Classical Electromagnetism Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Maxwell’s Equations 7 1.1 Introduction . .................................. 7 1.2 Maxwell’sEquations................................ 7 1.3 ScalarandVectorPotentials............................. 8 1.4 DiracDeltaFunction................................ 9 1.5 Three-DimensionalDiracDeltaFunction...................... 9 1.6 Solution of Inhomogeneous Wave Equation . .................... 10 1.7 RetardedPotentials................................. 16 1.8 RetardedFields................................... 17 1.9 ElectromagneticEnergyConservation....................... 19 1.10 ElectromagneticMomentumConservation..................... 20 1.11 Exercises....................................... 22 2 Electrostatic Fields 25 2.1 Introduction . .................................. 25 2.2 Laplace’s Equation . ........................... 25 2.3 Poisson’sEquation.................................. 26 2.4 Coulomb’sLaw................................... 27 2.5 ElectricScalarPotential............................... 28 2.6 ElectrostaticEnergy................................. 29 2.7 ElectricDipoles................................... 33 2.8 ChargeSheetsandDipoleSheets.......................... 34 2.9 Green’sTheorem.................................. 37 2.10 Boundary Value Problems . ........................... 40 2.11 DirichletGreen’sFunctionforSphericalSurface.................. 43 2.12 Exercises....................................... 46 3 Potential Theory 49 3.1 Introduction . .................................. 49 2 CLASSICAL ELECTROMAGNETISM 3.2 AssociatedLegendreFunctions........................... 49 3.3 SphericalHarmonics................................. 50 3.4 Laplace’s Equation in Spherical Coordinates .................... 51 3.5 Poisson’sEquationinSphericalCoordinates.................... 52 3.6 MultipoleExpansion................................. 53 3.7 AxisymmetricChargeDistributions......................... 56 3.8 DirichletProbleminSphericalCoordinates..................... 57 3.9 NewmannProbleminSphericalCoordinates.................... 58 3.10 Laplace’s Equation in Cylindrical Coordinates . ................ 59 3.11 Poisson’sEquationinCylindricalCoordinates................... 64 3.12 Exercises....................................... 69 4 Electrostatics in Dielectric Media 73 4.1 Polarization..................................... 73 4.2 Boundary Conditions for E and D .......................... 75 4.3 Boundary Value Problems with Dielectrics . .................... 75 4.4 EnergyDensityWithinDielectricMedium..................... 81 4.5 ForceDensityWithinDielectricMedium...................... 82 4.6 Clausius-MossottiRelation............................. 85 4.7 DielectricLiquidsinElectrostaticFields...................... 86 4.8 Exercises....................................... 89 5 Magnetostatic Fields 93 5.1 Introduction . .................................. 93 5.2 Biot-SavartLaw................................... 93 5.3 Continuous Current Distribution . .................... 93 5.4 CircularCurrentLoop................................ 95 5.5 LocalizedCurrentDistribution........................... 98 5.6 Exercises.......................................100 6 Magnetostatics in Magnetic Media 103 6.1 Magnetization....................................103 6.2 Magnetic Susceptibility and Permeability . ....................104 6.3 Ferromagnetism...................................105 6.4 Boundary Conditions for B and H ..........................107 6.5 PermanentFerromagnets..............................108 6.6 UniformlyMagnetizedSphere...........................110 6.7 SoftIronSphereinUniformMagneticField....................113 6.8 MagneticShielding.................................114 6.9 MagneticEnergy...................................116 6.10 Exercises.......................................117 7 Wave Propagation in Uniform Dielectric Media 119 CONTENTS 3 7.1 Introduction . ..................................119 7.2 FormofDielectricConstant.............................120 7.3 AnomalousDispersionandResonantAbsorption..................122 7.4 Wave Propagation in Conducting Media . ....................124 7.5 HighFrequencyLimit................................126 7.6 PolarizationofElectromagneticWaves.......................127 7.7 FaradayRotation...................................128 7.8 WavePropagationinMagnetizedPlasmas.....................131 7.9 WavePropagationinDispersiveMedia.......................132 7.10 Wave-FrontPropagation...............................136 7.11 SommerfeldPrecursor................................139 7.12 MethodofStationaryPhase.............................144 7.13 GroupVelocity....................................145 7.14 