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1 History and Perspective 1

1.1 Brief History of the Science of Electromagnetism 1

1.2 Electromagnetism in the Standard Model 5

2 Vector Calculus 9

2.1 Vector Algebra 10

2.1.1 Definitions 10

2.1.2 Addition and Multiplication of Vectors 13

2.1.3 Vector Product Identities 14

2.1.4 Geometric Meanings 16

2.2 Vector Differential Operators 18

2.2.1 Gradient of a Scalar Function 18

2.2.2 Divergence of a Vector Function 19

2.2.3 Curl of a Vector Function 20

2.2.4 DelIdentities 23

2.3 Integral Theorems 25

2.3.1 Gauss’s Theorem 26

2.3.2 Stokes’s Theorem 27

2.3.3 Vector Calculus in Fluid Mechanics 29

2.4 Curvilinear Coordinates 30

2.4.1 General Derivations 30

2.4.2 Cartesian, Cylindrical, and Spherical Coordinates 33

2.5 The Helmholtz Theorem 37

3 Basic Principles of Electrostatics 44

3.1 Coulomb’s Law 44

3.1.1 The Superposition Principle 46

3.2 The Electric Field 46

3.2.1 Definition 46

3.2.2 Charge as the Source of E 47

3.2.3 Field of a Charge Continuum 49

3.3 Curl and Divergence of E 54

3.3.1 Field Theory Versus Action at a Distance 56

3.3.2 Boundary Conditions of the Electrostatic Field 56

3.4 The Integral Form of Gauss’s Law 57

3.4.1 Flux and Charge 57

3.4.2 Proof of Gauss’s Law 57

3.4.3 Calculations Based on Gauss’s Law 59

3.5 Green’s Function and the Dirac delta Function 62

3.5.1 The Dirac delta Function 62

3.5.2 Another Proof of Gauss’s Law 65

3.6 The Electric Potential 65

3.6.1 Definition and Construction 65

3.6.2 Poisson’s Equation 68

3.6.3 Example Calculations of V （x） 69

3.7 Energy of the Electric Field 72

3.8 The Multipole Expansion 75

3.8.1 Two Charges 75

3.8.2 The Electric Dipole 77

3.8.3 Moments of a General Charge Distribution 78

3.8.4 Equipotentials and Field Lines 79

3.8.5 Torque and Potential Energy for a Dipole in an Electric Field 80

3.9 Applications 82

3.10 Chapter Summary 83

4 Electrostatics and Conductors 92

4.1 Electrostatic properties of conductors 93

4.2 Electrostatic Problems with Rectangular Symmetry 98

4.2.1 Charged Plates 98

4.2.2 Problems with Rectangular Symmetry and External Point Charges.The Method of Images 102

4.3 Problems with Spherical Symmetry 107

4.3.1 Charged Spheres 107

4.3.2 Problems with Spherical Symmetry and External Charges 113

4.4 Problems with Cylindrical Symmetry 116

4.4.1 Charged Lines and Cylinders 116

4.4.2 Problems with Cylindrical Symmetry and an External Line Charge 124

5 General Methods for Laplace’s Equation 133

5.1 Separation of Variables for Cartesian Coordinates 135

5.1.1 Separable Solutions for Cartesian Coordinates 136

5.1.2 Examples 138

5.2 Separation of Variables for Spherical Polar Coordinates 147

5.2.1 Separable Solutions for Spherical Coordinates 147

5.2.2 Legendre Polynomials 149

5.2.3 Examples with Spherical Boundaries 150

5.3 Separation of Variables for Cylindrical Coordinates 159

5.3.1 Separable Solutions for Cylindrical Coordinates 160

5.4 Conjugate Functions in 2 Dimensions 163

5.5 Iterative Relaxation: A Numerical Method 172

6 Electrostatics and Dielectrics 186

6.1 The Atom as an Electric Dipole 187

6.1.1 Induced Dipoles 187

6.1.2 Polar Molecules 189

6.2 Polarization and Bound Charge 191

6.3 The Displacement Field 195

6.3.1 Linear Dielectrics 197

