Fitzpatrick R. Maxwell's Equations and the Principles of Electromagnetism

450 Pages·2007·2.47 MB·English

MAXWELL’S EQUATIONS AND
THE PRINCIPLES OF
ELECTROMAGNETISM
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MAXWELL’S EQUATIONS AND
THE PRINCIPLES OF
ELECTROMAGNETISM
RICHARDFITZPATRICK,PH.D.
UniversityofTexasatAustin
InfinitySciencePressLLC
Hingham,Massachusetts
NewDelhi
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ForFaith
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C
ONTENTS
Chapter 1. Introduction 1
Chapter 2. Vectors and Vector Fields 5
2.1 Introduction 5
2.2 Vector Algebra 5
2.3 Vector Areas 8
2.4 The Scalar Product 10
2.5 The Vector Product 12
2.6 Rotation 15
2.7 The Scalar Triple Product 17
2.8 The Vector Triple Product 18
2.9 Vector Calculus 19
2.10 Line Integrals 20
2.11 Vector Line Integrals 23
2.12 Surface Integrals 24
2.13 Vector Surface Integrals 26
2.14 Volume Integrals 27
2.15 Gradient 28
2.16 Divergence 32
2.17 The Laplacian 36
2.18 Curl 38
2.19 Polar Coordinates 43
2.20 Exercises 45
Chapter 3. Time-Independent Maxwell Equations 49
3.1 Introduction 49
3.2 Coulomb’s Law 49
3.3 The Electric Scalar Potential 54
3.4 Gauss’ Law 57
3.5 Poisson’s Equation 66
3.6 Ampère’s Experiments 68
vii
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viii CONTENTS
3.7 The Lorentz Force 71
3.8 Ampère’s Law 75
3.9 Magnetic Monopoles? 76
3.10 Ampère’s Circuital Law 79
3.11 Helmholtz’s Theorem 86
3.12 The Magnetic Vector Potential 91
3.13 The Biot-Savart Law 95
3.14 Electrostatics aND Magnetostatics 97
3.15 Exercises 101
Chapter 4. Time-Dependent Maxwell Equations 107
4.1 Introduction 107
4.2 Faraday’s Law 107
4.3 Electric Scalar Potential? 112
4.4 Gauge Transformations 113
4.5 The Displacement Current 116
4.6 Potential Formulation 123
4.7 Electromagnetic Waves 124
4.8 Green’s Functions 132
4.9 Retarded Potentials 136
4.10 Advanced Potentials? 142
4.11 Retarded Fields 145
4.12 Maxwell’s Equations 149
4.13 Exercises 151
Chapter 5. Electrostatic Calculations 157
5.1 Introduction 157
5.2 Electrostatic Energy 157
5.3 Ohm’s Law 163
5.4 Conductors 165
5.5 Boundary Conditions on the Electric Field 171
5.6 Capacitors 172
5.7 Poisson’s Equation 178
5.8 The Uniqueness Theorem 179
5.9 One-Dimensional Solutions of Poisson’s
Equation 184
5.10 The Method of Images 186
5.11 Complex Analysis 195
5.12 Separation of Variables 202
5.13 Exercises 209
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CONTENTS ix
Chapter 6. Dielectric and Magnetic Media 215
6.1 Introduction 215
6.2 Polarization 215
6.3 Electric Susceptibility and Permittivity 217
6.4 Boundary Conditions For E and D 218
6.5 Boundary Value Problems with Dielectrics 220
6.6 Energy Density within a Dielectric Medium 226
6.7 Force Density within a Dielectric Medium 228
6.8 The Clausius-Mossotti Relation 230
6.9 Dielectric Liquids in Electrostatic Fields 233
6.10 Polarization Current 237
6.11 Magnetization 237
6.12 Magnetic Susceptibility and Permeability 240
6.13 Ferromagnetism 241
6.14 Boundary Conditions for B and H 243
6.15 Boundary Value Problems with Ferromagnets 244
6.16 Magnetic Energy 249
6.17 Exercises 251
Chapter 7. Magnetic Induction 255
7.1 Introduction 255
7.2 Inductance 255
7.3 Self-Inductance 257
7.4 Mutual Inductance 261
7.5 Magnetic Energy 264
7.6 Alternating Current Circuits 270
7.7 Transmission Lines 274
7.8 Exercises 280
Chapter 8. Electromagnetic Energy and Momentum 283
8.1 Introduction 283
8.2 Energy Conservation 283
8.3 Electromagnetic Momentum 287
8.4 Momentum Conservation 291
8.5 Angular Momentum Conservation 294
8.6 Exercises 297
Chapter 9. Electromagnetic Radiation 299
9.1 Introduction 299
9.2 The Hertzian Dipole 299
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x CONTENTS
9.3 Electric Dipole Radiation 306
9.4 Thompson Scattering 307
9.5 Rayleigh Scattering 310
9.6 Propagation in a Dielectric Medium 312
9.7 Dielectric Constant of a Gaseous Medium 313
9.8 Dispersion Relation of a Plasma 314
9.9 Faraday Rotation 318
9.10 Propagation in a Conductor 322
9.11 Dispersion Relation of a Collisional Plasma 324
9.12 Normal Reflection at a Dielectric Boundary 326
9.13 Oblique Reflection at a Dielectric Boundary 330
9.14 Total Internal Reflection 336
9.15 Optical Coatings 339
9.16 Reflection at a Metallic Boundary 342
9.17 Wave-Guides 343
9.18 Exercises 348
Chapter 10.Relativity and Electromagnetism 351
10.1 Introduction 351
10.2 The Relativity Principle 351
10.3 The Lorentz Transformation 352
10.4 Transformation of Velocities 357
10.5 Tensors 359
10.6 Physical Significance of Tensors 364
10.7 Space-Time 365
10.8 Proper Time 370
10.9 4-Velocity and 4-Acceleration 371
10.10 The Current Density 4-Vector 372
10.11 The Potential 4-Vector 374
10.12 Gauge Invariance 374
10.13 Retarded Potentials 375
10.14 Tensors and Pseudo-Tensors 377
10.15 The Electromagnetic Field Tensor 381
10.16 The Dual Electromagnetic Field Tensor 384
10.17 Transformation of Fields 386
10.18 Potential Due to a Moving Charge 387
10.19 Field Due to a Moving Charge 388
10.20 Relativistic Particle Dynamics 391
10.21 Force on a Moving Charge 393
10.22 The Electromagnetic Energy Tensor 394
10.23 Accelerated Charges 397
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