a100a117a145a62a226a102a27a132a221a135a76 a48a159a165a27a49a132a158,a48
a159a83a23a41a112a19a19a62a62a54a45a117a27a103a197a134a6a53a53a226a226a102a27a27a62a62a94a124a131a112a90a21a47a164a27.
361/384
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a131a212a133a197(Cerenkov)a203a19
vectorB(vectorr,t) =
integraldisplay ∞
∞
dω vectorBω(vectorr)e?iωt
vectorBω(vectorr) = 1
2pi
integraldisplay ∞
∞
dt vectorB(vectorr,t)eiωt = 12pi
integraldisplay ∞
∞
dt?×vectorA(vectorr,t)eiωt
= 12pi
integraldisplay ∞
∞
dt
integraldisplay ∞
∞
dωprime?×e?iωprimetvectorAωprime(vectorr)eiωt =?×vectorAω(vectorr)
=?×bracketleftbigeμvectorv04pir δ[(1?vectorn·vectorv0v )ω]eiωv(vectorr?vectorn·vectorr0)bracketrightbig = ieμωvectorn×vectorv04pivr δ[(1?vectorn·vectorv0v )ω]
dWω
d? =
4piv
μ r
2vectorBω·vectorB?ω = (vectorn×vectorv0)·(vectorn×vectorv0)ω
2e2μ
4pi δ
2[(1?vectorn·vectorv0
v )ω]
= v
2
0 sin
2θω2e2μ
8pi2 δ[(1?
vectorn·vectorv0
v )ω]
integraldisplay ∞
∞
dt eiω(1?vectorn·vectorv0v )t
= v0 sin
2θω2e2μ
8pi2 δ[(1?
vectorn·vectorv0
v )ω]
integraldisplay ∞
∞
dx ei ωv0(1?vectorn·vectorv0v )x
= sin
2θω2e2μ
8pi2 δ[(1?
vectorn·vectorv0
v )
ω
v0]
integraldisplay ∞
∞
dx
362/384
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a131a212a133a197(Cerenkov)a203a19
dWω
d? =
sin2θω2e2μ
8pi2 δ[(1?
vectorn·vectorv0
v )
ω
v0]
integraldisplay ∞
∞
dx
= Lsin
2θω2e2μ
8pi2 δ[(1?
vectorn·vectorv0
v )
ω
v0] vectorn·vectorv0≡v0 cosθ
dWω
dL d? =
sin2θω2e2μ
8pi2 δ[(1?
vectorn·vectorv0
v )
ω
v0]=
ω2e2μ
8pi2 (1?
v2
v20)δ[(1?
vectorn·vectorv0
v )
ω
v0]
dWω
dL =
ω2e2μ
8pi2 (1?
v2
v20)
integraldisplay pi
0
d?δ[ωv( vv
0
cosθ)] = ωe
2μv
4pi [1?
v2
v20]
a217a165,v = v(ω),Wωa144a51v(ω) <v0a164a137a209a27a170a227a107a1540a0a122.
a100a117cosθ = v(ω)v0,a216a211a170a199a62a94a197a27a203a19a14a189a216a131a211,a94a200a197a236a192a74a152a189a27a170
a145,a140a177a26a26a20a20a40a189a189a27a27θa138,a207a13a255a189θa210a140a177a189a209a226a102a27a132a221.
a121a51a131a212a133a197a203a19a50a141a65a94a117a226a102a79a234a236a165,a167a27a96a58a180a144a80a185a140a117a152a189a132a221a27
a226a102,a207a13a59a157a10a36a132a226a102a27a90a54,a13a133a140a177a79a40a47a255a209a226a102a27a36a196a132a221.
363/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94
a107a152a9a154a62a94a229a138a94a94a117a117a62a102,a166a217a51dta158a109a83a36a196a229a108dvectorl
vectorFa154a62a94 + vectorFa62a94 = mvectora
vectorFa62a94a147a76a62a102a23a41a27a27a62a62a94a124a233a103a67a23a41a27a140a85a27a138a94a156
a107a60a60a64a64a143a62a102a23a41a27a27a62a62a94a124a216a65a84a233a103a67a107a138a94
a249a144a180a183a142a218a33a132a36a196a158a27a40a216
a61a166a127a196a107a103a103a138a138a94a167a167a143a180a195a161a140a156
vectorFa62a94 =?e[vectorE +vectorv×vectorB]
a107a60a226a100a96a178a62a102a23a41a27a27a62a62a94a124a216a65a84a233a103a67a107a138a94!
