[PDF] Lectures on Electromagnetic Field Theory - Purdue Engineering

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Lectures on

Electromagnetic FieldTheory

Weng ChoCHEW

Fall2021,

1Purdue University

1

UpdatedF ebruary4,2022

iiElectromagneticFieldTheor y

Contents

Prefacexiii

Acknowledgementsxiv

I Fundamentals,ComplexMedia,Theorems andPrinciples 1

1 Introduction,Maxwell's Equations3

1.1 ImportanceofElectromagnetics .. .. .. .. .. .. .. .. .. .. .. .. .. 3

1.1.1 TheElectromagnetic Spectrum .... .. .. .. .. .. .. .. .. .. 6

1.1.2 ABrief Historyof Electromagnetics. .. .. .. .. .. .. .. .. .. 6

1.2 Maxwell'sEquationsin Inte gral Form...... .... .. .. .. .. .. .. 9

1.3 StaticE lectromagnetics.... .. .. .. .. .. .. .. .. .. .. .. .. .. 9

1.3.1 Coulomb'sLaw (Statics). ...... .. .. .. .. .. .. .. .. .. .10

1.3.2 ElectricField (Statics). .. .. .. .. .. .. .. .. .. .. .. .. .. 10

1.3.3 Gauss'sLawfor ElectricFlux(Statics). .. .. .. .. .. .. .. .. .13

1.3.4 DerivationofGauss's Law fromCoulom b'sLaw(Statics) .. .... .14

2 Maxwell'sEq uations,DierentialOperatorForm19

2.1 Gauss'sDiv ergenceTheorem.. .. .. .. .. .. .. .. .. .. .. .. .. .19

2.1.1 SomeDetails .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 21

2.1.2 PhysicalMeaningof Divergence Operator .... .... .. .. .. .. 23

2.1.3 Gauss'sLawin DierentialOperator Form .... .... .... .. .24

2.2 Stokes'sTheorem. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .24

2.2.1 PhysicalMeaningof Cu rlOp erator... ........ .. .. .. .. .27

2.2.2 Faraday'sLawin DierentialOp eratorF orm..... .... .... .28

2.3 Maxwell'sEquationsin Dierenti al OperatorForm... .... ...... .. 29

2.4 HistoricalNotes .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .29

3 ConstitutiveRel ations,WaveEquation, andStaticGreen'sFunction33

3.1 SimpleConstitu tiveRelations... .. .. .. .. .. .. .. .. .. .. .. .. 33

3.2 Emergenceof Wa vePhenomenon,

Triumphof Maxwell's Equations... .. .. .. .. .. .. .. .. .. .. .. 35

3.3 StaticE lectromagnetics{Revisted.... .. .. .. .. .. .. .. .. .. .. .38

iii ivElectromagneticFieldTheor y

3.3.1 Electrostatics. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .38

3.3.2 Electrostaticsand KVL. .. .. .. .. .. .. .. .. .. .. .. .. .39

3.3.3 Poisson'sEquation. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 39

3.3.4 StaticGreen's Function .... .. .. .. .. .. .. .. .. .. .. .. 40

3.3.5 Laplace'sEquation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .42

4 Magnetostatics,B oundaryConditions,andJump Conditions45

4.1 Magnetostatics. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 45

4.1.1 Moreon Coulomb Gauge... .. .. .. .. .. .. .. .. .. .. .. 47

4.1.2 Magnetostaticsand KCL .... .. .. .. .. .. .. .. .. .. .. .47

4.2 BoundaryConditions|1D Poisson's Equation... .. .. .. .. .. .. .. .47

4.3 BoundaryConditions|Maxw ell'sEquations.. .. .. .. .. .. .. .. .. .50

4.3.1 Faraday'sLaw. .... .. .. .. .. .. .. .. .. .. .. .. .. .. 50

4.3.2 Gauss'sLawfor ElectricFlux.. .. .. .. .. .. .. .. .. .. .. .51

4.3.3 Ampere'sLaw .... .. .. .. .. .. .. .. .. .. .. .. .. .. .53

4.3.4 Gauss'sLawfor MagneticFlux.. .. .. .. .. .. .. .. .. .. .. 54

5 Biot-Savartlaw,Conductiv eMediaInterface, InstantaneousPo ynting's

Theorem57

5.1 DerivationofBiot-Sa vart Law.... .... .. .. .. .. .. .. .. .. .. .58

5.2 Shieldingb yConductiveMedia .... .. .. .. .. .. .. .. .. .. .. .. 60

5.2.1 BoundaryConditions|Cond uctiveMediaCase.... .. .. .. .. .60

5.2.2 ElectricField Insidea Conductor. .. .. .. .. .. .. .. .. .. .. 61

5.2.3 MagneticField Insidea Conductor. .. .. .. .. .. .. .. .. .. .63

5.3 InstantaneousPo ynting'sTheorem..... .. .. .. .. .. .. .. .. .. .65

6 Time-HarmonicFields, ComplexP ow er71

6.1 Time-HarmonicFields|Linear Systems. .. .. .. .. .. .. .. .. .. .. 72

6.2 FourierTransform Technique. ...... .. .. .. .. .. .. .. .. .. .. 74

6.3 ComplexP ower..... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 75

7 Moreon Constitutive Relations,UniformPlaneW av e81

7.1 Moreon Constitutive Relations... .. .. .. .. .. .. .. .. .. .. .. .81

7.1.1 IsotropicF requencyDispersiv eMedia...... .. .. .. .. .. .. 81

7.1.2 AnisotropicMedia .. .. .. .. .. .. .. .. .. .. .. .. .. .. .83

7.1.3 Bi-anisotropicMedia .. .. .. .. .. .. .. .. .. .. .. .. .. .. 84

7.1.4 InhomogeneousMedia .. .. .. .. .. .. .. .. .. .. .. .. .. .85

7.1.5 Uniaxialand BiaxialMedia .. .. .. .. .. .. .. .. .. .. .. .. 85

7.1.6 NonlinearMedia .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 85

7.2 WavePhenomenonintheF requencyDomain .. .. .... .. .. .. .. .. 86

7.3 UniformPlane Wa vesin3D.... .. .. .. .. .. .. .. .. .. .. .. .88

Contentsv

8 LossyMedia, Lorentz ForceLaw, Drude-Lorentz-SommerfeldModel 93

8.1 PlaneW avesinLossyConductiveMedia .. .. .... .. .. .. .. .. .. .93

8.1.1 HighConductivit yCase.. .. .. .. .. .. .. .. .. .. .. .. .. 95

8.1.2 LowConductivity Case.. .... .. .. .. .. .. .. .. .. .. .. 95

8.2 LorentzForce Law.. .... .. .. .. .. .. .. .. .. .. .. .. .. .. .96

8.3 Drude-Lorentz-SommerfeldModel.... .... .. .. .. .. .. .. .. .. 96

8.3.1 ColdCollisionless PlasmaM ed ium....... .. .. .. .. .. .. .97

8.3.2 BoundElectron Case|Heuristics. .. .. .. .. .. .. .. .. .. .. 99

8.3.3 BoundElectron Case|SimpleMath Model .. .... .. .. .. .. .100

8.3.4 Dampingor Dissipative Case... .. .. .. .. .. .. .. .. .. .. 101

8.3.5 BroadApplicabilit yofDrude-Lorentz-Sommerfeld Model .... ...102

8.3.6 FrequencyDispersiv eMedia|AGeneralDiscus sion. .. .......103

8.3.7 PlasmonicNanoparticles .. .. .. .. .. .. .. .. .. .. .. .. .. 104

9 WavesinGyrotropicMedia,P olarization109

9.1 GyrotropicMedia andF araday Rotation..... .. .. .. .. .. .. .. .. 109

9.2 WavePolarization.. .... .. .. .. .. .. .. .. .. .. .. .. .. .. .112

9.2.1 GeneralPolarizations|Ellipticaland CircularPolarizations. .. .. .112

9.2.2 ArbitraryP olarizationCaseandAxial Ratio

1. .. .. .. .. .. .. .115

9.3 PolarizationandP ow erFlow..... .... .. .. .. .. .. .. .. .. .. 116

10 Momentum,ComplexP oyn ting'sTheorem,LosslessCondition,Energy

Density121

10.1 SpinAngular Momentum andCylindricalVector Beam. .. .... .. .. .122

10.2 MomentumDensity ofElectromagneticField. .. .. .. .. .. .. .. .. .123

10.3 ComplexP oynting'sTheoremandLosslessConditions. .. .. .. .. .. .. 124

10.3.1 ComplexP oynting'sTheorem.... .. .. .. .. .. .. .. .. .. .124

10.3.2 LosslessConditions .. .. .. .. .. .. .. .. .. .. .. .. .. .. 125

10.3.3 AnisotropicMedium Case. .. .. .. .. .. .. .. .. .. .. .. .. 126

10.4 EnergyDe nsityinDispersiveMedia

2. .. .. .. .. .. .. .. .. .. .. .. 127

11 UniquenessTheorem 131

11.1 TheDierence Solutionsto Sourc e-Free Maxwell'sEquations..... .... 131

11.2 Conditionsfor Uniqueness. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 134

11.2.1 IsotropicCase .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 135

11.2.2 GeneralAnisotropic Case. .. .. .. .. .. .. .. .. .. .. .. .. 135

11.3 HindSigh tUsingLinearAlgebr a. .. .... .. .. .. .. .. .. .. .. .. 136

11.4 Connectionto Poles ofaLinearSystem .. .. .. .. .. .. .. .. .. .. .137

11.5 Radiationfrom Antenna SourcesandRadiationCondition

3. .. .. .. .. .139 1

This sectionis mathematicallycomplicated. Itcan be skipped onrst reading.

