Foundations of Applied Electrodynamics

FOUNDATIONS\nOF APPLIED\nELECTRODYNAMICS......Page 5
Contents......Page 9
Preface......Page 17
1 Maxwell Equations......Page 21
1.1.1 Coulomb’s Law......Page 22
1.1.2 Ampère’s Law......Page 25
1.1.4 Law of Conservation of Charge......Page 29
1.2 Maxwell Equations, Constitutive Relation, and Dispersion......Page 30
1.2.1 Maxwell Equations and Boundary Conditions......Page 31
1.2.2 Constitutive Relations......Page 35
1.2.3 Wave Equations......Page 38
1.2.4 Dispersion......Page 40
1.3.1 Superposition Theorem......Page 42
1.3.3 Conservation of Electromagnetic Energy......Page 43
1.3.4 Conservation of Electromagnetic Momentum......Page 45
1.3.6 Uniqueness Theorems......Page 47
1.3.7 Equivalence Theorems......Page 52
1.3.8 Reciprocity......Page 56
1.4 Wavepackets......Page 59
1.4.1 Spatial Wavepacket and Temporal Wavepacket......Page 60
1.4.3 Energy Density for Wavepackets......Page 62
1.4.4 Energy Velocity and Group Velocity......Page 65
1.4.5 Narrow-band Stationary Stochastic Vector Field......Page 67
2 Solutions of Maxwell Equations......Page 69
2.1.1 Linear Space, Normed Space and Inner Product Space......Page 70
2.1.2 Linear and Multilinear Maps......Page 72
2.2 Classification of Partial Differential Equations......Page 74
2.2.1 Canonical Form of Elliptical Equations......Page 76
2.2.3 Canonical Form of Parabolic Equations......Page 77
2.3.1 Limitation of Classical Solutions......Page 78
2.3.2 Theory of Generalized Functions......Page 80
2.3.3 Sobolev Spaces......Page 86
2.3.4 Generalized Solutions of Partial Differential Equations......Page 87
2.4.1 Rectangular Coordinate System......Page 89
2.4.2 Cylindrical Coordinate System......Page 90
2.4.3 Spherical Coordinate System......Page 91
2.5.1 Fundamental Solutions of Partial Differential Equations......Page 93
2.5.2 Integral Representations of Arbitrary Fields......Page 94
2.5.3 Integral Representations of Electromagnetic Fields......Page 98
2.6.1 Vector Potential, Scalar Potential, and Gauge Conditions......Page 103
2.6.2 Hertz Vectors and Debye Potentials......Page 107
2.6.3 Jump Relations in Potential Theory......Page 109
2.7.1 Generalized Calculus of Variation......Page 113
2.7.2 Lagrangian Formulation......Page 115
2.7.3 Hamiltonian Formulation......Page 120
3 Eigenvalue Problems......Page 125
3.1.1 Compact Operators and Embeddings......Page 126
3.1.2 Closed Operators......Page 129
3.1.3 Spectrum and Resolvent of Linear Operators......Page 130
3.1.4 Adjoint Operators and Symmetric Operators......Page 132
3.1.5 Energy Space......Page 134
3.1.6 Energy Extension, Friedrichs Extension and Generalized Solution......Page 136
3.2.1 Positive-Bounded-Below Symmetric Operators......Page 140
3.2.2 Compact Symmetric Operators......Page 146
3.3.1 Mode Theory for Waveguides......Page 150
3.3.2 Mode Theory for Cavity Resonators......Page 160
3.4.1 Mode Theory for Spherical Waveguides......Page 165
3.4.2 Singular Functions and Singular Values......Page 169
3.5 Eigenfunctions of Curl Operator......Page 170
4 Antenna Theory......Page 173
4.1 Antenna Parameters......Page 174
4.1.1 Radiation Patterns and Radiation Intensity......Page 175
4.1.2 Radiation Efficiency, Antenna Efficiency and Matching Network Efficiency......Page 176
4.1.4 Input Impedance, Bandwidth and Antenna Quality Factor......Page 177
4.1.5 Vector Effective Length, Equivalent Area and Antenna Factor......Page 178
4.1.6 Polarization and Coupling......Page 181
4.2 Properties of Far Fields......Page 182
4.3.1 Field Expansions in Terms of Spherical Vector Wavefunctions......Page 185
4.3.2 Completeness of Spherical Vector Wavefunctions......Page 191
