1 1. Experiments 1A magnet and a loop of wire 2two loops of wire, a battery and a switch A current is observed in the loop as long as the magnetic flux through the loop is changing with time. § 21.1 Faraday’s law of electromagnetic induction 2 2. Electromotive force-emf ⊕← e F r ⊕← e F r ? + RK →⊕← k F r e F r e F r ⊕← e F r ⊕← R purpose of emf: supply a nonstatic electric force to move the charge, keep the potential difference of the two plates and the current in the circuit. § 21.1 Faraday’s law of electromagnetic induction Mechanism: The work done by opposing converse the energy in other forms into electric energy. e F r k F r ⊕← e F r ⊕← e F r ? + RK →⊕← k F r e F r Outside the circuit： e F r move +q from positive plate to negative plate Inside the circuit： ek FF rr > move +q from negative plate to positive plate § 21.1 Faraday’s law of electromagnetic induction 3 energy conversion： 0d =? ∫ lF L e rr ⊕← e F r ⊕← e F r ? + RK →⊕← k F r e F r ∫ + ? <? 0dlF e rr ∫ ? + >? 0dlF e rr e F r Outside the circuit： Inside the circuit： k F r How about ? 0 q F E k k r r =Define the nonstatic electric field 0d >?= ∫ lEA L k rr Not a conservative force § 21.1 Faraday’s law of electromagnetic induction lE L k rr demf ?= ∫ Define the emf ∫ + ? ?= ）（inside demf lE k rr emf is the measure of the capacity of transforming the other form energy to electric energy. + ? Direction of the emf § 21.1 Faraday’s law of electromagnetic induction 4 3. Faraday’s law of electromagnetic induction Faraday discovered that the induced emf around a closed path is equal to the negative of the time rate of change of the magnetic flux through the same path t Φ m d d emfinduced ?= § 21.1 Faraday’s law of electromagnetic induction ∫ ?= ABΦ m rr d Change by magnetic field Change by area of loop 0,0 d d ,0 <>> ε t Φ Φ m m 0,0 d d ,0 ><> ε t Φ Φ m m t Ψ t NΦ t Φ N mmm d d d )d( d d emf ?=?=?= § 21.1 Faraday’s law of electromagnetic induction emf emf + + Reference direction 5 Lenz’s law Is there a easy way to determine the direction of the induced current or emf? The induced current will always be directed so as to oppose the change in the magnetic flux that is taking place. § 21.1 Faraday’s law of electromagnetic induction emf emf + + ① Emf arising from moving conductor m f l + ? e f d c v r → B r ⊕ × U? × × × × × × × × × × × BlvV =? em FF = l V qqEqvB ? == Equilibrium state: Lorentz force is the nonstaticelectric force. BvqFF mK r r rr ×== Bv q F E m K r r r r ×== § 21.1 Faraday’s law of electromagnetic induction 6 y x × B r × × × + + ? d c ∫ + ? =?= ）（inside demf lE K rr ∫ + ? ?× ）（inside d)( lBv rr r lBv L rr r d)(emf ?×= ∫ or According to the definition of the emf § 21.1 Faraday’s law of electromagnetic induction 0> m F A 0< ′ m F A 0= m F A Does the Lorentz force do work? v r ' v r V r m F ' m F m F r Example 1: A metal rod of length L is rotated at angular speed about an end in a uniform magnetic field , as shown in Figure. Find the absolute value of the induced emf and indicated which end of the rod is at the higher electric potential. ω B r ???? ???? ???? ???? ???? ???? L ω B r § 21.1 Faraday’s law of electromagnetic induction 7 Solution 1: ???? ???? ???? ???? ???? ???? L ω B r O P § 21.1 Faraday’s law of electromagnetic induction lBl lBlB d dd)(d(emf) ω υυ = =?×= rr r 2 2 1 demf)(demf LBllB L o ωω ∫∫ === ld Choose wire segment l r d ωυ l=Its speed is Solution 2: The magnetic flux through the pie-shaped segment of the circle of angle θ 2 d 2 L BBSSB m θ ==?