2The derivationinthis sectionis complex,but worth thepain, sincethis knowle dgewas notdisco vered

untilthe1960s.

3Mayb eskippedon rstreading.

viElectromagneticFieldTheor y

12 ReciprocityTheorem143

12.1 MathematicalDeriv ation... .. .. .. .. .. .. .. .. .. .. .. .. .. .144

12.1.1 LorentzReciprocit yTheorem.... .. .. .. .. .. .. .. .. .. .146

12.1.2 ReactionReciprocit yTheorem.... .. .. .. .. .. .. .. .. .. 146

12.2 Conditionsfor Reciprocit y..... .. .. .. .. .. .. .. .. .. .. .. .. 147

12.3 Applicationto aTw o-Port NetworkandCircuitTheory ...... .. .. .. 148

12.4 VoltageSourcesin Electromagnetics. .. .. .. .. .. .. .. .. .. .. .. 151

12.5 HindSigh t... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .151

13 EquivalenceTheorems,Huygens' Principle155

13.1 EquivalenceTheoremsor Equivalence Principles. .... .. .. .. .. .. .155

13.1.1 Inside-OutCase .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 156

13.1.2 Outside-inCase .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .157

13.1.3 GeneralCase.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 158

13.2 ElectricCurr entonaPEC.. .. .. .. .. .. .. .. .. .. .. .. .. .. .158

13.3 MagneticCurr entonaPMC.. .. .. .. .. .. .. .. .. .. .. .. .. .159

13.4 Huygens'P rincipleandGreen'sTheorem .. .. .. .. .. .. .. .. .. .. 159

13.4.1 ScalarW avesCase.... .. .. .. .. .. .. .. .. .. .. .. .. .160

13.4.2 ElectromagneticW avesCase.... .. .. .. .. .. .. .. .. .. .163

IIT ransmissionLines,Wav esin LayeredMedia,Waveguides, and

CavityResonators169

14 CircuitTheory Revisited171

14.1 KirchhoCurrent Law.. .... .. .. .. .. .. .. .. .. .. .. .. .. .171

14.2 KirchhoVoltage Law.. .... .. .. .. .. .. .. .. .. .. .. .. .. .172

14.2.1 Faraday'sLawand theFluxLinkage Term .. .. .... .... .. .174

14.2.2 Inductor|FluxLink ageAmplier.. .. .. .. .. .. .. .. .. .. 176

14.2.3 Capacitance|DisplacementCurrentAmplier.. .... .. .. .. .177

14.3 Resistor. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 178

14.4 SomeRemarks .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .179

14.5 EnergyStorage Method forInductorandCapacitor .. .. .. .. .. .. .. .179

14.6 FindingClosed-F ormFormulas forInductanceandCapac itanc e. .. .....180

14.7 ImportanceofCircuit Theoryin ICDesign .. .. .. .. .. .. .. .. .. .183

14.8 DecouplingC apacitorsandSpiralInductors... .. .. .. .. .. .. .. .. 185

15 TransmissionLines189

15.1 TransmissionLineTheory .. .. .. .. .. .. .. .. .. .. .. .. .. .. .190

15.1.1 Time-DomainAnalysis. .. .. .. .. .. .. .. .. .. .. .. .. .. 191

15.1.2 Frequency-DomainAnalysis|theP ow erofPhasorTechniqueAgain! .194

15.2 LossyT ransmissionLine.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 196

Contentsvii

16 Moreon Transmission Lines201

16.1 TerminatedTransmission Lines... .. .. .. .. .. .. .. .. .. .. .. .201

16.1.1 ShortedT erminations... .. .. .. .. .. .. .. .. .. .. .. .. 204

16.1.2 OpenTerminations .... .. .. .. .. .. .. .. .. .. .. .. .. .205

16.2 SmithC hart.... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .206

16.3 VSWR(V oltageStandingW aveRatio) ........ .. .. .. .. .. .. .. 210

17 Multi-JunctionT ransmissionLines,Duality Principle217

17.1 Multi-JunctionT ransmissionLines.. .. .. .. .. .. .. .. .. .. .. .. 217

17.1.1 Single-JunctionT ransmissionLines.. .. .. .. .. .. .. .. .. .. 219

17.1.2 Two-JunctionTransmission Lines|GeneralizedRe

ectionCo ecient 220

17.1.3 RecursiveFormulaforGenerali zedRe

ectionCoecient .. ......221

17.1.4 StrayCapacitanceand Inductance .. .... .. .. .. .. .. .. .. 223

17.1.5 Multi-PortNetw ork..... .. .. .. .. .. .. .. .. .. .. .. .225

17.2 DualityPrinciple. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .225

17.2.1 UnusualSwaps

4. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .226

17.2.2 Left-HandedMaterials andDouble Negative Materials. .... .. .. 227

17.3 FictitiousM agneticCurrents. .... .. .. .. .. .. .. .. .. .. .. .. 228

18 Re ection,T ransmission,andInteresti ngPh ysicalPhenomena231

18.1 Re

ectionand Tran smission|SingleInterfaceCase.... .... .. .. .. .231

18.1.1 TEP olarization(Perpendicular orEPolarization)

5. .. .. .. .. .. 232

18.1.2 TMP olarization(Parallelor HPolarization)

6. .. .. .. .. .. .. .235

18.1.3 LensOptics andRa yT racing... .... .. .. .. .. .. .. .. .. 235

18.2 InterestingPhysicalPhenomena. .... .. .. .. .. .. .. .. .. .. .. .236

18.2.1 TotalInternal Re

ection... .. .. .. .. .. .. .. .. .. .. .. .237

19 BrewsterAngle, SPP, HomomorphismwithTransmission Lines243

19.1 Brewster'sAngle .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 243

19.1.1 SurfacePlasmon Polariton .... .. .. .. .. .. .. .. .. .. .. .246

19.2 Homomorphismof UniformPlane Wa ve sandTransmissionLinesEquations.248

19.2.1 TEor TE

zWaves... .. .. .. .. .. .. .. .. .. .. .. .. .. 249

19.2.2 TMor TM

zWaves... .. .. .. .. .. .. .. .. .. .. .. .. .. 250

20 WavesinLayered Media253

20.1 WavesinLayered Media. ...... .. .. .. .. .. .. .. .. .. .. .. .253

20.1.1 GeneralizedRe

ectionCo ecient forLayeredMedia. .. ...... .254

20.1.2 RaySeriesIn terpre tationofGeneralizedRe

ectionCoecien t. .. .255

20.2 PhaseV elocityandGroupVelocit y. .. ........ .. .. .. .. .. .. .256

20.2.1 PhaseV elocity..... .. .. .. .. .. .. .. .. .. .. .. .. .. 256

20.2.2 GroupV elocity..... .. .. .. .. .. .. .. .. .. .. .. .. .. 2574

This sectioncan be skippedonrst reading.

5These polarizationsarealso var io uslyknowasTEz, orthe sandppolarizations,a descendent fromthe

notations foracoustic wa veswheresandpstand forshear andpressure wa ves, respectively.

6Also knownasTM zpolarization.