4.4 Foster Theorems and Relationship Between Quality Factor and Bandwidth......Page 192
4.4.1 Poynting Theorem and the Evaluation of Antenna Quality Factor......Page 193
4.4.2 Equivalent Circuit for Transmitting Antenna......Page 196
4.4.3 Foster Theorems for Ideal Antenna and Antenna Quality Factor......Page 198
4.4.4 Relationship Between Antenna Quality Factor and Bandwidth......Page 202
4.5.1 Spherical Wavefunction Expansion for Antenna Quality Factor......Page 203
4.5.2 Minimum Possible Antenna Quality Factor......Page 205
4.6.1 Directive Antenna......Page 206
4.6.2 Omni-Directional Antenna......Page 209
4.6.3 Best Possible Antenna Performance......Page 212
4.7.1 Quality Factor for Arbitrary Antenna......Page 213
4.7.2 Quality Factor for Small Antenna......Page 214
4.7.3 Some Remarks on Electromagnetic Stored Energy......Page 220
5 Integral Equation Formulations......Page 223
5.1 Integral Equations......Page 224
5.2 TEM Transmission Lines......Page 225
5.3 Waveguide Eigenvalue Problems......Page 227
5.3.1 Spurious Solutions and their Discrimination......Page 228
5.3.2 Integral Equations Without Spurious Solutions......Page 230
5.4 Metal Cavity Resonators......Page 231
5.5.1 Three-Dimensional Scatterers......Page 233
5.5.2 Two-Dimensional Scatterers......Page 244
5.5.3 Scattering Cross-Section......Page 250
5.5.4 Low Frequency Solutions of Integral Equations......Page 251
5.6 Multiple Metal Antenna System......Page 253
5.7.1 Projection Method......Page 258
5.7.2 Moment Method......Page 259
5.7.3 Construction of Approximating Subspaces......Page 260
6.1 Transmission Line Theory......Page 265
6.1.1 Transmission Line Equations......Page 266
6.1.2 Signal Propagations in Transmission Lines......Page 269
6.2.1 One-Port Network......Page 270
6.2.2 Multi-Port Network......Page 272
6.3 Waveguide Junctions......Page 274
6.4.1 Impedance Matrix......Page 278
6.4.2 Scattering Matrix......Page 282
6.4.3 Antenna System with Large Separations......Page 283
6.5.1 Universal Power Transmission Formula......Page 287
6.5.2 Power Transmission Between Two Planar Apertures......Page 290
6.5.3 Power Transmission Between Two Antenna Arrays......Page 293
6.6.1 Compensation Theorem for Time-Harmonic Fields......Page 295
6.6.2 Scattering Parameters in a Scattering Environment......Page 296
6.6.3 Antenna Input Impedance in a Scattering Environment......Page 299
6.7.1 RLC Equivalent Circuit for a One-Port Microwave Network......Page 300
6.7.2 RLC Equivalent Circuits for Current Sources......Page 302
7 Fields in Inhomogeneous Media......Page 307
7.1.1 The Spectrum......Page 308
7.1.2 Spectral Theorem......Page 309
7.1.3 Generalized Eigenfunctions of Self-Adjoint Operators......Page 310
7.1.4 Bilinear Forms......Page 312
7.1.5 Min-Max Principle......Page 314
7.1.6 A Bilinear Form for Maxwell Equations......Page 315
7.2.1 Wave Equations in Inhomogeneous Media......Page 316
7.2.2 Waves in Slowly Varying Layered Media and WKB Approximation......Page 317
7.2.3 High Frequency Approximations and Geometric Optics......Page 318
7.2.4 Reflection and Transmission in Layered Media......Page 323
7.3.1 General Field Relationships......Page 325
7.3.2 Symmetric Formulation......Page 326
7.3.3 Asymmetric Formulation......Page 327
7.4.1 Circular Optical Fiber......Page 329
7.4.2 Guidance Condition......Page 332
7.4.3 Eigenvalues and Essential Spectrum......Page 333
7.5 Inhomogeneous Cavity Resonator......Page 339
7.5.1 Mode Theory......Page 340
7.5.2 Field Expansions......Page 345
8 Time-domain Theory......Page 349
8.1 Time-domain Theory of Metal Waveguides......Page 350
8.1.1 Field Expansions......Page 351
