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Classical Electrodynamics Part II Classical Electrodynamics Part II by Robert G. Brown Duke University Physics Department Durham, NC 27708-0305 [email protected] Acknowledgements I’d like to dedicate these notes to the memory of Larry C. Biedenharn. Larry was my Ph.D. advisor at Duke and he generously loaned me his (mostly handwritten or crudely typed) lecture notes when in the natural course of events I came to teach Electrodynamics for the first time. Most of the notes have been completely rewritten, typeset with latex, changed to emphasize the things that I think are important, but there are still important fragments that are more or less pure Biedenharn, in particular the lovely exposition of vector spherical harmonics and Hansen solutions (which a student will very likely be unable to find anywhere else). I’d also like to acknowledge and thank my many colleagues at Duke and elsewhere who have contributed ideas, criticisms, or encouragement to me over the years, in particular Mikael Ciftan (my “other advisor” for my Ph.D. and beyond), Richard Palmer and Ronen Plesser. Copyright Notice Copyright Robert G. Brown 1993, 2007 Notice This set of “lecture notes” is designed to support my personal teaching ac- tivities at Duke University, in particular teaching its Physics 318/319 series (graduate level Classical Electrodynamics) using J. D. Jackson’s Classical Elec- trodynamics as a primary text. However, the notes may be useful to students studying from other texts or even as a standalone text in its own right. It is freely available in its entirety online at http://www.phy.duke.edu/ rgb/Class/Electrodynamics.php ∼ as well as through Lulu’s “book previewer” at http://www.lulu.com/content/1144184 (where one can also purchase an inexpensive clean download of the book PDF in Crown Quarto size – 7.444 9.681 inch pages – that can be read using any PDF browser or locally printed).× In this way the text can be used by students all over the world, where each student can pay (or not) according to their means. Nevertheless, I am hoping that students who truly find this work useful will purchase either the PDF download or the current paper snapshot, if only to help subsidize me while I continue to write more inexpensive textbooks in physics or other subjects. These are real lecture notes, and they therefore have errors great and small, missing figures (that I usually draw from memory in class), and they cover and omit topics according to my own view of what is or isn’t important to cover in a one-semester course. Expect them to change without warning as I add content or correct errors. Purchasers of a paper version should be aware of its imperfection and be prepared to either live with it or mark up their own copies with corrections or additions as need be in the lecture note spirit, as I do mine. The text has generous margins, is widely spaced, and contains scattered blank pages for students’ or instructors’ own use to facilitate this. I cherish good-hearted communication from students or other instructors pointing out errors or suggesting new content (and have in the past done my best to implement many such corrections or suggestions). Contents Preface iii 0.1 The Interplay of Physics and Mathematics ............................. vii Links 3 0.1 PersonalContactInformation . 3 0.2 UsefulTextsandWebReferences . 3 I Mathematical Physics 5 1 Mathematical Prelude 7 2 Numbers 9 2.1 RealNumbers............................. 9 2.2 ComplexNumbers .......................... 10 3 Vectors and Vector Products 15 3.1 ScalarsandVectors.......................... 16 3.2 TheScalar,orDotProduct . 16 3.2.1 TheLawofCosines ..................... 18 3.3 TheVector,orCrossProduct . 18 3.4 TripleProductsofVectors. 20 3.5 δij and ǫijk .............................. 21 3.5.1 The Kronecker Delta Function and the Einstein Summa- tionConvention ....................... 21 3.5.2 The Levi-Civita Tensor . 22 3.5.3 The Epsilon-Delta Identity . 22 4 Tensors 25 4.1 The Dyad and N-adicForms .................... 25 4.2 CoordinateTransformations. 28 i 5 Group Theory 33 5.0.1 Subgroups........................... 34 5.0.2 Abelian (Commutative) Groups . 34 5.0.3 Lie(Continuous)Groups . 35 5.1 CoordinateTransformationGroups . 35 5.1.1 TheTranslationGroup . 36 5.1.2 TheRotationGroup ..................... 36 5.1.3 TheInversionGroup.. .. .. .. .. .. .. .. .. 37 6 Scalar and Vector Calculus 39 6.1 Scalar Differentiation . 40 6.2 VectorDifferentiation ........................ 41 6.2.1 The Partial Derivative . 41 6.3 TheGradient ............................. 42 6.4 VectorDerivatives .......................... 42 6.4.1 TheSumRules........................ 43 6.4.2 TheProductRules...................... 43 6.5 SecondDerivatives .......................... 44 6.6 ScalarIntegration........................... 