viiiElectromagneticFieldTheor y

20.3 WaveGuidanceinaLa yered Media. .. ...... .. .. .. .. .. .. .. .261

20.3.1 TransverseResonanceCondition. .. .. .. .. .. .. .. .. .. .. 261

21 DielectricSlab Wa veguides265

21.1 GeneralizedT ransverseResonanceCondition..... .. .. .. .. .. .. .265

21.1.1 ParallelPlateW av eguide....... .. .. .. .. .. .. .. .. .. 266

21.2 DielectricSlab Wa veguide...... .. .. .. .. .. .. .. .. .. .. .. .266

21.2.1 TECase .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .267

21.2.2 TMCase .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 273

21.2.3 ANote onCut-O ofDielectric Wa veguides .. ........ .. .. .274

22 HollowWa veguides277

22.1 GeneralInfor mationonHollowWa veguides .... ........ .. .. .. .277

22.1.1 Absenceof TEMMo dein aHollowW av eguide. .... ........ 278

22.1.2 TECase (Ez= 0,Hz6= 0,TE zcase) .. .. .. .. .. .. .. .. .. .279

22.1.3 TMCase (

E z6= 0,Hz= 0,TM zCase) .. .. .. .. .. .. .. .. .. 281

22.2 RectangularW aveguides..... .. .. .. .. .. .. .. .. .. .. .. .. .282

22.2.1 TEMo des(Hz6= 0,H Modes orTEzModes). .. .. .. .. .. .. .282

23 Moreon Hollow Waveguides 287

23.1 RectangularW aveguides,Contd.... .... .. .. .. .. .. .. .. .. .. 288

23.1.1 TMMo des(Ez6= 0,E Modes orTMzModes). .. .. .. .. .. .. 288

23.1.2 BouncingW avePicture.... .. .. .. .. .. .. .. .. .. .. .. 289

23.1.3 FieldPlots .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .290

23.2 CircularW aveguides..... .. .. .. .. .. .. .. .. .. .. .. .. .. .292

23.2.1 TECase .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .292

23.2.2 TMCase .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 295

24 Moreon Wa veguidesandTransmissionLines301

24.1 CircularW aveguides,Contd.... .... .. .. .. .. .. .. .. .. .. .. 301

24.1.1 AnApplication ofCircular Wa veguide ........ .. .. .. .. .302

24.2 Remarkson Quasi-TEMMo des,Hybrid Modes,andSurface PlasmonicMo des305

24.2.1 Quasi-TEMMo des... .. .. .. .. .. .. .. .. .. .. .. .. .. 306

24.2.2 HybridMo des{Inhomogeneously-FilledWav eguides....... .. .. 307

24.2.3 Guidanceof Modes .... .. .. .. .. .. .. .. .. .. .. .. .. .308

24.3 Homomorphismof Wa veguidesandTransmissionLines.. .. .... .. .. .309

24.3.1 TECase .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .309

24.3.2 TMCase .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 311

24.3.3 ModeConv ersion..... .. .. .. .. .. .. .. .. .. .. .. .. .312

25 CavityResonators317

25.1 TransmissionLineMo delof aResonator.. .. .. .. .. .. .. .. .. .. .317

25.2 CylindricalW aveguideResonators...... .. .. .. .. .. .. .. .. .. 320

25.2.1 LowestModeof aRectangularCavit y. .. .. ...... .. .. .. .322

25.3 SomeAppl icationsofResonators... .. .. .. .. .. .. .. .. .. .. .. 324

25.3.1 Filters. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .325

Contentsix

25.3.2 ElectromagneticSources .. .. .. .. .. .. .. .. .. .. .. .. .. 327

25.3.3 FrequencySensor. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .329

26 QualityFactor ofCavities, Mode Orthogonality333

26.1 TheQuali tyFactorofa Cavity|GeneralConcept .. .. ...... .. .. .333

26.1.1 Analoguewith anLC Tank Circuit. .... .. .. .. .. .. .. .. .334

26.1.2 Relationto Bandwidthand Pole Location .... .... .. .. .. .. 337

26.1.3 WallLossand Qfor aMetallic Cavit y|AP erturbationConcept .. .339

26.1.4 Example:The Qof TM

110Mode. .. .. .. .. .. .. .. .. .. .. 342

26.2 ModeOrthogonality andMatrixEigenv alueProblem .. ...... .. .. .. 343

26.2.1 MatrixEigen valueProblem(EVP) ...... .. .. .. .. .. .. .. 343

26.2.2 HomomorphismwiththeW av eguide ModeProblem....... ...344

26.2.3 ProofofOrthogonalit yof Wave guide Modes

7. .. .. .. .. .. .. .346

IIIRadiation,High-F requencyAppro ximation,Computational

Electromagnetics, QuantumTheoryof Light 349

27 Scalarand Vector Potentials 351

27.1 Scalaran dVector PotentialsforTime-HarmonicFields ...... .. .. .. .351

27.2 Scalaran dVector PotentialsforStatics|AReview ...... .. .. .. .. .352

27.2.1 Scalarand Vector Potentialsfor Electrodynamics.... .. .... .. 353

27.2.2 Degreeof Freedom inMaxwell'sEquations .. .... .. .. .. .. .355

27.2.3 Moreon Scalarand Vector Poten tials... ...... .. .. .. .. .356

27.3 Wheni sStaticElectromagneticTheory Valid ?. .. ...... .. .. .. .. 357

27.3.1 CuttingThrough TheChaste .. .. .. .. .. .. .. .. .. .. .. .358

27.3.2 DimensionalAnalysis Approach andCoordinate Stretching

8. .. .. .359

27.3.3 Quasi-StaticElectromagnetic Theory. .. .. .. .. .. .. .. .. .. 363

28 Radiationb yaHertzianDip ole367

28.1 History. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 367

28.2 Approximationby aPoint Source. ...... .. .. .. .. .. .. .. .. .. 369

28.2.1 CaseI. NearField, r1. .. .. .. .. .. .. .. .. .. .. .. .371

28.2.2 CaseI I.FarField (RadiationField),r1. .. .. .. .. .. .. .373

28.3 Radiation,P ower,andDirectiveGainP atterns. .... .. .... .. .. .. 373

28.3.1 RadiationResistance .. .. .. .. .. .. .. .. .. .. .. .. .. .. 376

29 RadiationFi elds,DirectiveGain, EectiveAperture 383

29.1 RadiationFields orF ar-FieldAppro ximation... .... .. .. .. .. .. .384

29.1.1 Far-FieldApproximation .... .. .. .. .. .. .. .. .. .. .. .385

29.1.2 LocallyPlane Wave Approximation...... .... .. .. .. .. .386

29.1.3 DirectiveGainP atter nRevisited...... .. .. .. .. .. .. .. .389

29.2 EectiveAperture andDirectiveGain .. .... .. .. .. .. .. .. .. .. 3907

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29.2.1 TheElectromagnetic Spectrum .... .. .. .. .. .. .. .. .. .. 392

