Foundations of Applied Electrodynamics

511 Pages·2010·6.883 MB·English

by  Wen Geyi(auth.)

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FOUNDATIONS
OF APPLIED
ELECTRODYNAMICS
Foundations of Applied Electrodynamics Wen Geyi
© 2010 John Wiley & Sons Ltd. ISBN: 978-0-470-68862-5
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FOUNDATIONS
OF APPLIED
ELECTRODYNAMICS
WenGeyi
Waterloo,Canada
A John Wiley and Sons, Ltd., Publication
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Thiseditionfirstpublished2010
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Wen,Geyi.
Foundationsofappliedelectrodynamics/GeyiWen.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-470-68862-5(cloth)
1.Electrodynamics–Mathematics. I.Title.
QC631.3.W46 2010
537.601(cid:2)51–dc22
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN 978-0-470-68862-5(H/B)
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Contents
Preface xv
1 MaxwellEquations 1
1.1 ExperimentalLaws 2
1.1.1 Coulomb’sLaw 2
1.1.2 Ampe`re’sLaw 5
1.1.3 Faraday’sLaw 9
1.1.4 LawofConservationofCharge 9
1.2 MaxwellEquations,ConstitutiveRelation,andDispersion 10
1.2.1 MaxwellEquationsandBoundaryConditions 11
1.2.2 ConstitutiveRelations 15
1.2.3 WaveEquations 18
1.2.4 Dispersion 20
1.3 TheoremsforElectromagneticFields 22
1.3.1 SuperpositionTheorem 22
1.3.2 CompensationTheorem 23
1.3.3 ConservationofElectromagneticEnergy 23
1.3.4 ConservationofElectromagneticMomentum 25
1.3.5 ConservationofElectromagneticAngularMomentum 27
1.3.6 UniquenessTheorems 27
1.3.7 EquivalenceTheorems 32
1.3.8 Reciprocity 36
1.4 Wavepackets 39
1.4.1 SpatialWavepacketandTemporalWavepacket 40
1.4.2 SignalVelocityandGroupVelocity 42
1.4.3 EnergyDensityforWavepackets 42
1.4.4 EnergyVelocityandGroupVelocity 45
1.4.5 Narrow-bandStationaryStochasticVectorField 47
2 SolutionsofMaxwellEquations 49
2.1 LinearSpaceandLinearOperator 50
2.1.1 LinearSpace,NormedSpaceandInnerProductSpace 50
2.1.2 LinearandMultilinearMaps 52
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2.2 ClassificationofPartialDifferentialEquations 54
2.2.1 CanonicalFormofEllipticalEquations 56
2.2.2 CanonicalFormofHyperbolicEquations 57
2.2.3 CanonicalFormofParabolicEquations 57
2.3 ModernTheoryofPartialDifferentialEquations 58
2.3.1 LimitationofClassicalSolutions 58
2.3.2 TheoryofGeneralizedFunctions 60
2.3.3 SobolevSpaces 66
2.3.4 GeneralizedSolutionsofPartialDifferentialEquations 67
2.4 MethodofSeparationofVariables 69
2.4.1 RectangularCoordinateSystem 69
2.4.2 CylindricalCoordinateSystem 70
2.4.3 SphericalCoordinateSystem 71
2.5 MethodofGreen’sFunction 73
2.5.1 FundamentalSolutionsofPartialDifferentialEquations 73
2.5.2 IntegralRepresentationsofArbitraryFields 74
2.5.3 IntegralRepresentationsofElectromagneticFields 78
2.6 PotentialTheory 83
2.6.1 VectorPotential,ScalarPotential,andGaugeConditions 83
2.6.2 HertzVectorsandDebyePotentials 87
2.6.3 JumpRelationsinPotentialTheory 89
2.7 VariationalPrinciples 93
2.7.1 GeneralizedCalculusofVariation 93
2.7.2 LagrangianFormulation 95
2.7.3 HamiltonianFormulation 100
3 EigenvalueProblems 105
3.1 IntroductiontoLinearOperatorTheory 106
3.1.1 CompactOperatorsandEmbeddings 106
3.1.2 ClosedOperators 109
3.1.3 SpectrumandResolventofLinearOperators 110
3.1.4 AdjointOperatorsandSymmetricOperators 112
3.1.5 EnergySpace 114
3.1.6 EnergyExtension,FriedrichsExtensionand
GeneralizedSolution 116
3.2 EigenvalueProblemsforSymmetricOperators 120
3.2.1 Positive-Bounded-BelowSymmetricOperators 120
3.2.2 CompactSymmetricOperators 126
3.3 InteriorElectromagneticProblems 130
3.3.1 ModeTheoryforWaveguides 130
3.3.2 ModeTheoryforCavityResonators 140
3.4 ExteriorElectromagneticProblems 145
