Electromagnetism

Tamer Bécherrawy received a Doctorate from the University of Paris and a PhD in theoretical physics from the University of Rochester, New York. He has taught physics at the Faculty of Science of the Lebanese University in Beirut, the University of Savoy in Chambery, the IUFM and the University of Nancy in France. He was head of the Physics Department at the Lebanese University and is the author of a number of research articles on High Energy Particle Physics.

Preface xi

List of Symbols xv

Chapter 1. Prologue 1

1.1. Scalars and vectors 2

1.2. Effect of rotations on scalars and vectors 5

1.3. Integrals involving vectors 7

1.4. Gradient and curl, conservative field and scalar potential 8

1.5. Divergence, conservative flux, and vector potential 10

1.6. Other properties of the vector differential operator 10

1.7. Invariance and physical laws 11

1.8. Electric charges in nature 14

1.9. Interactions in nature 18

1.10. Problems 19

Chapter 2. Electrostatics in Vacuum 23

2.1. Electric forces and field 23

2.2. Electric energy and potential 25

2.3. The two fundamental laws of electrostatics 26

2.4. Poisson’s equation and its solutions 29

2.5. Symmetries of the electric field and potential 31

2.6. Electric dipole 34

2.7. Electric field and potential of simple charge configurations 38

2.8. Some general properties of the electric field and potential 39

2.9. Electrostatic energy of a system of charges 42

2.10. Electrostatic binding energy of ionic crystals and atomic nuclei 48

2.11. Interaction-at-a-distance and local interaction* 50

2.12. Problems 52

Chapter 3. Conductors and Currents 61

3.1. Conductors in equilibrium 61

3.2. Conductors with cavities, electric shielding 64

3.3. Capacitors 66

3.4. Mutual electric influence of conductors 72

3.5. Electric forces between conductors 73

3.6. Currents and current densities 76

3.7. Classical model of conduction, Ohm’s law and the Joule effect 79

3.8. Resistance of conductors 81

3.9. Variation of resistivity with temperature, superconductivity 82

3.10. Band theory of conduction, semiconductors* 84

3.11. Electric circuits 90

3.12. Problems 92

Chapter 4. Dielectrics 97

4.1. Effects of dielectric on capacitors 97

4.2. Polarization of dielectrics 99

4.3. Microscopic interpretation of polarization 100

4.4. Polarization charges in dielectric 102

4.5. Potential and field of polarized dielectrics 103

4.6. Gauss’s law in the case of dielectrics, electric displacement 105

4.7. Electrostatic equations in dielectrics 106

4.8. Field and potential of permanent dielectrics 109

4.9. Polarization of a dielectric in an external field 113

4.10. Energy and force in dielectrics 115

4.11. Action of an electric field on a polarized medium 116

4.12. Electric susceptibility and permittivity 118

4.13. Variation of polarization with temperature 120

4.14. Nonlinear dielectrics and non-isotropic dielectrics 122

4.15. Problems 124

Chapter 5. Special Techniques and Approximation Methods 127

5.1. Unicity of the solution 128

5.2. Method of images 130

5.3. Method of analytic functions 134

5.4. Method of separation of variables 135

5.5. Laplace’s equation in Cartesian coordinates 136

5.6. Laplace’s equation in spherical coordinates 138

5.7. Laplace’s equation in cylindrical coordinates143

5.8. Multipole expansion 146

5.9. Other methods 147

5.10. Problems 149

Chapter 6. Magnetic Field in Vacuum 153

6.1. Force exerted by a magnetic field on a moving charge 153

6.2. Force exerted by a magnetic field on a current, Laplace’s force 155

6.3. Magnetic flux and vector potential 157

6.4. Magnetic field of particles and currents, Biot-Savart’s law 159

6.5. Magnetic moment 161

6.6. Symmetries of the magnetic field 165

6.7. Ampère’s law in the integral form 167

6.8. Field and potential of some simple circuits 169

6.9. Equations of time-independent magnetism in vacuum, singularities of B 174

6.10. Magnetic energy of a circuit in a field B 178

6.11. Magnetic forces 180

6.12. Question of magnetic monopoles* 186

6.13. Problems188

Chapter 7. Magnetism in Matter 195

7.1. Types of magnetism 195

7.2. Diamagnetism and paramagnetism 197

7.3. Magnetization current 201

7.4. Magnetic field and vector potential in the presence of magnetic matter 203

7.5. Ampère’s law in the integral form in the presence of magnetic matter 204

7.6. Equations of time-independent magnetism in the presence of matter 206

