Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
Chapter 22
Conductor,Dielectrics,Electric
Energy Storage
Capacitance
Dielectrics
Energy Stored in an Electric Field
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
New words and expressions
capacitance 电容
capacitor (condenser) 电容器
dielectric 电介质
farad 法拉
series 串联
parallel 并联
bound charges 束缚电荷
polarization 极化
electric displacement 电位移矢量
orientation polarization 位移 极化
displacement polarization 取向极化
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
22-1,Capacitor(condenser) P525-526
a
+Q
b
-Q
Capacitor,two conductors (separated by an
insulator) with equal and opposite charges,
Two conductors called plates（极板）,
1,The Definition of Capacitor
2,Symbol of Capacitors
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
3,The use of capacitors
电容器是一储能元件 (Storehouses for potential energy)
纸质电容器 陶瓷电容器电解电容器 钽电容器可变电容器
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
Capacitors are widely used in electronic circuits:
Store charges,camera flash; computer if power
fails; part of tuner of a radio,sever as memory in
RAM,key on keyboard.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
22-2,Capacitance P526-529
1,Definition of capacitance of isolated conductors
+ ++
+
+ +
+++QV
电荷电势
q
V
3q
3v
2q
2v
4q
4v
nq
nv
….
….
v
q
c o n s tVQ?/
V
QC?
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
The value of capacitance depends only on the
size,shape and relative position of the two
conductors,also on the material that separates
them.
The capacitance is a measure of how much
charge must be put on the plates to produce a
certain V between them,The greater the
capacitance,the more charge is required.
Its value (固有容电本领 ) depends only on the
geometry (几何因素 ) of plates,dielectric property
and not on their Q or V.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2,Unit of capacitance
R
R
Q
Q
V
Q
C 0
0
π4
π4
R
Q
3,The capacitance of an isolated conducting sphere
1微法 （?F） =10-6F
1皮法 （ pF） = 10-6?F = 10-12 F
SI unit,coulomb per volt farad (F) (法拉 )
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
4,The capacitance of capacitors (电容器的电容量）
孤立导体的电容很小，用它作电容器不适合。用两个导体组成的电容器可实现较大的电容。
电容器电容
ABBA U
Q
UU
QC?
-
AVBV
Q- Q?
ABUq?
ABCUq?
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
实际中屏蔽不一定要求很高故在工艺上用两无限大平板代替 AB导体。
绝缘纸金色铂
C
Two plates are rolled
into the form of a
cylinder with paper or
other insulator
separating the plates.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
5.Calculating the Capacitance
Determining capacitance (求电容步骤 )：
1) Assuming a charge q to have been placed on the
plates (让导体带等量异号电荷）；
2) Finding E due to this charge (求两极板间电场 )；
3) Evaluating V (求两极板间电压 )；
4) Calculating C from definition (用定义 C=q/V求 C)。
Assume q E? )(VV
AB? V
qC?
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
1) A Parallel-Plate Capacitor (p527)
Assume? (charge per unit area),S (area) and d,
0?
E
ABAB ldEV
d E d l0 Ed?
0?
d?
ABV
QC? 0/
d
S?
d
S0
C 与 Q 无关。
d
S
+
+
+
+
+
+
Q Q-
-
-
-
-
-
-
The field between
the plates is
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2) A Spherical Capacitor (球形电容器 ),P529
1R
2R
＋ ＋＋
＋
＋ ＋ ＋
＋
-
-
--
--
--
r
r2
0π4
e
r
QE
)( 21 RrR
P
*
A spherical capacitor consist of two thin
concentric spherical conducting shells.
Determine the capacitance of the two shells.
Solution,The inner shell carries a uniformly
distributed charge Q on its surface,and the outer
shell an equal but opposite
charge –Q.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2
1
2
0
d
π4
d
R
Rl r
rQlEV
)11(
π4 210 RR
Q -?
2
04 r
qE
1R
2R
＋ ＋＋
＋
＋ ＋ ＋
＋
-
-
--
--
--
r
P
*
abV
QC?
