This text advances from the basic laws of electricity and magnetism to classical electromagnetism in a quantum world. Suitable for first-year graduate students in physics who have taken an undergraduate course in electromagnetism, it focuses on core concepts and related aspects of math and physics.
Progressing from the basic laws of electricity and magnetism and their unification by Maxwell and Einstein, the treatment culminates in a survey of the role of classical electromagnetism in a quantum world. Each stage of the theory is carefully developed in a clear and systematic approach that integrates mathematics and physics so that readers are introduced to the theory and learn the mathematical skills incontext of real physics applications. Topics include methods of solution in electrostatics, Green's functions, electrostatics in matter, magnetism and ferromagnetism, electromagnetic waves in matter, special relativity, and the electrodynamics of moving bodies. Newly revised by author Jerrold Franklin, the book includes the new section Answers to Odd-Numbered Problems.
Author(s): Jerrold Franklin
Edition: 2
Publisher: Dover
Year: 2017
Language: English
Pages: 639
City: Mineola, New York
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