30 ArrayAntennas, FresnelZone, RayleighDistance395

30.1 LinearAr rayofDipoleAn tennas. .... .... .. .. .. .. .. .. .. .. 395

30.1.1 Far-FieldApproximation ofaLinearArra y. .. .. .... .. .. .. 397

30.1.2 RadiationP atternofanArra y. .. .... .. .. .. .. .. .. .. .397

30.2 ValidityoftheF ar-FieldAppro ximation. .... .... .. .. .. .. .. .. 400

30.2.1 RayleighDistance. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 402

30.2.2 NearZone, Fresnel Zone,andFar Zone. .. .... .. .. .. .. .. 403

31 DierentTyp esofAntennas|Heuristics405

31.1 ResonanceT unnelinginAntenna .. .... .. .. .. .. .. .. .. .. .. .406

31.2 HornAn tennas... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 411

31.3 Quasi-OpticalA ntennas.... .. .. .. .. .. .. .. .. .. .. .. .. .. 413

31.4 SmallAn tennas... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 415

32 Shielding,Ima geTheory421

32.1 Shielding. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .421

32.1.1 ANote onElectrostatic Shielding. .. .. .. .. .. .. .. .. .. .. 421

32.1.2 RelaxationTime .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 422

32.2 ImageTheory .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 424

32.2.1 ElectricCharges andElectric Dipole s. ...... .. .. .. .. .. .424

32.2.2 MagneticCharges andMagnetic Dipoles .. .... .. .. .. .. .. .426

32.2.3 PerfectMagneticConductor (PMC)Surfaces .. .. .. .. .. .. .. 428

32.2.4 MultipleImages .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 429

32.2.5 SomeSp ecialCases|Spheres,Cylinders,and DielectricIn ter faces .. 430

33 HighF requencySolutions,GaussianBeams 433

33.1 TangentPlaneApproximations .. .... .. .. .. .. .. .. .. .. .. .. 434

33.2 Fermat'sPrinciple. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 435

33.2.1 GeneralizedSnell'sLa w. .... .. .. .. .. .. .. .. .. .. .. .437

33.3 GaussianBeam .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .438

33.3.1 Derivationofthe Paraxial/P arabolic WaveEquation.. ........ 438

33.3.2 Findinga ClosedF ormSolution .... .. .. .. .. .. .. .. .. .439

33.3.3 Othersolutions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .442

34 Scatteringof ElectromagneticField 445

34.1 RayleighScattering. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 445

34.1.1 Scatteringb yaSmallSpherical Particle .. .. .... .. .. .. .. .447

34.1.2 ScatteringCross Section. .. .. .. .. .. .. .. .. .. .. .. .. .450

34.1.3 SmallConductiv eParticle. .... .. .. .. .. .. .. .. .. .. .. 453

34.2 MieScatterin g... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 454

34.2.1 OpticalTheorem .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 455

34.2.2 MieScattering by SphericalHarmonicExpansions .... .. .. .. .456

Contentsxi

34.2.3 Separationof Variables inSphericalCoordinates

9. .. .. .. .. .. .456

35 SpectralExpansionsof SourceFields|Sommerfeld Integrals 461

35.1 SpectralRepresentations ofSources.. .. .. .. .. .. .. .. .. .. .. .. 461

35.1.1 AP ointSource|FourierExpansionan dContourIn tegration. .... 462

35.2 ASou rceonTop ofa LayeredMediu m. .. ...... .... .. .. .. .. .467

35.2.1 ElectricDip oleFields{SpectralExpansion .... .. .. .. .. .. .. 468

35.3 StationaryPhase Method|F ermat'sPrinciple... .... .. .. .. .. .. .471

36 ComputationalElectrom agnetics,NumericalMethods 477

36.1 ComputationalElectromagnetic s,NumericalMethods .. .... .. .. .. .479

36.2 Examplesof Dierential Equations... .. .. .. .. .. .. .. .. .. .. .479

36.3 Examplesof Integral Equations... .. .. .. .. .. .. .. .. .. .. .. .480

36.3.1 VolumeIntegral Equation... .. .. .. .. .. .. .. .. .. .. .. 480

36.3.2 SurfaceIn tegralEquation.. .. .. .. .. .. .. .. .. .. .. .. .482

36.4 Functionasa Ve ctor. ...... .. .. .. .. .. .. .. .. .. .. .. .. .483

36.5 Operatorasa Map. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 484

36.5.1 Domainand RangeSpaces .. .. .. .. .. .. .. .. .. .. .. .. 484

36.6 ApproximatingOperator EquationswithMatrix Equations... .. .. .. .485

36.6.1 SubspacePro jectionMethods. .... .. .. .. .. .. .. .. .. .. 485

36.6.2 DualSpaces .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .486

36.6.3 Matrixand Vector Representations.. .... .. .. .. .. .. .. .. 486

36.6.4 MeshGeneration .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 487

36.6.5 DierentialEquationSolv ersv ersusIntegr alEquationSolvers .....488

36.7 MatrixSolution by Matrix-FreeMethod .... .... .. .. .. .. .. .. .489

36.7.1 Gradientofa Functional .. .... .. .. .. .. .. .. .. .. .. .. 490

37 FiniteDi erenceMethod,Y eeAlgorithm495

37.1 Finite-DierenceTime-Domain Method .... .. .. .. .. .. .. .. .. .495

37.1.1 TheFinite-Dierence Approximation .... .. .. .. .. .. .. .. .496

37.1.2 TimeStepping orTime Marching .. .... .. .. .. .. .. .. .. .498

37.1.3 StabilityAnalysis. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .500

37.1.4 Grid-DispersionError. .. .. .. .. .. .. .. .. .. .. .. .. .. 502

37.2 TheY eeAlgorithm.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .504

37.2.1 Finite-DierenceF requencyDomainMethod... .. .... .. .. .508

37.3 AbsorbingBoundary Conditions. .. .. .. .. .. .. .. .. .. .. .. .. 508

38 QuantumTheoryof Light 513

38.1 HistoricalBac kgroundonQuantum Theory. .... .. .. .. .. .. .. .. 513

38.2 ConnectingE lectromagneticOscillationtoSim ple Pendulum ........ .516

38.2.1 SimpleElectromagnetics Oscillators. .. .. .. .. .. .. .. .. .. 517

38.3 HamiltonianTheory .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 518

38.4 SchrodingerEquation(1925). .. .. .. .. .. .. .. .. .. .. .. .. .. 520

38.5 BeautifyingSc hrodingerEquation... .. .. .. .. .. .. .. .. .. .. .. 5239

Mayb eskippedon rstreading.

xiiElectromagneticFieldTheor y

38.6 SomeSalien tFeaturesof QuantumTheory .. ...... .. .. .. .. .. .. 526

38.6.1 Randomnessof Quantum Observables.. .... .. .. .. .. .. .. 526

38.6.2 Wave-ParticleDuality... .... .. .. .. .. .. .. .. .. .. .. 527

38.7 UncertaintyPrincipleandEigen values ofan Operator.... .. .. .... .527

38.8 QuantumInformationScience andQuan tumIn terpretation. .... .... .528

38.8.1 WavesfromCoupledHarm onic Oscillators{HamiltonianTheory ...530

38.8.2 Maxwell'sEquationsfrom Energy Conservation{Hamiltonian Theory.531

38.8.3 HeisenbergPictureversusSchrodinger Picture. .... ...... .. 532

38.9 PhotonCarrying PlaneW av e....... .. .. .. .. .. .. .. .. .. .. 535

38.10WaveofArbitraryPolarization .. .. .... .. .. .. .. .. .. .. .. .. 537

39 QuantumCoherent StateofLight 541

39.1 TheQuan tumCoherentState .... .. .. .. .. .. .. .. .. .. .. .. .541

39.2 SomeW ordsonQuantum Randomnessand QuantumObservables .. .. ..542

39.3 TheCoheren tStates.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 543

39.3.1 TimeEv olutionofaQuan tumState .. .... .. .. .. .. .. .. .544

39.4 Moreon theCreation andAnnihilation Operator .. .. .... .. .. .. .. 546

39.4.1 TheCorresp ondencePrinciplefora Pendulum .. .. .... .. .. .546

39.4.2 ConnectingQuan tumPendulumto ElectromagneticOscillator.. .. 549

39.5 Epilogue. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 551

Preface

This setof lecturenotes isfrom my teaching ofECE 604,ElectromagneticFieldTheory ,at ECE, PurdueUniv ersity,WestLafayette. Itisintended forentrylevelgraduate students. Because dierentuniversities havedieren tundergraduaterequirementsinelectromagnetic eld theory,thisis acourse intended to\lev elthe playingeld".F romthis pointon ward, hopefully,alls tuden tswillhavethefundamental background inelectromagneticeldtheory needed totak eadvancelev elcoursesanddo researchatPur due. In developingthiscourse, Iha ve drawn heavilyuponknowledge ofourpredecessorsin this area. Manyofthe textbo oksand papersused,Ihav elisted theminther ef erence list.Being a practitionerin thiseld foro ver 40y ears,Ihaveseen electromagnetictheory impacting moderntec hnologydevelopment unabated.Despiteitsage,theset ofMaxw ell'sequations has enduredand contin uedtobeimportan t,from staticstooptics,from classicaltoquantum, and fromnanometerlengthscales togalactic lengths cales.T he applicationsofelectromagnetic technologiesha vealsobeentremendous andwide-ranging: fromgeophysicale xploration, re- mote sensing,bio-sensing, electricalmac hinery, renewableandcleanenergy, biomedicalen- gineering, opticsand photonics,computer ch ipdesign, computersystem,quantumcomputer designs, quantumcommunication andmanymore. Electromagneticeld theoryisnotev ery- thing, butit remainsan importan tcomp onentof moderntechnologydevelopments. The challengeinteac hingthis courseisonho wto teach ov er150 yearsofkno wledgein one semester:Ofcourse thisis missionimp ossib le! To dothis,weusethetraditionalwisdom of engineeringeducation: Distillth ekno wledge,makeit assimpleasp ossible,andteach the fundamentalbig ideasin oneshort semester. Becauseof this,y oumayn dthe ow ofthe lectures erratic.Some times, Ifeeltheneed totouc hon certainbig ideasb eforemo vingon, resulting inthe choppiness ofthecurriculum. Also, inthis course,I exploitmathem atical\homomorphism" asm uchasp ossibl eto simplify theteac hing.Afteryears ofpractising inthisarea,I ndthat somec omple xand advancedconcepts become simplerifmathematicalhomomorphism isexploited bet ween the advancedconcepts andsimpler ones.An example ofthis ison waves inla yered media.The problem ishomomorphic tothe transmissionline problem:Hence, usingtransm is sionline theory,one cansimplify thederiv ationsof somecomplicated formulas. A largepart ofmo dernelectromagnetic technologiesisbased onheuristics. Thisissome- thing dicultto teach, asitrelieson physical insight andexp erience.Mo derncommercial softwarehasr eshap edthislandscape:Theeldofmath-ph ysicsmo delingthroughnumeri- cal simulations,known ascomputationalelectromagnetic(CEM), has maderapid advances in recentyears. Manycu t-and-trylab oratoryexperiments,basedonheuristics,hav ebeen xiii xivLectures onElectr omagneticField Theory replaced bycut-and-trycomputer ex perimen ts,whicharealotcheaper. An excitingmo derndevelopment istheroleofelectromagneticsand Maxwe ll's equations in quantumtechnologies. Wewillconnect Maxwell'sequationsto quantum electromagnetics towardtheend ofthis course.This isa challenge, asit hasnev erb eendone beforeatan entryl evelcoursetomy knowledge. The rstv acuumtubecomputer, ENIACwas builtaround 1945.Afterthat,in the1950s, a seriesof vacuum tubeplustransistor computerswerebuilt includingthe ILLIAC series at Uof Illinois.Those computerscan ll awhole room. Aftersome70y ears,with the compoundingeect ofnanotec hnologies,w ecannow buildap oc ket-sizecellphone packed with billionsof transistors.A change inmodusop erandiis thatengineering designsare done increasingly morewith softw aretoreducecostratherthan cut-and-tryexp eriments. Thus, an importanteldofcomputational electromagnetics(CEM) hasemerged inrecen ty ears. Virtual prototypingofengineering designscan be donewith softw areratherthanhardw are. In fact,95 percen tofacomputerch ipdesign isno wdone withsoftwaresimulation greatly reducing thedesign cost.Unfortunately ,w ecanonlys pendtw olectures onCEM toconvey some ofthe big ideasacrosstothe students ofelectromagnetics. Thedevil isin thedetails in theimplemen tationsofthese bigideas, whichcan be pursued inothercourses.

Weng ChoC HEW

February4, 2022Pur dueUniversity

Acknowledgements

I liketothank DanJiao forsharing herlec ture notesin thiscourse, asw ellasAndyWei ner for sharingh isexperiencein teachingthiscourse inthe beginning.MarkLundstrom gav eme useful feedbackonChapters 38and 39on thequan tumtheory ofligh t. I liketothank DanJiao forsharing herrecen tstellar contributions tofast algorithmsin computational electromagnetics.Thanks alsoto AndyW einerand MahdiHosseini forsharing their fascinatingadv ancesinquan tumopticsfromtheir researchgroup. Ilik eto thankErhan Kudeki ofIlli noiswhoalwa ystak esaninterestonm ywritingonthis subject matter. Also, Iam thankful toDong-YeopNa forhelping teachpartof thiscourse asw ellas collaboratingon advances inquantumelectromagnetics. Thanksalso areduetheteac hing assistants,Bo yuanZhang,Jie Zhu, andIvanOkhmato vskiiforsu pporting thiscourseand also toRob ertHsueh-YungChao whotooktime toread thelecture notesandgav eme some veryuseful feedback. Thomas Roth,Dong-Y eopNa,andI recently taught ashort courseonquantum electro- magnetics atAP-S/URSI, Singapore 2021.Someofthe materialsfor theshort courseare factored intothelast tw oc haptersofthelecturenotes.W ehave alsocollab oratedon research in thisexciting emergingtopic.