12
2104
RR
RR
-
可看出 C只与几何尺寸有关，而与 q 无关。
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
3) A Cylindrical Capacitor (圆柱形电容器 ),P528
1
2
00
ln
2
d
2
d
2
1
R
Rr
r
rEV
R
R
B
A
r
E
02
In the region of,
21 RrR
AR
BR
l
BRl
+
+
+
+
-
-
-
-
1
2
0
1
2
0
ln
2
ln
2 R
R
L
R
R
L
V
q
C
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
— depends only on geometrical
factors,in this case L,R1 and R2.
12
0
ln
2
RRL
CC o
Capacitance of unit length
AR
BR
l
BRl
+
+
+
+
-
-
-
-
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
22-3,Capacitor in series and parallel 529-532
1,Capacitors in series (电容器的串联 )
features：
nqqqq21
nUUUU21
1C 2C nC
1U 2U nU
U
1q 2q nq
由 CqU?
n
n
C
q
C
q
C
q
C
q
2
2
1
1有
nCCC
1111
21
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
discussion
1.电容越串容量越小。
The equivalent capacitance C is smaller than the
smallest contributing capacitance.
2.可提高电容耐压程度，外加电压由各电容器分压。
若面积 S相同，相当于将极板间距增大。
d
SC 0
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2,Capacitors in parallel (电容器的并联 ):
features
nUUUU21
nqqqq21
1C
2C
nC
1q
2q
nq
由 CUq?
nn UCUCUCCU2211
nCCCC21
电容越并越大，若极板间距 d相同，电容并联相当增加面积 S 。 The net effect of connecting capacitors in
parallel is thus to increase the capacitance.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
22-5,Dielectric (P533-535)
If you fill the space between the plates of a
capacitor with a dielectric,what happens to the
capacitance?
1,Dielectric--- insulating material（绝缘材料）
Q
Q-
+ + + + + + +
- - - - - - -
0U
0C C
U
r?
Q
Q-
+ + + + + + +
- - - - - - -
实际的电容两板间充满了电介质，电介质对电容器内的电场有什么影响呢？
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
r
0
E
E?
0
r
1
UU
1r
介电常数
The capacitance increased by a numerical factor
k,which he called the dielectric constant of
insulating material (or?r).
0r CC
r0
permittivity of material
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2,Molecular Description of Dielectrics (P536-537):
1) Polar (有极分子 ) Dielectrics,The molecules of
some dielectrics,like H2O,have permanent
electric dipole moments.
介质中的电偶极子排列杂乱,宏观不显极性。 neutralized (电中性 )
有极分子的取向极化 orientation polarization：
电偶极子在外场作用下发生转向,沿着外电场方向取向 。 The
electric dipoles tends to line up
with an external electric field.
0E
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2),Nonpolar (无极分子 ) Dielectrics,The dielectrics
(such as H2,N2) have no permanent electric
dipoles.
无极分子的位移极化,displacement polarization,
正负电荷中心拉开，形成电偶极子。
0E
介质表面出现极化电荷 。 appearing surface
charges on the slab faces.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
Common effects:
Effects of dielectrics polarizing——generating
polarization charges or bound charges (束缚电荷 )
and polarization field ( 产生退极化场 ) within
medium.
Non-polar ~ displacement polarization (位移 极化 )
Polar molecule ~ orientation polarization (取向极化 )
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
+ + + + + +
- - - - - -+ + + + + + + + + + +
- - - - - - - - - - -
d r
r
0
0?
E
EEE D?-?
0
r
r 1 EE
D?
-
0E
'E? E?
外场为极化电荷场
0
'
DE
0
0
0?
E
3.电介质中的电场强度
0
r
r 1'?
-?
0
r
r 1' QQ
-?
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
The net field
inside the dielectric
has the same
direction with but
smaller in
So the effect of
dielectric is to
weaken the electric
field.
0E
DE
magnitude.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
4,Dielectrics and Gauss’ Law:
'0 EEE
'?P?
