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ONE VECTOR ANALYSIS 7

1.1 VECTOR ALGEBRA 7

1.1.1 Vector Operations 7

1.1.2 Vector Algebra：Component Form 10

1.1.3 Triple Products 13

1.1.4 How Vectors Transform 14

1.2 DIFFERENTIAL CALCULUS 16

1.2.1 “Ordinary” Derivatives 16

1.2.2 Gradient 17

1.2.3 The Operator ▽ 20

1.2.4 Divergence 21

1.2.5 The Curl 22

1.2.6 Product Rules 24

1.2.7 Second Derivatives 26

1.3 INTEGRAL CALCULUS 28

1.3.1 “Ordinary” Integration 28

1.3.2 The Fundamental Theorem for Gradients 29

1.3.3 The Fundamental Theorem for Divergences 31

1.3.4 The Fundamental Theorem for Curls 35

1.3.5 Relations Among the Fundamental Theorems 38

1.3.6 Divergence-Less and Curl-Less Fields 40

1.4 CURVILINEAR COORDINATES 40

1.4.1 Spherical Polar Coordinates 40

1.4.2 Cylindrical Coordinates 45

1.5 THE ROLE OF VECTOR CALCULUS IN ELECTRODYNAMICS 46

TWO ELECTROSTATICS 49

2.1 THE ELECTROSTATIC FIELD 49

2.1.1 Introduction 49

2.1.2 Coulomb’s Law 50

2.1.3 The Electric Field 51

2.1.4 Continuous Charge Distributions 52

2.2 DIVERGENCE AND CURL OF ELECTROSTATIC FIELDS 56

2.2.1 Field Lines and Gauss’s Law 56

2.2.2 The Divergence of E 60

2.2.3 Applications of Gauss’s Law 61

2.2.4 The Curl of E 66

2.3 ELECTRIC POTENTIAL 68

2.3.1 Introduction to Potential 68

2.3.2 Comments on Potential 69

2.3.3 Poisson’s Equation and Laplace’s Equation 73

2.3.4 Potential of a Charge Distribution 74

2.3.5 Summary； Electrostatic Boundary Conditions 77

2.4 WORK AND ENERGY IN ELECTROSTATICS 79

2.4.1 The Work Done in Moving a Charge 79

2.4.2 The Energy of a Point Charge Distribution 80

2.4.3 The Energy of a Continuous Charge Distribution 82

2.4.4 Comme？ts on Electrostatic Energy 83

2.5 CONDUCTORS 85

2.5.1 Basic Properties of Conductors 85

2.5.2 Induced Charges 87

2.5.3 The Surface Charge on a Conductor； the Force on a Surface Charge 90

2.5.4 Capacitors 91

THREE SPECIAL TECHNIQUES FOR CALCULATING POTENTIALS 96

3.1 LAPLACE’S EQUATION AND UNIQUENESS THEOREMS 96

3.1.1 Introduction 96

3.1.2 Laplace’s Equation in One Dimension 97

3.1.3 Laplace’s Equation in Two Dimensions 98

3.1.4 Laplace’s Equation in Three Dimensions 100

3.1.5 Boundary Conditions for Laplace’s Equation 101

3.1.6 Conductors and the Second Uniqueness Theorem 103

3.2 THE METHOD OF IMAGES 106

3.2.1 The Classic Image Problem 106

3.2.2 The Induced Surface Charge 108

3.2.3 Force and Energy 108

3.2.4 Other Image Problems 109

3.3 SEPARATION OF VARIABLES 112

3.3.1 Cartesian Coordinates 112

3.3.2 Spherical Coordinates 121

3.4 MULTIPOLE EXPANSION 129

3.4.1 Approximate Potential at Large Distances 129

3.4.2 The Monopole and Dipole Terms 131

3.4.3 Origin of Coordinates in Multipole Expansions 133

3.4.4 The Electric Field of a Dipole 134

FOUR ELECTROSTATIC FIELDS IN MATTER 138

4.1 POLARIZATION 138

4.1.1 Dielectrics 138

4.1.2 Induced Dipoles 139

4.1.3 Alignment of Polar Molecules 141

4.1.4 Polarization 144

4.2 THE FIELD OF A POLARIZED OBJECT 144

4.2.1 Bound Charges 144

4.2.2 Physical Interpretation of Bound Charge 147

4.2.3 The Field Inside a Dielectric 150

4.3 THE ELECTRIC DISPLACEMENT 152

4.3.1 Gauss’s Law in the Presence of Dielectrics 152

4.3.2 A Deceptive Parallel 155

4.4 LINEAR DIELECTRICS 156

4.4.1 Susceptibility，Permittivity，Dielectric Constant 156

4.4.2 Special Problems Involving Linear Dielectrics 161

4.4.3 Force and Energy in Dielectric Systems 166

4.4.4 Polarizability and Susceptibility 170

FIVE MAGNETOSTATICS 174

5.1 THE LORENTZ FORCE LAW 174

5.1.1 Magnetic Fields 174

5.1.2 Magnetic Forces 176

5.1.3 Currents 180

5.2 THE BIOT-SAVART LAW 184

5.2.1 Steady Currents 184

5.2.2 The Magnetic Field of a Steady Current 185

5.3 THE DIVERGENCE AND CURL OF B 190

5.3.1 Straight-Line Currents 190

5.3.2 The Divergence of B 192

5.3.3 The Curl of B 193

5.3.4 Ampere’s Law 194

5.3.5 Comparison of Magnetostatics and Electrostatics 201

5.4 MAGNETIC VECTOR POTENTIAL 203