BrillouinPrecursor.................................147 7.15 SignalArrival....................................149 7.16 Exercises.......................................150 8 Wave Propagation in Inhomogeneous Dielectric Media 153 8.1 Introduction . ..................................153 8.2 LawsofGeometricOptics..............................153 8.3 FresnelRelations..................................155 8.4 TotalInternalReflection...............................161 8.5 Reflection by Conducting Surfaces . ....................166 8.6 Ionospheric Radio Wave Propagation . ....................167 8.7 WKBApproximation................................168 8.8 Reflection Coefficient................................172 8.9 ExtensiontoObliqueIncidence...........................173 8.10 Ionospheric Pulse Propagation ...........................177 8.11 Measurement of Ionospheric Electron Density Profile . .........179 8.12 Ionospheric Ray Tracing . ...........................180 8.13 AsymptoticSeries..................................183 8.14 WKBSolutionasAsymptoticSeries........................188 8.15 StokesConstants...................................189 8.16 WKB Reflection Coefficient.............................194 8.17 JeffriesConnectionFormula.............................197 8.18 Exercises.......................................198 9 Radiation and Scattering 201 9.1 Introduction . ..................................201 9.2 BasicAntennaTheory................................201 9.3 Antenna Directivity and EffectiveArea.......................204 9.4 AntennaArrays...................................208 9.5 ThomsonScattering.................................210 4 CLASSICAL ELECTROMAGNETISM 9.6 RayleighScattering.................................212 9.7 Exercises.......................................213 10 Resonant Cavities and Waveguides 215 10.1 Introduction . ..................................215 10.2 Boundary Conditions . ...........................215 10.3 Cavities with Rectangular Boundaries . ....................217 10.4 Quality Factor of a Resonant Cavity . ....................218 10.5 Axially Symmetric Cavities . ...........................219 10.6 Cylindrical Cavities . ...........................222 10.7 Waveguides.....................................223 10.8 DielectricWaveguides................................225 10.9 Exercises.......................................230 11 Multipole Expansion 231 11.1 Introduction . ..................................231 11.2 MultipoleExpansionofScalarWaveEquation...................231 11.3 Angular Momentum Operators ...........................233 11.4 MultipoleExpansionofVectorWaveEquation...................234 11.5 PropertiesofMultipoleFields............................237 11.6 Solution of Inhomogeneous Helmholtz Equation . ................240 11.7 SourcesofMultipoleRadiation...........................242 11.8 RadiationfromLinearCentre-FedAntenna.....................245 11.9 SphericalWaveExpansionofVectorPlaneWave..................248 11.10MieScattering....................................250 11.11Exercises.......................................254 12 Relativity and Electromagnetism 255 12.1 Introduction . ..................................255 12.2 RelativityPrinciple.................................255 12.3 LorentzTransformation...............................256 12.4 Transformation of Velocities . ...........................260 12.5 Tensors........................................261 12.6 PhysicalSignificanceofTensors..........................265 12.7 Space-Time . ..................................266 12.8 ProperTime.....................................270 12.9 4-Velocity and 4-Acceleration . ...........................271 12.10CurrentDensity4-Vector..............................272 12.11Potential4-Vector..................................273 12.12GaugeInvariance..................................274 12.13RetardedPotentials.................................274 12.14 Tensors and Pseudo-Tensors . ...........................276 12.15ElectromagneticFieldTensor............................278 CONTENTS 5 12.16DualElectromagneticFieldTensor.........................281 12.17TransformationofFields..............................283 12.18PotentialDuetoaMovingCharge.........................284 12.19FieldDuetoaMovingCharge...........................285 12.20RelativisticParticleDynamics............................286 12.21ForceonaMovingCharge.............................288 12.22ElectromagneticEnergyTensor...........................289 12.23 Accelerated Charges . ...........................292 12.24LarmorFormula...................................295 12.25RadiationLosses...................................299 12.26 Angular Distribution of Radiation . ....................300 12.27SynchrotronRadiation................................301