a40a220a85a254a61a122a218a197a240a189a198a53a63a216a140a85a27a103a103a138a138a94
364/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94,a85a254a61a122a134a197a240a189a198; a218a238a238a189a189a198
a85a254a61a122a134a197a240a189a198,a107a152a9a154a62a94a229a138a94a94a117a117a62a102,a166a217a51dta158a109a83a36a196a229a108dvectorl,
vectorFa154a62a94 + vectorFa62a94 = mvectora?
contintegraldisplay
V
dvectorσ·vectorS =
integraldisplay
V
dV vectorfa62a94·vectorv + ddt
integraldisplay
V
dVw
(vectorFa154a62a94?mvectora)·dvectorl =?vectorFa62a94·dvectorl =?
integraldisplay
dV vectorfa62a94·dvectorl = dWa103+
contintegraldisplay
dvectorσ·vectorSdt
dWa103 =
integraldisplay
∞
dV 12(epsilon10E2 + 1μ
0
B2) dWa203≡
contintegraldisplay
∞
dσ·vectorSdt =
integraldisplay t2
t1
dt I
dWa62a94≡dWa103 +dWa203a180a145a62a62a27a27a27a62a62a102a36a196a23a41a27a27a62a62a94a124a69a164a27a85a254a67a122.
a63a216a157a185a10a145a62a226a102a27a103a131a112a138a94a163a103a67a201a103a67a23a41a27a124a27a138a94a229a156a156a164a164
a233a216a145a62a226a102,dWa62a94 = 0 →(vectorFa154a62a94?mvectora)·dvectorl = 0→vectorFa154a62a94 = mvectora
a233a145a62a226a102,dWa62a94negationslash= 0 →(vectorFa154a62a94?mvectora)·dvectorl = dWa62a94 = dWa103+dWa203
365/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94,a62a102a27a178a59a36a196a144a167
a28a46,a62a214a254a33a169a217a51a140a187a143r0a27a165a165,r0a161a143a62a102a27a27a62a62a94a140a187,a62a214a151a221ρ =?e4
3pir
30
4pir2E =
braceleftbigg ρ
epsilon10
4
3pir
3 r≤r0
eepsilon1
0
r>r0 →vectorE =
braceleftbigg? er
4piepsilon10r30vectorer r≤r0
e4piepsilon1
0r2
vectorer r>r0
vectorB=
integraldisplay
dτprimeμ04piρvectorv×
vectorR
R3 =
μ0
4pivectorv×
integraldisplay
dτprimeρ
vectorR
R3 =μ0epsilon10vectorv×
integraldisplay
dτprime ρ
vectorR
4piepsilon10R3 =
vectorv
c2×
vectorE
Wa103 = We + Wm
We =
integraldisplay
dτ epsilon102E2 = epsilon1024pi[
integraldisplay r0
0
dr r2( er4piepsilon1
0r30
)2 +
integraldisplay ∞
r0
dr r2( e4piepsilon1
0r2
)2]
= e
2
8piepsilon10[
integraldisplay r0
0
dr r
4
r60 +
integraldisplay ∞
r0
dr 1r2] = e
2
8piepsilon10(
1
5r0 +
1
r0) =
3
20piepsilon10
e2
r0
Wm=
integraldisplay
dτprime 12μ
0
B2=
integraldisplay
dτ 12μ
0c4
(vectorv×vectorE)·(vectorv×vectorE)=
integraldisplay
dτ 12μ
0c4
[v2E2?(vectorv·vectorE)2]
=
integraldisplay
dτ E2 v
2
2μ0c4(1?
1
3) =
v2
3μ0epsilon10c4We =
2v2
3c2We
366/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94,a62a102a27a178a59a36a196a144a167
integraldisplay
dτEiEj=13δij
integraldisplay
dτE2
integraldisplay
dτ(vectorv·vectorE)2=
3summationdisplay
i,j=1
vivj
integraldisplay
dτEiEj=13v2
integraldisplay
dτE2
vectorB = vectorv
c2×
vectorE Wa103 = We + Wm We = 3
20piepsilon10
e2
r0 Wm =
2v2
3c2We
Wa103 = We + Wm = We(1 + 2v
2
3c2) dWa103 =
4
3c2Wevectorv·vectoradt
Wa203 =
integraldisplay t2
t1
dt μ06pice2a2 =
integraldisplay t2
t1
μ0
6pice
2vectora·dvectorv = μ0e
2
6pic[vectora·vectorv
vextendsinglevextendsinglet2
t1?
