Table of Contents

1 Force and Energy in Electrostatics

1.1 Electric Charge 2

1.2 The Electric Field 6

1.3 Electric Fields in Matter 10

1.3.1 The Atomic Charge Density 10

1.3.2 The Atomic Electric Field 11

1.3.3 The Macroscopic Electric Field 13

1.4 Gauss' Law 16

1.4.1 The Flux of a Vector Field 17

1.4.2 The Flux of the Electric Field out of a Closed Surface 19

1.4.3 The Divergence of a Vector Field 24

1.4.4 The Differential Form of Gauss' Law 26

1.5 Electrostatic Energy 28

1.5.1 The Electrostatic Potential 28

1.5.2 The Electric Field as the Gradient of the Potential 31

1.5.3 The Dipole Potential 35

1.5.4 Energy Changes Associated with the Atomic Field 38

1.5.5 Capacitors, and Energy in Macroscopic Fields 40

* 1.5.6 Energy Stored by a Number of Charged Conductors 44

Problems 1 46

2 Dielectrics

2.1 Polarization 49

2.2 Relative Permittivity and Electric Susceptibility 55

2.2.1 The Local Field 59

2.2.2 Polar Molecules 60

2.2.3 Non-polar Liquids. 67

2.3 Macroscopic Fields in Dielectrics 70

2.3.1 The Volume Density of Polarization Charge. 71

2.3.2 The Electric Displacement Vector 73

2.3.3 Boundary Conditions for D and E. 76

2.4 Energy in the Presence of Dielectrics 79

* 2.4.1 Some Further Remarks about Energy and Forces. 80

Problems 2 82

3 Electrostatic Field Calculations

3.1 Poisson's Equation and Laplace's Equation 85

3.1.1 The Uniqueness Theorem 88

3.1.2 Electric Fields ·in the Presence of Free Charge 89

3.2 Boundaries Between Different Regions 91

* 3.3 Boundary Conditions and Field Patterns 93

* 3.3.1 Electrostatic Images 93

* 3.3.2 Spheres and Spherical Cavities in Uniform External Field 97

3.4 Electrostatic Lenses 100

* 3.5 Numerical Solutions of Poisson's Equation 103

3.6 Summary of Electrostatics 107

Problems 3 109

4 Steady Currents and Magnetic Fields

4.1 Electromotive Force and Conduction 112

4.1.1 Current and Resistance 112

* 4.1.2 The Calculation of Resistance 116

4.2 The Magnetic Field 119

4.2.1 The Lorentz Force 119

4.2.2 Magnetic Field Lines 123

4.3 The Magnetic Dipole 127

4.3.1 Current Loops in External Fields 127

4.3.2 Magnetic Dipoles and Magnetic Fields 130

4.4 Ampere's Law 132

4.4.1 The Field of a Large Current Loop 132

4.4.2 The Biot−Savart Law 137

4.4.3 Examples of the Calculation of Magnetic Fields 139

4.5 The Differential Form of Ampere's Law 144

4.5.1 The Operator Curl. 144

4.5.2 The Vector Curl B 148

4.5.3 The Magnetic Vector Potential 148

4.6 Forces and Torques on Coils 150

4.6.1 Magnetic Flux 151

4.7 The Motion of Charged Particles in Electric and Magnetic Fields 154

4.7.1 The Motion of a Charged Particle in a Uniform Magnetic Field 155

4.7.2 Magnetic Mirrors and Plasmas 157

* 4.7.3 Magnetic Quadrupole Lenses 159

Problems 4 163

5 Magnetic Materials

5.1 Magnetization 166

5.1.1 Diamagnetism 169

5.1.2 Paramagnetism 173

5.1.3 Ferromagnetism 175

5.2 The Macroscopic Magnetic Field Inside Media 176

5.2.1 The Surface Currents on a Uniformly Magnetized Body 178

5.2.2 The Distributed Currents Within a Magnetized Body 179

5.2.3 Magnetic Susceptibility and Atomic Structure 183

5.3 The Field Vector H 186

5.3.1 Ampere's Law for the Field H 186

5.3.2 The Boundary Conditions on the Field Band H 191

5.4 Magnets 194

5.4.1 Electromagnets 194

* 5.4.2 Permanent Magnets 204

5.5 Summary of Magnetostatics 208

Problems 5 209

6 Electromagnetic Induction and Magnetic Energy

6.1 Electromagnetic Induction 212

6.1.1 Motional Electromotive Force 214

6.1.2 Faraday's Law 218

6.1.3 Examples of Induction 221

6.1.4 The Differential Form of Faraday's Law 228

6.2 Self-inductance and Mutual Inductance 230

6.2.1 Self-inductance 230

6.2.2 Mutual Inductance 232

6.3 Energy and Forces in Magnetic Fields 234

6.3.1 The Magnetic Energy Stored in an Inductor 234

6.3.2 The Total Magnetic Energy of a System of Currents 235

