Buch, Englisch, 608 Seiten, Format (B × H): 156 mm x 234 mm, Gewicht: 910 g
Buch, Englisch, 608 Seiten, Format (B × H): 156 mm x 234 mm, Gewicht: 910 g
ISBN: 978-0-19-879813-2
Verlag: OXFORD UNIV PR
An Introduction to Electrodynamics provides an excellent foundation for those undertaking a course on electrodynamics, providing an in-depth yet accessible treatment of topics covered in most undergraduate courses, but goes one step further to introduce advanced topics in applied physics, such as fusions plasmas, stellar magnetism and planetary dynamos.
Some of the central ideas behind electromagnetic waves, such as three-dimensional wave propagation and retarded potentials, are first explored in the introductory background chapters and explained in the much simpler context of acoustic waves. The inclusion of two chapters on magnetohydrodynamics provides the opportunity to illustrate the basic theory of electromagnetism with a wide variety of physical applications of current interest. Davidson places great emphasis on the pedagogical
development of ideas throughout the text, and includes many detailed illustrations and well-chosen exercises to complement the material and encourage student development.
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PART I MATHEMATICAL PRELIMINARIES
1 Vector Calculus 3
1.1 Vectors and Vector Algebra 3
1.1.1 Elementary Definitions and Properties ofVectors 3
1.1.2 Vectors in Cartesian Coordinate Systems 6
1.1.3 Mirror Reflections: Polar Versus Axial Vectors Revisited 9
1.1.4 From Point Vectors to Vector Fields 10
1.2 Line and Surface Integrals 11
1.2.1 Line Integrals, Conservative Fields and the Scalar Potential 11
1.2.2 Surface Integrals and Solenoidal Fields 13
1.3 The Gradient, Divergence and Curl 15
1.3.1 The Gradient and the Scalar Potential Revisited 15
1.3.2 The Divergence, Gauss' Theorem and the Laplacian 17
1.3.3 The Curl 20
1.4 Stokes'Theorem and Conservative Fields Revisited 22
1.5 Solenoidal Fields Revisited: The Vector Potential and Helmholtz's
Decomposition 23
1.6 An Aside: The Troubled History ofVector Calculus 24
References 29
2 The Physical Signatures of Three Important Partial Differential
Equations 31
2.1 Three Important Second-Order Partial Differential Equations 31
2.1.1 The Classification ofLinear Second-Order Partial Differential Equations 31
2.1.2 A Key Property ofthe Diffusion Equation: The Idea ofDiffusion Length 33
2.1.3 A Key Property of the Wave Equation: dAlembert s Solution 35
2.1.4 A Key Property of Poisson's Equation: Greens Inversion Formula 37
2.2 The Diffusion Equation Revisited 38
2.2.1 More on Self-Similar Solutions 38
2.2.2 The Diffusion of Oscillatory Fields 40
2.2.3 Spherically Symmetric Diffusion, Kelvin and the Core of the Earth 41
2.3 Waves and the Wave Equation Revisited 43
2.3.1 From Waves on a String to Alfven Waves in a Plasma 43
2.3.2 Spherically Symmetric Solutions of the Wave Equation 44
2.3.3 Some Wave-Bearing Systems Not Governed by the Wave Equation 45
2.3.4 Dispersive Versus Non-Dispersive Waves and d'Alembert Explained 47
2.3.5 The Concept of Group Velocity for Dispersive Waves 48
2.4 More on Inverting Poisson's Equation 52
2.4.1 An Example ofthe Use of Greens Function 52
2.4.2 A Glimpse at Gauss' Law of Electrostatics 53
2.4.3 A Glimpse at the Biot-Savart Law ofMagnetostatics 55
2.4.4 Information Propagation and Hidden Physics