6.3.2 The Clausius-Mossotti Formula 198

6.3.3 Poisson’s Equation in a Uniform Linear Dielectric 200

6.4 Dielectric Material in a Capacitor 201

6.4.1 Design of Capacitors 203

6.4.2 Microscopic Theory 204

6.4.3 Energy in a Capacitor 205

6.4.4 A Concrete Model of a Dielectric 207

6.5 Boundary Value Problems with Dielectrics 208

6.5.1 The Boundary Conditions 208

6.5.2 A Dielectric Sphere in an Applied Field 209

6.5.3 A Point Charge above a Dielectric with a Planar Bound-ary Surface 211

6.5.4 A Capacitor Partially Filled with Dielectric 212

7 Electric Currents 222

7.1 Electric Current in a Wire 222

7.2 Current Density and the Continuity Equation 224

7.2.1 Local Conservation of Charge 226

7.2.2 Boundary Condition on J（x, t） 226

7.3 Current and Resistance 228

7.3.1 Ohm’s Law 228

7.3.2 Fabrication of Resistors 233

7.3.3 The Surface Charge on a Current Carrying Wire 234

7.4 A Classical Model of Conductivity 236

7.5 Joule’s Law 238

7.6 Decay of a Charge Density Fluctuation 239

7.7 Ⅰ-Ⅴ Characteristic of a Vacuum-Tube Diode 241

7.8 Chapter Summary 246

8 Magnetostatics 252

8.1 The Magnetic Force and the Magnetic Field 253

8.1.1 Force on a Moving Charge 253

8.1.2 Force on a Current-Carrying Wire 255

8.2 Applications of the Magnetic Force 255

8.2.1 Helical or Circular Motion of q in Uniform B 255

8.2.2 Cycloidal Motion of q in Crossed E and B 258

8.2.3 Electric Motors 260

8.3 Electric Current as a Source of Magnetic Field 262

8.3.1 The Biot-Savart Law 262

8.3.2 Forces on Parallel Wires 266

8.3.3 General Field Equations for B（x） 267

8.4 Ampere’s Law 270

8.4.1 Ampere Law Calculations 271

8.4.2 Formal Proof of Ampere’s Law 277

8.5 The Vector Potential 280

8.5.1 General Solution for A（x） 281

8.6 The Magnetic Dipole 284

8.6.1 Asymptotic Analysis 284

8.6.2 Dipole Moment of a Planar Loop 286

8.6.3 Torque and Potential Energy of a Magnetic Dipole 287

8.6.4 The Magnetic Field of the Earth 291

8.7 The Full Field of a Current Loop 291

9 Magnetic Fields and Matter 307

9.1 The Atom as a Magnetic Dipole 307

9.1.1 Diamagnetism 310

9.1.2 Paramagnetism 313

9.2 Magnetization and Bound Currents 314

9.2.1 Examples 316

9.2.2 A Geometric Derivation of the Bound Currents 320

9.3 Ampere’s Law for Free Currents, and H 323

9.3.1 The Integral Form of Ampere’s Law 326

9.3.2 The Constitutive Equation 326

9.3.3 Magnetic Susceptibilities 326

9.3.4 Boundary Conditions for Magnetic Fields 329

9.4 Problems Involving Free Currents and Magnetic Materials 331

9.5 A Magnetic Body in an External Field: The Magnetic Scalar Potentialφm （x） 335

9.6 Ferromagnetism 342

9.6.1 Measuring Magnetization Curves: The Rowland Ring 343

9.6.2 Magnetization Curves of Ferromagnetic Materials 345

9.6.3 The Permeability of a Ferromagnetic Material 346

10 Electromagnetic Induction 355

10.1 Motional EMF 356

10.1.1 Electromotive Force 356

10.1.2 EMF from Motion in B 357

10.1.3 The Faraday Disk Generator 358

10.2 Faraday’s Law of Electromagnetic Induction 360

10.2.1 Mathematical Statement 361

10.2.2 Lenz’s Law 363

10.2.3 Eddy Currents 364

10.3 Applications of Faraday’s Law 368

10.3.1 The Electric Generator and Induction Motor 369

10.3.2 The Betatron 371

10.3.3 Self-Inductance 372

10.3.4 Classical Model of Diamagnetism 375

10.4 Mutual Inductance 376

