Classical Electromagnetism.Second edition

Preface to the second edition xiii
Preface to the first edition xiii
1 Foundations of Electrostatics 1
1.1 Coulomb’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Electric Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Potential gradient . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Gauss’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.1 Examples of Gauss’s law . . . . . . . . . . . . . . . . . . 15
1.4.2 Spherically symmetric charge (and mass) distributions . 16
1.5 The Variation of E . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5.2 Dirac delta function . . . . . . . . . . . . . . . . . . . . 24
1.5.3 Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.6 Summary of Vector Calculus . . . . . . . . . . . . . . . . . . . . 31
1.6.1 Operation by ∇ . . . . . . . . . . . . . . . . . . . . . . . 31
1.6.2 Integral theorems . . . . . . . . . . . . . . . . . . . . . . 34
1.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2 Further Development of Electrostatics 41
2.1 Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 Electrostatic Energy . . . . . . . . . . . . . . . . . . . . . . . . 45
2.3 Electric Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.1 Fields due to dipoles . . . . . . . . . . . . . . . . . . . . 51
2.3.2 Forces and torques on dipoles . . . . . . . . . . . . . . . 54
2.3.3 Dipole singularity at r=0 . . . . . . . . . . . . . . . . . 57
2.4 Electric QuadrupoleMoment . . . . . . . . . . . . . . . . . . . 58
2.4.1 Dyadics . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.4.2 Quadrupole dyadic . . . . . . . . . . . . . . . . . . . . . 60
2.4.3 Multipole expansion . . . . . . . . . . . . . . . . . . . . 65
2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
iii
iv CONTENTS
3 Methods of Solution in Electrostatics 71
3.1 Differential Formof Electrostatics . . . . . . . . . . . . . . . . . 71
3.1.1 Uniqueness theorem . . . . . . . . . . . . . . . . . . . . 72
3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2.1 Infinite grounded plane . . . . . . . . . . . . . . . . . . . 77
3.2.2 Conducting sphere . . . . . . . . . . . . . . . . . . . . . 79
3.3 Separation of Variables for Cartesian Coordinates . . . . . . . . 82
3.3.1 Hollow conducting box . . . . . . . . . . . . . . . . . . . 82
3.3.2 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3.3 Open conducting channel . . . . . . . . . . . . . . . . . . 88
3.3.4 Fourier sine integral . . . . . . . . . . . . . . . . . . . . 89
3.4 Surface Green’s Function . . . . . . . . . . . . . . . . . . . . . . 92
3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4 Spherical and Cylindrical Coordinates 101
4.1 General Orthogonal Coordinate Systems . . . . . . . . . . . . . 101
4.2 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.1 Separation of variables in spherical coordinates . . . . . 105
4.2.2 Azimuthal symmetry, Legendre polynomials . . . . . . . 106
4.2.3 Boundary value problems with azimuthal symmetry . . . 112
4.2.4 Multipole expansion . . . . . . . . . . . . . . . . . . . . 116
4.2.5 Spherical harmonics . . . . . . . . . . . . . . . . . . . . 122
4.3 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . 130
4.3.1 Separation of variables in cylindrical coordinates . . . . . 132
4.3.2 2-dimensional cases (polar coordinates) . . . . . . . . . . 132
4.3.3 3-dimensional cases, Bessel functions . . . . . . . . . . . 136
4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5 Green’s Functions 149
5.1 Application of Green’s Second Theorem. . . . . . . . . . . . . . 149
5.2 Green’s Function Solution of Poisson’s Equation . . . . . . . . . 150