2.5 Inverting the Inhomogeneous Wave Equation: Retarded Potentials 57
2.5.1 Retarded Potentials in Acoustics 57
2.5.2 Retarded Potentials and Moving Sources 58
References ^
PART II THE FUNDAMENTALS OF ELECTRODYNAMICS
3 ABirdsEyeViewofElectromagnetism 65
3.1 The Lorentz Force, the Electric Field and Charge Conservation 65
3.2 Ampere's Law, the Biot-Savart Law and the Ampere-Maxwell Equation 69
3.3 Faraday's Law in Differential and Integral Form 71
3.4 The Vector Potential 74
3.5 Maxwell's Equations, Electromagnetic Energy and the Poynting Vector 75
3.6 Electromagnetic Waves, Gauges and Retarded Potentials Revisited 77
Reference 81
4 The Foundations ofElectrostatics 83
4.1 Coulomb's Law, the Electrostatic Field and Poisson's Potential 83
4.1.1 Coulomb s Law and the Electric Field 83
4.1.2 The Physical Interpretation of the Electrostatic Potential 86
4.2 Gauss'Law in Differential and Integral Form 88
4.3 Electric Fields Associated with Particularly Simple Charge Distributions 89
4.4 The Electric Dipole 91
4.5 The Elimination of Electrostatic Fields within the Interior of Conductors 93
4.5.1 AMacroscopic Approach to Charges and Electric Fields in Matter 93
4.5.2 There is No Electrostatic Field Inside a Conductor 93
4.5.3 Surface Charges and Boundary Conditions 95
4.6 Simple Examples ofElectric Fields in the Presence of Conductors 96
4.6.1 Electrostatic Shielding 96
4.6.2 Sharp Conductors and Lightning Rods 97
References 101
5 Solving for Electrostatic Fields, the Multipole Expansion and
Electrostatic Energy 103
5.1 Methods for Finding Electrostatic Fields 103
5.1.1 Equations, Boundary Conditions and a Uniqueness Theorem 103
5.1.2 The Method of Images: ASimple Example 105
5.1.3 The Method of Complex Potentials for Two-Dimensional Fields 106
5.2 Examples ofAxisymmetnc Fields 110
5.2.1 A Point Charge Near a Grounded Conducting Sphere 111
5.2.2 A Conducting Sphere m a Uniform Electric Field 112
5.2.3 Charged Cones and Sharp Conductors Revisited 114
5.3 The Average Electric Field in a Spherical Control Volume 1 Dielectrics 139
6.1 The Polarisation of Dielectrics 140
6.1.1 An Experiment with Capacitors 140
6.1.2 Mechanisms ofPolarisation: Electronic Versus Orientation Polarisation 142
6.2 The Polarisation Vector and Bound Charge Distributions 144
6.2.1 The Polarisation Vector, Bound Charges and the Exterior Electric Field 144
6.2.2 The Electric Field Induced by a Uniformly Polarised Sphere 146
6.2.3 The Polarisation Current Density 148
6.2.4 The Macroscopic Electric Field at an Interior Point m a Dielectric 148
6.3 The Combined Electric Field, Gauss' Law and
the Electric Displacement 150
6.4 Relating the Polarisation Vector to the Local Electric Field 151
6.4.1 The Concept of a Local Field 152
6.4.2 Linear Dielectrics and the Relative Permittivity Revisited 153
6.4.3 The Parallel-Plate Capacitor Revisited 154
References 156
7.3 Amperes Law and the Biot-Savart Law 165
7.3.1 Amperes Law
7.3.2 The Biot-Savart Law 166
7.4 Some Simple Examples of Static Magnetic Fields 168
7.4.1 Current Sheets, Current Rings and Solenoids 168
7.4.2 Free-Space Magnetic Fields and Potential Theory Revisited 170
7.4.3 An Aside: Isolated Flux Tubes, Sunspots and Solar Flares 172
7.5 The Kinematics ofAxisymmetric Magnetic Fields 174
7.6 The Magnetic Dipole 176
7.6.1 The Current Ring Revisited 176
7.6.2 The Dipole Moment as a Local Measure of the Mean Magnetic Field 179