6.3.3 The Potential Energy of a Coil in a Field and the Force on the Coil 237

6.3.4 The Total Magnetic Energy in Terms of the Fields B and H 239

6.3.5 Non-linear Media 241

* 6.3.6 Further Comments on Energy in Magnetic Fields. 243

6.4 The Measurement of Magnetic Fields and Susceptibilities 246

6.4.1 The Measurement of Magnetic Fields 246

6.4.2 The Measurement of Magnetic Susceptibilities 248

Problems 6 250

7 Alternating Currents and Transients

7.1 Alternating Current Generators 253

7.2 Amplitude, Phase and Period 256

7.3 Resistance, Capacitance and Inductance in A.C. Circuits 257

7.4 The Phasor Diagram and Complex Impedance 260

7.5 Power in A.C. Circuits 266

7.6 Resonance 268

7.7 Transients 274

Problems 7 280

8 Linear Circuits

8.1 Networks 282

8.1.1 Kirchhoffs Rules 283

8.1.2 Loop Analysis, Node Analysis and Superposition 286

8.1.3 A.C. Networks 288

8.2 Audio-frequency Bridges 291

8.3 Impedance and Admittance 293

8.3.1 Input Impedance 296

8.3.2 Output Impedance and Thevenin's Theorem 297

8.4 Filters 299

8.4.1 Ladder Networks 301

8.4.2 Higher Order Filters and Delay Lines 303

8.5 Transformers 307

8.5.1 The Ideal Transformer 308

8.5.2 Applications of Transformers. 311

8.5.3 Real Transformers 312

* Problems 8 318

9 Transmission Lines

9.1 Propagation of Signals in a Lossless Transmission Line 324

9.2 Practical Types of Transmission Line 329

9.2.1 The Parallel Wire Transmission Line 329

9.2.2 The Coaxial Cable 331

9.2.3 Parallel Strip Lines 333

9.3 Reflections 335

* 9.4 The Input Impedance of a Mismatched Line 338

* 9.5 Lossy Lines 342

Problems 9 345

10 Maxwell's Equations

10.1 The Equation of Continuity 348

10.2 Displacement Current 350

10.3 Maxwell's Equations 356

10.4 Electromagnetic Radiation 359

* 10.5 The Microscopic Field Equations 360

Problems 10 362

11 Electromagnetic Waves

11.1 Electromagnetic Waves in Free Space 365

11.2 Plane Waves and Polarization 368

11.2.1 Plane Waves in Free Space 373

11.2.2 Plane Waves in Isotropic Insulating Media 375

11.3 Dispersion 379

11.4 Energy in Electromagnetic Waves 383

11.5 The Absorption of Plane Waves in Conductors and the Skin Effect 388

11.6 The Reflection and Transmission of Electromagnetic Waves 391

11.6.1 Boundary Conditions on Electric and Magnetic Fields 392

11.6.2 Reflection at Dielectric Boundaries 396

11.6.3 Reflection at Metallic Boundaries 399

* 11.6.4 Polarization by Reflection 401

* 11.7 Electromagnetic Waves and Photons 404

Problems 11 406

12 Waveguides

12.1 The Propagation of Waves Between Conducting Plates. 409

12.2 Rectangular Waveguides. 415

12.2.1 The TE 01 Mode. 420

* 12.2.2 Further Comments on Waveguides 423

12.3 Cavities 426

Problems 12 430

13 The Generation of Electromagnetic Waves

13.1 The Retarded Potentials 433

13.2 The Hertzian Dipole 436

13.3 Antennas 443

Problems 13 450

14 Electromagnetism and Special Relativity

14.1 Introductory Remarks 451

14.2 The Lorentz Transformation 452

14.3 Charges and Fields as seen by Different Observers 455

14.4 Four-vectors 458

14.5 Maxwell's Equations in Four-vector Form 461

14.6 Transformation of the Fields 464

14.7 Magnetism as a Relativistic Phenomenon 469

14.8 Retarded Potentials From the Relativistic Standpoint 473

Problems 14 476

Appendix A Units

A.1 Electrical Units and Standards 477

A.l.l The Definition of the Ampere 477

A.l.2 Calibration and Comparison of Electrical Standards 479

A.2 Gaussian Units 482

A.3 Conversion between SI and Gaussian Units 485

Appendix B Fields and Differential Operators

B.l The Operators div, grad and curl 487

B.2 Formulae in Different Coordinate Systems 489

B.3 Identities 493

Appendix C The Derivation of the Blot-Savart Law Solutions to Problems 497

Solutions to Problems 497

Further Reading 518

Index 519

About the Author

I. S. Grant and W. R. Phillips are the authors of Electromagnetism, 2nd Edition, published by Wiley.