5.3 Surface Green’s Function . . . . . . . . . . . . . . . . . . . . . . 151
5.4 Symmetry of Green’s Function . . . . . . . . . . . . . . . . . . . 152
5.5 Green’s Reciprocity Theorem . . . . . . . . . . . . . . . . . . . 152
5.6 Green’s Functions for Specific Cases . . . . . . . . . . . . . . . . 154
5.7 Constructing Green’s Functions . . . . . . . . . . . . . . . . . . 155
5.7.1 Construction of Green’s function from eigenfunctions . . 155
5.7.2 Reduction to a one-dimensional Green’s function . . . . 156
5.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
CONTENTS v
6 Electrostatics in Matter 165
6.1 Polarization Density . . . . . . . . . . . . . . . . . . . . . . . . 165
6.2 The Displacement Vector D . . . . . . . . . . . . . . . . . . . . 167
6.3 Uniqueness Theorem with Polarization . . . . . . . . . . . . . . 170
6.4 Boundary Value Problems with Polarization. . . . . . . . . . . . 171
6.4.1 Boundary conditions on D, E, and φ . . . . . . . . . . . 171
6.4.2 Needle or lamina . . . . . . . . . . . . . . . . . . . . . . 173
6.4.3 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.4.4 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.4.5 Dielectric sphere in a uniform electric field . . . . . . . . 177
6.4.6 Dielectric sphere and point charge . . . . . . . . . . . . . 179
6.5 Induced Dipole-Dipole Force, the Van derWaals Force . . . . . 181
6.6 Molecular Polarizability . . . . . . . . . . . . . . . . . . . . . . 182
6.6.1 Microscopic electric field . . . . . . . . . . . . . . . . . . 182
6.6.2 Clausius-Mossotti relation . . . . . . . . . . . . . . . . . 184
6.6.3 Models formolecular polarization . . . . . . . . . . . . . 185
6.7 Electrostatic Energy in Dielectrics . . . . . . . . . . . . . . . . . 187
6.8 Forces on Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . 188
6.9 Steady State Currents . . . . . . . . . . . . . . . . . . . . . . . 192
6.9.1 Current density and continuity equation . . . . . . . . . 192
6.9.2 Ohm’s law . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.9.3 Relaxation constant . . . . . . . . . . . . . . . . . . . . . 194
6.9.4 Effective resistance . . . . . . . . . . . . . . . . . . . . . 195
6.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7 Magnetostatics 201
7.1 Magnetic Forces Between Electric Currents . . . . . . . . . . . . 201
7.2 Units of Electricity andMagnetism . . . . . . . . . . . . . . . . 204
7.3 The Magnetic Field B . . . . . . . . . . . . . . . . . . . . . . . 206
7.4 Applications of the Biot-Savart Law . . . . . . . . . . . . . . . . 207
7.5 Magnetic Effects on Charged Particles . . . . . . . . . . . . . . 212
7.6 Magnetic Effects of Current Densities . . . . . . . . . . . . . . . 215
7.6.1 Volume current density j . . . . . . . . . . . . . . . . . . 215
7.6.2 Surface current density K . . . . . . . . . . . . . . . . . 215
7.6.3 Magnetic effects ofmoving charges . . . . . . . . . . . . 216
7.7 Differential FormofMagnetostatics . . . . . . . . . . . . . . . . 218
7.8 The Vector Potential A . . . . . . . . . . . . . . . . . . . . . . . 219
7.8.1 Gauge transformation . . . . . . . . . . . . . . . . . . . 220
7.8.2 Poisson’s equation for A . . . . . . . . . . . . . . . . . . 221
7.9 Ampere’s Circuital Law . . . . . . . . . . . . . . . . . . . . . . 222
vi CONTENTS
7.10 Magnetic Scalar Potential . . . . . . . . . . . . . . . . . . . . . 226
7.10.1 Magnetic field of a current loop . . . . . . . . . . . . . . 227
7.11 Magnetic DipoleMoment . . . . . . . . . . . . . . . . . . . . . . 230
7.11.1 Magnetic multipole expansion . . . . . . . . . . . . . . . 230
7.11.2 Magnetic dipole scalar potential of a current loop . . . . 231