7.7 Another Aside: Planetary Magnetic Fields and the Dipole-Rotation
ScalingLaw 180
7.8 The Mathematical Origins of the Biot-Savart Law 184
7.8.1 The Biot-Savart Law as a Solution of Poisson's Equation 184
7.8.2 Confirmation that the Biot-Savart Law Yields the Correct Values of
V • B and V x B 185
7.9 From Statics to Dynamics: A Second Glimpse at Retarded Potentials 186
References 193
8 Magnetostatics II: Dipoles, Force Distributions and Energy 195
8.1 The Dipole Revisited: Localised Current Distributions and their Far Field 195
8.2 The Torque, Force and Energy of a Dipole Sitting in an External
Magnetic Field 198
8.2.1 The Magnetic Torque Acting on a Dipole in an External Magnetic Field 198
8.2.2 The Net Lorentz Force Acting on a Dipole in an External Magnetic Field 200
8.2.3 The Potential Energy of a Dipole in an External Magnetic Field 201
8.3 The Average Magnetic Field in a Spherical Control Volume 202
8.4 The Potential Energy ofa Steady Distribution of Currents 204
8.4.1 The Potential Energy of a Dipole Revisited 204
8.4.2 The Mutual Potential Energy ofSteady Currents 207
8.4.3 The Total Magnetic Energy of Steady Currents 208
8.5 How to Create the Perfect Dipole: The Rotating Charged Sphere 209
8.6 Maxwell's Stresses and Faraday s Tension 210
8.7 An Aside: The Difficulty of Confining Fusion Plasmas with
Magnetic Fields 214
8.8 Force-Free Fields and Magnetic Helicity Revisited 216
References 219
9 Magnetic Fields in Matter 221
9.1 Three Types ofMagnetism in Matter 221
9.1.1 Diamagnetism, Paramagnetism and Ferromagnetism 221
9.1.2 ANon-Quantum Cartoon on Diamagnetism 223
9.2 The Magnetisation Vector and Bound Current Distributions 225 9.3 The Uniformly Magnetised Sphere 229
9.4 Amperes Law Applied to Magnetic Materials: The H Field 230
9.5 The Magnetic Susceptibility and Relative Permeability 232
9.6 Magnetic Circuits and Electromagnets 234
9.7 Magnetic Shielding 235
9.8 Magnetisation Curves, Hysteresis and Permanent Magnets 236
References 240
10 Faraday's Law ofElectromagnetic Induction 241
10.1 Electromagnetic Induction: One Physical Law or Two? 242
10.2 The Definition of the Electromotive Force 244
10.3 From Faraday's Law in Differential Form to Faraday s Flux Rule 246
10.3.1 Faraday's Differential Law Versus Faraday's Flux Rule 246
10.3.2 A Useful Kinematic Result 247
10.3.3 Deriving the Flux Rule from Faraday's Law 248
10.4 Applications of Faraday s Flux Rule 249
10.5 The Flux Rule Applied to Material Circuits: Lenz's Law and
Alfven's Theorem 252
10.5.1 Lenz's Law 253
10.5.2 Alfven's Theorem 253
10.6 The Oscillating Slide—A Poor Man's Alfven Wave 256
10.7 Faraday's Law Expressed in Terms ofthe Vector Potential 258
References 260
11 Quasi-Static Magnetic Fields: Magnetic Energy and Inductance .... 261
11.1 The Idea of a Quasi-Static Magnetic Field 261
11.2 Magnetic Energy 262
11.2.1 The Rate ofWorking of the Electric Field on a Conductor 262
11.2.2 A Quasi-Steady Magnetic Energy Equation 264
11.2.3 What Happened to the Electric Energy? 265
11.2.4 Allowing for External Energy Sources 266
11.2.5 AnAmbiguity in the Poynting Energy Flux 267
11.3 Inductance 268
11.3.1 Mutual Inductance and Neumann's Equation 268
11.3.2 Faraday's Flux Rule Revisited and Self-Inductance 270
11.3.3 Magnetic Energy in Terms of Current and Inductance 273
11.3.4 Magnetic Forces in Terms of Inductance 275
11.4 Magnetic Diffusion Revisited 277
11.5 Quasi-magnetostatics versus Quasi-electrostatics 278