要回避 q'的影响
0
0 '
- qqSdE
S
0E
E
0?-0? '?- '?
'E
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
电介质中的高斯定理 Gauss’ law with dielectric
0
r
r' 1 QQ
-?
)(1d '0
0
QQSE
S
-
0?
0?-
'?-
'? + + + + + +
- - - - - -+ + + + + + + + + + +
- - - - - - - - - - -
r?
S
r0
r0
0d
QSE
S
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
EED r0
电位移矢量 （均匀各相同性介质）
有介质 时的 高斯 定理
i
iS QSD 0d
electric displacement
穿过闭合面的电位移通量，等于面内自由电荷的代数和。
The flux of the electric displacement through a
closed surface of any shape equals to the
algebraic sum of free charges enclosed within the
surface.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
D?(1) is only a supplementary physical quantity,
and has no real physical meaning like,If a test
charge is put in electric field,the force acted on
it is determined by but,D?
E?
E?
Descriptions:?
(2) The sum in the right side does not include
bounded (induced) charge,but free charge only !
(taken fully into account by introducing k &,D?
(3) The unit of is C/m2D?
roo
DDE
k?
有介质时先求 UED
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
例 1,将电荷 q 放置于半径为 R
相对电容率为?r 的介质球中心，
求,I 区,II区的 D,E及 V。
r? I II
Rq
A charge q metal is immersed
uniform dielectric material
sphere with radius R.
Calculate the distribution
of,and,PE?
PD
pV
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
解,在介质球内、外各作半径为 r 的高斯球面。
r?
I II
R
q
r
0qSdDS
0c o s qD dSS?
r
球面上各点 D大小相等，
2
0
4 r
qD
I区,21 4 r
qD
II区,22 4 r
qD
024 qrD?,// SdD 1c o s
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
由 ED r 0?
I区：
r
DE
0
1
1?
II区：
2
04 r
q
r
r
DE
0
2
2? 2
04 r
q
r
E
0?
0E?
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
aa ldEV
由a E d r
RRr drEdrEV 211 R
r?
q
I II
r dr
r
qdr
r
q
R
R
r
r
2
0
2
0 44
I区：
R
q
Rr
q
r 00 4
11
4
-?
R
r?
q
I II
r
r drEV 22
dr
r
q
r
2
04 r
q
04
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
例 2：平行板电容器极板间距为 d,极板面积为 S，面电荷密度为?0,其间插有厚度为 d’,电容率为?r 的电介质，求,P1,P2点的场强 E。
0?-0?
d
'd
r?
1P 2P
①,解,过 P1 点作高斯柱面,左右底面分别经过导体和 P1 点。
D?
侧右底左底 DDDD
高斯面
D
0qSdDS
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
例 2：平行板电容器极板间距为 d,极板面积为 S，面电荷密度为?0,其间插有厚度为 d’,电容率为?r 的电介质，求,P1,P2点的场强 E。
0?-0?
d
'd
r?
1P 2P
①,解,过 P1 点作高斯柱面,左右底面分别经过导体和 P1 点。
D?
侧右底左底 DDDD
高斯面
D
0qSdDS
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
0?左底D?
0?-0?
d
'd
r?
1P 2P高斯面
0?侧D?
导体内 D=0
SdD
右底DD
右底?c o s1 dSD
SD 1? Sq 00
01D
r
DE
0
1
1?
0
0
真空中 1?r?
D
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
0?-0?
d
'd
r?
1P 2P高斯面过 P2点作高斯柱面,左右底面分别经过导体和 P2点。
侧右底左底 DDDD
00 右底DD
同理
0q
SD2 S0
02D
21 DD 0?
D
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
0?-0?
d
'd
r?
1P 2P高斯面
r
DE
0
2
2?
r
0
0?
I区：,01D
II区：,02D
0
0
1?
E
r
E
0
0
2?