7.11.3 Magnetic dipole vector potential of a current loop . . . . 232
7.11.4 Magnetic dipolemoment of a current density . . . . . . . 234
7.11.5 Intrinsic magneticmoments . . . . . . . . . . . . . . . . 235
7.11.6 Magnetic dipole force, torque and energy . . . . . . . . . 236
7.11.7 Gyromagnetic ratio . . . . . . . . . . . . . . . . . . . . . 239
7.11.8 The Zeeman effect . . . . . . . . . . . . . . . . . . . . . 240
7.11.9 Fermi-Breit interaction between magnetic dipoles . . . . 241
7.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
8 Magnetization and Ferromagnetism 249
8.1 Magnetic Field IncludingMagnetization . . . . . . . . . . . . . 249
8.2 The H Field, Susceptibility and Permeability . . . . . . . . . . . 251
8.3 Comparison ofMagnetostatics and Electrostatics . . . . . . . . 254
8.4 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
8.5 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
8.6 Permanent Magnetism . . . . . . . . . . . . . . . . . . . . . . . 258
8.7 The Use of the H Field for a PermanentMagnet . . . . . . . . . 260
8.8 BarMagnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
8.9 Magnetic Images . . . . . . . . . . . . . . . . . . . . . . . . . . 265
8.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
9 Time Varying Fields, Maxwell’s Equations 269
9.1 Faraday’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
9.2 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
9.3 Displacement Current, Maxwell’s Equations . . . . . . . . . . . 275
9.4 Electromagnetic Energy . . . . . . . . . . . . . . . . . . . . . . 276
9.4.1 Potential energy inmatter . . . . . . . . . . . . . . . . . 278
9.5 Magnetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.6 ElectromagneticMomentum, Maxwell Stress Tensor . . . . . . . 280
9.6.1 Momentum in the polarization and magnetization fields . 283
9.7 Application of the Stress Tensor . . . . . . . . . . . . . . . . . . 285
9.8 MagneticMonopoles . . . . . . . . . . . . . . . . . . . . . . . . 287
9.8.1 Dirac charge quantization . . . . . . . . . . . . . . . . . 288
9.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
CONTENTS vii
10 Electromagnetic Plane Waves 295
10.1 Electromagnetic Waves from Maxwell’s Equations . . . . . . . . 295
10.2 Energy andMomentum in an ElectromagneticWave . . . . . . 298
10.2.1 Radiation pressure . . . . . . . . . . . . . . . . . . . . . 300
10.3 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
10.3.1 Polarized light . . . . . . . . . . . . . . . . . . . . . . . . 300
10.3.2 Circular basis for polarization . . . . . . . . . . . . . . . 302
10.3.3 Birefringence . . . . . . . . . . . . . . . . . . . . . . . . 303
10.3.4 Partially polarized light . . . . . . . . . . . . . . . . . . 305
10.4 Reflection and Refraction at a Planar Interface . . . . . . . . . . 307
10.4.1 Snell’s law . . . . . . . . . . . . . . . . . . . . . . . . . . 308
10.4.2 Perpendicular polarization . . . . . . . . . . . . . . . . . 309
10.4.3 Parallel polarization . . . . . . . . . . . . . . . . . . . . 310
10.4.4 Normal incidence . . . . . . . . . . . . . . . . . . . . . . 312
10.4.5 Polarization by reflection . . . . . . . . . . . . . . . . . . 312
10.4.6 Total internal reflection . . . . . . . . . . . . . . . . . . . 314
10.4.7 Non-reflective coating . . . . . . . . . . . . . . . . . . . . 316
10.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
11 Electromagnetic Waves in Matter 321
11.1 Electromagnetic Waves in a Conducting Medium . . . . . . . . 321
11.1.1 Poor conductor . . . . . . . . . . . . . . . . . . . . . . . 323
11.1.2 Good conductor . . . . . . . . . . . . . . . . . . . . . . . 324
11.2 Electromagnetic Wave at the Interface of a Conductor . . . . . . 324
