Chapter 7 Electrodynamics 7.0 Introduction 7.1 Electromotive Force 7.2 Electromagnetic Induction 7.3 Maxwell’s Equations

7.0 Introduction electrostatic static magnetostatic = magnetostatic = conservation of charge

7.0 (2) Maxwell’s equations:

7.0 (3) = Magnetic flux Induced electric field (force) induce

7.0 (4) e.g. , ~ E,B fields propagate in vacuum wave

7.0 (5) A.C. current can generate electromagnetic wave antenna cyclotron mass free electron laser …..

7.1 Electromotive Force 7.1.1 Ohm’s Law 7.1.2 Electromotive Force 7.1.3 Motional emf

7.1.1 Ohm’s Law Current density conductivity force per unit charge of the medium for perfect conductors resistivity ( a formula based on experience) usually true but not in plasma; especially, hot. Ohm’s Law

7.1.1 (2) Total current flowing from one electrode to the other V=I R Ohm’s Law (based on experience) Potential current resistance [ in ohm (Ω) ] Note : for steady current and uniform conductivity

7.1.1 (3) I=? R=? uniform V Ex. 7.1 sol:

7.1.1 (4) A=const Ex. 7.3 Prove the field is uniform V=V0 V=0 =const

7.1.1 (5) Ex. 7.2 V

7.1.1 (6) The physics of Ohm’s Law and estimation of microscopic s the charge will be accelerated by before a collision time interval of the acceleration is mean free path typical case for very strong field and long mean free path

7.1.1 (7) The net drift velocity caused by the directional acceleration is = mass of the molecule e charge molecule density free electrons per molecule Power is dissipated by collision Joule heating law

7.1.2 Electromotive Force electrostatic force electromotive force The current is the same all the way around the loop. Produced by the charge accumulation due to Iin > Iout electrostatic force electromotive force outside the source

7.1.3 motional emf

7.1.3 (2) = Work is done by the pull force, not .

7.1.3 (3) magnetic flux for the loop flux rule for motional emf

7.1.3 (4) a general proof

7.1.3 (5) Ex. 7.4 =?

7.2 Electromagnetic Induction 7.2.1 Faraday’s Law 7.2.2 The Induced Electric Field 7.2.3 Inductance 7.2.4 Energy in Magnetic Fields

7.2.1 Faraday’s Law M. Faraday’s experiments Induce induce induce loop moves B moves Induce induce induce Faraday’s Law (integral form) Faraday’s Law (differential form)

7.2.1 (2) A changing magnetic field induces an electric field. (b) & (c) induce that causes drive Lenz’s law : Nature abhors a change in flux ( the induced current will flow in such a direction that the flux it produces tends to cancel the change. )

7.2.1 (3) Ex. 7.5 Induced ? sol: at center , spread out near the ends

7.2.1 (4) Ex. 7.6 Plug in, induces Plug in, why ring jump? ring jump.

7.2.2 The Induced Electric Field

7.2.2 (2) Ex. 7.7 induced = ? sol: =

7.2.2 (3) Ex. 7.8. The charge ring is at rest sol: What happens? torque on the angular momentum on the wheel

7.2.2 (4) Induced sol: quasistatic

7.2.2 (5) = Constant K( s , t ) s << c t t = I / (dI/dt)

7.2.3 Inductance mutual inductance

7.2.3 (2) Neumann formula The mutual inductance is a purely geometrical quantity M21 = M12 = M F1 = M12 I2 F1 = F2 if I1 = I2

7.2.3 (3) Ex. 7.10 1 2 sol: B1 is too complicated… 2 = ? n2 turns per unit length Ex. 7.10 1 n1 turns per unit length I given 2 sol: assume I too. B1 is too complicated… 2 = ? Instead, assume I running through solenoid 2

7.2.3 (4) changing current in loop1, induces current in loop2 self inductance self-inductance (or inductance ) [ unit: henries (H) ] back emf

7.2.3 (5) Ex. 7.11 L(self-inductance)=? b a N turns sol:

7.2.3 (6) Ex. 7.12 sol: general solution particular solution

7.2.4 Energy in Magnetic Fields In E.S. test charge From the work done, we find the energy in , But, does no work. WB = ? In back emf

7.2.4 (2) In volume

7.2.4 (3) Ex. 7.13 sol: ＜

7.3 Maxwell’s Equations 7.3.1 Electrodynamics before Maxwell 7.3.2 How to fix Ampere’s Law 7.3.3 Maxwell’s Equations 7.3.4 Magnetic Charge 7.3.5 Maxwell’s Equation in Matter 7.3.6 Boundary Conditions

7.3.1 Electrodynamics before Maxwell (Gauss Law) (no name) (Faraday’s Law) (Ampere’s Law) but =0 Ampere’s Law fails because

7.3.1 an other way to see that Ampere’s Law fails for nonsteady current loop 1 2 For loop 1, Ienc = 0 For loop 2, Ienc = I they are not the same.

7.3.2 How to fix Ampere’s Law continuity equations, charge conservation such that, Ampere’s law shall be changed to Jd displacement current A changing electric field induces a magnetic field.

7.3.2 = loop 1 2 for the problem in 7.3.1 between capacitors

7.3.3 Maxwell’s equations Gauss’s law Faraday’s law Ampere’s law with Maxwell’s correction Force law continuity equation ( the continuity equation can be obtained from Maxwell’s equation )

7.3.3 Since , produce ,

7.3.4 Magnetic Charge Maxwell equations in free space ( i.e., , ) symmetric With and , the symmetry is broken. If there were ,and . symmetric and So far, there is no experimental evidence of magnetic monopole.

7.3.5 Maxwell’s Equation in Matter bound charge bound current Q surface charge

7.3.5 (2) Ampere’s law ( with Maxwell’s term )

7.3.5 (3) In terms of free charges and currents, Maxwell’s equations become displacement current one needs constitutive relations:

7.3.5 (4) for linear dielectric. or

7.3.6 Boundary Condition Maxwell’s equations in integral form Over any closed surface S for any surface bounded by the S closed loop L

7.3.6 = = =