D
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
r
例 3 常用的圆柱形电容器，是由半径为 的长直圆柱导体和同轴的半径为 的薄导体圆筒组成，
并在直导体与导体圆筒之间充以相对电容率为 的电介质,设直导体和圆筒单位长度上的电荷分别为和,求（ 1） 电介质中的电场强度、电位移矢量
（２） 电介质内、外表面的电势差。
1R
2R
r?
-
1R
2R
-
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
lSDS d
解（ 1）
lrlDπ2
r
D
π2
r
DE
r0r0 π2
)( 21 RrR
1R
2R
r
-
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
（２） 由上题可知
1r0
1 π2 RE
)( 1Rr?
2r0
2 π2 RE
)( 2Rr?
r
DE
r0r0 π2
1R
2R
r
-
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
（３） 由（１）可知
r
E
r0π2
)( 21 RrR
2
1 r0π2
dd R
R r
rrEU
1
2
0
lnπ2 RR
r
1R
2R
r
-
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
Example,A parallel-plate capacitor of plate area A
and plate separation d,A potential difference V0 is
applied between the plates,The battery is then
disconnected,and a dielectric slab of thickness b
and dielectric constant k is placed between the
plates.
(a) What is the capacitance C0 before the
dielectric slab is inserted?
(b) What free charge appears on the plates?
k
db
A
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
Solution:
d
AC o
o
(a)
ooVCq?(b)
Note that the free charge remains unchanged as
the slab is put into place since charging battery
was disconnected before the slab was introduced.
k
db
A
(c) What is the electric field E0 in the gaps
between the plates and the dielectric slab?
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
(c) What is the electric field E0 in the gaps
between the plates and the dielectric slab?
E0 does not change as the slab is introduced
since q is constant.
qAEAdE oo
I oo
A
qE
o
o b d
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
(d) What is the electric field E1 in the
dielectric slab?
AdEAdD o 1II k?
kk?
o
o
E
A
qE
1
介质中,先求 D，再求 E！
qAEo -?-? 1k?
(e) What is the potential difference V between
the plates after the slab has been introduced?
bEbdEV o 1)(?-?
It is less than the original potential difference.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
(f) What is the capacitance with the slab in place?
The Key Idea now is that the capacitance C is just
related to the free charge q and the potential
difference V.
This is greater than the
original capacitance.
bEbdE
q
V
qC
o 1)(?-
b d
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
22-4,Electric Energy Storage (P532-533)
电容器带电可看成从一个极板移动电荷到另一个极板，外力作功使电容器带电。 q-q?
dq
E?
移动 dq 作的元功 u d qdW?
V
极板带电量从 0到 Q作功,
1 Electric Energy
dWW Q 0
V d qQ 0
The work required to bring
the total capacitor charge up
to a final value Q is:
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
V d qW Q
0
dqCqQ 0 CQ
2
2
1?C
qV?
This work is stored as potential energy U in the
capacitor,
QVCV
C
QU
2
1
2
1
2
2
2
电容器 (Q)储能
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2 Electric Energy Density
电容器充电后具有能量,有电荷就伴生电场,电荷与电场是不可分的,电容器的能量可以说是电场的能量 。
2
2
1 CVU?
As an example let us calculate the energy
stored in a parallel-plate capacitor,以充满介质的平行板电容器为例
,0 d SC r EdV?
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2
2
1 CVU?
20 )(
2
1 Ed
d
Sr SdE
r
2
02
1
VVE
2
2
1
由
ED
VE DVU 2
1?有体VEU
2
2
1 体
ED V21? 体VD
2
2
1?
VV
D
2
2
1?
(where Sd is,the volume of
capacitor)
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
The energy for unit-volume:
energy density (电场能量密度 )
u is proportion to the square of E,So the
more the electric field E,the greater is the
energy of electric field.
体V
Uu? 2
2
1 Eu
ED21?
