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1 Vector Analysis 1

1.1 Vector Algebra 1

1.1.1 Vector Operations 1

1.1.2 Vector Algebra:Component Form 4

1.1.3 Triple Products 7

1.1.4 Position,Displacement,and Separation Vectors 8

1.1.5 How Vectors Transform 10

1.2 Differential Calculus 13

1.2.1 "Ordinary"Derivatives 13

1.2.2 Gradient 13

1.2.3 The Operator 16

1.2.4 The Divergence 17

1.2.5 The Curl 19

1.2.6 Product Rules 20

1.2.7 Second Derivatives 22

1.3 Integral Calculus 24

1.3.1 Line,Surface,and Volume Integrals 24

1.3.2 The Fundamental Theorem of Calculus 28

1.3.3 The Fundamental Theorem for Gradients 29

1.3.4 The Fundamental Theorem for Divergences 31

1.3.5 The Fundamental Theorem for Curls 34

1.3.6 Integration by Parts 37

1.4 Curvilinear Coordinates 38

1.4.1 Spherical Polar Coordinates 38

1.4.2 Cylindrical Coordinates 43

1.5 The Dirac Delta Function 45

1.5.1 The Divergence of ？/r2 45

1.5.2 The One-Dimensional Dirac Delta Function 46

1.5.3 The Three-Dimensional Delta Function 50

1.6 The Theory of Vector Fields 52

1.6.1 The Helmholtz Theorem 52

1.6.2 Potentials 53

2 Electrostatics 58

2.1 The Electric Field 58

2.1.1 Introduction 58

2.1.2 Coulomb's Law 59

2.1.3 The Electric Field 60

2.1.4 Continuous Charge Distributions 61

2.2 Divergence and Curl of Electrostatic Fields 65

2.2.1 Field Lines,Flux,and Gauss's Law 65

2.2.2 The Divergence of E 69

2.2.3 Applications of Gauss's Law 70

2.2.4 The Curl of E 76

2.3 Electric Potential 77

2.3.1 Introduction to Potential 77

2.3.2 Comments on Potential 79

2.3.3 Poisson's Equation and Laplace's Equation 83

2.3.4 The Potential of a Localized Charge Distribution 83

2.3.5 Summary;Electrostatic Boundary Conditions 87

2.4 Work and Energy in Electrostatics 90

2.4.1 The Work Done to Move a Charge 90

2.4.2 The Energy of a Point Charge Distribution 91

2.4.3 The Energy of a Continuous Charge Distribution 93

2.4.4 Comments on Electrostatic Energy 95

2.5 Conductors 96

2.5.1 Basic Properties 96

2.5.2 Induced Charges 98

2.5.3 Surface Charge and the Force on a Conductor 102

2.5.4 Capacitors 103

3 Special Techniques 110

3.1 Laplace's Equation 110

3.1.1 Introduction 110

3.1.2 Laplace's Equation in One Dimension 111

3.1.3 Laplace's Equation in Two Dimensions 112

3.1.4 Laplace's Equation in Three Dimensions 114

3.1.5 Boundary Conditions and Uniqueness Theorems 116

3.1.6 Conductors and the Second Uniqueness Theorem 118

3.2 The Method of Images 121

3.2.1 The Classic Image Problem 121

3.2.2 Induced Surface Charge 123

3.2.3 Force and Energy 123

3.2.4 Other Image Problems 124

3.3 Separation of Variables 127

3.3.1 Cartesian Coordinates 127

3.3.2 Spherical Coordinates 137

3.4 Multipole Expansion 146

3.4.1 Approximate Potentials at Large Distances 146

3.4.2 The Monopole and Dipole Terms 149

3.4.3 Origin of Coordinates in Multipole Expansions 151

3.4.4 The Electric Field of a Dipole 153

4 Electric Fields in Matter 160

4.1 Polarization 160

4.1.1 Dielectrics 160

4.1.2 Induced Dipoles 160

4.1.3 Alignment of Polar Molecules 163

4.1.4 Polarization 166

4.2 The Field of a Polarized Object 166

4.2.1 Bound Charges 166

4.2.2 Physical Interpretation of Bound Charges 170

4.2.3 The Field Inside a Dielectric 173

4.3 The Electric Displacement 175

4.3.1 Gauss's Law in the Presence of Dielectrics 175

4.3.2 A Deceptive Parallel 178

4.3.3 Boundary Conditions 178

4.4 Linear Dielectrics 179

4.4.1 Susceptibility,Permittivity,Dielectric Constant 179

4.4.2 Boundary Value Problems with Linear Dielectrics 186

4.4.3 Energy in Dielectric Systems 191

4.4.4 Forces on Dielectrics 193

5 Magnetostatics 202

5.1 The Lorentz Force Law 202

5.1.1 Magnetic Fields 202

5.1.2 Magnetic Forces 204

5.1.3 Currents 208

5.2 The Biot-Savart Law 215

5.2.1 Steady Currents 215

5.2.2 The Magnetic Field of a Steady Current 215

5.3 The Divergence and Curl of B 221

5.3.1 Straight-Line Currents 221

5.3.2 The Divergence and Curl of B 222

5.3.3 Applications of Ampère's Law 225

5.3.4 Comparison of Magnetostatics and Electrostatics 232

5.4 Magnetic Vector Potential 234

