Introduction to Classical Electrodynamics 3 presents the applications of electromagnetic theory in modern physics.
This book covers waveguides, resonant cavities and transmission lines. It also analyzes electromagnetic radiation from localized and extended oscillating sources, as well as radiation from (non-)relativistic point charges. The book examines particles in circular motion, Thomson scattering, Cherenkov radiation, and Kirchhoff scattering, and presents in detail the problems of diffraction by circular apertures and conducting spheres. Finally, it deals with the principles of special relativity, the covariant formulation of electromagnetism, as well as the concepts of electromagnetic mass, radiation reaction force and the Abraham–Lorentz model.
This book is intended for physics and mathematics students, as well as engineering students interested in the challenges of electromagnetic theory. The discussion is supplemented with numerous applications derived from the theoretical concepts presented.
Table of Contents:
Preface ix
Chapter 1 Waveguides 1
1.1. The different types of waveguides 1
1.2. Description of a guided wave 2
1.2.1. Propagation equation and field expressions 2
1.2.2. Classification of propagation modes 6
1.3. TE and TM modes in a hollow conducting waveguide 8
1.3.1. Rectangular waveguide 9
1.3.2. Attenuation coefficient in a lossy waveguide 24
1.3.3. Propagation between two parallel planes 29
1.3.4. Cylindrical waveguide 33
1.4. Dielectric waveguides 37
1.4.1. Propagation between two parallel dielectric planes 37
1.4.2. Step index optical fiber 44
1.4.3. Modes with azimuthal symmetry 45
1.4.4. Modes with azimuthal dependence 48
1.4.5. Propagation modes in weakly guiding fibers 54
1.5. Resonant cavities 59
1.5.1. Rectangular metallic resonant cavity 60
1.5.2. Cylindrical resonant cavity 65
Chapter 2 Transmission Lines 71
2.1. Parallel-plate transmission line 71
2.2. General formulation 73
2.3. Modeling of lossless transmission lines 78
2.4. Modeling of a transmission line with losses 79
2.5. Wave equations and sinusoidal solutions 82
2.6. Effects of losses in transmission lines 86
2.6.1. Low-loss limit 87
2.6.2. Distortionless lines 87
2.6.3. Solutions with arbitrary impedance at the line output 88
2.7. Resonance effect in a transmission line 94
2.8. Energy aspect in a transmission line 95
Chapter 3 Electromagnetic Radiation 99
3.1. Retarded potentials 99
3.1.1. Inhomogeneous wave equations 99
3.1.2. Solutions of the inhomogeneous wave equations 101
3.1.3. Solutions by Fourier analysis 102
3.2. Radiation from a localized oscillating source 106
3.2.1. Radiation from an electric dipole 113
3.2.2. Radiations from the magnetic moment and the electric quadrupole 119
3.2.3 Multipole radiation of order l ≥ 2 127
3.2.4. Energy and angular momentum of multipolar fields 132
3.3. Radiation from an extended source 135
Chapter 4 Radiation from Point Charges 145
4.1. Lińard–Wiechert potentials 145
4.2. Fields of a moving point charge 147
4.3. Radiation from a charge in uniform motion 152
4.4. Radiation from a slowly moving charge 153
4.5. Radiation from a relativistic charge 154
4.5.1. Velocity and acceleration parallel 155
4.5.2. Acceleration perpendicular to velocity 158
4.5.3. General case 166
4.6. Frequency distribution of the radiated energy 170
4.7. Charge in instantaneous circular motion 177
4.8. Particle in circular motion 183
4.9. Thomson scattering of radiation 188
4.10. Cherenkov radiation 196
Chapter 5 Kirchhoff Diffraction 201
5.1. Initial value problem: Kirchhoff’s integral representation 201
5.1.1. Green’s function for the time-dependent wave equation 201
5.1.2. Cauchy problem 204
5.1.3. Poisson solution 205
5.1.4. Kirchhoff representation of the field 206
5.2. Kirchhoff integral for diffraction 207
5.3. Vector form of the Kirchhoff integral 212
5.4. Babinet’s principle of complementary screens 217
5.5. Diffraction by a circular aperture 220
5.6. Scattering by a conducting sphere in the short-wavelength limit 231
Chapter 6 Theory of Special Relativity 241
6.1. Postulates of special relativity and Lorentz transformation 241
6.2. FitzGerald–Lorentz length contraction, time dilation and proper time 248
6.3. Law of addition of velocities 250
6.4. Four-dimensional Formalism 255
6.4.1. Brehme–Lorentz diagram 255
6.4.2. Transformations and Lorentz group 259
6.4.3. Contravariant coordinates 262
6.4.4. Covariant coordinates 263
6.4.5. Change of basis 264
6.4.6. Four-velocity, four-momentum and four-acceleration 266
6.4.7. Tensors 269
6.4.8. Differentiation, vector analysis and integration 272
6.5. Relativistic dynamics 274
6.5.1. Relativistic Lagrangian for a free particle 274
6.5.2. Four-momentum 275
6.5.3. Application to collisions 278
6.5.4. Minkowski four-force 280
Chapter 7 Covariant Formulation 285
7.1. A brief remark on field theory 285
7.1.1. Transition from a discrete system to a continuous one 285
7.1.2. Continuous system 286
7.1.3. Euler–Lagrange equations 287
7.1.4. Hamiltonian formalism 288
7.1.5. Gauge invariance 290
7.1.6. Infinitesimal symmetry 291
7.1.7. Energy–momentum tensor of a field 292
7.1.8. Angular momentum tensor 293
7.1.9. Electromagnetic interaction 293
7.2. Free particle in a field 294
7.2.1. Four-potential 294
7.2.2. Equation of relativistic dynamics 296
7.2.3. Electromagnetic field tensor 298
7.2.4. Lorentz force 300
7.2.5. Change of reference frames for the field 302
7.2.6. Invariants of the electromagnetic field 305
7.2.7. First group of Maxwell equations 306
7.3. Sources of the field: second group of Maxwell equations 307
7.3.1. Interaction between a field and a current 308
7.3.2. Free field action (field self-interaction) 309
7.3.3. Magnetization and polarization 311
7.4. Energy–momentum tensor 312
7.4.1. Conservation law 314
7.4.2. Time component: conservation of energy 315
7.4.3. Spatial components: conservation of momentum 316
7.5. Electromagnetic mass 318
7.6. Radiation reaction force from energy balance 321
7.7. Abraham–Lorentz model 325
7.8. Abraham–Lorentz equation of motion 332
7.9. Dirac radiation equation 334
7.10 And the story continues 335
References 337
Index 343
About the Author :
Boucif Abdesselam is a professor in the Physics Department of the Faculty of Science and Technology at the Université Ain Temouchent Belhadj Bouchai, Algeria. He is also a member of the Theoretical Physics Laboratory at the University of Oran 1, Algeria.