2
2
1 D?
Read example 22-7 on page 533
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
Homework:p541,10,12,19;
p543,46,48,50; p543,56,57,58,
59; 65
Chapter 22
Conductor,Dielectrics,Electric
Energy Storage
Capacitance
Dielectrics
Energy Stored in an Electric Field
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
New words and expressions
capacitance 电容
capacitor (condenser) 电容器
dielectric 电介质
farad 法拉
series 串联
parallel 并联
bound charges 束缚电荷
polarization 极化
electric displacement 电位移矢量
orientation polarization 位移 极化
displacement polarization 取向极化
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
22-1,Capacitor(condenser) P525-526
a
+Q
b
-Q
Capacitor,two conductors (separated by an
insulator) with equal and opposite charges,
Two conductors called plates（极板）,
1,The Definition of Capacitor
2,Symbol of Capacitors
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
3,The use of capacitors
电容器是一储能元件 (Storehouses for potential energy)
纸质电容器 陶瓷电容器电解电容器 钽电容器可变电容器
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
Capacitors are widely used in electronic circuits:
Store charges,camera flash; computer if power
fails; part of tuner of a radio,sever as memory in
RAM,key on keyboard.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
22-2,Capacitance P526-529
1,Definition of capacitance of isolated conductors
+ ++
+
+ +
+++QV
电荷电势
q
V
3q
3v
2q
2v
4q
4v
nq
nv
….
….
v
q
c o n s tVQ?/
V
QC?
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
The value of capacitance depends only on the
size,shape and relative position of the two
conductors,also on the material that separates
them.
The capacitance is a measure of how much
charge must be put on the plates to produce a
certain V between them,The greater the
capacitance,the more charge is required.
Its value (固有容电本领 ) depends only on the
geometry (几何因素 ) of plates,dielectric property
and not on their Q or V.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2,Unit of capacitance
R
R
Q
Q
V
Q
C 0
0
π4
π4
R
Q
3,The capacitance of an isolated conducting sphere
1微法 （?F） =10-6F
1皮法 （ pF） = 10-6?F = 10-12 F
SI unit,coulomb per volt farad (F) (法拉 )
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
4,The capacitance of capacitors (电容器的电容量）
孤立导体的电容很小，用它作电容器不适合。用两个导体组成的电容器可实现较大的电容。
电容器电容
ABBA U
Q
UU
QC?
-
AVBV
Q- Q?
ABUq?
ABCUq?
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
实际中屏蔽不一定要求很高故在工艺上用两无限大平板代替 AB导体。
绝缘纸金色铂
C
Two plates are rolled
into the form of a
cylinder with paper or
other insulator
separating the plates.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
5.Calculating the Capacitance
Determining capacitance (求电容步骤 )：
1) Assuming a charge q to have been placed on the
plates (让导体带等量异号电荷）；
2) Finding E due to this charge (求两极板间电场 )；
3) Evaluating V (求两极板间电压 )；
4) Calculating C from definition (用定义 C=q/V求 C)。
Assume q E? )(VV
AB? V
qC?
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
1) A Parallel-Plate Capacitor (p527)
Assume? (charge per unit area),S (area) and d,
0?
E
ABAB ldEV
d E d l0 Ed?
0?
d?
ABV
QC? 0/
d
S?
d
S0
C 与 Q 无关。
d
S
+
+
+
+
+
+
Q Q-
-
-
-
-
-
-
The field between
the plates is
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2) A Spherical Capacitor (球形电容器 ),P529
1R
2R
＋ ＋＋
＋
＋ ＋ ＋
＋
-
-
--
--
--
r
r2
0π4
e
r
QE
)( 21 RrR
P
*
A spherical capacitor consist of two thin
concentric spherical conducting shells.
Determine the capacitance of the two shells.
Solution,The inner shell carries a uniformly
distributed charge Q on its surface,and the outer
shell an equal but opposite
charge –Q.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2
1
2
0
d
π4
d
R
Rl r
rQlEV
)11(
π4 210 RR
Q -?
2
04 r
qE
1R
2R
＋ ＋＋
＋
＋ ＋ ＋
＋
-
-
--
--
--
r
P
*
abV
QC?