8.1.2 Solution of the Modified Klein–Gordon Equation......Page 354
8.1.3 Excitation of Waveguides......Page 358
8.2.1 Field in Arbitrary Cavities......Page 362
8.2.2 Fields in Waveguide Cavities......Page 369
8.3.1 Transverse Field Equations......Page 380
8.3.2 Spherical Transmission Line Equations......Page 381
8.4.1 Radiation from an Arbitrary Source......Page 383
8.4.2 Radiation from Elementary Sources......Page 385
8.4.3 Enhancement of Radiation......Page 387
8.4.4 Time-domain Integral Equations......Page 389
9 Relativity......Page 399
9.1.1 Tensor Algebra......Page 400
9.1.2 Tangent Space, Cotangent Space and Tensor Space......Page 404
9.1.3 Metric Tensor......Page 407
9.2.1 Galilean Relativity Principle......Page 408
9.3.1 Intervals......Page 409
9.3.2 Derivation of the Lorentz Transformation......Page 411
9.3.3 Properties of Space–Time......Page 413
9.4.2 Four-Momentum Vector......Page 415
9.4.3 Relativistic Equation of Motion......Page 416
9.4.4 Angular Momentum Tensor and Energy-Momentum Tensor......Page 418
9.5.1 Covariance of Continuity Equation......Page 420
9.5.2 Covariance of Maxwell Equations......Page 421
9.5.3 Transformation of Electromagnetic Fields and Sources......Page 422
9.5.4 Covariant Forms of Electromagnetic Conservation Laws......Page 423
9.6 General Theory of Relativity......Page 424
9.6.1 Principle of Equivalence......Page 425
9.6.2 Manifolds......Page 426
9.6.3 Tangent Bundles, Cotangent Bundles and Tensor Bundles......Page 427
9.6.4 Riemannian Manifold......Page 429
9.6.5 Accelerated Reference Frames......Page 430
9.6.6 Time and Length in Accelerated Reference Frame......Page 433
9.6.7 Covariant Derivative and Connection......Page 435
9.6.8 Geodesics and Equation of Motion in Gravitational Field......Page 438
9.6.9 Bianchi Identities......Page 441
9.6.11 Einstein Field Equations......Page 442
9.6.12 The Schwarzschild Solution......Page 445
9.6.13 Electromagnetic Fields in an Accelerated System......Page 446
10 Quantization of Electromagnetic Fields......Page 449
10.1.1 Basic Postulates of Quantum Mechanics......Page 450
10.1.2 Quantum Mechanical Operators......Page 451
10.1.3 The Uncertainty Principle......Page 452
10.1.4 Quantization of Classical Mechanics......Page 454
10.1.5 Harmonic Oscillator......Page 455
10.1.6 Systems of Identical Particles......Page 457
10.2 Quantization of Free Electromagnetic Fields......Page 458
10.2.1 Quantization in Terms of Plane Wave Functions......Page 459
10.2.2 Quantization in Terms of Spherical Wavefunctions......Page 465
10.3.1 Statistical States......Page 468
10.3.2 Most Probable Distributions......Page 469
10.3.3 Blackbody Radiation......Page 470
10.4.1 The Hamiltonian Function of the Coupled System......Page 471
10.4.2 Quantization of the Coupled System......Page 473
10.4.3 Perturbation Theory......Page 475
10.4.4 Induced Transition and Spontaneous Transition......Page 478
10.4.5 Absorption and Amplification......Page 481
10.4.6 Quantum Mechanical Derivation of Dielectric Constant......Page 482
10.5.1 The Klein–Gordon Equation......Page 485
10.5.2 The Dirac Equation......Page 486
A.2 Set Operations......Page 489
A.3 Set Algebra......Page 490
B.1 Formulas from Vector Analysis......Page 491
B.2 Vector Analysis in Curvilinear Coordinate Systems......Page 492
B.2.1 Curvilinear Coordinate Systems......Page 493
B.2.2 Differential Operators......Page 496
B.2.3 Orthogonal Systems......Page 498
C.1 Bessel Functions......Page 501
C.2 Spherical Bessel Functions......Page 502
C.3 Legendre Functions and Associated Legendre Functions......Page 503
Appendix D: SI Unit System......Page 505
Bibliography......Page 507
Index......Page 517