integraldisplay t2
t1
dtvectorv·˙vectora]
a63a216vectorv·vectoravextendsinglevextendsinglet
1
= vectorv·vectoravextendsinglevextendsinglet
2
a27a156a47:
Wa203 =?μ0e
2
6pic
integraldisplay t2
t1
dtvectorv·˙vectora dWa203 =?μ0e
2
6picvectorv·
˙vectoradt
(vectorFa154a62a94?mvectora)·dvectorl = 43c2Wevectorv·vectoradt?μ0e
2
6picvectorv·
˙vectoradt
vectorFa154a62a94 + vectorFs = (m + 4
3c2We)vectora
vectorFs = μ0e2
6pic
˙vectora
367/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94,a62a94a159a254; a203a19a123a90a229
vectorF + vectorFs = (m + 4
3c2We)vectora
vectorFs = μ0e2
6pic
˙vectora
a107a8a159a254me = m + 43c2Wea165 43c2Wea53a103a62a94a131a112a138a94a27a94a124a85a233a159a254a27a0a122,a88a219a110
a41a186 a23a62a102a27a62a94a159a254,a167a100a64a220a169a195a123a134a62a102a169a109a27a62a94a124a47a164,a249a220a169a85a254a195a123
a108a62a102a27a36a196a85a254a165a169a108a209a22.
a127a196a20a162a8a254a42a9a20a20a27a27a27a62a62a102a159a254a65a65a107a107a131a8a220a169a180a62a94a159a254
me～Wec2 = 3μ020pi e
2
r0 →re～
μ0e2
pime ～10
15a146
a233a39a229a216a102a27a186a221a14310?13a146,a6a102a27Bohra140a187a1430.5×10?10a146,a62a102a180a154a126a2a27a226a102.
a62a102a203a19a62a94a124,a103a28a65a201a20a135a192,a135a192a229a143vectorFs.
μ0e2
6pic
˙vectora = mevectora→vectorr(t) =vectorr(0) +vectorv(0)t + μ0e2
6picmevectora(0)e
6picme
μ0e2 t
a62a102a242a195a129a92a132a36a196!
368/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94,a204a130a27a103a44a176a221
a140a178a59a59a27a27a6a102a117a49a28a46,a62a102a254a33a169a217a117a6a102a83,a6a102a216a63a117a62a102a31a165a37a160
a152,a8 a6a102a201a9a46a138a94a45a117a0,a6a102a216a160a108a62a102a31a165a37a160a152vectorr0(t),a6a102a216a134a62a102a31a165
a37a131a109a23a41a152a6a53a161a69a229
vectorf =?evectorE =?eρvectorr0(t)
3epsilon10 ≡?meω
2
0vectorr0(t) ω
2
0 =
eρ
3epsilon10me =
e2
4epsilon10pimer20
a166a62a102a138a92a132a36a196(a20a8a196)a13a203a19a117a49,a162a8a254a119a20a27a117a49a216a180a252a170a199,a13a180a107a152a189
a27a170a199a169a217a176a221,a51a49a204a165a76a121a143a6a102a117a49a27a204a130a107a152a189a103a44a176a221.
meω20vectorr0(t) + μ0e
2
6pic
···
vectorr0 (t) = me¨vectorr0(t)
a193a38a41,vectorr0(t) =vectorr0(0)e?iωt→?meω20 + μ0e
2
6pic(?i)
3ω3 =?meω2
ω2 = ω20? iμ0e
2
6picmeω
3 ω =
braceleftbiggω
0 a216a127a196a20a203a19a135a192
ω0?iΓ2 a127a196a20a203a19a135a192
Γ≡μ0e
2ω2
0
6picme,a203a19a135a192a8a65
369/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94,a204a130a27a103a44a176a221
meω20vectorr0(t) + μ0e
2
6pic
···
vectorr0 (t) = me¨vectorr0(t)
a193a38a41,vectorr0(t) =vectorr0(0)e?iωt ω =
braceleftbiggω
0 a216a127a196a20a203a19a135a192
ω0?iΓ2 a127a196a20a203a19a135a192
Γ≡μ0e
2ω2
0
6picme,a203a19a135a192a8a65
vectorr0(t) =vectorr0(0)e?Γ2te?iω0t vectora(t) =vectora(0)e?Γ2te?iω0t vectora(0) =?ω2vectorr0(0)
vectorB(vectorr,t) = eμ0
4picrvectora(t
)×vectorn
vectorE(vectorr,t) = cvectorB(vectorr,t)×vectorn = eμ0
4pir[vectora(t
)×vectorn]×vectorn = eμ0
4pir[vectora(t
)·vectornvectorn?vectora(t?)]