References 284
12 Transient and AC Circuits 285
12.1 Idealised Components in Circuit Theory 285
12.2 An Example of Transient Response: The RLC Damped Oscillator 288
12.3 Multi-loop Networks: Kirchhoff s Rules for Ideal Circuits 289
12.4 AC Networks: Impedances and Equivalent Circuits 290
12.5 Ladder Networks: A Glimpse at Filters and Transmission Lines 292
12.6 Leaving the Quasi-Static Regime
12.6.1 AParallel-Plate Capacitor at High Frequencies
12.6.2 A Solenoid at High Frequencies
r. C ^01
References
13 Static Versus Dynamic Fields: Maxwell's Equations 303
13.1 A Recapitulation of the Laws ofElectrostatics and
Quasi-Magnetostatics
13.2 Charge Conservation, the Displacement Current and Maxwell's
Equations
13.3 How the Displacement Current Works
13.4 Maxwell's Equations in Vacuum and the Speed
of Light
13.5 Energy Relationships and the Poynting Flux 312
13.6 The Lorenz Gauge and Wave Equations for the Potentials 314
13.7 Inverting the Wave Equations: Retarded Potentials 316
13.8 From Retarded Potentials to Retarded Fields 317
13.9 A Summary of the Equations of Electrodynamics 319
13.10 The Electromagnetic Stress Tensor and Field Momentum 321
References 326
PART III SPECIAL TOPICS
14 Confined Waves: Transmission Lines, Waveguides
and Resonant Cavities 329
14.1 The Transmission of Energy and Information at High Frequencies 329
14.2 Transmission Lines and TEM Waves 331
14.2.1 The Equivalent Circuit for Ideal Transmission Lines: Heavisides
Equations 331
14.2.2 Deriving the Ideal Transmission Line Equations from Maxwell's
Equations 334
14.2.3 Generalising the Ideal Transmission Line Equations to Arbitrary
Cross-sections 335
14.2.4 Footnotes on Transmission Lines: Ohmic Losses, Energy and
Reflections 338
14.3 Waveguides 340
14.3.1 Two More Types ofWaves: TM and TE Waves 340
14.3.2 Mode Shapes in Rectangular and Circular Waveguides 343
14.3.3 The Group Velocity in Waveguides 344
14.4 Resonant Cavities 345
14.5 An Aside: Optical Fibres 347
References 352
15 Maxwell's Equations in Free Space I: The Propagation ofWaves 353
15.1 The Electromagnetic Spectrum 353
15.2 Maxwell's Equations in Vacuum 354
15.2.1 Retarded Potentials and Wave Propagation 354
15.2.2 Symmetries in the Free-Space Maxwell Equations 356
15.3 Plane Waves 356
15.3.1 TEM Waves and D'Alembert Revisited 356
15.3.2 Monochromatic Plane Waves, Polarisation and the Poynting Flux 358
15.4 Spherical Waves Far from Their Sources: A Glimpse at Radiation 360
15.4.1 Radially Propagating Waves Far from an Oscillating Electric Dipole 360
15.4.2 Radially Propagating Waves Far from an Oscillating Magnetic Dipole 363
15.5 Three-Dimensional Waves: The Potentials of Hertz and Debye 364
15.5.1 The Hertz Vectors 364
15.5.2 The Poloidal-Toroidal Decomposition of Solenoidal Vector Fields 367
15.5.3 The P-T Decomposition Applied to the Free-Space Equations 369
15.5.4 From the P-T Decomposition to the Debye Potentials 370
References 373
16 Maxwells Equations in Free Space II: Radiation 375
16.1 What is a Radiation Field in Electrodynamics? 375
16.2 The Magnetic Dipole 376
16.2.1 The Far-Field Potential 377
16.2.2 The Electromagnetic Field and Poynting Energy Flux 379
16.3 The Hertzian Dipole 380
16.3.1 The Far-Field Potentials 380
16.3.2 The Electromagnetic Field and Poynting Energy Flux 382
16.4 The Far Field of an Arbitrary Compact Source 384
16.4.1 The Far-Field Potentials 385