=Φ ∫ rr The induced emf 2d d 2d d emfinduced 22 BL t BL t m ωθ == Φ = § 21.1 Faraday’s law of electromagnetic induction ???? ???? ???? ???? ???? ???? L ω B r O P θ 8 ② Induced current and induced electric field A current exists in the wire loop with the galvanometer as long as the magnetic flux through the loop is changing with time. Such current is called induced current. Maxwell: induced electric field—changing magnetic field flux through a loop induces an electric field, that causes the charge to move and produce the electric current. What makes the charge move and causes the current? EqF rr = § 21.1 Faraday’s law of electromagnetic induction t Φ m d d emfinduced ?= ∫ ?= ABΦ m rr d A t B t Φ m r r d d d d d emfinduced ??=?= ∫ lE L k rr demf ?= ∫ ∫ + ? ?= ）（inside demf lE k rr § 21.1 Faraday’s law of electromagnetic induction Maxwell: new effect 9 Note: 1induced electric field is different from the static electric field. 2charges present in the conducting wire loop detect the presence of the induced electric field. if the conductor is absent, the induced electric field (caused by the changing magnetic flux)still is present in space. What is the direction of the induced electric field? § 21.1 Faraday’s law of electromagnetic induction The direction of the electric field induced by the changing magnetic flux through the loop is around the circumference of the loop in the direction of the current induced. B r increasing A r induced E r B r decreasing A r induced E r Reference direction The results of experiment: § 21.1 Faraday’s law of electromagnetic induction 10 (iii)The work done by the static electrical force around a closed path is zero, but the work done by the electrical force due to induced electric field around a closed path is not zero. (i)The electric field lines representing the induced electric field form closed contours. § 21.1 Faraday’s law of electromagnetic induction (ii)The static electric field arises from stationary electric charge, the induced electric field is produced from changing magnetic flux. 4. The characters of the induced electric field 1The static electric field lines do not form closed loops, they begin on positive charge and end on negative charge. But the induced electric field lines form closed loops, it means that the flux of the induced electric field through any closed surface is zero. 2The electric force produced by the induced electric field is not a conservative force! The work done by the electrical force due to the induced electric field around a closed path is not zero. § 21.1 Faraday’s law of electromagnetic induction 0d =? ∫ sE r r 0d ≠? ∫ lE rr 11 ∫∫ ?=?= pathclsdpathclsd elec dd lEqlFW kk rrrr )2(dd pathclsdpathclsd elec rElElE q W kkk π==?= ∫∫ rr Define the induced electromotive force(emf) q W lE k elec d = ?= ∫ rr Induced emf § 21.1 Faraday’s law of electromagnetic induction These electric and magnetic fields are mutually perpendicular to each other. § 21.1 Faraday’s law of electromagnetic induction t Br E t B rrElE t B r t lE rBBAAB m m d d 2 d d )2(d d d d d d d induced 2 inducedinduced 2 induced 2 ?= ?==? ?=?=? ==?= ∫ ∫ ∫ ππ π Φ πΦ rr rr rr 12 If the radius of the solenoid is a, the time rate of the change of magnetic field is , the sides of the trapezoid is a, a, a, and 2a respectively, find the emf for each sides of the trapezoid and the total emf. 0 d d > t B Solution： ODOA , connect 0==== CDABODOA εεεε For loop OAD 4 3 2 BaSBΦ OADm =?= ? B r × o D CB A a a a2 § 21.1 Faraday’s law of electromagnetic induction Exercise 1: Choose clockwise as + direction For loop OBC, the flux B a SBΦ OADm 6 2 π =?= CB t Ba t Φ m OBCBC → ?=?== d d 6d d 2 π εε ? For loop ABCD, the emf DACDBCAB εεεεε +++= t B a t B a t B a d d ) 4 3 6 ( d d 4 3 d d 6 222 ??