= eμ04pir[vectora(0)·vectornvectorn?vectora(0)]eiωrc e?iωt = vectorE(vectorr,0)e?iωt
vectorE(vectorr,0)≡ eμ0
4pir[vectora(0)·vectornvectorn?vectora(0)]e
iωrc = eμ0ω
2
4pir [?vectorr0(0)·vectornvectorn +vectorr0(0)]
vectorE(vectorr,t) =
braceleftbigg 0 t<0
vectorE(vectorr,0)E?iωt t≥0 =
integraldisplay ∞
∞
dωprime vectorEωprimee?iωprimet
370/384
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a145a62a226a102a27a27a62a62a94a124a233a226a102a29a28a27a135a138a94,a204a130a27a103a44a176a221
vectorEωprime = 1
2pi
integraldisplay ∞
∞
dt vectorE(vectorr,t)eiωprimet = 12pivectorE(vectorr,0)
integraldisplay ∞
0
dt ei(ωprime?ω)t = i
vectorE(vectorr,0)
2pi(ωprime?ω)
a152a135a6a102a117a209a27a54a76a252a160a31a161a27a49a85a254 =
integraldisplay ∞
∞
dt S =12epsilon10c
integraldisplay ∞
∞
dt vectorE(vectorr,t)·vectorE?(vectorr,t)
= 12epsilon10c
integraldisplay ∞
∞
dt
integraldisplay ∞
∞
dωprime
integraldisplay ∞
∞
dω” vectorEωprimee?iωprimet·(vectorEω”e?iω”t)?
= piepsilon10c
integraldisplay ∞
∞
dωprime
integraldisplay ∞
∞
dω” δ(ωprime?ω”)vectorEωprime·vectorE?ω” = piepsilon10c
integraldisplay ∞
∞
dωprime vectorEωprime·vectorE?ωprime
a49a114 = a252a160a158a109a252a160a31a161a254a54a76a27a49a85a254 = n0
integraldisplay ∞
∞
dt S≡
integraldisplay ∞
∞
dωprime Iωprime
n0,a252a160a158a109a117a49a27a6a102a234
Iωprime = pin0cepsilon10vectorEωprime·vectorE?ωprime = n0cepsilon10
vectorE(vectorr,0)·vectorE(vectorr,0)?
4pi(ωprime?ω)(ωprime?ω?) =
n0cepsilon10vectorE(vectorr,0)·vectorE(vectorr,0)?
4pi[(ωprime?ω0)2+Γ2]
λ =||?(2picωprime )||= 2picω2
0
ωprime = 2picω2
0
Γ = μ0e
2
3me ～10
4?A
371/384
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a62a94a197a27a209a19a134a225a194,a48a159a27a218a218a209a209,a103a100a62a102a233a233a62a62a94a197a27a209a19
a152a189a170a199a27a9a62a94a197a221a19a20a62a102a254a166a62a102a177a131a211a170a199a137a114a189a8a196a191a149a9a203a19a209a62
a94a197a23a62a94a197a27a209a19.
a63a216a62a102vlessmuchca36a196a27a156a47,a62a102a36a196a27a130a221～a8a204～vTlessmuchcT = λ,Ta143a92a19a197a177a207,
λa143a92a19a197a27a197a127.a183a130a140a67a113a242a9a62a124a119a164a134vectorra195a39,a62a102a201a27a94a229～evB～evEca131
a233a233a62a62a229a143a140a3a209.