PartI

Fundamentals,ComplexMedia,

Theorems andPrinciples

1

Chapter 1

Introduction,Maxwell's

Equations

In theb eginning,thiseldis eitherkno wnas theeld ofelectricit yandmagnetismor the eld of optics.Butlater, asw eshall discuss, thesetwo eldsare foundtobe basedon thesame set equationskno wnasMaxwell's equations.Maxw ell'sequationsuniedthese tw oelds, and itis commonto callthe studyof electromagneticth eory basedon Maxwell's equations electromagnetics.It haswide-rangingapplications fromstatics toultra-violet light inthe presentw orldwithimpacton many di erent technologies.

1.1 ImportanceofElectromagnetics

Wewill explainwh yelectromagnetics issoimportan t,and itsimpact onv erymanydierent areas. Thenw ewillgive abrief historyofelectromagnetics,and how ithas evolv edin the modernw orld.Nextwe willgo brie yover Maxwell's eq uationsin theirfullglory.Butw e will beginthestudy ofelectromagnetics by focussing onstatic problemswhichare validin the long-wavelengthlimitoratzero frequency. Electromagnetnics hasb eenbasedonMaxw ell'sequations, which aretheresultof the seminal workofJames ClerkMaxw ellcompleted in1865, afterhispresentation tothe British RoyalSociet yin1864.Hewas very muc hinspired bytheexp erimentallymotivated Faraday's law,Amp ere'slaw,Coulom b'slawas wellasGauss's law.Ithas been over150 years agono w, and thisis along timecompared tothe leapsand bounds progre ssw eha ve madeintech- nological advancements.Nevertheless,researc hinelectromagneticshas continuedunab ated despite itsage. Thereason isthat electromagneticsis extremely usefulas itis perv asive,and has impacteda largesec torof moderntechnologies. Tounderstand why electromagneticsissouseful, we hav eto understanda fewpoints aboutMaxw ell'sequations.

ˆMaxwell'sequations arev alido verav astlengthscalefromsubatomicdimensions togalactic dimensions.Hence, theseequations arev alido ver avastrange ofwavelengths:

3

4ElectromagneticFieldTheory

from staticto ultra-violetw av elengths. 1 ˆMaxwell'sequations arerelativistic inv ariant intheparlanceofspecialrelativity [1].In fact, Einsteinw asmotivatedwith thetheoryofsp ecialrelativ ity in1905 by Maxwell's equations [2].These equationslo okthe same,irrespective ofwhat inertialreference frame

2one isin.

ˆMaxwell'sequations arev alidin thequantum regime,as itw asdemonstrate db yP aul Dirac in1927 [3].Hence, many methods ofcalculatingtheresp onseofamediu mto classical eldcan be appliedinthequan tumregime also.When electromagnetictheory is combinedwithquan tumth eory, theeldofquantumopticscameabout. Roy Glauber wona Nobel prizein2005b ecauseof hisw orkin thisarea[4].

ˆMaxwell'sequations andthe pertinen tgauge theoryhasinspiredYang-Mills theory(1954) [5],whic hisalsokno wnas ageneralized electromagnetictheory.Y ang-Mills

theory ismotivated bydierential formsindierential geometry[6].T oquote from Misner, Thorne,and Wheeler,\Dieren tialforms illuminateelectromagnetictheory, and electromagnetictheory illuminatesdieren tial forms."[7,8] ˆMaxwell'sequations aresome ofthe mostaccurate physical equationsthat hav eb een validatedb yexperiments. In1985,RichardFeynman wrotethat electromagnetictheory had beenvalidated toonepartin abillion.

3Now,it hasb een validatedtoonepartin

a trillion(Ao yamaetal,Sty er,2012). 4 ˆAs aconsequence, electromagneticshas permeated many technologies,andhas atremen- dous impactin sci enceandtechnology.Thisis manifestedin electricalengineering, optics, wirelessand opticalcomm unications,computers, remotesensing,subsurface sensing, bio-medicalengineering etc.It isexp ectedthat quantum electromagnetics(the quantumextension ofelectromagnetics) willgro win importance asquantumtec hnolo- gies develop.1 Currentlithograp hyprocessisw orkingwith usingdeep ultra-violetlightwith awavelengthof 193nm.

2An inertialreference frameis aco ordinateframe thatis travelingat av elocit yv.

3This meansthat ifa jetis to

y fromNew York toLos Angeles,an errorof onepart inabillionmeans an errorof afew millmeters.

4This meansan er rorofahairline,ifone were tosho ota light beam fromtheearth tothe moon.

Introduction,Maxwell's Equations5Figure 1.1:The impactof electromagneticsin many technologies. Theareas inblueare

prevalentareasimpacted by electromagneticssome 20yearsago [9],and theareas in

brownar emodernemerging areasimpactedby electromagnetics.Figure 1.2:Kn owledgegrowslike atree.Engineeringkno wledgeandre al-world applica-

tions aredriv enbyfundamen talknowledgefrom mathandthesciences. Atau niversit y, wedo science-basedengineering research thatcan impactwide-rangingreal-world ap- plications. Buteveryone isequallyimportantin transformingour societ y.Justlikethe parts ofthe human body, noonecanclaimthatoneismore importan tthan theothers.

6ElectromagneticFieldTheory

Figure 1.2sho wshowkno wledgearedriven bybasicmath andscience knowledge.Its growthis like atree.Dueto thev asto ceanof knowledge thatw eareimmersedin, itis importantthatw ecollab oratetodevelop technologiesthatcan transformthis world.

1.1.1 TheElectromagnetic Spectrum

The electromagneticeld has beenusedfrom verylow fre que ncie stovery highfrequencies. Atv erylowfrequencies, ultra-lowfrequency(ULF) <3Hz, extremely-lowfrequency (ELF)

3-3000 Hz,v erylowfrequency (VLF)3KHz to30 KHzha ve beenusedto probe theearth

surface, andsubmarine communication becauseoftheir deeperpenetration depths.The AM radio stationop eratinginthesev eral100 KHzhas wavelength ofsev eral100 m.FM radio are inthe 100MHz range,while TVstations operate inthe several 100MHz. Microwav es havewa velengthoforderofcm,andinfra-red light rangesfrom 1000m to1 m. Thevisibl e spectrumranges from700 nmto 400nm. Ultra-violet(UV) light rangesfrom 400nm to100 nm, whileX-ra yisgenerallyb elow 100nm to1nm.Gammaray isgenerally belo w1 nm. UV lightof193 nmare no wused for nano-lithography.X-rayis importan tforimaging, while gammar ayisusedfors omemedical applications.The lights above UVare gen eral ly harmful toth ehumanb ody.

1.1.2 ABrief Historyof Electromagnetics

Electricityand magnetismha ve beenknownto mankindforalon gtime.Also,thephysical propertiesof light haveb eenknown.But theeldofelectricit yandmagnetism,nowtermed electromagnetics inthe modern world,hasb eenthoughtto be governedb ydierentph ysical lawsa sopposed tothoseforoptics.Thisis understandableas theph ysicsof electricity and magnetism isquite dierent ofthephysics ofoptics asthey wereknown toh umansthen. Forexample, lode stonewaskno wnto theancientGreek andChinesearound600 BCto

400 BC.Compass was usedinChinasince 200B C.Static electricity was reported bythe

Greek asearly as400 BC.But thesecuriosities didnot make animpact until theage of telegraphy.Thecoming about oftelegraph ywasdue tothe invention ofthevoltaic cellor the galvaniccellin thelate 1700's,b yLuigi Galvani andAlesandro Volta[10].It was soon discoveredthattwopieces ofwire, connectedtoavoltaiccell, cantran smit informationat a distance. So bythe early1800'sthisp ossibility hads purred thedevelopmentoftelegraph y. Both Andre-MarieAmp ere(1823)[11,12]andMic haelF araday (1838)[13] didexp erimentsto betterunderstand theprop ertiesof electricityandmagnetism. Andhence, Ampere'slaw and Faradaylaw arenamedafterthem. Kirc hhov oltageand current lawswerealsodev eloped in 1845to helpb etterun derstandtelegraphy[14, 15].Despitethesela ws,thetechnology of telegraphyw aspoorly understood.Forinstance, itwasnotknown astowhy thetelegraph y signal wasdistorted. Ideally,thesignal shouldbea digitalsignal switchin gb etweenone's and zero's,but thedigital si gn allostitsshaperapidlyalong atelegraph yline. 55
As aside note,in 1837,Morse inv ented theMorse codefortelegraphy[16].There were crossp ollination

of ideasacross theA tlantic oceandespitethedistance.In fact,BenjaminFranklin associated lightning with

electricityin thelatter partof the18-th century .Also, noticethat electrical machinerywasinv ented in1832

eventhough electromagnetictheory was notfully understood.