12
2104
RR
RR
-
可看出 C只与几何尺寸有关，而与 q 无关。
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
3) A Cylindrical Capacitor (圆柱形电容器 ),P528
1
2
00
ln
2
d
2
d
2
1
R
Rr
r
rEV
R
R
B
A
r
E
02
In the region of,
21 RrR
AR
BR
l
BRl
+
+
+
+
-
-
-
-
1
2
0
1
2
0
ln
2
ln
2 R
R
L
R
R
L
V
q
C
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
— depends only on geometrical
factors,in this case L,R1 and R2.
12
0
ln
2
RRL
CC o
Capacitance of unit length
AR
BR
l
BRl
+
+
+
+
-
-
-
-
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
22-3,Capacitor in series and parallel 529-532
1,Capacitors in series (电容器的串联 )
features：
nqqqq21
nUUUU21
1C 2C nC
1U 2U nU
U
1q 2q nq
由 CqU?
n
n
C
q
C
q
C
q
C
q
2
2
1
1有
nCCC
1111
21
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
discussion
1.电容越串容量越小。
The equivalent capacitance C is smaller than the
smallest contributing capacitance.
2.可提高电容耐压程度，外加电压由各电容器分压。
若面积 S相同，相当于将极板间距增大。
d
SC 0
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2,Capacitors in parallel (电容器的并联 ):
features
nUUUU21
nqqqq21
1C
2C
nC
1q
2q
nq
由 CUq?
nn UCUCUCCU2211
nCCCC21
电容越并越大，若极板间距 d相同，电容并联相当增加面积 S 。 The net effect of connecting capacitors in
parallel is thus to increase the capacitance.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
22-5,Dielectric (P533-535)
If you fill the space between the plates of a
capacitor with a dielectric,what happens to the
capacitance?
1,Dielectric--- insulating material（绝缘材料）
Q
Q-
+ + + + + + +
- - - - - - -
0U
0C C
U
r?
Q
Q-
+ + + + + + +
- - - - - - -
实际的电容两板间充满了电介质，电介质对电容器内的电场有什么影响呢？
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
r
0
E
E?
0
r
1
UU
1r
介电常数
The capacitance increased by a numerical factor
k,which he called the dielectric constant of
insulating material (or?r).
0r CC
r0
permittivity of material
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2,Molecular Description of Dielectrics (P536-537):
1) Polar (有极分子 ) Dielectrics,The molecules of
some dielectrics,like H2O,have permanent
electric dipole moments.
介质中的电偶极子排列杂乱,宏观不显极性。 neutralized (电中性 )
有极分子的取向极化 orientation polarization：
电偶极子在外场作用下发生转向,沿着外电场方向取向 。 The
electric dipoles tends to line up
with an external electric field.
0E
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2),Nonpolar (无极分子 ) Dielectrics,The dielectrics
(such as H2,N2) have no permanent electric
dipoles.
无极分子的位移极化,displacement polarization,
正负电荷中心拉开，形成电偶极子。
0E
介质表面出现极化电荷 。 appearing surface
charges on the slab faces.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
Common effects:
Effects of dielectrics polarizing——generating
polarization charges or bound charges (束缚电荷 )
and polarization field ( 产生退极化场 ) within
medium.
Non-polar ~ displacement polarization (位移 极化 )
Polar molecule ~ orientation polarization (取向极化 )
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
+ + + + + +
- - - - - -+ + + + + + + + + + +
- - - - - - - - - - -
d r
r
0
0?
E
EEE D?-?
0
r
r 1 EE
D?
-
0E
'E? E?
外场为极化电荷场
0
'
DE
0
0
0?
E
3.电介质中的电场强度
0
r
r 1'?
-?
0
r
r 1' QQ
-?
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
The net field
inside the dielectric
has the same
direction with but
smaller in
So the effect of
dielectric is to
weaken the electric
field.
0E
DE
magnitude.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
4,Dielectrics and Gauss’ Law:
'0 EEE
'?P?