16.4.2 The Electromagnetic Field and Poynting Energy Flux 387
16.4.3 A Comparison of Sources 389
16.5 Larmor's Radiation Equation 389
16.6 An Aside: Why Don't Atoms Collapse? 390
16.7 The Centre-Fed Linear Antenna 391
References 395
17 Maxwell's Equations in Free Space III: The Fields
ofMoving Charges 397
17.1 The Potentials for a Point Charge Executing Arbitrary Motion 398
17.1.1 Moving Sources in Acoustics Revisited 398
17.1.2 The Lienard-Wiechert Potentials for a Moving Point Charge 399
17.2 From the Lienard-Wiechert Potentials to the Electromagnetic Field 400
17.2.1 Kinematic Preliminaries 400
17.2.2 The Electric Field for a Moving Point Charge 402
17.2.3 The Magnetic Field and Poynting Vector 403
17.2.4 The Case of Uniform Motion and a Glimpse at the Lorentz
Transformation 405
17.2.5 The Sudden Deceleration of a Moving Charge: Bremsstrahlung 407
17.3 The Heaviside-Feynman Equations for the Field of a Point Charge 408
17.4 Radiation Patterns and a Second Look at Larmor s Radiation Equation 411
References 415
18 Maxwell's Equations in Dielectric and Magnetic Materials 417
18.1 A Summary ofthe Behaviour of Static Fields in Matter 417
18.2 A Second Look at the Polarisation Current Density 420
18.3 The Macroscopic Maxwell Equations for Polarised
and Magnetised Matter 421
18.4 WhyAre There No Magnetic Charges? 424
18.5 Boundary Conditions for the Macroscopic Maxwell Equations 424
References All
19 Plane Waves in Stationary Dielectrics and Conductors 429
19.1 Electromagnetic Waves in Stationary Dielectrics 429
19.1.1 Governing Equations and Boundary Conditions 429
19.1.2 The Transmission and Reflection of Normal Waves at Plane Interfaces 430
19.1.3 The Refraction and Reflection of Oblique Waves at Plane Interfaces 432
19.1.4 Total Internal Reflection 436
19.1.5 Dispersive Waves and Rainbows 437
19.2 An Aside: The Scattering ofLight 440
19.2.1 Rayleigh Scattering ofLight and Why the Sky Is Blue 440
19.2.2 Thomson Scattering by Free Charges 442
19.3 Waves in Stationary Conductors 442
19.3.1 Damped Waves and the Skin Depth 442
19.3.2 Reflection at the Surface of a Good Conductor 445
References 447
20 Magnetohydrodynamics I: Governing Equations
and Kinematic Theorems 449
20.1 What is Magnetohydrodynamics (MHD)? 449
20.2 A Crash Course in Fluid Dynamics 451
20.2.1 An Eulerian Description ofMotion and the Convective Derivative 451
20.2.2 More Kinematics: Mass Conservation, Spin and Deformation in a Fluid 452
20.2.3 Dynamics at Last: The Navier-Stokes Equation and Energy Dissipation 454
20.2.4 A Transport Equation for Vorticity 457
20.2.5 Helmholtz's Laws oflnviscid Vortex Dynamics and Kinetic Helicity 460 20.2.6 Boundary Layers and Turbulence 453
20.3 Stripping Down Maxwells Equations for MHD 454
20.3.1 The Charge Density and Electrostatic Energy are Negligible 464 20.3.2 The Reduced Form ofMaxwells Equations for MHD 466 20.3.3 An Evolution Equation for the Magnetic Field 467 20.4 Some Simple Kinematic Theorems 453 20.4.1 The Analogy Between the Evolution ofMagnetic Fields and Vorticity 468 20.4.2 Perfect Conductors LAlfvens Theorem Revisited 469 20.4.3 Perfect Conductors 2: The Conservation ofMagnetic Helicity 471 20.4.4 Azimuthal Field Generation by Differential Rotation 472 20.4.5 The Phenomenon ofFlux Expulsion 474 20.5 The Governing Equations ofMHD 476 20.5.1 Governing Equations and Dimensionless Groups 475 20.5.2 Energy Relationships
