Electromagnetism Zhu Jiongming Department of Physics Shanghai Teachers University

Electromagnetism Chapter 1 Electric Field Chapter 2 Conductors Chapter 3 Dielectrics Chapter 4 Direct-Current Circuits Chapter 5 Magnetic Field Chapter 6 Electromagnetic InductionElectromagnetic Induction Chapter 7 Magnetic Materials Chapter 8 Alternating Current Chapter 9 Electromagnetic Waves

Chapter 6 Electromagnetic Induction §1. Electromagnetic InductionElectromagnetic Induction §2. Lenz’s LawLenz’s Law §3. Motional Electromotive Force (emf)Motional Electromotive Force (emf) §4. Induced emf Induced Electric FieldInduced emf Induced Electric Field §5. Self InductionSelf Induction §6. Mutual InductionMutual Induction §7. Eddy CurrentEddy Current §8. Transient State of RL CircuitTransient State of RL Circuit §9. Transient State of RC CircuitTransient State of RC Circuit §10. Magnetic EnergyMagnetic Energy

§1. Electromagnetic Induction 1. Experiment A current can be induced in a loop as the magnetic flux through the loop changes with timeExperiment 2. Faraday’s Law of Induction A Induced emf = K  time rate of change of flux in SI unit （ V 、 Wb 、 s ） K = 1 （ Direction --- Lenz’s Law ）

§2. Lenz’s Law 1. Two statementsTwo statements ● Induced current always tends to oppose the flux change ● Magnetic force on induced current always tends to oppose the motion of conductor 2. Expression of Faraday’s LawExpression of Faraday’s Law 3. ExamplesExamples

1. Two Statements (1) F Induced current always tends to oppose the flux change —— flux of induced current ● in the opposite direction for  > 0 ● in the same direction for  < 0 N S A N S A

1. Two Statements (2) F Magnetic force on induced current always tends to oppose the motion of conductor ∵ v towards right v R I f安f安 ∴ f Ampere left （ f Ampere =  Idl  B ） ∴ I counterclockwise ( or ：  > 0 ∴ flux of I in opposite direction ) F Work and Energy ： Work by external force  I 2 Rt ( heat energy ) （ constant velocity ， kinetic energy not changed ） If f Ampere towards right ， energy conservation violated

2. Expression of Faraday’s Law Positive directions for  and  ： right hand rule then          

Example （ p.276 / ） A small N turns coil A of radius r is placed coaxially in a long solenoid S of n turns per unit length. Current I in S decreases from 1.5A to - 1.5A at a steady rate in 0.05 s. Find emf in A. Sol. ： B =  0 nI Direction: same with I and not changed

Exercises p.276 / , 3

§3. Motional emf Three ways to change the flux  ： F change the area ( move the loop ) —— motional emfmotional emf F change the field B —— induced emf F change both of them

§3. Motional emf 1. Motional emf and Lorentz ForceMotional emf and Lorentz Force 2. Calculation of Motional emfCalculation of Motional emf 3. ExamplesExamples 4. Alternating-Current GeneratorAlternating-Current Generator

1. Motional emf and Lorentz Force Magnetic force on electrons in conductor ab f = - ev  B （ downward ）  I counterclockwise direction （ caused by emf on ab ） emf = work on unit charge by Lorentz force as moving it from b to a v I f b a c d ∵ vl is the rate at which the area changes ∴ vBl is the rate at which the flux changes ， = d  / dt

Motional emf F suitable also for unclosed conductor ， but no I ， only emf v  o General formula for Motional emf ： F B in the formula not change with time

2. Calculation of Motional emf Two ways to calculate Motional emf ： where v 、 B might not be uniform ， so could not be brought out in front of the integral sign If not closed ， add an imaginary curve to make it closed or get d  / dt by the area swept per unit time v