11.2.1 Perfect conductor . . . . . . . . . . . . . . . . . . . . . . 324
11.2.2 Radiation pressure . . . . . . . . . . . . . . . . . . . . . 325
11.2.3 Interface with a good conductor . . . . . . . . . . . . . . 327
11.3 Frequency Dependence of Permittivity . . . . . . . . . . . . . . 330
11.3.1 Molecular model for permittivity . . . . . . . . . . . . . 330
11.3.2 Dispersion and absorption . . . . . . . . . . . . . . . . . 331
11.3.3 Conduction electrons . . . . . . . . . . . . . . . . . . . . 332
11.4 Causal Relation Between D and E . . . . . . . . . . . . . . . . 333
11.5 Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
11.5.1 Natural line width . . . . . . . . . . . . . . . . . . . . . 338
11.6 Wave Propagation in a Dispersive Medium . . . . . . . . . . . . 339
11.6.1 Group velocity and phase velocity . . . . . . . . . . . . . 339
11.6.2 Spread of a wave packet . . . . . . . . . . . . . . . . . . 340
11.6.3 No electromagnetic wave travels faster than c . . . . . . 342
11.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
viii CONTENTS
12 Wave Guides and Cavities 349
12.1 CylindricalWave Guides . . . . . . . . . . . . . . . . . . . . . 349
12.1.1 Phase and group velocities in a wave guide . . . . . . 350
12.2 Eigenmodes in aWave Guide . . . . . . . . . . . . . . . . . . 351
12.2.1 TEMwaves . . . . . . . . . . . . . . . . . . . . . . . 352
12.2.2 TMwaves . . . . . . . . . . . . . . . . . . . . . . . . 354
12.2.3 TE waves . . . . . . . . . . . . . . . . . . . . . . . . . 355
12.2.4 Summary of TMand TEmodes . . . . . . . . . . . . 356
12.2.5 Rectangular wave guides . . . . . . . . . . . . . . . . 356
12.2.6 Circular wave guides . . . . . . . . . . . . . . . . . . 358
12.3 Power Transmission and Attenuation inWave Guides . . . . . 359
12.3.1 Power transmitted . . . . . . . . . . . . . . . . . . . . 359
12.3.2 Losses and attenuation . . . . . . . . . . . . . . . . . 361
12.4 Cylindrical Cavities . . . . . . . . . . . . . . . . . . . . . . . . 362
12.4.1 Resonant modes of a cavity . . . . . . . . . . . . . . . 362
12.4.2 Rectangular cavity . . . . . . . . . . . . . . . . . . . . 364
12.4.3 Circular cylindrical cavity . . . . . . . . . . . . . . . . 364
12.4.4 Electromagnetic energy in a cavity . . . . . . . . . . . 365
12.4.5 Power loss, quality factor . . . . . . . . . . . . . . . . 367
12.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
13 Electromagnetic Radiation and Scattering 373
13.1 Wave Equation with Sources . . . . . . . . . . . . . . . . . . . 373
13.2 The Lorenz Gauge . . . . . . . . . . . . . . . . . . . . . . . . 374
13.3 Retarded Solution of theWave Equation . . . . . . . . . . . . 375
13.4 Radiation Solution of theWave Equation . . . . . . . . . . . . 379
13.5 Center-Fed Linear Antenna . . . . . . . . . . . . . . . . . . . . 382
13.6 Electric Dipole Radiation . . . . . . . . . . . . . . . . . . . . . 385
13.7 Radiation by Atoms . . . . . . . . . . . . . . . . . . . . . . . . 387
13.8 Larmor Formula for Radiation by an Accelerating Charge . . . 389
13.9 Magnetic Dipole Radiation . . . . . . . . . . . . . . . . . . . . 391
13.10 Electric Quadrupole Radiation . . . . . . . . . . . . . . . . . . 393
13.11 Scattering of Electromagnetic Radiation . . . . . . . . . . . . 397
13.11.1 Electric dipole scattering . . . . . . . . . . . . . . . . 397
13.11.2 Scattering by a conducting sphere, magnetic
dipole scattering . . . . . . . . . . . . . . . . . . . . . 400
13.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
14 Special Relativity 407
14.1 The Need for Relativity . . . . . . . . . . . . . . . . . . . . . . 407
14.2 The Lorentz Transformation . . . . . . . . . . . . . . . . . . . 411