5.4.1 The Vector Potential 203

5.4.2 Summary； Magnetestatic Boundary Conditions 208

5.4.3 Multipole Expansion of the Vector Potential 210

SIX MAGNETOSTATIC FIELDS IN MATTER 218

6.1 MAGNETIZATION 218

6.1.1 Diamagnets，Paramagnets，Ferromagnets 218

6.1.2 Torques and Forces on Magnetic Dipoles 219

6.1.3 Effect of Magnetic Field on Atomic Orbits 222

6.1.4 Magnetization 224

6.2 THE FIELD OF A MAGNETIZED OBJECT 225

6.2.1 Bound Currents 225

6.2.2 Physical Interpretation of Bound Currents 227

6.2.3 The Magnetic Field Inside Matter 229

6.3 THE AUXILIARY FIELD H 230

6.3.1 Ampere’s Law in Magnetized Materials 230

6.3.2 A Deceptive Parallel 233

6.4 LINEAR AND NONLINEAR MEDIA 234

6.4.1 Magnetic Susceptibility and Permeability 234

6.4.2 Ferromagnetism 237

SEVEN ELECTRODYNAMICS 243

7.1 ELECTROMOTIVE FORCE 243

7.1.1 Ohm’s Law 243

7.1.2 Electromotive Force 250

7.1.3 Motional emf 252

7.2 FARADAY’S LAW 257

7.2.1 Electromagnetic Induction 257

7.2.2 Inductance 263

7.2.3 Energy in Magnetic Fields 268

7.3 MAXWELL’S EQUATIONS 273

7.3.1 Electrodynamics Before Maxwell 273

7.3.2 How Maxwell Fixed Up Ampere’s Law 274

7.3.3 Maxwell’s Equations and Magnetic Charge 276

7.3.4 Maxwell’s Equations Inside Matter 277

7.3.5 Boundary Conditions 280

7.4 POTENTIAL FORMULATION OF ELECTRODYNAMICS 282

7.4.1 Scalar and Vector Pater？ials 282

7.4.2 Gauge Transformation？ 283

7.4.3 Coulomb Gauge and Lorentz Gauge 284

7.4.4 Lorentz Force Law in Patential Form 286

7.5 ENERGY AND MOMENTUM IN ELECTRODYNAMICS 287

7.5.1 Newton’s Third Law in Electrodynamics 287

7.5.2 Poynting’s Theorem 288

7.5.3 Maxwell’s Stress Tensor 291

EIGHT ELECTROMAGNETIC WAVES 295

8.1 THE WAVE EQUATION 295

8.1.1 Introduction 295

8.1.2 The Wave Equation in One Dimension 297

8.1.3 Sinusoidal Waves 300

8.1.4 Polarization 304

8.1.5 Boundary Conditions：Reflection and Transmission 306

8.2 ELECTROMAGNETIC WAVES IN NONCONDUCTING MEDIA 309

8.2.1 Monochromatic Pl？？？Waves in Vacuum 309

8.2.2 Energy and Momentum of Electromagnetic Waves 313

8.2.3 Propagation Through Linear Media 315

8.2.4 Reflection and Transmission at Normal Incidence 316

8.2.5 Reflection and Transmission at Oblique Incidence 318

8.3 ？CTROMAGNETIC WAVES IN CONDUCTORS 324

8.3.1 The Modified Wave E？？？tion 324

8.3.2 Monochromatic Plane Waves in Conducting Media 327

8.3.3 Reflection and Trans？？？ssion at a Conducting Surface 330

8.4 DISPERSION 333

8.4.1 The Frequency Dependence of ？，μ，and σ 333

8.4.2 Dispersion in Nonconductors 335

8.4.3 Free Electrons in Conductors and Plasmas 340

NINE ELECTROMAGNETIC RADIATION 345

9.1 DIPOLE RADIATION 345

9.1.1 Retarded Potentials 345

9.1.2 Electric Dipole Radiation 350

9.1.3 Magnetic Dipole Radiation 356

9.1.4 Radiation from an Arbitrary Distribution of Charges and Currents 360

9.2 RADIATION FROM A POINT CHARGE 365

9.2.1 Lienard-Wiechert Potentials 365

9.2.2 The Fields of a Point Charge in Motion 370

9.2.3 Power Radiated by a Point Charge 375

9.3 RADIATION REACTIQN 380

9.3.1 The Abraham-Lorentz Formula 380

9.3.2 The Physical Origin of the Radiation Reaction 384

TEN ELECTRODYNAMICS AND RELATIVITY 388

10.1 THE SPECIAL THEORY OF RELATIVITY 388

10.1.1 Einstein’s Postulates 388

10.1.2 The Geometry of Relativity 396

10.1.3 The Lorentz Transformations 406

10.1.4 The Structure of Spacetime 412

10.2 RELATIVISTIC MECHANICS 420

10.2.1 Proper Time and Prop？r Velocity 420

10.2.2 Relativistic Energy and Momentum 422

10.2.3 Relativistic Kinematics 426

10.2.4 Relativistic Dynamics 430

10.3 RELATIVISTIC ELECTRODYNAMICS 435

10.3.1 Magnetism as a Relativistic Phenomenon 435

10.3.2 How the Fields Transform 437

10.3.3 The Field Tensor 445

10.3.4 Electrodynamics in Tensor Notation 448

10.3.5 Potential Formulation of Relativistic Electrodynamics 451

APPENDIX A VECTOR CALCULUS IN CURVILINEAR COORDINATES 454

INTRODUCTION 454

NOTATION 454

GRADIENT 455

DIVERGENCE 456

CURL 458

LAPLACIAN 460

APPENDIX B UNITS 467

INDEX 467