m¨vectorr0(t) = μ0e
2
6pic
···
vectorr0 (t) + evectorE0e?iωt
a193a38a41,vectorr0(t) =vectorr0(0)e?iωt→?mω2vectorr0(0) = μ0eiω
3
6pic vectorr0(0) + e
vectorE0
vectorr0(0) =?e
vectorE0
mω2?iμ0e2ω36pic =
evectorE0
mω2(1?2iree2ω3pic )
re<vTlessmuchλ→ωrec lessmuch2pi
→?e
vectorE0
mω2
vectora(t) =vectora(0)e?iωt vectora(0) =?ω2vectorr0(0) = e
vectorE0
m
vectorE(vectorr,t) = vectorE(vectorr,0)e?iωt vectorE(vectorr,0) = eμ0ω2
4pir e
iωcr[vectorr0(0)?vectorr0(0)·vectornvectorn]
372/384
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a62a94a197a27a209a19a134a225a194,a48a159a27a218a218a209a209,a103a100a62a102a233a233a62a62a94a197a27a209a19
vectorE(vectorr,t) = vectorE(vectorr,0)e?iωt vectorE(vectorr,0) = eμ0ω2
4pir e
iωcr[vectorr0(0)?vectorr0(0)·vectornvectorn]
vectorS(vectorr,t) = epsilon10c
2
vectorE(vectorr,t)·vectorE?(vectorr,t)vectorn = μ0e2ω4
32pi2r2c[r
2
0(0)?(vectorn·vectorr0(0))
2]
= μ0e
4E2
0
32pi2m2er2c(1?cos
2α) = I0r
2
e
r2 sin
2α
vectorn·vectorE0 = E0 cosα I0 = E
2
0
2μ0c re =
μ0e2
4pime
dI
d? = I0r
2
e sin
2α
I =
contintegraldisplay
dvectorσ·vectorS =
integraldisplay
d? r2vectorS·vectorn = 2piI0r2e
integraldisplay pi
0
dαsin2α = 83pir2eI0 = σI0
σ = 83pir2ea180Thomsona209a19a31a161.
373/384
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a62a94a197a27a209a19a134a225a194,a48a159a27a218a218a209a209,a103a100a62a102a233a233a62a62a94a197a27a209a19
vectorn·vectorE0 = E0 cosα I0 = E
2
0
2μ0c re =
μ0e2
4pime
dI
d? = I0r
2
e sin
2α I = σI0 σ = 8
3pir
2
e
a45
a54
a8a8
a8a25
a8a8
a8a8a8a42
a72a72a72a89
y
z
x
vectorE0 vectorrφ
θ
a101a101a92a92a19a197a8a204a145a197a169a217,a75a140a177a233a217a18a178a254,a23a92a19a144a149
a143za144a149,vectorra160a117 xza178a161vectorE0a134xa182a89a14φ,vectorra134za144a149a27a89a14θ,
nEx = cosφ nEz = 0 nrx = sinθ nry = 0
sin2α = 1?cos2α = 1?(vectornE·vectornr)2 = 1?(nExnrx)2 = 1?cos2φsin2θ
sin2α = 12pi
integraldisplay 2pi
0
dφsin2α= 12pi
integraldisplay 2pi
0
dφ[1?12 sin2θ(1+cos2φ)]=12(1+cos2θ)
dI
d? =
1
2I0r
2
e(1 + cos
2θ) dσ
d? =
d(I/I0)
d? =
1
2r
2
e(1 + cos
2θ)
Thomsona135a169a31a161.
374/384
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a62a94a197a27a209a19a134a225a194,a48a159a27a218a218a209a209,a229a80a62a102a233a233a62a62a94a197a27a209a19
m¨vectorr0(t) =?mγ˙vectorr0(t)?ω20mvectorr0(t) + evectorE0e?iωt?mγ˙vectorr0(t),a202a162a229a145
a193a38a41,vectorr0(t)=vectorr0(0)e?iωt→?mω2vectorr0(0)=iωmγvectorr0(0)?ω20mvectorr0(0)+evectorE0
vectorr0(0) = e