10.5 Magnetic Field Energy 382

10.5.1 Energy in a Ferromagnet 386

11 The Maxwell Equations 397

11.1 The Maxwell Equations in Vacuum and the Displacement Current 398

11.1.1 The Displacement Current 399

11.2 Scalar and Vector Potentials 405

11.2.1 Gauge Transformations and Gauge Invariance 406

11.2.2 Gauge Choices and Equations for A（x,t） and V（x,t） 407

11.3 The Maxwell Equations in Matter 410

11.3.1 Free and Bound Charge and Current 410

11.3.2 Boundary Conditions of Fields 413

11.4 Energy and Momentum of Electromagnetic Fields 415

11.4.1 Poynting’s Theorem 416

11.4.2 Field Momentum 421

11.5 Electromagnetic Waves in Vacuum 423

11.5.1 Derivation of the Wave Equation 424

11.5.2 An Example of a Plane Wave Solution 425

11.5.3 Derivation of the General Plane Wave Solution 431

11.5.4 A Spherical Harmonic Wave 434

11.5.5 The Theory of Light 437

12 Electromagnetism and Relativity 445

12.1 Coordinate Transformations 446

12.1.1 The Galilean Transformation 446

12.1.2 The Lorentz Transformation 448

12.1.3 Examples Involving the Lorentz Transformation 450

12.2 Minkowski Space 452

12.2.1 4-vectors, Scalars, and Tensors 452

12.2.2 Kinematics of a Point Particle 455

12.2.3 Relativistic Dynamics 457

12.3 Electromagnetism in Covariant Form 458

12.3.1 The Lorentz Force and the Field Tensor 458

12.3.2 Maxwell’s Equations in Covariant Form 460

12.3.3 The 4-vector Potential 462

12.4 Field Transformations 463

12.5 Fields Due to a Point Charge in Uniform Motion 468

12.6 Magnetism from Relativity 474

12.7 The Energy-Momentum Flux Tensor 477

13 Electromagnetism and Optics 485

13.1 Electromagnetic Waves in a Dielectric 485

13.2 Reflection and Refraction at a Dielectric Interface 488

13.2.1 Wave Vectors 490

13.2.2 Reflectivity for Normal Incidence 494

13.2.3 Reflection for Incidence at Arbitrary Angles: Fresnel’s Equations 498

13.3 Electromagnetic Waves in a Conductor 505

13.3.1 Reflectivity of a Good Conductor 509

13.4 A Classical Model of Dispersion: The Frequency Dependence of Material Properties 511

13.4.1 Dispersion in a Dielectric 512

13.4.2 Dispersion in a Plasma 514

14 Wave Guides and Transmission Lines 523

14.1 Electromagnetic Waves Between Parallel Conducting Planes 524

14.1.1 The TEM Solution 526

14.1.2 TE Waves 528

14.1.3 TM Waves 537

14.1.4 Summary 540

14.2 The Rectangular Wave Guide 540

14.2.1 Transverse Electric Modes TE（m, n） 541

14.2.2 Transverse Magnetic Modes TM（m, n） 547

14.3 Wave Guide of Arbitrary Shape 549

14.4 The TEM Mode of a Coaxial Cable 551

14.5 Cavity Resonance 555

15 Radiation of Electromagnetic Waves 560

15.1 The Retarded Potentials 561

15.1.1 Green’s Functions 561

15.2 Radiation from an Electric Dipole 567

15.2.1 The Hertzian Dipole 571

15.2.2 Atomic Transitions 574

15.2.3 Magnetic Dipole Radiation 575

15.2.4 Complete Fields of a Hertzian Dipole 577

15.3 The Half-Wave Linear Antenna 579

15.4 The Larmor Formula: Radiation from a Point Charge 584

15.5 Classical Electron Theory of Light Scattering 589

15.6 Complete Fields of a Point Charge: The Lienard-Wiechert Potentials 593

15.6.1 A Charge with Constant Velocity 596

15.6.2 The Complete Fields 598

15.6.3 Generalization of the Larmor Formula 599

A Electric and Magnetic Units 607

B The Helmholtz Theorem 610

Index 613