ELECTROMAGNETIC RADIATION Electromagnetic Radiation Richard Freeman, James King, and Gregory Lafyatis 1 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©RichardFreeman,JamesKing,GregoryLafyatis2019 Themoralrightsoftheauthorshavebeenasserted FirstEditionpublishedin2019 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2018953425 ISBN 978–0–19–872650–0 DOI:10.1093/oso/9780198726500.001.0001 Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. Contents Part I Introductory Foundations 1 EssentialsofElectricityandMagnetism 3 1.1 Maxwell’sstaticequationsinvacuum 3 1.1.1 Electrostaticequations 4 1.1.2 Magnetostaticequations 5 1.1.3 Lorentzforce 6 1.2 Maxwell’sstaticequationsinmatter 6 1.2.1 Responseofmaterialtofields 7 1.2.2 Boundchargesandcurrents 9 1.2.3 Macroscopicfields 10 1.2.4 PolarizabilityandSusceptibility 11 1.2.5 Thecanonicalconstitutiverelations 13 1.2.6 Electricfieldsandfreechargesinmaterials 13 1.3 Energyofstaticchargeandcurrentconfigurations 14 1.3.1 Electrostaticfieldenergy 14 1.3.2 Magneticfieldenergy 16 1.4 Maxwell’sdynamicequationsinvacuum 18 1.4.1 Faraday’scontribution 19 1.4.2 Conservationofchargeandthecontinuityequation 20 1.4.3 Maxwell’scontribution 21 1.5 Maxwell’sdynamicequationsinmatter 22 1.5.1 Originofmaterialcurrents 22 1.6 Planewavepropagationinvacuum 24 1.6.1 Polarizationofplanewaves 26 1.7 E&Mpropagationwithinsimplemedia 29 1.8 Electromagneticconservationlaws 30 1.8.1 Energydensity 30 1.8.2 Poynting’sTheorem 31 1.8.3 Linearmomentumdensity 31 1.8.4 Maxwellstresstensor 33 1.9 Radiationinvacuum 34 1.9.1 Fieldamplitudeasafunctionofdistancefromthesource 35 1.9.2 Decouplingofradiationfieldsfromthesource 35 1.9.3 Illustrationofcoupledanddecoupledfieldsfromanacceleratedcharge 36 Exercises 38 1.10 Discussions 39 vi Contents 2 ThePotentials 43 2.1 Themagneticandelectricfieldsintermsofpotentials 43 2.2 Gaugeconsiderations 44 2.3 ThewaveequationsprescribingthepotentialsusingtheLorenzgauge 45 2.4 Retardedtime 46 2.4.1 Potentialswithretardedtime 48 2.5 Momentsoftheretardedpotential 49 2.5.1 Potentialzones 49 2.5.2 Generalexpansionoftheretardedpotential 51 Exercises 55 2.6 Discussions 55 Part II Origins of Radiation Fields 3 GeneralRelationsbetweenFieldsandSources 63 3.1 Relatingretardedpotentialstoobservablefields 63 3.1.1 Spatialderivativesofretardedpotentials 65 3.2 Jefimenko’sequationsfromtheretardedpotentials 67 3.3 Graphicalrepresentationoftransversefieldsarisingfromacceleration 69 3.4 Jefimenko’sequationswithoutregardtoretardedpotentials: GreenFunctions 71 3.4.1 Fieldcharacteristics 74 3.4.2 Example:fieldsdirectlyfromJefimenko’sequations 75 Exercises 79 3.5 Discussions 80 4 FieldsinTermsoftheMultipoleMomentsoftheSource 85 4.1 MultipoleradiationusingJefimenko’sequations 85 4.1.1 Approximatespatialdependence 85 4.1.2 Radiationfromzerothordermoments 87 4.1.3 Radiationfromfirstordermoments 89 4.2 Multipoleradiationfromthescalarexpansionofthevectorpotential 91 4.2.1 Fieldsfromanelectricdipolemoment 92 4.2.2 Fieldsfrommagneticdipolemoment 94 4.2.3 Fieldsfromelectricquadrupolemoment 98 4.3 Powerradiatedintermsofmultipolemomentsofthesource 99 4.3.1 Powerradiatedbyelectricdipolemoment 99 4.3.2 Powerradiatedbymagneticdipolemoment 100 4.3.3 Powerradiatedbyelectricquadrupolemoment 101 Exercises 103 4.4 Discussions 106 Contents vii Part III Electromagnetism and Special Relativity 5 IntroductiontoSpecialRelativity 113 5.1 Historicalintroduction–1666to1905 115 5.1.1 Thenatureofspaceandtime 115 5.1.2 Thenatureoflight 117 5.1.3 Michelson–Morleyexperiments 121 5.2 EinsteinandtheLorentztransformation 121 5.2.1 Einstein’sapproach 122 5.2.2 TheLorentztransformation:covarianceamonginertialframes 125 5.3 Theinvariantintervalandthegeometryofspace-time 130 5.3.1 Minkowskispace-timediagrams 131 5.3.2 Physicalconsequencesofspecialrelativity 135 5.4 Vectorspaceconcepts 139 5.4.1 Contravariantandcovariantvectors 141 5.4.2 Themetrictensor 148 5.4.3 Generationofother4-vectorsand4-tensors 149 5.5 Someimportantgeneral4-vectors 150 5.5.1 The4-gradientoperator 151 5.5.2 The4-vectorvelocity 153 5.5.3 The4-vectormomentum 154 5.5.4 The4-vectorforce 157 5.6 Someimportant“E&M”4-vectors 159 5.6.1 The4-wavevector 159 5.6.2 