vectorE0
(ω20?ω2)m?iωmγ =
evectorE0
mradicalbig(ω20?ω2)2 +ω2γ2e
iδ =vectorr0eiδ
vectora(t) =vectora(0)e?iωt vectora(0) =?ω2vectorr0(0)
vectorr0 = e
vectorE0
mradicalbig(ω20?ω2)2 +ω2γ2 tanδ =
ωγ
ω20?ω2
vectorE(vectorr,t) = vectorE(vectorr,0)e?iωt vectorE(vectorr,0) = eμ0ω2
4pir e
iωcr[vectorr0(0)?vectorr0(0)·vectornvectorn]
vectorS(vectorr,t) = epsilon10c
2
vectorE(vectorr,t)·vectorE?(vectorr,t)vectorn = μ0e2ω4
32pi2r2c[r
2
0?(vectorn·vectorr0)
2]
= μ0e
2ω4
32pi2r2c(
e
m)
2 E
2
0 sin
2α
(ω20?ω2)2 +ω2γ2 =
μ0e4E20
32pi2m2r2c
ω4 sin2α
(ω20?ω2)2 +ω2γ2
= I0r
2
e
r2 sin
2α ω
4
(ω20?ω2)2 +ω2γ2
375/384
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a62a94a197a27a209a19a134a225a194,a48a159a27a218a218a209a209,a229a80a62a102a233a233a62a62a94a197a27a209a19
vectorS(vectorr,t) = I0r2e
r2 sin
2α ω
4
(ω20?ω2)2 +ω2γ2 I0 =
E20
2μ0c re =
μ0e2
4pim
dσ
d? =
1
2r
2
e(1 + cos
2θ) ω
4
(ω20?ω2)2 +ω2γ2 I = σI0
σ = 83pir2e ω
4
(ω20?ω2)2 +ω2γ2
ωlessmuchω0→ dσd? = 12r2e(1 + cos2θ)(ωω
0
)4 σ = 83pir2e(ωω
0
)4
Rayleigha209a19,a51a140a132a49a204a165,a249a249a49a49a209a19a129a8,a98a49a209a19a129a245,a160a108a92a19a19a229a229a144a149a26a194a27
a49,a217a112a170(a55)a220a169a211a27a39a173a140a117a92a19a229a49a204a169a217a165 a27a112a170a164a169,a13a51a223a19a229a49a204
a164a169a165a36a170(a249)a220a169a27a39a173a79a92a10,a211a158a223a19a19a229a229a111a114a221a126a8a10.a85a152a165a55a218.a70a209a189a73
a19a165a249a218.a103a85a20a204a78a180a232a231.a193a85a49a130a216a118a209a180a100a167a218a229a27.
ωgreatermuchω0→ dσd? = 12r2e(1+cos2θ) σ = 83pir2e
a103a100a62a102a209a19.
376/384
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a62a94a197a27a209a19a134a225a194,a48a159a27a218a218a209a209,a229a80a62a102a233a233a62a62a94a197a27a209a19
dσ
d? =
1
2r
2
e(1+cos
2θ) ω
4
(ω20?ω2)2+ω2γ2 σ =
8
3pir
2
e
ω4
(ω20?ω2)2+ω2γ2
ω = ω0→ dσd? = 12r2e(1 + cos2θ)(ω0γ )2 σ = 83pir2e(ω0γ )2
a1a8,a92a19a197a85a254a26a114a17a27a225a194a50a186a152,a23a92a19a197a252a160a170a199a109a133a92a19a20a252a160a161a200a27
a85a254a143 I0(ω),a75a252a160a158a109a229a80a62a102a203a19a27a111a85a254
W = 83pir2e
integraldisplay ∞
∞
dω ω
4
(ω20?ω2)2 +ω2γ2I0(ω)
≈ 83pir2eI0(ω0)
integraldisplay ∞
∞
dω ω
4
0
4ω20(ω0?ω)2 +ω20γ2
= 23pir2eω20I0(ω0)
integraldisplay ∞
∞
dω ω
4
0
(ω0?ω)2 + (γ2)2
= 4pi3γr2eω20I0(ω0)
integraldisplay ∞
∞
du 1u2 + 1 = 4pi
2
3γ r
2
eω
2
0I0(ω0)
a100a100a117a117a85a254a197a240,a167a143a180a62a102a252a160a158a109a108a92a19a197a225a194a27a111a85a254.