5.4.1 The Vector Potential 234

5.4.2 Summary;Magnetostatic Boundary Conditions 240

5.4.3 Multipole Expansion of the Vector Potential 242

6 Magnetic Fields in Matter 255

6.1 Magnetization 255

6.1.1 Diamagnets,Paramagnets,Ferromagnets 255

6.1.2 Torques and Forces on Magnetic Dipoles 255

6.1.3 Effect of a Magnetic Field on Atomic Orbits 260

6.1.4 Magnetization 262

6.2 The Field of a Magnetized Object 263

6.2.1 Bound Currents 263

6.2.2 Physical Interpretation of Bound Currents 266

6.2.3 The Magnetic Field Inside Matter 268

6.3 The Auxiliary Field H 269

6.3.1 Ampère's law in Magnetized Materials 269

6.3.2 A Deceptive Parallel 273

6.3.3 Boundary Conditions 273

6.4 Linear and Nonlinear Media 274

6.4.1 Magnetic Susceptibility and Permeability 274

6.4.2 Ferromagnetism 278

7 Electrodynamics 285

7.1 Electromotive Force 285

7.1.1 Ohm's Law 285

7.1.2 Electromotive Force 292

7.1.3 Motional emf 294

7.2 Electromagnetic Induction 301

7.2.1 Faraday's Law 301

7.2.2 The Induced Electric Field 305

7.2.3 Inductance 310

7.2.4 Energy in Magnetic Fields 317

7.3 Maxwell's Equations 321

7.3.1 Electrodynamics Before Maxwell 321

7.3.2 How Maxwell Fixed Ampère's Law 323

7.3.3 Maxwell's Equations 326

7.3.4 Magnetic Charge 327

7.3.5 Maxwell's Equations in Matter 328

7.3.6 Boundary Conditions 331

8 Conservation Laws 345

8.1 Charge and Energy 345

8.1.1 The Continuity Equation 345

8.1.2 Poynting's Theorem 346

8.2 Momentum 349

8.2.1 Newton's Third Law in Electrodynamics 349

8.2.2 Maxwell's Stress Tensor 351

8.2.3 Conservation of Momentum 355

8.2.4 Angular Momentum 358

9 Electromagnetic Waves 364

9.1 Waves in One Dimension 364

9.1.1 The Wave Equation 364

9.1.2 Sinusoidal Waves 367

9.1.3 Boundary Conditions:Reflection and Transmission 370

9.1.4 Polarization 373

9.2 Electromagnetic Waves in Vacuum 375

9.2.1 The Wave Equation for E and B 375

9.2.2 Monochromatic Plane Waves 376

9.2.3 Energy and Momentum in Electromagnetic Waves 380

9.3 Electromagnetic Waves in Matter 382

9.3.1 Propagation in Linear Media 382

9.3.2 Reflection and Transmission at Normal Incidence 384

9.3.3 Reflection and Transmission at Oblique Incidence 386

9.4 Absorption and Dispersion 392

9.4.1 Electromagnetic Waves in Conductors 392

9.4.2 Reflection at a Conducting Surface 396

9.4.3 The Frequency Dependence of Permittivity 398

9.5 Guided Waves 405

9.5.1 Wave Guides 405

9.5.2 TE Waves in a Rectangular Wave Guide 408

9.5.3 The Coaxial Transmission Line 411

10 Potentials and Fields 416

10.1 The Potential Formulation 416

10.1.1 Scalar and Vector Potentials 416

10.1.2 Gauge Transformations 419

10.1.3 Coulomb Gauge and Lorentz Gauge 421

10.2 Continuous Distributions 422

10.2.1 Retarded Potentials 422

10.2.2 Jefimenko's Equations 427

10.3 Point Charges 429

10.3.1 Liénard-Wiechert Potentials 429

10.3.2 The Fields of a Moving Point Charge 435

11 Radiation 443

11.1 Dipole Radiation 443

11.1.1 What is Radiation? 443

11.1.2 Electric Dipole Radiation 444

11.1.3 Magnetic Dipole Radiation 451

11.1.4 Radiation from an Arbitrary Source 454

11.2 Point Charges 460

11.2.1 Power Radiated by a Point Charge 460

11.2.2 Radiation Reaction 465

11.2.3 The Physical Basis of the Radiation Reaction 469

12 Electrodynamics and Relativity 477

12.1 The Special Theory of Relativity 477

12.1.1 Einstein's Postulates 477

12.1.2 The Geometry of Relativity 483

12.1.3 The Lorentz Transformations 493

12.1.4 The Structure of Spacetime 500

12.2 Relativistic Mechanics 507

12.2.1 Proper Time and Proper Velocity 507

12.2.2 Relativistic Energy and Momentum 509

12.2.3 Relativistic Kinematics 511

12.2.4 Relativistic Dynamics 516

12.3 Relativistic Electrodynamics 522

12.3.1 Magnetism as a Relativistic Phenomenon 522

12.3.2 How the Fields Transform 525

12.3.3 The Field Tensor 535

12.3.4 Electrodynamics in Tensor Notation 537

12.3.5 Relativistic Potentials 541

A Vector Calculus in Curvilinear Coordinates 547

A.1 Introduction 547

A.2 Notation 547

A.3 Gradient 548

A.4 Divergence 549

A.5 Curl 552

A.6 Laplacian 554

B The Helmholtz Theorem 555

C Units 558

Index 562