Introduction,Maxwell's Equations7

It wasnotun til1865 thatJamesClerkMaxw ell[17] putin themissing termi nAm- pere'sla w,thedisplacement current term,onlythenthe mathematicaltheoryforelectricit y and magnetismw ascomplete.Ampere's law isnowkno wnasgen er alizedAmpere's law. The completeset ofequations areno wnamed Maxwell's equationsinhonorof JamesClerk

Maxwell.

The rousings uccessofMaxwell's theoryw asthatit predictedw avephenomena, asthey havebeen observedalongtelegraph ylines. Butit wasnotun til23yearslaterthat Heinrich Hertz in1888 [18]did experimen tto provethatelectromagneticeld canpropagate through space acrossa room. Thisillustratesthedicult yof knowledge disseminati onwhennew knowledgeis discov ered.Moreover,fromexp erimentalmeasurementofthe pe rmittivityand permeabilityofmatter, itw asdecided thatelectromagnetic wave mov esat atremendous speed.But thev elocit yoflighthasbeenkno wnfor alongwhilefrom astronomicalobserv a- tions (Roemer,1676)[19]. Thein terferencep henomen ainlight hasbeenobservedinNewton's ring (1704)[20]. Whenthese piecesof informationw erepieced together,it was decidedthat electricityand magnetism,and optics,are actuallygo verned by thesame physicallawor

Maxwell'sequations. Andoptics andelectromagnetics areunied into oneeld! Figure 1.3:A briefhistory ofe lectromagneticsand opticsas depictedinthisgure. In theearly days ,itwasthought thatoptics isadierent disciplinefromelectricity

and magnetism.Then after1865, thet wo eldsare uniedandgovernedb yMaxw ell's equations. In Figure1.3,a briefhistory ofelectromagnetics andoptics isdepicted. Inthe be ginning, it wasthought thatelectricityand magnetism,and opticswerego verned by dierent physical laws.Lo wfrequencyelectr omagn eticswasgovernedbytheunderstandingof eldsandtheir

8ElectromagneticFieldTheory

interactionwith media.Optical phenomenaw erego verned byray optics,re ectionand refraction ofligh t.Buttheadv ent ofMaxw ell'sequationsin1865revealed thatthey canbe unied underelectromagnetic theory. ThensolvingMaxwell's equationsb ecomesa rewarding mathematical endeavor. The photoelectriceect[21, 22],and Planck radiationla w[23] pointto thefact that electromagnetic energyis manifestedin termsof pack etsof energy, indicatingthecorpuscular nature ofligh t.Eachunit ofthisenergy isno wkno wnasthe photon.A photon carriesan energy packetequalto~!, where!is theangular frequencyof thephoton andthe Planc k constant~= 6:6261034J s,whic hisav erysmall constant. Hence,thehigherthe frequency, the easieri tistodetect thispac ket ofe ne rgy,orfeelthe graininessof electromagneticenergy. Eventually,in1927[3], quantum theoryw asincorp oratedintoelectromagnetics, andthe quantumnatu reoflight gives risetothe eldofquantum optics.Recen tly, evenmicro wa ve photons havebeenmeasured [24,25].Theyare dicultto detectbecauseof thelo wfrequency of microwave(10

9Hz) comparedto optics(10 15Hz): amicro wavephotoncarriesapacket of

energy aboutamillion timessmaller thanthat ofan opticalphoton. The progressin nano-fabrication[26] allows oneto makeoptic alcomp onents thataresub- optical wavelengthasthewa velength ofblue lightisabout450nm.

6As aresult, interaction

of lightwithnano-scale opticalcomp onents requiresthe solutionofMaxwell'sequationsin its full glory,whereastrad itionally, rayopticswereused todescribe many opticalphenomena. In theearly days ofquantumtheory ,there weretw oprevailingtheories ofquan tumin- terpretation. Quantummeasurements werefoundto berandom.In orderto explainthe probabilistic natureof quantum measurements,Einsteinp ositedthatarandom hiddenv ari- able causedtheran domoutcome ofanexperimen t.On theother hand,the Copenhagenschool of interpretationledb yNiels Bohr,assertedthatthe outcomeof aquan tummeasuremen tis not knownuntil afterameasurement [27]. In 1960s,Bell's theorem(b yJohn StewardBell)[28] saidthat aninequalityshould be satised ifEinstein's hiddenv ariable theorywascorrect.Otherwise, theviolation ofthe inequalityimplies thatthe Copenhagen scho olofinterpretation shouldprevail.Howev er , experimentalmeasurement showedthat theinequalitywasviolated, fav oringthe Copenhagen schoolofquan tumin terpretation[27].Thisin terpretationsaysthat aquan tumstate isina linear superpositionofstatesb eforea measurement. Butafterameasuremen t,a quantum state collapsesto thes tatethat ismeasured.This impliesthat quantum informationcan behidden incognitoin aquan tumstate.Hence,a quantum particle,s uc hasaphoton ,its state isunkno wnuntilafter itsmeasurement. Inother words, quantumtheoryis\spo oky"or \weird".This leadsto growing interest inquantuminformation andquantumcomm unication using photons.Quan tumtechnologywith theuseofph ot ons, anelectromagnetic quantum particle, isa subject ofgrowingin terest.This alsohasthep rofoundandbeautiful implic ation that \ourk armaisnotwritten onour foreheadwhen we were born, ourfuture isin ourown hands!"6 Size ofthe smallesttra nsisto rnowisabout5nm, whilethesizeof thecorono virusisabout 50to 140nm.

Introduction,Maxwell's Equations9

1.2 Maxwell'sEquationsin Integra lF orm

Eventhough experimen tallymotivated,Maxwell'sequations canbepresen tedasfundamental postulates.

7Wewill present themintheirin tegralforms, butwill notb elaborthemun til

later. ? C

E
dl=ddt

? S

B
dSFaraday'sLaw (1.2.1)

? C

H
dl=ddt

? S

D
dS+IAmpere'sLa w(1.2.2)

? S

D
dS=QGauss's orCoulom b'sLaw(1.2.3)

? S

B
dS= 0Gauss's Law (1.2.4)

The unitsof thebasic quantities abo vearegiv enas: E : V/mH: A/m D : C/m

2B: W/m2

I: AQ: C

where V=volts,A=amperes, C=coulombs,andW=w ebers. In thiscour se,weusea boldfacetodenote av ector,and ahattodenote aunit vector. Hence, av ectorcanbe writtenas E= ^xEx+^yEy+^zEz. where^ x, ^y, and^ zare unitv ectorsin Cartesian coordinates.Insome bo oks,alternativ ely,avectoris writtenasE= (Ex;Ey;Ez).

1.3 StaticElectromagnetics

In statics,the eldis assumedto be non-time-varying. Henceall thetimedependence terms can beremo vedfromMaxwell'sequations,andw eha ve? C

E
dl= 0Faraday'sLaw(1.3.1)

? C

H
dl=IAmpere'sLa w(1.3.2)

? S

D
dS=QGauss's orCoulom b'sLaw(1.3.3)

? S

B
dS= 0Gauss's Law(1.3.4)7

Postulatesin phy sicsaresimilartoaxiomsinmathematics. Theyare assumptionstha tneed notb e proved.

10ElectromagneticFieldTheory

The rstequation abo ve,whichisthe staticformofFaraday'slaw alsogiv esrise toKirc hho voltagela w.Thesecon dequation istheoriginalformofAm pere's law wheredisplacemen t currentw asignored.Thethird andthe fourthequations remainunc hangedcompared tothe time-varying(dynamic)form ofMaxw ell'sequations.

1.3.1 Coulomb'sLaw (Statics)

This law,develop edin1785[29],expressesthe forceb etw eent wo charges q1andq2. Ifthese chargesare positiv e,theforceisrepulsive andit isgiv enb y f 1 !

2=q1q24"r

2^r12(1.3.5)Figure 1.4:The forceb etw eentwochargesq1andq2. Theforce isrepulsiv eif thetwo

chargesha vethesamesign. where theunits are:f(force): newton q(charge):coulom bs "(permittivity):farads/meter r(distance betweenq1andq2): m ^r

12= unitv ectorpointing fromcharge1toc harge2

^r

12=r2r1jr

2r1j;r=jr2r1j(1.3.6)

Since theunit vector canbedened inthe above, theforce bet ween twochargescanalsob e rewritten as f 1 !2=q1q2(r2r1)4"jr

2r1j3;(r1;r2are positionvectors) (1.3.7)

1.3.2 ElectricField (Statics)

The electriceldEis denedas thef orce perunitcharge[30]. Fortw oc harges,oneofc harge qand theother oneof incremental charge
q, theforce bet weenthetwocharges, according

Introduction,Maxwell's Equations11

to Coulomb'slaw (1.3.5),is f=q
q4"r

2^r(1.3.8)

where ^ris aun itvectorp ointingfromc hargeqto theincrem entalcharge
q. Thenthe electric eldE, whichisthe forcep erunit charge, isgivenb y E=f4q ;(V/m) (1.3.9) This electriceld Efrom ap ointchargeqat theorgin ishence

E=q4"r

2^r(1.3.10)

Therefore, ingeneral, theelectric eldE(r) atlo cationrfrom ap ointchargeqatr0is given by E ( r) =q(rr0)4"jrr0j3(1.3.11) where theunit vector ^r=rr0jrr0j(1.3.12)Figure 1.5:E manatingEeld froman electricp oint chargeasdepictedby (1.3.11)and (1.3.10).