45 6.6.1 TheFundamentalTheoremofCalculus . 45 6.7 VectorIntegration .......................... 45 6.8 The Fundamental Theorem(s) of Vector Calculus . 46 6.8.1 AScalarFunctionofVectorCoordinates. 46 6.8.2 TheDivergenceTheorem . 47 6.8.3 Stokes’Theorem ....................... 48 6.9 IntegrationbyParts ......................... 49 6.9.1 ScalarIntegrationbyParts . 49 6.9.2 VectorIntegrationbyParts . 49 6.10 IntegrationByPartsinElectrodynamics . 51 7 Coordinate Systems 55 7.1 Cartesian ............................... 57 7.2 SphericalPolar ............................ 58 7.3 Cylindrical .............................. 60 8 The Dirac δ-Function 63 9 Math References 67 II Non-Relativistic Electrodynamics 69 10 Maxwell’s Equations 71 10.1 TheMaxwellDisplacementCurrent. 71 10.2Potentials ............................... 75 10.2.1 GaugeTransformations . 77 10.2.2 TheLorenzGauge .. .. .. .. .. .. .. .. .. 79 10.2.3 TheCoulomborTransverseGauge . 81 10.3 Poynting’sTheorem,WorkandEnergy. 84 10.4 MagneticMonopoles . .. .. .. .. .. .. .. .. .. .. 88 10.4.1 DiracMonopoles ....................... 89 11 Plane Waves 93 11.1 TheFreeSpaceWaveEquation . 93 11.1.1 Maxwell’s Equations . 93 11.1.2 TheWaveEquation . .. .. .. .. .. .. .. .. 95 11.1.3 PlaneWaves ......................... 96 11.1.4 Polarization of Plane Waves . 99 11.2 Reflection and Refraction at a PlaneInterface ............................ 101 11.2.1 Kinematics and Snell’s Law . 102 11.2.2 Dynamics and Reflection/Refraction . 103 11.3Dispersion............................... 110 11.3.1 StaticCase .......................... 110 11.3.2 DynamicCase ........................ 112 11.3.3 ThingstoNote ........................ 113 11.3.4 Anomalous Dispersion, and Resonant Absorption . 114 11.3.5 Attenuation by a complex ǫ .................115 11.3.6 LowFrequencyBehavior. 116 11.3.7 High Frequency Limit; Plasma Frequency . 117 11.4 Penetration of Waves Into a Conductor – Skin Depth . 119 11.4.1 Wave Attenuation in Two Limits . 119 11.5 Kramers-KronigRelations . 121 11.6 PlaneWavesAssignment. 123 12 Wave Guides 125 12.1 Boundary Conditions at a Conducting Surface: Skin Depth . 125 12.2 Mutilated Maxwell’s Equations (MMEs) . 131 12.3TEMWaves.............................. 134 12.4TEandTMWaves.......................... 135 12.4.1 TMWaves .......................... 137 12.4.2 SummaryofTE/TMwaves . 138 12.5 RectangularWaveguides . 139 12.6 ResonantCavities . .. .. .. .. .. .. .. .. .. .. 140 12.7 WaveGuidesAssignment . 141 13 Radiation 143 13.1 Maxwell’s Equations, Yet Again . 143 13.1.1 Quickie Review of Chapter 6 . 143 13.2 Green’sFunctionsfortheWaveEquation . 145 13.2.1 PoissonEquation. 147 13.2.2 Green’s Function for the Helmholtz Equation . 147 13.2.3 Green’s Function for the Wave Equation . 149 13.3 SimpleRadiatingSystems . 152 13.3.1 TheZones........................... 153 13.3.2 TheNearZone ........................ 154 13.3.3 TheFarZone......................... 155 13.4 TheHomogeneousHelmholtzEquation. 156 13.4.1 Properties of Spherical Bessel Functions . 157 13.4.2 JL(r), NL(r), and HL±(r) ..................159 13.4.3 GeneralSolutionstotheHHE. 159 13.4.4 Green’s Functions and Free Spherical Waves. 160 13.5 Electric Dipole Radiation . 161 13.5.1 Radiation outside the source . 162 13.5.2 Dipole Radiation .....................162 13.6 Magnetic Dipole and Electric Quadrupole Radiation Fields . 166 13.6.1 Magnetic Dipole Radiation . 167 13.6.2 Electric Quadrupole Radiation . 168 13.7 RadiationAssignment . 171 14 Vector Multipoles 175 14.1 Angular momentum and spherical harmonics . 175 14.2 Magnetic and Electric Multipoles Revisited . 177 14.3 Vector Spherical Harmonics and Multipoles . 179 15 The Hansen Multipoles 187 15.1 TheHansenMultipoles . 187 15.1.1 The Basic Solutions . 187 15.1.2 Their Significant Properties . 188 15.1.3 Explicit Forms . 188 15.2 Green’s Functions for the Vector Helmholtz Equation . 189 15.3 Multipolar Radiation, revisited . 190 15.4 ALinearCenter-FedHalf-WaveAntenna. 197 15.5 Connection to Old (Approximate) Multipole Moments . 200 15.6 AngularMomentumFlux . 202 15.7 Concluding Remarks About Multipoles . 205 15.8 TableofPropertiesofVectorHarmonics . 206 16 Optical Scattering 209 16.1 Radiation Reaction of a Polarizable Medium . 209 16.2 Scattering from a Small Dielectric Sphere . 212 16.3 ScatteringfromaSmallConducting Sphere . 217 16.4 ManyScatterers ........................... 220 III Relativistic Electrodynamics 225 17 Special Relativity 227 17.1 Einstein’sPostulates . 227 17.2 The ElementaryLorentzTransformation . 228 17.34-Vectors ............................... 232 17.4 Proper Time and Time Dilation . 237 17.5 Addition of Velocities . 238 17.6 RelativisticEnergyandMomentum . 240 18 The Lorentz Group 245 18.1 TheGeometryofSpace–Time . 245 18.2 Tensorsin4Dimensions . 247
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