20.5.3 The Lorentz Force Revisited: Maxwells Stresses Versus
Faraday's Tension 479
20.6 Some Simple Dynamics: Alfven Waves and Conservation of Cross
Helicity 480
References 485
21 Magnetohydrodynamics II: An Introduction to Plasma Containment;
Planetary Dynamos and Stellar Magnetism 487
21.1 The Magnetic Confinement of Fusion Plasmas 487
21.1.1 Toroidal Confinement in Tokomaks 487
21.1.2 Sausage-Mode and Kink Instabilities 489
21.1.3 Interchange and Ballooning Instabilities 491
21.2 Alfven Waves Revisited 493
21.2.1 Small-Amplitude Alfven Waves 493
21.2.2 Finite-Amplitude Waves, Colliding Wave Packets, and Alfvenic
Turbulence 495
21.3 Planetary Dynamos 496
21.3.1 The Need for a Dynamo Theory for the Earth 497
21.3.2 The Structure ofthe Planets, Force Balances and Planetary
Magnetic Fields 498
21.3.3 Geophysical Constraints on Planetary Dynamo Theories 501
21.3.4 Four Kinematic Theorems 502
21.3.5 Parker's Lift-and-Twist Mechanism and Mean-Field Electrodynamics 5 07
21.3.6 Two Popular Kinematic Dynamo Cartoons for Planets and Stars 510
21.3.7 Where Does the Helicity Come From and Whyis it Spatially Segregated? 511
21.4 Stellar MHD 514
21.4.1 The Structure of the Sun 514
21.4.2 Sunspots, the 22-Year Solar Cycle and the Solar Dynamo 516
21.4.3 The Solar Atmosphere: Prominences, Flares and Coronal Mass Ejections 517
21.4.4 The Solar Wind 519
References 522
22 An Introduction to Special Relativity
22.1 On the Curious Nature of Faraday's Flux Law
22.2 A Re-examination of Space and Time
22.2.1 Space and Time According to Newton and Galileo
22.2.2 Maxwell, the Aether and a Crisis in the Newtonian Paradigm
22.2.3 Einstein's Postulates
22.2.4 The Lorentz Transformation
22.2.5 The Downfall of Simultaneity
22.2.6 Time Dilation and the Lorentz-FitzGerald Contraction
22.2.7 The Relativistic Transformation ofVelocity
22.3 Space-Time, Relativistic Invariants and Four-Vectors
22.4 Relativistic Dynamics
22.4.1 Proper Time, Proper Velocity and Relativistic Momentum
22.4.2 Newton's Second Law and Relativistic Energy
References 523
23 Electromagnetism and Special Relativity 541
23.1 The Four-Current Density and Charge Conservation
in Relativistic Form 541
23.1.1 The Transformation of Charge, Volume and Charge Density 541
23.1.2 The Four-Current Density and Charge Conservation Revisited 542
23.2 The Four-Potential 544
23.3 The Four-Potential for a Charge Moving at Constant Velocity 545
23.4 How the E and B Fields Transform 547
23.4.1 Some Simple Examples 547
23.4.2 The General Transformation Laws 551
References 553
Appendix I Some Useful Mathematical Relationships 555
AI.l Vector Identities and Theorems 555
AI.1.1 Scalar and Vector Triple Products 555
AI.1.2 Grad, Div and Curl in Cartesian Coordinates 555
AI.1.3 Vector Identities 555
AI.l.4 Integral Theorems 556
AI.l.5 A Useful Vector Relationship 556
AI.l.6 The HelmholtzDecomposition 556
AI.2 Cylindrical and Spherical Polar Coordinates 556
AI.3 Legendre Functions 55g
AI.4 The Delta Function 550
Appendix II SI Versus Gaussian Units 562
Appendix III The Pioneers of Electromagnetism: Brief Historical
Sketches
AIII.l Ampere and the Birth ofElectrodynamics 565
AIII.2 Faraday Prince ofExperimentalists 557
AIII.3 Maxwell: The End of One Scientific Epoch and the Beginning ofAnother
AIII.4 Heaviside: The Maverick Maxwellian 572
AIII.5 Hertz: Maxwell Vindicated 574
References and Further Reading 576
Index
577