=+?= ππ + § 21.1 Faraday’s law of electromagnetic induction B r × o D CB A a a a2 Choose clockwise as + direction DA t B a t Φ m OADAD →?=?== d d 4 3 d d 2 ? εε 13 If vba tII r , , cos 0 ω= Find ?=ε Exercise 2: Solution： r rIa SBΦ m d 2 dd 0 π μ =?= rr r I B π μ 2 0 = raS dd = , Choose an area element v r → b a o I x Choose clockwise as + direction r dr § 21.1 Faraday’s law of electromagnetic induction Solution by applying Faraday’s law directly then ∫∫ + == bx x mm r rIa ΦΦ d 2 d 0 π μ x bxIa + = ln 2 0 π μ x bx tI a + ?= lncos 2 0 0 ω π μ t Φ m d d ?=ε ] d d )( coslnsin[ 2 00 t x xbx b t x bx t aI + ?+ + ?= ωωω π μ ]cos )( lnsin[ 2 00 t xbx bv x bx t aI ωωω π μ + + + ?= m ε In ε v r → b a o I x Choose clockwise as + direction r dr § 21.1 Faraday’s law of electromagnetic induction 14 § 21.1 Faraday’s law of electromagnetic induction Professor Donald Kerst built the world's first magnetic induction accelerator at the University of Illinois in 1940. ① Betatron 5. Applications of the induced electric field § 21.1 Faraday’s law of electromagnetic induction The cross section of a betatron 15 ② An AC generator tBA t t B t Φ tBA BAABΦ m ωω ω ω θ sin d dcos A d d -emfinduced cos cos m = ?= = = =?= rr tNBA t Ψ m ωωsin d d emfinduced =?= § 21.1 Faraday’s law of electromagnetic induction θ A r B r ω r The basic mechanism of an alternating- current generator § 21.1 Faraday’s law of electromagnetic induction 16 ③ Application of the induced current --the electric Guitar § 21.1 Faraday’s law of electromagnetic induction K R ε ~ × × × × × × × × × × § 21.1 Faraday’s law of electromagnetic induction ④ Induction furnace Induction heating ⑤ Magnetic damping 17 § 21.2 The Maxwell equations of electromagnetism and the electromagnetic wave 1. The Maxwell equations 1Gauss’s law for the electric field 0 byenclosednet surfaceclsd d ε S S Q SE =? ∫ rr 2Gauss’s law for the magnetic field 0d surfaceclsd =? ∫ S SB rr 3The Ampere-Maxwell law t IIIlB D L d d )(d elec 0000 pathclsd Φ εμμμ +=+=? ∫ rr 4Faraday’s law of electromagnetic induction t lE L d d d m pathclsd Φ ?=? ∫ rr 2. The remarkable aspects of Maxwell equations 1Maxwell was able to show that electric and magnetic fields can propagate themselves through space according to the classical wave equation. The speed of the electromagnetic wave is the speed of the light. m/s1000.3)(1 8 21 00 ×== μεc 2The equations are relativistically correct. The relativity is no effect on the equations. It means that light is electromagnetic wave. § 21.2 The Maxwell equations of electromagnetism and the electromagnetic wave 18 3. The electromagnetic wave electromagnetic spectrum § 21.2 The Maxwell equations of electromagnetism and the electromagnetic wave x x y y z z + ++ + + ++ + ? ? ? ? ? ? ? ? D I j ? ),( txE y j ? ),( txE y k ? ),( txB z O O The magnetic field and the electric field are perpendicular each other. § 21.2 The Maxwell equations of electromagnetism and the electromagnetic wave 19 1How do we know that various kinds of light really are electromagnetic waves? 2 How are such waves produced? 3 Are the waves longitudinal or transverse? 4 What is oscillating in an electromagnetic wave? Some significant questions: Electromagnetic waves: Radio waves, microwaves, visible light, X-rays, ··· § 21.2 The Maxwell equations of electromagnetism and the electromagnetic wave In free space(vacuum): 0d surfaceclsd =? ∫ S SE rr Gauss’s law for the electric field 0d surfaceclsd =? ∫ S SB rr Gauss’s law for the magnetic field t lB L d d d elec 00 pathclsd Φ εμ=? ∫ rr Ampere-Maxwell law for the magnetic field t lE L d d d m pathclsd Φ ?