12ElectromagneticFieldTheory

If onekno wsEdue toa poin tcharge,onewill knowEdue toan ychargedistribution becausean ychargedistribution canbed ecomp osedin tosumofp ointcharges.Forinstance,if there areNpointcharges eachwithamplitude qi, thenb ytheprincipleof linearsup erposition assuming thatlinearit yholds,thetotal eldpro ducedb ythese Nchargesis E ( r ) = NX i =1q i(rri)4"jrrij3(1.3.13) whereqi=%(ri)
Viis theincremental chargeatrienclosed inthe volume
Vi. Inth e continuumlimit,one gets E ( r ) = ? V%(r

0)(rr0)4"jrr0j3dV(1.3.14)

In otherw ords,thetotaleld, by theprinciple oflinear superposition,is thein tegralsum- mation ofthe contributions fromthedistributedc hargedensit y%(r).

Introduction,Maxwell's Equations13

1.3.3 Gauss'sLa wforElectricFlux (Statics)

This lawisalso known asCoulom b'slawas theyare closelyrelatedtoeac hother.Apparently , this simplela wwasrst expressedbyJoseph LouisLagrange [31]and later,reexpressedby

Gauss in1813 (Wikipedia).

This lawcanb eexpresse das

? S

D
dS=Q(1.3.15)

whereDis electric ux density withunitC/m 2andD="E,dSis anincremen talsurface at thep ointonSgivenb ydS^nwhere^nis theunit normalp ointing outwardawa yfromthe surface, andQis totalc hargeenclosedby thes urfaceS.Figure 1.6:Electric uxthrough anincremen talsurface dSwhere ^nis theunit normal, andDis theelectric uxdensit ypassing throughtheincremental surface. The left-handside of(1.3.15) represents asurface integralov era closedsur faceS. To understand it,one canbreak thesurface into asum ofincremen talsurfaces
Si, witha localunit normal ^niassociatedwith it.The surfacein tegralcan thenb eapproximatedb ya summation ? S

D
dSX

iD i
^ni
Si=X iD i

Si(1.3.16) where onehas denedthe incre mental surface
Si=^ni
Si. Inthe limitwhen
Sibecomes innitesimally small,the summationb ecomesa surfaceintegral.

14ElectromagneticFieldTheory

1.3.4 DerivationofGaus s'sLa wfromCoulomb'sLaw (Statics)

FromCoulom b'slaw,the ensuingelectricelddue toa poin tc harge,the electric ux is

D="E=q4r

2^r(1.3.17)

When aclosed sp hericalsurfaceSis drawnaroundthe poin tc hargeq, bysymmetry, the electric uxthough every pointof thesurfaceisthesame.Moreo ver, thenormal vector ^n on thesurface isjust ^r. Consequently,D
^n=D
^r=q=(4r2), whichisa constant ona spherical ofradius r. Hence,we concludethatfora poin tc harge q, andthe pertinen telectric uxDthat itpro ducesonaspherical surfacesatises, ? S

D
dS= 4r2D
^n= 4r2Dr=q(1.3.18)

Therefore, Gauss'sla wissatisedb ya poin tcharge.Figure 1.7:Electric uxfrom ap oint charge satisesGauss'slaw. Evenwhen theshap eof thesphericalsurfaceSis distortedfrom asphere toan arbitrary shapesurface S, itcan be shownthatthe total uxthroughSis stillq. Inother words, the total uxthr oughsufacesS1andS2in Figure1.8 arethe same. This canb eappreciatedby takinga sliverofthe angularsector assho wninFigure1.9.

Here,

S

1and
S2are twoincrementalsurfaces interceptedby thissliv erofangularsector.

The amountof

ux passingthrough thisincremen talsurface isgiv en bydS
D=^n
D
S= ^ n
^rDr
S. Here,D=^rDris pointinginthe^rdirection. In
S1, ^nis pointinginthe^r direction. Butin
S2, theincremen talareahasb eenenlarged by that^nnot alignedwith D . Butthis enlargement iscompensatedb y ^n
^r. Also,
S2has grownbigger,but the ux at
S

2has grownweak erbytheratio of(r2=r1)2. Finally,thet wo

uxesareequalinthe limit thatthe sliver ofangularsector becomes innitesimallysmall. Thisproves theassertion that thetotal uxes throughS1andS2are equal.Since thetotal uxfrom ap oint charge q through aclosed surfacei sindep endentofits shape,butalways equalto q, thenif we havea total chargeQwhichcan be expressedasthesum ofp oint charges, namely. Q=X iq i(1.3.19) Introduction,Maxwell's Equations15Figure 1.8:Same amount ofelectric uxfrom ap oint charge passesth rought wosurfaces S

1andS2. Thisallo wsGauss'slaw for electric

uxtob ederivable fromCoulom b'sla w for statics.

Then thetotal

ux throughaclosedsurface equalsthe totalc hargeenclosed by it,whic his

the statementofGauss's law orCoulom b'slaw.Figure 1.9:When as liv erofangularsectoristaken,sam eamoun tof electric

ux from a pointcharge passesthroughtw oincremen talsurfaces
S1and
S2at dierentdistances from thep ointcharge.

16ElectromagneticFieldTheory

Exercises forLecture 1

Problem 1-1:

(i) Explainwh ytheelectric ux goingthrough
S

1and
S2are thesame inFigure 1.9.

(ii) Findthe elddue toa ringof charges withline char gedensit y%C/m assho wninthe

gure (courtesyof Ramo,Whinnery ,and VanDuzer).Hint:Use symmetry.Figure 1.10:Electric eldof aring ofc harge(courtesy ofRamo, Whinnery ,andVan

Duzer [32]).

(iii) Whatis theelectric eldb etw eencoaxial cylindersofunitlengthinacoaxialcable?

Hint:Use symmetryand cylindricalco ordinatesto expressE= ^Eand applyGauss'sFigure 1.11:Figure forProblem 1-2for acoaxial cylinder.

law.

Introduction,Maxwell's Equations17

(iv) Thegu reshowsa sphereof uniform chargedensity.Find theelectric eldEas a function ofdistance rfrom thec enterofthesphere.Hint:Again, usesymmetry an d

spherical coordinatestoexpress E= ^rErand applyGauss 'slaw.Figure 1.12:Figure forProblem 1-3for asph ere withuniform charge density.

(v) Givenaninnitely longcylindrical circularwire carryinga DCcurren tI, ndthe magnetic eldaround thewire us ing symmetryargument,andAmpere'slaw.

18ElectromagneticFieldTheory

Chapter 2

Maxwell'sEquations,

DierentialOp eratorForm

Maxwell'sequations were originallywrittenin integral formas hasb eenshownintheprevious lecture. Integralformshave niceph ysicalmeaningandcanb eeas ilyrelatedto experime ntal measurements.Ho wever,thedierentialoperator form

1can beeasilycon verted todierential

equations orpartial dierential equationswhereawhole sleuthof mathematicalmetho ds and numericalmetho dscanbe deploy ed.Therefore,itis prudenttoderivethe dierential operator form ofMaxw ell'sequations.

2.1 Gauss'sDiv ergenceTheorem

Wewil lrstprov eGauss's divergencetheorem.

2The divergencetheoremis oneof themost

importanttheoremsin vector calculus[ ? ,32,34,64].Itsays that: ? V dVr
D=? S

D
dS(2.1.1)

The right-handsideof theab ov eis thetotalelectric uxDthat comesout ofthe surfaceS.

In theab ove,r
Dis denedas

r
D= lim
V!0?
S

D
dS
V

(2.1.2)

The aboveimpliesthatthediv ergenceof theelectric

uxD, orr
Dis givenby rstcomputing the uxcoming (oro ozing)out ofasmallv olume
Vsurrounded byasmall surface
Sand taking theirratio assho wnon theright-handside ofthe abo ve.Asshall be shown, theratio 1 Wecaution ourselves nottousethe term\dieren tialforms" which hasa dierent meaningusedin dierentialgeometry foranother formof Maxwell's equations.

2Named afterCarl Friedric hGauss,aprecociousgenius wholiv edb etween1777-1855.

19

20ElectromagneticFieldTheory

has an itelimitandev entually ,w ewillndasimpliedexpressionforit. Weknow thatif

V0 orsmall, thenthe abo ve impliesthat,

Vr
D?

S

D
dS(2.1.3)

First, weassumethat av olumeVhas beendiscretized3intoa sumof smallcub oids,where thei-th cuboidhasa volume of
Vias showninFigure 2.1.Then VNX i =1
V i(2.1.4)Figure 2.1:The discretizationof av olumeVintoa sumof smallv olumes
Vieachof whichis as mallcub oid.Stair-casingerroro ccursneartheb oundaryof thev olumeV but theerror diminishesas
V i!0.3 Other termsused are \tesselated",\meshed",or\gridded".