The4-currentdensity 161 5.6.3 The4-potential(inLorenzGauge) 161 5.7 Othercovariantandinvariantquantities 162 5.7.1 Theangularmomentum4-tensor 162 5.7.2 Space-timevolume 163 5.7.3 Space-timedeltafunction 164 5.8 Summaryof4-vectorresults 164 5.9 Maxwell’sequationsandspecialrelativity 165 5.9.1 ManifestcovarianceofMaxwell’sequations 165 5.9.2 Theelectromagneticfieldtensor 166 5.9.3 Simplefieldtransformationexamples 169 5.10 TheEinsteinstress-energytensor 173 Exercises 175 5.11 Discussions 176 6 RadiationfromChargesMovingatRelativisticVelocities 184 6.1 Lienard–Wiechertpotentials 185 6.1.1 Derivationbyintegraltransform 187 6.1.2 Derivationbygeometricconstruction 188 viii Contents 6.2 Radiationfieldsfromasinglechargeundergoingacceleration 190 6.2.1 Movingchargegeneralfieldcharacteristics 195 6.3 Powerradiatedfromanacceleratedcharge 196 6.3.1 LowvelocitiesandclassicalLarmor’sformula 197 6.3.2 Radiatedpowerforrelativisticparticles 198 6.4 Accelerationparallelandperpendiculartovelocity 200 6.4.1 Angulardistributionforacceleration(cid:2)tovelocity 200 6.4.2 Angulardistributionforacceleration⊥tovelocity 202 6.4.3 Totalradiatedpowerforacceleration(cid:2)and⊥tovelocity 203 6.5 Spectraldistributionofradiationfromanacceleratedcharge 205 6.6 Synchrotronradiation 209 6.7 Fieldsfromasinglechargemovingwithconstantvelocity 214 6.7.1 Parametrizationofthefields 218 6.7.2 Spectralenergydensityofthefields 220 6.7.3 Numberofphotonsassociatedwithfieldsofapassingcharge 222 6.8 Bremsstrahlung 223 Exercises 227 6.9 Discussions 228 7 RelativisticElectrodynamics 229 7.1 Dynamicsusingactionprinciples:LagrangianandHamiltonianmechanics 229 7.1.1 Conceptofaction 230 7.2 Relativisticmechanicsofsinglepoint-likeparticles 234 7.2.1 Therelativisticmechanicsofafreeparticle 234 7.2.2 Freeparticlecanonical4-momentum 236 7.2.3 Freeparticleangularmomentum4-tensor 237 7.2.4 Achargedparticleinanexternalelectromagneticfield 239 7.3 Theactionprincipledescriptionoftheelectromagneticfield 243 7.3.1 Equationsofmotion 245 7.3.2 Lagrangiandensityfunction 247 7.3.3 RecoveryofMaxwell’sequations 249 7.3.4 Gaugeinvariance 250 7.3.5 TheProcaLagrangian 252 7.4 TheHamiltoniandensityandcanonicalstress-energytensor 254 7.4.1 FromtheMaxwellstresstensortothe4Dstress-energytensor 254 7.4.2 Hamiltoniandensity:the“00”canonicalstress-energytensorcomponent 255 7.4.3 Canonicalstress-energytensorandconservationlaws 256 7.4.4 Canonicalelectromagneticstress-energytensor 257 7.4.5 Symmetricelectromagneticstress-energytensor 258 7.4.6 Angularmomentumdensityoffields 259 7.4.7 Electromagneticstress-energytensorincludingsourceterms 261 Exercises 261 7.5 Discussions 263 Contents ix 8 FieldReactionstoMovingCharges 267 8.1 Electromagneticfieldmasses 268 8.2 Fieldreactionasaself-force 269 8.2.1 Lorentzcalculationoftheself-force 270 8.2.2 Somequalitativeargumentsfortheself-force 274 8.3 Abraham–Lorentzformulaandtheequationsofmotion 276 8.3.1 Theequationsofmotion 278 8.3.2 Landau–Lifshitzapproximation 282 8.3.3 Characteristictime 283 8.4 The4/3problem,instability,andrelativity 284 8.5 InfinitemassoftheAbraham–Lorentzmodel 291 Exercises 294 8.6 Discussions 296 Part IV Radiation in Materials 9 PropertiesofElectromagneticRadiationinMaterials 303 9.1 Polarization,magnetization,andcurrentdensity 304 9.2 Apracticalconventionformaterialresponse 306 9.3 E&Mpropagationwithinsimplemedia 307 9.4 Frequencydependence 310 9.4.1 ω→∞ 310 9.4.2 ω→0 312 9.4.3 Planewavesversusdiffusion 313 9.4.4 Transientresponseinaconductor 316 9.4.5 Temporalwave-packet 317 9.4.6 Groupvelocityversusphasevelocity 319 9.4.7 Pulsebroadening 320 9.5 Planewavesatinterfaces 322 9.5.1 Boundaries 322 9.5.2 Fresneltransmissionandreflectionamplitudecoefficients 325 9.5.3 Totalinternalreflection 328 9.5.4 Fresneltransmissionandreflectionintensitycoefficients 332 9.5.5 Fresneltransmissionandreflection:vacuum/materialinterface 333 9.6 Somepracticalapplications 335 9.6.1 Thetwo-surfaceproblem 335 9.6.2 Lossydielectricsandmetals 338 9.7 Frequencyandtimedomainpolarizationresponsetothefields 339 9.7.1 Example 342 9.8 Kramers–Kronigrelationships 343 9.9 Measuringtheresponseofmattertofields 346 9.9.1 Measuringtheopticalconstantsofamaterial 347