要回避 q'的影响
0
0 '
- qqSdE
S
0E
E
0?-0? '?- '?
'E
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
电介质中的高斯定理 Gauss’ law with dielectric
0
r
r' 1 QQ
-?
)(1d '0
0
QQSE
S
-
0?
0?-
'?-
'? + + + + + +
- - - - - -+ + + + + + + + + + +
- - - - - - - - - - -
r?
S
r0
r0
0d
QSE
S
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
EED r0
电位移矢量 （均匀各相同性介质）
有介质 时的 高斯 定理
i
iS QSD 0d
electric displacement
穿过闭合面的电位移通量，等于面内自由电荷的代数和。
The flux of the electric displacement through a
closed surface of any shape equals to the
algebraic sum of free charges enclosed within the
surface.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
D?(1) is only a supplementary physical quantity,
and has no real physical meaning like,If a test
charge is put in electric field,the force acted on
it is determined by but,D?
E?
E?
Descriptions:?
(2) The sum in the right side does not include
bounded (induced) charge,but free charge only !
(taken fully into account by introducing k &,D?
(3) The unit of is C/m2D?
roo
DDE
k?
有介质时先求 UED
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
例 1,将电荷 q 放置于半径为 R
相对电容率为?r 的介质球中心，
求,I 区,II区的 D,E及 V。
r? I II
Rq
A charge q metal is immersed
uniform dielectric material
sphere with radius R.
Calculate the distribution
of,and,PE?
PD
pV
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
解,在介质球内、外各作半径为 r 的高斯球面。
r?
I II
R
q
r
0qSdDS
0c o s qD dSS?
r
球面上各点 D大小相等，
2
0
4 r
qD
I区,21 4 r
qD
II区,22 4 r
qD
024 qrD?,// SdD 1c o s
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
由 ED r 0?
I区：
r
DE
0
1
1?
II区：
2
04 r
q
r
r
DE
0
2
2? 2
04 r
q
r
E
0?
0E?
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
aa ldEV
由a E d r
RRr drEdrEV 211 R
r?
q
I II
r dr
r
qdr
r
q
R
R
r
r
2
0
2
0 44
I区：
R
q
Rr
q
r 00 4
11
4
-?
R
r?
q
I II
r
r drEV 22
dr
r
q
r
2
04 r
q
04
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
例 2：平行板电容器极板间距为 d,极板面积为 S，面电荷密度为?0,其间插有厚度为 d’,电容率为?r 的电介质，求,P1,P2点的场强 E。
0?-0?
d
'd
r?
1P 2P
①,解,过 P1 点作高斯柱面,左右底面分别经过导体和 P1 点。
D?
侧右底左底 DDDD
高斯面
D
0qSdDS
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
例 2：平行板电容器极板间距为 d,极板面积为 S，面电荷密度为?0,其间插有厚度为 d’,电容率为?r 的电介质，求,P1,P2点的场强 E。
0?-0?
d
'd
r?
1P 2P
①,解,过 P1 点作高斯柱面,左右底面分别经过导体和 P1 点。
D?
侧右底左底 DDDD
高斯面
D
0qSdDS
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
0?左底D?
0?-0?
d
'd
r?
1P 2P高斯面
0?侧D?
导体内 D=0
SdD
右底DD
右底?c o s1 dSD
SD 1? Sq 00
01D
r
DE
0
1
1?
0
0
真空中 1?r?
D
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
0?-0?
d
'd
r?
1P 2P高斯面过 P2点作高斯柱面,左右底面分别经过导体和 P2点。
侧右底左底 DDDD
00 右底DD
同理
0q
SD2 S0
02D
21 DD 0?
D
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
0?-0?
d
'd
r?
1P 2P高斯面
r
DE
0
2
2?
r
0
0?
I区：,01D
II区：,02D
0
0
1?
E
r
E
0
0
2?