Maxwell'sEqu ations,DierentialOp eratorForm21Figure 2.2:Fluxes fromadjacen tcub oidscanceleach otherleavingonly the

uxes atthe boundarythat re mainuncancelled.Pleaseimaginethatth ere isa thirddimension ofthe cuboidsin thispicture whereit comesout ofthe paper.

Then from(2.1.2) and(2.1.3), forthe i-th cuboid,

V ir
Di?
S iD i
dSi(2.1.5) By summingthe abo veoverallthecub oids,oroveri, onegets X i
V ir
DiX i?
S iD i
dSi? S

D
dS(2.1.6)

The lastappro ximationfollows,b ecauseitiseasily seenthatthe uxesout ofthe innersurfaces of thecub oidscanceleach other,lea vingonly uxe s o wingoutofthe cuboids attheedgeof the volumeVas explainedi nFigure2.2.The right-hand sideof theab oveequation (2.1.6) becomesa surfacein tegralo verthesurface Sexcept forthe stair-casingappro ximation(see Figure 2.1).Ho wever,thisapproximationb ecomesincreasingly goodas
V i!0. Moreover, the left-handside becomes avolumein tegral,and wehav e ? V dV r
D=? S

D
dS(2.1.7)

The aboveisGauss'sdivergence theorem.

2.1.1 SomeDetails

Next, wewillderiv ethe detailsofthedenition emb odi ed in(2.1.2). To thisend,weevaluate the numeratoroftheright-hand sidecarefully ,in accordancetoFigure2.3.

22ElectromagneticFieldTheoryFigure 2.3:F iguretoillustratethe calculationof

uxesfrom asmall cuboid wherea corner ofthe cuboid islocatedat (x0;y0;z0). Thereis athird zdimension ofthe cuboid not shown,andcoming outof thepap er.Hence, thiscub oid,unlik ethatshown inthe gure, hassix faces.

Accountingfor the

uxes goingthrough allthe six faces,assigning theappropriate signs in accordancewith the uxes leavi ngand enteringthecuboid,one arrives atthefollowing six terms?
S

D
dS Dx(x0;y0;z0)
y
z+ Dx(x0+
x;y0;z0)
y
z

D y(x0;y0;z0)
x
z+ Dy(x0;y0+
y;z0)
x
z D z(x0;y0;z0)
x
y+ Dz(x0;y0;z0+
z)
x
y(2.1.8) Factoringout thev olumeof thecuboid
V=
x
y
zin theab ove,onegets ?
S

D
dS
Vf[Dx(x0+
x;:::)Dx(x0;:::)]=
x

+[Dy(:::;y0+
y;:::)Dy(:::;y0;:::)]=
y +[Dz(:::;z0+
z)Dz(:::;z0)]=
zg(2.1.9)

Or that

?D
dS
V @Dx@x +@Dy@y +@Dz@z (2.1.10)

In thelimit when

V!0, then

lim
V!0?

D
dS
V

=@Dx@x +@Dy@y +@Dz@z =r
D(2.1.11)

Maxwell'sEqu ations,DierentialOp eratorForm23

where r= ^x@@x + ^y@@y + ^z@@z (2.1.12)

D= ^xDx+ ^yDy+ ^zDz(2.1.13)

The aboveisthedenitionof thed ivergence operator inCartesian coordinates. Thediver- gence operatorr
has itscomplicated representations incylindrical andsphericalcoordinates, a subjectthatw ew ouldnotdelve intointhis course.But theycanbe derived,andare best lookedupat thebac kof sometextb ooksonelectromagnetics. Consequently,oneob tainsGauss's divergencetheoremgiv enb y ? V dVr
D=? S

D
dS(2.1.14)

2.1.2 PhysicalMeaningof Divergence Operator

The physicalmeaningof diverge nc eisthatifr
D6= 0at ap oint inspace,itimpliesthatthere are uxeso ozingorexudingfrom thatp oint inspace [48].On theotherhand,ifr
D= 0, it impliesno uxo ozingout fromthatpoin tin space.In otherw ords,whatever uxthat goesin tothepoin tm ustcomeoutofit.The uxistermeddiv er gen ce free.Th us,r
Dis a measure ofho wmuch sourcesorsinksexistforthe uxat ap oint. Thesum ofthese sources or sinksgiv estheamoun tof uxlea vingorentering thesurface thatsurrounds thesources or sinks.

Moreover,ifone were toin tegrateadivergence-free

uxo verav olumeV, andin voking

Gauss's divergencetheorem,one gets

? S

D
dS= 0(2.1.15)

In suchascenerio, whatever

uxthat entersthesurface Smustlea veit.Inotherw ords,what comes inm ustgooutof thev olumeV, orthat uxis conserve d. Thisistrueofincompressible uid ow[164],ele ctric ux owina sourcefree region,aswell asmagnetic ux o w,where the uxis conserved.

24ElectromagneticFieldTheoryFigure 2.4:In anincompressible

ux o w, ux isconse rved:whatev er uxthat entersa volumeVmustlea vethevolumeV.

2.1.3 Gauss'sLa winDierential Operator Form

By furtherusin gGauss'sorCoulom b'sla wimplies that ? S

D
dS=Q=?

dV %(2.1.16) Wecan replacethe left-handside ofthe abo ve by (2.1.14)to arriveat ? V dVr
D=? V dV %(2.1.17) WhenV!0, wearrive atthepoin twise relationship,a relationshipatanarbitrarypointin space. Therefore, r
D=%(2.1.18)

2.2 Stokes'sTheorem

The mathematicaldescription of

uid ow was wellestablished beforethee stablishment of electromagnetic theory[ ? ]. Hence,m uchmathematicaldescription ofelectromagnetic theory uses thelanguage of uid. Inmathematical notations,Stok es'stheorem is 4 ? C

E
dl=?

S r E
dS(2.2.1) In theab ove,thecontourCis aclosed contour, whereasthesurfaceSis notclosed. 54 Named afterGeorge GabrielStok eswho livedbet ween 1819to1903.

5In otherw ords,Chas nob oundarywhereasShas boundary.Aclosedsurface Shas nob oundarylike

when wewere provingGauss'sdiv ergencetheorempreviously.

Maxwell'sEqu ations,DierentialOp eratorForm25

First, applyingStok es'stheoremtoa smallsurface
S, wedenea curlop erator6rat a pointtobe measuredas ( r E)
^n= lim
S!0?
C

E
dl
S

(2.2.2) In theab ove,Eis af orceperunitcharge,and r Eis av ector.Taking?

CE
dlas a

measure ofthe torqueor rotationof theeld Earound asmall loop
C, theratio ofthis rotation tothe areaof thelo op
Shas alimit when
Sbecomesinnitesimally small.Th is ratio isrelated to( r E)
^nwhere ^nis aunit normalto thesurface
S. Asin angular

momentum,the directionof the torqueis alongtherotationaxis ofthe force.Figure 2.5:In proving Stokes'stheorem,a closedcontourCis assumedto enclosean

open at surfaceS. Thenthe surfaceSis tessellatedin tosumofsmall rectsas shown. Stair-casing errorat the boundaryCvanishesin thelimit whenthe rects aremade vanishinglysmall.

First, the

at surfac eSenclosed byCis tessellated(alsocalled meshed,gridded, or discretized) intosumof smallrects (rectangles)as shown inFigure 2.5.Stok es'stheorem is then appliedto oneof these smallrects toarriveat ?
C iE i
dli= (r Ei)

Si(2.2.3) where onedenes
Si= ^n
S. Next,w esumtheab ov eequation overior overallthesmall rects toarriv eat X i?
C iE i
dli=X ir Ei

Si(2.2.4)6

Sometimes calleda rotationop erator.

26ElectromagneticFieldTheory

Again, onthe left-handside ofthe abo ve, allthe contourintegralsoverthe smallrects cancel eachother internal toSsaveforthose onthe boundary .In thelimit when
Si!0, the left-hand sideb ecomesacontour integral overthe largercontourC, andthe right-hand side becomesa surfacein tegralo verS. Onearriv esatStokes's theorem,whic his? C

E
dl=?

S (

r E)
dS(2.2.5)Figure 2.6:W eapproximatethe integrationov era smallrectusingthis gure.Thereare fouredges tothis smallrect.

Next, weneedto prov ethe detailsofdenition(2.2.2)usingFigure 2.6.P erformingthe integralo verthesmallrect,one gets?
C

E
dl=Ex(x0;y0;z0)
x+Ey(x0+
x;y0;z0)
y

Ex(x0;y0+
y;z0)
xEy(x0;y0;z0)
y =
x
yEx(x0;y0;z0)
y Ex(x0;y0+
y;z0)
y

Ey(x0;y0;z0)
x

+Ey(x0+
x;y0;z0)
x  (2.2.6)

Weha vepicked thenormaltotheincrementalsurface

Sto be^zin theab oveexample,

and hence,the abo vegivesrisetothe identitythat lim
S!0?

SE
dl
S

=@@x

Ey@@y

Ex= ^z
r E(2.2.7)

Maxwell'sEqu ations,DierentialOp eratorForm27

Pickingdieren t

Swith dierentorientations andnormal s^ nwhere ^n= ^xor ^n= ^y, one gets @@y

Ez@@z

Ey= ^x
r E(2.2.8)

@@z

Ex@@x

Ez= ^y
r E(2.2.9)

The abovegivesthex,y, andzcomponentsofr E. Itis tob enoted