=? ∫ rr Faraday’s law for the induced electric field A time-varying electric flux gives rise to a magnetic field and a time-varying magnetic flux gives rise to an electric field. § 21.2 The Maxwell equations of electromagnetism and the electromagnetic wave 20 t lE L d d d m pathclsd Φ ?=? ∫ rr Faraday’s law for the induced electric field 00)d(d ++++?=? ∫ lEElElE yyy rr )d( d d d d xlB tt z m ?=? Φ )1( d d d d y y t B x E t B x E z z ? ? ?= ? ? ?= § 21.2 The Maxwell equations of electromagnetism and the electromagnetic wave z y y y l t lB L d d d elec 00 pathclsd Φ εμ=? ∫ rr Ampere-Maxwell law for the magnetic field 00)d(d +++?=? ∫ lBBlBlB zzz rr )d( d d d d elec xlE tt y = Φ )2( d d d d 00 00 t E x B t E x B y z y z ? ? = ? ? ? =? με με l z zz y § 21.2 The Maxwell equations of electromagnetism and the electromagnetic wave 21 )2( 00 t E x B y z ? ? = ? ? ? με)1( y t B x E z ? ? ?= ? ? )1( 2 2 2 xt B x E z y ?? ? ?= ? ? )2( 2 2 00 2 t E tx B y z ? ? = ?? ? ? με 2 2 00 2 2 t E x E yy ? ? = ? ? με 0 2 2 00 2 2 = ? ? ? ? ? t E x E yy με In like manner: § 21.2 The Maxwell equations of electromagnetism and the electromagnetic wave )2( 00 t E x B y z ? ? = ? ? ? με)1( y t B x E z ? ? ?= ? ? )1( 2 2 2 t B tx E z y ? ? ?= ?? ? )2( 2 00 2 2 xt E x B y z ?? ? = ? ? ? με 2 2 00 2 2 t B x B zz ? ? = ? ? με 0 2 2 00 2 2 = ? ? ? ? ? t B x B zz με Compare with the classical wave equation 0 1 2 2 22 2 = ? ? ? ? ? tvx ΨΨ 00 1 με =v § 21.2 The Maxwell equations of electromagnetism and the electromagnetic wave 22 )cos(),( )cos(),( 0 0 tkxBtxB tkxEtxE z y ω ω ?= ?= If the charges move sinusoidally with time Conclusions: 1Light is electromagnetic wave; 2electromagnetic waves are transverse waves; 3acceleration of charges produce electromagnetic waves that travel at speed of light. § 21.2 The Maxwell equations of electromagnetism and the electromagnetic wave § 21.3 Self-inductance, mutual inductance and energy of the magnetic field 1. Self-inductance t I L t m d d d d emfinduced ?=?= Φ LI m =Φ IB ∝ Self- inductance I L m Φ = define I I I I 23 0 μ n I l nIB 0 μ= IlAnnlBANBANΦΨ mm 2 0 μ==== lAnlAn I Ψ L m 2 0 2 0 μμ === Exercise1: Find the self-inductance of a long solenoid of length l, with n coils per meter of its length, each with cross-sectional area A. Solution: According to Ampere’s law ∑ ∫ =? i i IlB 0 L d μ rr § 21.3 Self-inductance, mutual inductance and energy of the magnetic field 2. Mutual inductance t I M t I M IM d d d d emfinduced 1 21 21 1 21 21 12121 ?=?= = = Φ Φ Φ The magnetic field of current I 1 in the first coil produces magnetic flux Φ 21 through the second coil : Mutual inductance § 21.3 Self-inductance, mutual inductance and energy of the magnetic field 24 t I M t I M IM d d d d emfinduced 2 12 12 2 12 12 21212 ?= Φ ?= Φ = =Φ The magnetic field of current I 2 in the second coil produces magnetic flux Φ 12 through the first coil: Mutual inductance Since the geometry and the medium between the coils are same MMM == 2112 § 21.3 Self-inductance, mutual inductance and energy of the magnetic field 1 N 2 N l 2 2R 1 2R 1 L 2 L Exercise 2: Find the mutual inductance two long solenoids. 222 111 ? ? ? ? ? ? ? ? lLNR lLNR 、、、 、、、 Assume: Solution: ? ? ? ? ? > <= = )(0 )( 2 22 2 0220 2 Rr RrI l N In B μμ The magnetic field in the coil 2 § 21.3 Self-inductance, mutual inductance and energy of the magnetic field 25 2 22 21 0 2in212112 d 1 RI l NN ABNSBN A πμ Φ = ?=?= ∫ rr 2 2 21 0 2 12 R l NN I M πμ Φ == The mutual inductance The magnetic flux through the coil 1 § 21.3 Self-inductance, mutual inductance and energy of the magnetic field General case: 21 LLKM = 10 ≤≤ K 2 1 2 12 1 21 1 2 11 )( R l N Rl l N VnL πμπμμ =?