3.4.1 ModeTheoryforSphericalWaveguides 145
3.4.2 SingularFunctionsandSingularValues 149
3.5 EigenfunctionsofCurlOperator 150
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Contents ix
4 AntennaTheory 153
4.1 AntennaParameters 154
4.1.1 RadiationPatternsandRadiationIntensity 155
4.1.2 RadiationEfficiency,AntennaEfficiencyandMatching
NetworkEfficiency 156
4.1.3 DirectivityandGain 157
4.1.4 InputImpedance,BandwidthandAntennaQualityFactor 157
4.1.5 VectorEffectiveLength,EquivalentAreaandAntennaFactor 158
4.1.6 PolarizationandCoupling 161
4.2 PropertiesofFarFields 162
4.3 SphericalVectorWavefunctions 165
4.3.1 FieldExpansionsinTermsofSphericalVectorWavefunctions 165
4.3.2 CompletenessofSphericalVectorWavefunctions 171
4.4 FosterTheoremsandRelationshipBetweenQualityFactorand
Bandwidth 172
4.4.1 PoyntingTheoremandtheEvaluationofAntennaQualityFactor 173
4.4.2 EquivalentCircuitforTransmittingAntenna 176
4.4.3 FosterTheoremsforIdealAntennaandAntennaQualityFactor 178
4.4.4 RelationshipBetweenAntennaQualityFactorandBandwidth 182
4.5 MinimumPossibleAntennaQualityFactor 183
4.5.1 SphericalWavefunctionExpansionforAntennaQualityFactor 183
4.5.2 MinimumPossibleAntennaQualityFactor 185
4.6 MaximumPossibleProductofGainandBandwidth 186
4.6.1 DirectiveAntenna 186
4.6.2 Omni-DirectionalAntenna 189
4.6.3 BestPossibleAntennaPerformance 192
4.7 EvaluationofAntennaQualityFactor 193
4.7.1 QualityFactorforArbitraryAntenna 193
4.7.2 QualityFactorforSmallAntenna 194
4.7.3 SomeRemarksonElectromagneticStoredEnergy 200
5 IntegralEquationFormulations 203
5.1 IntegralEquations 204
5.2 TEMTransmissionLines 205
5.3 WaveguideEigenvalueProblems 207
5.3.1 SpuriousSolutionsandtheirDiscrimination 208
5.3.2 IntegralEquationsWithoutSpuriousSolutions 210
5.4 MetalCavityResonators 211
5.5 ScatteringProblems 213
5.5.1 Three-DimensionalScatterers 213
5.5.2 Two-DimensionalScatterers 224
5.5.3 ScatteringCross-Section 230
5.5.4 LowFrequencySolutionsofIntegralEquations 231
5.6 MultipleMetalAntennaSystem 233
5.7 NumericalMethods 238
5.7.1 ProjectionMethod 238
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5.7.2 MomentMethod 239
5.7.3 ConstructionofApproximatingSubspaces 240
6 NetworkFormulations 245
6.1 TransmissionLineTheory 245
6.1.1 TransmissionLineEquations 246
6.1.2 SignalPropagationsinTransmissionLines 249
6.2 ScatteringParametersforGeneralCircuits 250
6.2.1 One-PortNetwork 250
6.2.2 Multi-PortNetwork 252
6.3 WaveguideJunctions 254
6.4 MultipleAntennaSystem 258
6.4.1 ImpedanceMatrix 258
6.4.2 ScatteringMatrix 262
6.4.3 AntennaSystemwithLargeSeparations 263
6.5 PowerTransmissionBetweenAntennas 267
6.5.1 UniversalPowerTransmissionFormula 267
6.5.2 PowerTransmissionBetweenTwoPlanarApertures 270
6.5.3 PowerTransmissionBetweenTwoAntennaArrays 273
6.6 NetworkParametersinaScatteringEnvironment 275
6.6.1 CompensationTheoremforTime-HarmonicFields 275
6.6.2 ScatteringParametersinaScatteringEnvironment 276
6.6.3 AntennaInputImpedanceinaScatteringEnvironment 279
6.7 RLCEquivalentCircuits 280
6.7.1 RLCEquivalentCircuitforaOne-PortMicrowaveNetwork 280
6.7.2 RLCEquivalentCircuitsforCurrentSources 282
7 FieldsinInhomogeneousMedia 287
7.1 FoundationsofSpectralAnalysis 288
7.1.1 TheSpectrum 288
7.1.2 SpectralTheorem 289
7.1.3 GeneralizedEigenfunctionsofSelf-AdjointOperators 290
7.1.4 BilinearForms 292
7.1.5 Min-MaxPrinciple 294
7.1.6 ABilinearFormforMaxwellEquations 295
7.2 PlaneWavesinInhomogeneousMedia 296
7.2.1 WaveEquationsinInhomogeneousMedia 296
7.2.2 WavesinSlowlyVaryingLayeredMediaandWKBApproximation 297
7.2.3 HighFrequencyApproximationsandGeometricOptics 298
7.2.4 ReflectionandTransmissioninLayeredMedia 303