7.7. Discontinuities of the magnetic field 209

7. 8. Examples of calculation of the field of permanent magnets 211

7.9. Magnetization of a body in an external field 214

7.10. Magnetic susceptibility, nonlinear mediums and non-isotropic mediums 216

7.11. Action of a magnetic field on a magnetic body 218

7.12. Magnetic energy in matter 220

7.13. Variation of magnetization with temperature 221

7.14. Ferromagnetism 224

7.15. Magnetic circuits 227

7.16. Problems 229

Chapter 8. Induction 233

8.1. Induction due to the variation of the flux, Faraday’s and Lenz’s laws 233

8.2. Neumann’s induction 235

8.3. Lorentz induction 236

8.4. Lorentz induction and the Galilean transformation of fields 239

8.5. Mutual inductance and self-inductance 240

8.6. LR circuit 244

8.7. Magnetic energy 247

8.8. Magnetic forces acting on circuits 249

8.9. Some applications of induction 252

8.10. Problems 256

Chapter 9. Maxwell’s Equations 263

9.1. Fundamental laws of electromagnetism 263

9.2. Maxwell’s equations 267

9.3. Electromagnetic potentials and gauge transformation 270

9.4. Quasi-permanent approximation 272

9.5. Discontinuities on the interface of two mediums 276

9.6. Electromagnetic energy and Poynting vector 277

9.7. Electromagnetic pressure, Maxwell’s tensor 278

9.8. Problems 280

Chapter 10. Electromagnetic Waves 283

10.1. A short review on waves 284

10.2. Electromagnetic waves in infinite vacuum and dielectrics 291

10.3. Polarization of electromagnetic waves 295

10.4. Energy and intensity of plane electromagnetic waves 299

10.5. Momentum and angular momentum densities, radiation pressure 301

10.6. A simple model of dispersion 304

10.7. Electromagnetic waves in conductors 308

10.8. Electromagnetic waves in plasmas 314

10.9. Quantization of electromagnetic waves 320

10.10. Electromagnetic spectrum 321

10.11. Emission of electromagnetic radiations 323

10.12. Spontaneous and stimulated emissions 325

10.13. Problems 328

Chapter 11. Reflection, Interference, Diffraction and Diffusion 337

11.1. General laws of reflection and refraction 337

11.2. Reflection and refraction on the interface of two dielectrics 340

11.3. Total reflection 346

11.4. Reflection on a conductor 349

11.5. Reflection on a plasma 352

11.6. Interference of two electromagnetic waves 353

11.7. Superposition of several waves, conditions for observable interference 355

11.8. Huygens-Fresnel’s principle and diffraction by an aperture 357

11.9. Diffraction by an obstacle, Babinet’s theorem 363

11.10. Diffraction by several randomly distributed identical apertures 364

11.11. Diffraction grating 365

11.12. X-ray diffraction 368

11.13. Diffusion of waves* 370

11.14. Cross-section* 375

11.15. Problems 378

Chapter 12. Guided Waves 389

12.1. Transmission lines 390

12.2. Guided waves 394

12.3. Waveguides formed by two plane and parallel plates 397

12.4. Guided electromagnetic waves in a hollow conductor 400

12.5. Energy propagation in waveguides 404

12.6. Cavities 406

12.7. Applications of waveguides 407

12.8. Problems 409

Chapter 13. Special Relativity and Electrodynamics 413

13.1. Galilean relativity in mechanics 414

13.2. Galilean relativity and wave theory* 415

13.3. The 19th Century experiments on the velocity of light 420

13.4. Special theory of relativity 421

13.5. Four-dimensional formalism 424

13.6. Elements of relativistic mechanics 427

13.7. Special relativity and wave theory* 430

13.8. Elements of relativistic electrodynamics 434

13.9. Problems 438

Chapter 14. Motion of Charged Particles in an Electromagnetic Field 443

14.1. Motion of a charged particle in an electric field 443

14.2. Bohr model for the hydrogen atom* 447

14.3. Rutherford’s scattering * 450

14.4. Motion of a charged particle in a magnetic field 451

14.5. Motion in crossed electric and magnetic fields 457

14.6. Magnetic moment in a magnetic field 459

14.7. Problems 461

Chapter 15. Emission of Radiation 467

15.1. Retarded potentials and fields 467

15.2. Dipole radiation 469

15.3. Electric dipole radiation 470

15.4. Magnetic dipole radiation 474

15.5. Antennas 476

15.6. Potentials and fields of a charged particle* 479

15.7. Case of a charged particle with constant velocity * 482

15.8. Radiated energy by a moving charge 484

15.9. Problems 486

Answers to Some Problems 491

Appendix A. Mathematical Review 511

Appendix B. Units in Physics 527

Appendix C. Some Physical Constants 533

Further Reading 535

Index 537