== 2 2 2 2 2 R l N L πμ= 21 1 2 LL R R M =∴ 2 2 21 0 2 12 R l NN I M πμ Φ ==Q § 21.3 Self-inductance, mutual inductance and energy of the magnetic field 26 Example: A toroidal with rectangular cross section and N turns carries a time-varying current .On the axis of the toroidal there is another infinite straight conductor line. Find the mutual inductance and the induced emf in the straight current line. tII ωcos 0 = h 2 R 1 R h Solution I： ∫ ∫ = ?=Φ= 2 1 d 2 d 10 12121 R R S r rhIN SBNN π μ ψ rr r I B π μ 2 10 1 = § 21.3 Self-inductance, mutual inductance and energy of the magnetic field rhS dd = Assume a current in the straight line 1 I 1 210 21 ln 2 R RhIN π μ Ψ = 1 20 1 21 ln 2 R RNh I M π μΨ == t R RhNI tI tR RNh t I M ω π ωμ ω π μ sinln 2 )cos( d d ln 2 d d emf induced 1 200 0 1 20 2 = ?= ?= h 2 R 1 R h § 21.3 Self-inductance, mutual inductance and energy of the magnetic field induced emf in the straight current line 27 Solution II： according to the Faraday’s law 0, 2 ext. 20 toroidalin == B r NI B π μ h 2 R 1 R h The straight line closed at infinite, the flux through this loop is tI R RNh r drNhI SBSBSBSB R R R R R R m ω π μ π μ Φ cosln 22 dddd 0 1 2020 ext.in 0 ext. 2 1 2 2 1 1 ?== ?+?+?=?= ∫ ∫∫∫∫ ∞ rrrrrrrr rhS dd = t R RNhI t m ω π ωμΦ sinln 2d d emfinduced 1 200 ?=?= § 21.3 Self-inductance, mutual inductance and energy of the magnetic field 3. The application of mutual inductance Transformers: ΦNΦΨ ΦNΦΨ m m 2total22 1total11 == == 2222 1111 d d )( d d emf d d )( d d emf V t NN t V t NN t ?=?=?= ?=?=?= Φ Φ Φ Φ 2 1 2 1 V V N N = § 21.3 Self-inductance, mutual inductance and energy of the magnetic field 28 4. Energy stored in a magnetic field UIR t I L ==?= d d emfinduced t L R I I tI I d d 0 0 ∫∫ ?= t L R I I ?= 0 ln LRtLRt e R eII ?? == ε 0 Integral the both sides of the equation § 21.3 Self-inductance, mutual inductance and energy of the magnetic field ? R L 12 ε K ILItI t I LtUIqUW dd d d ddd ?=??=== 2 0 2 1 d d LIILIWW I =?== ∫∫ 2 2 1 LIWE m == lA B n B lAnE m 0 2 2 0 2 0 2 ))(( 2 1 μμ μ == n B IlAnL 0 2 0 μ μ ==For a long solenoid § 21.3 Self-inductance, mutual inductance and energy of the magnetic field 29 Magnetic energy density 0 2 2μ B Al E w m m == V B VwE VV mm d 2 d 0 2 ∫∫ == μ The magnetic energy It is valid for any magnetic field. § 21.3 Self-inductance, mutual inductance and energy of the magnetic field Capacitor C Q QVCV 22 1 2 1 2 2 == Self-inductance coil 2 2 1 LI 2 0 2 1 Ew e ε= 0 2 2μ B w m = Electric energy Magnetic energy Electric energy density Magnetic energy density Compare(contrast) of the electric and magnetic field energy § 21.3 Self-inductance, mutual inductance and energy of the magnetic field 30 § 21.4 the oscillation of LC circuit and the production of electromagnetic waves ? R L 12 ε K C L 12 ε K 0 d d =+ C Q t I L 0 d d d d 2 2 =+ = LC Q t Q t Q I LC tQtI tQtQ 1 )sin()( )cos()( 0 0 = +?= += ω φωω φω § 21.4 the oscillation of LC circuit and the production of electromagnetic waves The energy in the circuit oscillates between the capacitor and the inductor. C Q tLQt C Q tLQLI t C Q C Q 2 0 222 0 2 2 0 222 0 2 2 2 0 2 2 1 )(sin 2 1 )(cos 2 1 )(sin 2 1 2 1 )(cos 2 1 2 1 =+++ += += φωωφω φωω φω 31 The oscillation of LC circuit: § 21.4 the oscillation of LC circuit and the production of electromagnetic waves § 21.4 the oscillation of LC circuit and the production of electromagnetic waves 32 § 21.4 the oscillation of LC circuit and the production of electromagnetic waves § 21.4 the oscillation of LC circuit and the production of electromagnetic waves