D
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
r
例 3 常用的圆柱形电容器，是由半径为 的长直圆柱导体和同轴的半径为 的薄导体圆筒组成，
并在直导体与导体圆筒之间充以相对电容率为 的电介质,设直导体和圆筒单位长度上的电荷分别为和,求（ 1） 电介质中的电场强度、电位移矢量
（２） 电介质内、外表面的电势差。
1R
2R
r?
-
1R
2R
-
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
lSDS d
解（ 1）
lrlDπ2
r
D
π2
r
DE
r0r0 π2
)( 21 RrR
1R
2R
r
-
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
（２） 由上题可知
1r0
1 π2 RE
)( 1Rr?
2r0
2 π2 RE
)( 2Rr?
r
DE
r0r0 π2
1R
2R
r
-
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
（３） 由（１）可知
r
E
r0π2
)( 21 RrR
2
1 r0π2
dd R
R r
rrEU
1
2
0
lnπ2 RR
r
1R
2R
r
-
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
Example,A parallel-plate capacitor of plate area A
and plate separation d,A potential difference V0 is
applied between the plates,The battery is then
disconnected,and a dielectric slab of thickness b
and dielectric constant k is placed between the
plates.
(a) What is the capacitance C0 before the
dielectric slab is inserted?
(b) What free charge appears on the plates?
k
db
A
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
Solution:
d
AC o
o
(a)
ooVCq?(b)
Note that the free charge remains unchanged as
the slab is put into place since charging battery
was disconnected before the slab was introduced.
k
db
A
(c) What is the electric field E0 in the gaps
between the plates and the dielectric slab?
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
(c) What is the electric field E0 in the gaps
between the plates and the dielectric slab?
E0 does not change as the slab is introduced
since q is constant.
qAEAdE oo
I oo
A
qE
o
o b d
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
(d) What is the electric field E1 in the
dielectric slab?
AdEAdD o 1II k?
kk?
o
o
E
A
qE
1
介质中,先求 D，再求 E！
qAEo -?-? 1k?
(e) What is the potential difference V between
the plates after the slab has been introduced?
bEbdEV o 1)(?-?
It is less than the original potential difference.
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
(f) What is the capacitance with the slab in place?
The Key Idea now is that the capacitance C is just
related to the free charge q and the potential
difference V.
This is greater than the
original capacitance.
bEbdE
q
V
qC
o 1)(?-
b d
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
22-4,Electric Energy Storage (P532-533)
电容器带电可看成从一个极板移动电荷到另一个极板，外力作功使电容器带电。 q-q?
dq
E?
移动 dq 作的元功 u d qdW?
V
极板带电量从 0到 Q作功,
1 Electric Energy
dWW Q 0
V d qQ 0
The work required to bring
the total capacitor charge up
to a final value Q is:
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
V d qW Q
0
dqCqQ 0 CQ
2
2
1?C
qV?
This work is stored as potential energy U in the
capacitor,
QVCV
C
QU
2
1
2
1
2
2
2
电容器 (Q)储能
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2 Electric Energy Density
电容器充电后具有能量,有电荷就伴生电场,电荷与电场是不可分的,电容器的能量可以说是电场的能量 。
2
2
1 CVU?
As an example let us calculate the energy
stored in a parallel-plate capacitor,以充满介质的平行板电容器为例
,0 d SC r EdV?
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
2
2
1 CVU?
20 )(
2
1 Ed
d
Sr SdE
r
2
02
1
VVE
2
2
1
由
ED
VE DVU 2
1?有体VEU
2
2
1 体
ED V21? 体VD
2
2
1?
VV
D
2
2
1?
(where Sd is,the volume of
capacitor)
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
The energy for unit-volume:
energy density (电场能量密度 )
u is proportion to the square of E,So the
more the electric field E,the greater is the
energy of electric field.
体V
Uu? 2
2
1 Eu
ED21?
2
2
1 D?
Read example 22-7 on page 533
Chapter 22 Capacitance,Dielectrics,Electric Energy Storage
Homework:p541,10,12,19;
p543,46,48,50; p543,56,57,58,
59; 65