7.3 InhomogeneousMetalWaveguides 305
7.3.1 GeneralFieldRelationships 305
7.3.2 SymmetricFormulation 306
7.3.3 AsymmetricFormulation 307
7.4 OpticalFibers 309
7.4.1 CircularOpticalFiber 309
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7.4.2 GuidanceCondition 312
7.4.3 EigenvaluesandEssentialSpectrum 313
7.5 InhomogeneousCavityResonator 319
7.5.1 ModeTheory 320
7.5.2 FieldExpansions 325
8 Time-domainTheory 329
8.1 Time-domainTheoryofMetalWaveguides 330
8.1.1 FieldExpansions 331
8.1.2 SolutionoftheModifiedKlein–GordonEquation 334
8.1.3 ExcitationofWaveguides 338
8.2 Time-domainTheoryofMetalCavityResonators 342
8.2.1 FieldinArbitraryCavities 342
8.2.2 FieldsinWaveguideCavities 349
8.3 SphericalWaveExpansionsinTime-domain 360
8.3.1 TransverseFieldEquations 360
8.3.2 SphericalTransmissionLineEquations 361
8.4 RadiationandScatteringinTime-domain 363
8.4.1 RadiationfromanArbitrarySource 363
8.4.2 RadiationfromElementarySources 365
8.4.3 EnhancementofRadiation 367
8.4.4 Time-domainIntegralEquations 369
9 Relativity 379
9.1 TensorAlgebraonLinearSpaces 380
9.1.1 TensorAlgebra 380
9.1.2 TangentSpace,CotangentSpaceandTensorSpace 384
9.1.3 MetricTensor 387
9.2 Einstein’sPostulatesforSpecialRelativity 388
9.2.1 GalileanRelativityPrinciple 388
9.2.2 FundamentalPostulates 389
9.3 TheLorentzTransformation 389
9.3.1 Intervals 389
9.3.2 DerivationoftheLorentzTransformation 391
9.3.3 PropertiesofSpace–Time 393
9.4 RelativisticMechanicsinInertialReferenceFrame 395
9.4.1 Four-VelocityVector 395
9.4.2 Four-MomentumVector 395
9.4.3 RelativisticEquationofMotion 396
9.4.4 AngularMomentumTensorandEnergy-MomentumTensor 398
9.5 ElectrodynamicsinInertialReferenceFrame 400
9.5.1 CovarianceofContinuityEquation 400
9.5.2 CovarianceofMaxwellEquations 401
9.5.3 TransformationofElectromagneticFieldsandSources 402
9.5.4 CovariantFormsofElectromagneticConservationLaws 403
9.5.5 TotalEnergy-MomentumTensor 404
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9.6 GeneralTheoryofRelativity 404
9.6.1 PrincipleofEquivalence 405
9.6.2 Manifolds 406
9.6.3 TangentBundles,CotangentBundlesandTensorBundles 407
9.6.4 RiemannianManifold 409
9.6.5 AcceleratedReferenceFrames 410
9.6.6 TimeandLengthinAcceleratedReferenceFrame 413
9.6.7 CovariantDerivativeandConnection 415
9.6.8 GeodesicsandEquationofMotioninGravitationalField 418
9.6.9 BianchiIdentities 421
9.6.10 PrincipleofGeneralCovarianceandMinimalCoupling 422
9.6.11 EinsteinFieldEquations 422
9.6.12 TheSchwarzschildSolution 425
9.6.13 ElectromagneticFieldsinanAcceleratedSystem 426
10 QuantizationofElectromagneticFields 429
10.1 FundamentalsofQuantumMechanics 430
10.1.1 BasicPostulatesofQuantumMechanics 430
10.1.2 QuantumMechanicalOperators 431
10.1.3 TheUncertaintyPrinciple 432
10.1.4 QuantizationofClassicalMechanics 434
10.1.5 HarmonicOscillator 435
10.1.6 SystemsofIdenticalParticles 437
10.2 QuantizationofFreeElectromagneticFields 438
10.2.1 QuantizationinTermsofPlaneWaveFunctions 439
10.2.2 QuantizationinTermsofSphericalWavefunctions 445
10.3 QuantumStatistics 448
10.3.1 StatisticalStates 448
10.3.2 MostProbableDistributions 449
10.3.3 BlackbodyRadiation 450
10.4 InteractionofElectromagneticFieldswiththeSmall
ParticleSystem 451
10.4.1 TheHamiltonianFunctionoftheCoupledSystem 451
10.4.2 QuantizationoftheCoupledSystem 453
10.4.3 PerturbationTheory 455
10.4.4 InducedTransitionandSpontaneousTransition 458
10.4.5 AbsorptionandAmplification 461
10.4.6 QuantumMechanicalDerivationofDielectricConstant 462
10.5 RelativisticQuantumMechanics 465
10.5.1 TheKlein–GordonEquation 465
10.5.2 TheDiracEquation 466
AppendixA: SetTheory 469
A.1 BasicConcepts 469
A.2 SetOperations 469
A.3 SetAlgebra 470