CONTENTS ix
14.3 Consequences of the Lorentz Transformation . . . . . . . . . . 414
14.3.1 Relativistic addition of velocities . . . . . . . . . . . . 415
14.3.2 Lorentz contraction . . . . . . . . . . . . . . . . . . . 416
14.3.3 Time dilation . . . . . . . . . . . . . . . . . . . . . . 417
14.4 Mathematics of the Lorentz Transformation . . . . . . . . . . 420
14.4.1 Three-dimensional rotations . . . . . . . . . . . . . . 421
14.4.2 Four-dimensional rotations in space-time . . . . . . . 424
14.5 Relativistic Space-Time . . . . . . . . . . . . . . . . . . . . . . 429
14.5.1 The light cone . . . . . . . . . . . . . . . . . . . . . . 430
14.5.2 Proper time . . . . . . . . . . . . . . . . . . . . . . . 431
14.6 Relativistic Kinematics . . . . . . . . . . . . . . . . . . . . . . 432
14.6.1 Four-velocity . . . . . . . . . . . . . . . . . . . . . . . 433
14.6.2 Energy-momentum four-vector . . . . . . . . . . . . . 434
14.6.3 E=mc2 . . . . . . . . . . . . . . . . . . . . . . . . . . 435
14.7 Doppler Shift and Stellar Aberration . . . . . . . . . . . . . . 437
14.8 Natural Relativistic Units, No More c . . . . . . . . . . . . . . 439
14.9 Relativistic “Center ofMass” . . . . . . . . . . . . . . . . . . 440
14.10 Covariant Electromagnetism . . . . . . . . . . . . . . . . . . . 442
14.10.1 Charge-current four-vector jμ . . . . . . . . . . . . . . 442
14.10.2 Lorentz invariance of charge . . . . . . . . . . . . . . 443
14.10.3 The four-potential Aμ . . . . . . . . . . . . . . . . . . 444
14.10.4 The electromagnetic field tensor Fμν . . . . . . . . . . 445
14.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
15 The Electrodynamics of Moving Bodies 453
15.1 Relativistic Electrodynamics . . . . . . . . . . . . . . . . . . . 453
15.1.1 Covariant extension of F = ma . . . . . . . . . . . . . 453
15.1.2 Motion in amagnetic field . . . . . . . . . . . . . . . 455
15.1.3 Linear accelerator . . . . . . . . . . . . . . . . . . . . 456
15.2 Lagrange’s and Hamilton’s Equations for Electrodynamics . . 457
15.2.1 Non-relativistic Lagrangian . . . . . . . . . . . . . . . 457
15.2.2 Relativistic Lagrangian . . . . . . . . . . . . . . . . . 458
15.2.3 Hamiltonian for electrodynamics . . . . . . . . . . . . 460
15.3 Fields of a ChargeMoving with Constant Velocity . . . . . . . 462
15.3.1 Energy loss of a moving charge . . . . . . . . . . . . . 463
15.3.2 Interaction between moving charges . . . . . . . . . . 465
15.4 Electromagnetic Fields of aMoving Charge . . . . . . . . . . . 468
15.4.1 Covariant solution of the wave equation . . . . . . . . 468
15.4.2 Lienard-Wiechert potentials and fields of
amoving charge . . . . . . . . . . . . . . . . . . . . . 471
15.4.3 Constant velocity fields . . . . . . . . . . . . . . . . . 474
x CONTENTS
15.5 Electromagnetic Radiation by aMoving Charge . . . . . . . . 475
15.5.1 Radiation with acceleration parallel to velocity . . . . 476
15.5.2 Radiation with acceleration perpendicular to velocity 479
15.5.3 Radiation from a circular orbit . . . . . . . . . . . . . 480
15.5.4 Relativistic Larmor formula . . . . . . . . . . . . . . . 483
15.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
16 Classical EM in a Quantum World 487
16.1 Looking Back . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
16.2 Electromagnetism as a Gauge Theory . . . . . . . . . . . . . . 489
16.3 Local Gauge Invariance as the Grand Unifier of Interactions . 493
16.4 Classical Electromagnetism and Quantum Electrodynamics . . 495
16.5 Natural Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
16.6 α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
APPENDIX A 503
APPENDIX B 505
BIBLIOGRAPHY 507