西南交通大学：《大学物理》课程教学资源（讲稿，双语）CHAPTER 21 Faraday's Law and Electromagnetic Induction

UNIVERSITY PHYSICS II CHAPTER 21 Faraday's Law and Electromagnetic Induction 821.1 Faraday's law of electromagnetic induction 1. Experiments ① A magnet and a ② two loops of wire,a loop of wire battery and a switch A current is observed in the loop as long as the magnetic flux through the loop is changing with time

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

821.1 Faraday's law of electromagnetic induction 2. Electromotive force-emr ←⊕ F←田 K 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 821.1 Faraday's law of electromagnetic induction Mechanisn F ×/ak4F田 The work done by Fk opposing Fe converse the energy in other forms into electric K R energy Outside the F move +g from positive plate to negative plate nside the e circul el>e move +q from negative plate to positive plate

2 2. Electromotive force-emf Fe← ⊕ r ← ⊕ Fe r + − K R ←⊕→ Fk r Fe r Fe r ← ⊕ Fe 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. Fe r Fk ← ⊕ r Fe r ← ⊕ Fe r + − K R ←⊕→ Fk r Fe r Outside the circuit： Fe r move +q from positive plate to negative plate Inside the circuit： Fk Fe r r > move +q from negative plate to positive plate §21.1 Faraday’s law of electromagnetic induction

821.1 Faraday's law of electromagnetic induction energy conversion Outside the circuit: ←田 F4F←田 ←田→ F Fd/>0 Inside the circuits F d/o Not a conservative force 821.1 Faraday's law of electromagnetic induction Define the emf emf=F Ed emf is the measure of the capacity of transforming the emf= Edi other form energy to electric energy. (inside) Direction of the emf

3 energy conversion： ⋅d = 0 ∫ F l L e r r Fe← ⊕ r ← ⊕ Fe r + − K R ←⊕→ Fk r Fe r ∫ + − Fe ⋅dl 0 r r Fe r Outside the circuit： Inside the circuit： Fk r How about ? 0 q F E k k r r Define the nonstatic electric field = = ⋅d > 0 ∫ A E l L k r r Not a conservative force §21.1 Faraday’s law of electromagnetic induction E l L k r r emf= ⋅d ∫ Define the emf ∫ + − = ⋅ （inside） emf E dl k r r 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

821.1 Faraday's law of electromagnetic induction 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 d induced emf dt Change by magnetic field 更=B.dA Change by area of loop s 21.1 Faraday's law of electromagnetic induction dgp emf=-N“m= d(nm) dy dt dt emf em Reference direction n>0, >0,0,Q空 0

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 Φm d d induced emf = − §21.1 Faraday’s law of electromagnetic induction ∫ Φm = B⋅ A r r d Change by magnetic field Change by area of loop 0, 0 d d > 0, > ε 0, t Φ Φ m m t Ψ t NΦ t Φ N m m m d d d d( ) d d emf = − = − = − §21.1 Faraday’s law of electromagnetic induction emf emf + + Reference direction

821.1 Faraday's law of electromagnetic induction enz’slaw 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. mf em s 21.1 Faraday's law of electromagnetic induction Emf arising from moving conductor Equilibrium state: B f △V 4 vB=gE=q △V=B Lorentz force is the nonstaticelectric force qν×BE v×B q

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 + + ①Emf arising from moving conductor mf l + − e f d c v r → B r ⊕ × ∆U × × × × × × × × × × × ∆V = Blv Fm = Fe l V qvB qE q ∆ = = Equilibrium state: Lorentz force is the nonstaticelectric force. FK Fm qv B r r r r = = × v B q F E m K r r r r = = × §21.1 Faraday’s law of electromagnetic induction

821.1 Faraday's law of electromagnetic induction According to the definition of the emf emf= inside) or emf= YL (vx B).d/ Does the lorentz force do work? A,>0AF,<0 B d s 21.1 Faraday's law of electromagnetic induction Example 1: A metal rod of length l is rotated at angular speed a about an end in a uniform magnetic field B, 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 8/88 ⑧⑧ ② B 6

6 y x × B r × × × + + − d c ∫ + − = ⋅ = （inside） emf E dl K r r ∫ + − × ⋅ （inside ） (v B) dl r r r v B l L r r r emf = ( × )⋅d ∫ or According to the definition of the emf §21.1 Faraday’s law of electromagnetic induction > 0 Fm A ′ < 0 Fm A = 0 Fm A Does the Lorentz force do work? v r ' v r V Fm r ' Fm Fm 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

821.1 Faraday's law of electromagnetic induction Solution 1 Choose wire segment dI Its speed is U=la ②68 d(em=(×B)d= DBd/988B Blade emf=d(emf= BaldI=BoL s 21.1 Faraday's law of electromagnetic induction Solution 2 PO ② The magnetic flux through e the pie-shaped segment of the circle of angle 6 e2a88 B·dS=Bs=B 2 B The induced emf d o bL d8 oBL induced emf= dt 2

7 Solution 1: ⊗⊗⊗⊗ ⊗⊗⊗⊗ ⊗⊗⊗⊗ ⊗⊗⊗⊗ ⊗⊗⊗⊗ ⊗⊗⊗⊗ L ω B r O P §21.1 Faraday’s law of electromagnetic induction Bl l B l B l d d(emf) ( ) d d ω υ υ = = × ⋅ = r r r 2 2 1 emf d(emf) B ldl B L L o = ∫ ∫ = ω = ω d l Choose wire segment l r d Its speed is υ = lω Solution 2: The magnetic flux through the pie-shaped segment of the circle of angle θ 2 d 2 L m B S BS B θ Φ = ⋅ = = ∫ r r The induced emf d 2 d d 2 d induced emf 2 2 BL t BL t m θ ω = = Φ = §21.1 Faraday’s law of electromagnetic induction ⊗⊗⊗⊗ ⊗⊗⊗⊗ ⊗⊗⊗⊗ ⊗⊗⊗⊗ ⊗⊗⊗⊗ ⊗⊗⊗⊗ L ω B r O P θ

821.1 Faraday's law of electromagnetic induction @2Induced 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 What makes the charge move and causes the current? F=ge 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. s 21.1 Faraday's law of electromagnetic induction Maxwell: new effect d④ induced emf= B·d4 d④ induced emf= ∫ dA dt dt emfE·dcmf=「E4d (inside) 8

8 ②Induced 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? F qE r r = §21.1 Faraday’s law of electromagnetic induction t Φm d d induced emf = − ∫ Φm = B⋅ A r r d A t B t Φm r r d d d d d induced emf = − = − ⋅ ∫ E l L k r r emf= ⋅d ∫ ∫ + − = ⋅ （inside） emf E dl k r r §21.1 Faraday’s law of electromagnetic induction Maxwell: new effect

821.1 Faraday's law of electromagnetic induction Note Induced electric field is different from the static electric field @charges 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? 821.1 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 The results of experiment: B Increas sing fA B decreasing↑A E Reference direction

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 Einduced r B r decreasing A r Einduced r Reference direction The results of experiment: §21.1 Faraday’s law of electromagnetic induction

821.1 Faraday's law of electromagnetic induction (iThe electric field lines representing the induced electric field form closed contours iThe static electric field arises from stationary electric charge, the induced electric field is produced from changing magnetic flux. (iliThe 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 821.1 Faraday's law of electromagnetic induction 4. The characters of the induced electric field @The static electric field lines do not form closed loops, they begi An n positive chars 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. Eds=0 @The 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 Ed≠0

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 ⋅d = 0 ∫ E s r r ⋅d ≠ 0 ∫ E l r r