Example （ p.229/ [Ex.1] ） (1) A straight wire of length L is rotating at angular velocity  in a uniform magnetic field B. Find emf  ab and potential difference U ab. a b dldl v l  a ： low potential ， negative charges b ： high potential ， positive charges Sol. ： (1) take dl at a distance l from a v  B same direction with dl, v =  l

Example （ p.229/ [Ex.1] ） (2) (2) dd a b L LdLd direction ： imaginary loop counterclockwise ， a  b assume rotating d  in d t the area swept is

4. Alternating-Current Generator Loop plane is perpendicular to magnetic field at t = 0 （ S is in the same direction with B ，  = 0 at t = 0 ）  B S N

Exercises p.277 / , 2

§4. Induced emf Induced Field 1. Induced Electric FieldInduced Electric Field 2. Properties of Induced Electric FieldProperties of Induced Electric Field 3. Induced Electric Field Due to the ChangingInduced Electric Field Due to the Changing Magnetic Field in a Solenoid 4. ExamplesExamples

1. Induced Electric Field Loop does not move ， magnetic field is changing B changing  flux  =  B·dS changing   induced current  force F on q  F / q = E F Coulomb's law  Coulomb's electric field E C F changing magnetic field  induced electric field E I Total electric field E = E C + E I or

2. Properties of Induced Electric Field Total electric field E = E C + E I Potential field Source field Sourceless field Eddy field

3. Induced Electric Field in a Solenoid F symmetry （ infinite long ）， E I is in the plane perpendicular to the axis —— no axial component F symmetry （ circle ）， E I —— no radial component S Conclusion ： E I has only tangent component ， E I lines are concentric circles

Example 1 （ p.236/ [Ex.1] ） In a long solenoid of radius R, dB / dt is known, find E I. Sol. ： R r

Example 2 （ p.237/ [Ex.2] ） In a long solenoid of radius R, dB / dt, h, L are known. Find  MN. Sol. ： o h r Another way ：   OMN = BS L M N emfs are zero on OM and ON

Exercises p.279 /

§5. Self Induction 1. Self InductionSelf Induction 2. Self InductanceSelf Inductance 3. ExamplesExamples

1. Self Induction I  B   I B An induced emf appears in any coil in which the current is changing —— Self-induced emf

2. Self Inductance F tightly wound coil ： every turn as a loop ， the same flux  S passes through all the turns F N - turn coil ： flux linkage  S = N  S  S   S  B  I Definition ：  S = L I L ： Self inductance Unit ： 1 Henry

Example （ p.245/ [Ex.] ） A long solenoid of volume  ， n turns per unit length. Find it’s L. Sol. ： B =  0 nI  S = N  S = nl  S =  0 n 2 IS l =  0 n 2  I = LI  L =  0 n 2 

Exercises p.280 / , 2, 3

§6. Mutual Induction 1. Mutual InductionMutual Induction 2. Mutual InductanceMutual Inductance 3. Two Coils in SeriesTwo Coils in Series ● connect in the same direction ● connect in the opposite direction

1. Mutual Induction F I 1   11 ，  12  12 ： flux through coil 2 by the field of I 1 I1I1 B 1 2  11  12 Self induced emf Mutual induced emf F in the same way  21 ： flux through coil 1 by the field of I 2 total flux through coil 1  1 =  11   21 + for I 1, I 2 in same direction - for opposite direction

2. Mutual Inductance F flux linkage ：  11 = N 1  11,  21 = N 1  21,  1 = N 1  1   1 =  11   21 （  1 =  11   21 ）  11 = L 1 I 1 ，  21 = M 21 I 2 M 21 ： flux through coil 1 by the field of I 2  I 2 M 12 ： flux through coil 2 by the field of I 1  I 1 F can be prove ： M 12 = M 21 = M Mutual Inductance M ： interaction of two coils

3. Two Coils in Series F Connect in the same directionConnect in the same direction F Connect in the opposite directionConnect in the opposite direction