377/384
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a62a94a197a27a209a19a134a225a194,a48a159a27a218a218a209a209,a48a159a27a218a218a209a209
a23a48a159a159a165a165a252a160a78a200a62a102a234a143N.a122a135a62a102a177a27a107a170a199ω0a8a196,a48a159a159a165a165a27a27a62a62a124a143
vectorE = vectorE0e?iωt vectorP = epsilon10χevectorE = Nevectorr0(t) = Ne evectorE0e?iωt
(ω20?ω2)m?iωm?γ
χe = Ne
2
mepsilon10[ω20?ω2?iω?γ] →epsilon1 = epsilon10(1 +χe) = epsilon10 +
Ne2
m[ω20?ω2?iω?γ]
epsilon1r = epsilon1primer + iepsilon1”r epsilon1primer = 1 + Ne
2
epsilon10m
ω20?ω2
(ω20?ω2)2 +ω2?γ2
epsilon1”r = Ne
2
epsilon10m
ω?γ
(ω20?ω2)2 +ω2?γ2?γ,a202a162a88a234 γ,a62a62a19a19a199
k2?k2R?k2I + 2ivectorkR·vectorkI = ω2μepsilon1+ iωμγ a23a181μ,γ,ωa143a162
k2R?k2I =ω2μepsilon10epsilon1primer 2vectorkR·vectorkI=ω2μepsilon10epsilon1”r+ωμγ=ωμ(γ+ωepsilon10epsilon1”r)≡ωμγprime
epsilon1a27a74a220a229a27a138a94a180a242a6a53a27γa67a143γprime,a69a164a62a94a197a27a225a194.
a101γprime = 0,a75epsilon1a162a220a134ωa107a39a191a155a88a242a19a199a134ωa107a39,a61a216a211
a170a199a27a27a62a62a94a197a27a242a19a14a221a216a211,a249a135a53a159a23a23a48a48a159a27a218a218a209a209.
a88a88a92a92a19a49a180a120a49,a218a218a209a209a69a164a209a19a49a143a245a218a49.
378/384
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a62a196a229a198
a51a121a147a212a110a198a165a27a27a47a47a160
379/384
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trianglerightsldtrianglerightsld
triangleleftsld
trianglerightsld
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a111a140a229a198
a169a219a229a198
a218a79a229a198
a254a102a229a198
a62a62a196a196a229a198
380/384
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trianglerightsld
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a229a198a78a88
a178a59a229a198
a178a59a218a79a229a198
a254a102a229a198
a254a102a218a79a229a198
381/384
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trianglerightsldtrianglerightsld
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a131a112a138a94a229
a62a229
a94a229
a62a94a229a127a127a167a167a229α～
1
137
a89a169a1023×10?7m 0.4eV
a149a6a1025×10?11m 0.2keV
a102a229<10?18m G～ 1300GeV2
a62a102< 10?18m 1√G～300GeV
a62a102a218a152
a114a229< 10?15m gs～1?10
a159a102,a165a102,pia48a102～10?15m 100MeV
a140a218a152
······
······
a218a229 a127a127a167a167a229 g～ 15×1019GeV2
10?35 m 1√g～5×1019GeV
...,..,..
TOE(a135a117?)···
TOE,Theory Of Everything !
382/384
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a136a171a229a198
a131a112a138a94 a178a59(a218a79) a254a102(a218a79)
======== ========== ================
a218a229 a50a194a131a233a216 a254a102a218a229(a107a129a167a221)
a62a94
a62
a94
a62a62a196a196a229a198 a62a62a196a196a229a198 (a107a129a167a221)
a102 × a155a196a229a198(a107a129a167a221)
a114 × a218a196a229a198(a107a129a167a221)
383/384
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a19a155a155a173a173a86a212a110a198a252a140a226a187
a212a159:
a226a102 a254a102
a197 a178a59a124
→ a254a102a124 a228a107a197a226a19a148a53
a143a159a111a135a107a124a186
a254a102a229a198a254a102a21
a131a233a216a196a254a21a169a97
a226a102a23a41a7a171a78a121a143a254a102a124→a131a233a216a254a102a124a216
a123a161a254a102a124a216
384/384
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a62a94a138a94
a62a94a138a94a180a134a60a97a70a126a41a185a129a151a131a131a39a27a196a29a131a112a138a94
a62a94a138a94a180a60a97a131a233a110a41a129a208a27a196a29a131a112a138a94
a2a167a51a212a159a173a46a27a196a29a8a164a165a53a189a144a85a229a57a207a138a94a156
a207a143a144a130a62a214a47a216a164a152a135a17a23a33a103a84a27a196a29a173a189a40a8:
–a58a58a62a62a214a209a121a195a161a140
–a107a129a169a217a62a214a195a123a130a62a94a138a94a242a62a214a118a224a51a152a229
a62a102a28a46a111a180a45a20a40a74
a103a44a46a73a135a217a167a27a85a47a164a196a29a173a189a40a8a27a196a29a131a112a138a94!
–a140a186a221,a218a229a131a112a138a94
–a2a186a221,a114a131a112a138a94
–a88a219a110a41a102a131a112a138a94a27a138a94?