Connect in the Same Direction  1 =  11 +  21  2 =  22 +  12 I I

Connect in the Opposite Direction  1 =  11 -  21  2 =  22 -  12 I I

Exercises p.280 / , 2

§7. Eddy Current F Applications F Harmfulness F Electromagnetic Damping F Skin Effect

§8. Transient State of RL Circuit 1. Connect RL Circuit with an emfConnect RL Circuit with an emf 2. Remove the emf from RL CircuitRemove the emf from RL Circuit F Steady state ： I = 0 （ K open ） and I =  / R （ K close ） F Transient State ： i ： 0  I capitals ： I 、 U steady quantities little ： i 、 u varying quantities Ohm’s Law, Kirchhoff’s Rules are workable, as long as the quasi-steady condition satisfied R L  K

1. Connect RL Circuit with an emf （ 1 ） Equation ：（ Kirchhoff’s loop rule ） R L  K SS i —— deferential equation of i

Connect RL Circuit with an emf （ 2 ） 0 t i (t) R L  K 自自 i

Discussion ： (1) if t   ， then i  I ， actually need only little timelittle time (2) Inductive time constant  Connect RL Circuit with an emf （ 3 ） 0t i (t) rate governed by （ time constant ） The less the time constant  is ， the more rapidly the current goes to it’s equilibrium value. 0.63I 11 22

2. Remove the emf from RL Circuit （ 1 ） The switch closes at t = 0 ， consider the uper part of the circuit ( loop ) 自自 i R L  R’

Remove the emf from RL Circuit （ 2 ） Discussion ： (1) if t   ， then i  0 ， actually need only little timelittle time (2) time constant ： 0t i (t) 0.37I 0 11 I0I0 22 The less the time constant  is ， the more rapidly the current goes to it’s equilibrium value 0.

Exercises p.280 / , 5

§9. Transient State of RC Circuit 1. Connect RC Circuit with an emfConnect RC Circuit with an emf 2. Remove the emf from RC CircuitRemove the emf from RC Circuit 3. SummarySummary

1. Connect RC Circuit with an emf （ 1 ） Charging a capacitor ： u c ： 0   Equation （ Kirchhoff’s loop rule ） R  K C uCuC uRuR i

Connect RC Circuit with an emf （ 2 ） Discussion ： (1) if t   ， then u C   ， actually need only little time (2) Capacitive time constant ：  = RC The less the , the more rapidly it goes to equilibrium. F R smaller ， current larger F C smaller ， need less charge 0 t uC (t)uC (t)  Need less time

2. Remove the emf from RC Circuit The switch 2  1 at t = 0 （ discharging a capacitor ） i 1 R  K C 2 0 t uC (t)uC (t) 

3. Summary F Steps to treat transient process ：  differential equation  general solution （ in exponential fashion ）  initial condition  constant F Features ：  C  voltage can not change suddenly  L  current can not change suddenly F energy （ L 、 C can store energy ， need time ）  C ： L ： （ R does not store energy ， I, U change suddenly ）

Exercises p.281 / , 2

§10. Magnetic Energy 1. Magnetic Energy of a Self-InductorMagnetic Energy of a Self-Inductor 2. Magnetic Energy of Mutual-InductorsMagnetic Energy of Mutual-Inductors

1. Magnetic Energy of a Self-Inductor F Steady ： I constant ，  = IR  I = I 2 R R L  K Power of emf Thermal energy F transient ：  idt ：  idt = i 2 R dt + Lidi work of emf = thermal energy+ energy to build up field in L i ： 0  I （ I lager ， B stronger ， W m lager ） （ this is magnetic energy ）

2. Magnetic Energy of Mutual-Inductors work of emfs = thermal energy in Rs+ magnetic energy W m in Ls M L1L1 11 R1R1 K1K1 L2L2 22 R2R2 K2K2 ① × i 1 dt + ② × i 2 dt ： ① ②

Exercises p.282 /