Electrodynamics - Duke University

1 Classical ElectrodynamicsPart IIbyRobert G. BrownDuke University Physics DepartmentDurham, NC d like to dedicate these notes to the memory of Larry C. was my advisor at Duke and he generously loaned mehis (mostlyhandwritten or crudely typed) lecture notes when in the natural course of eventsI came to teach Electrodynamics for the first time. Most of thenotes have beencompletely rewritten, typeset with latex, changed to emphasize the things thatI think are important, but there are still important fragments that are moreor less pure Biedenharn, in particular the lovely exposition of vector sphericalharmonics and Hansen solutions (which a student will very likely be unable tofind anywhere else).I d also like to acknowledge and thank my many colleagues at Duke andelsewhere who have contributed ideas, criticisms, or encouragement to me overthe years, in particular Mikael Ciftan (my other advisor for my andbeyond), Richard Palmer and Ronen NoticeCopyright Robert G.

2 Brown 1993, 2007 NoticeThis 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 sClassical Elec-trodynamicsas a primary text. However, the notes may be useful to studentsstudying from other texts or even as a standalone text in its own is freely available in its entirety online rgb/ well as through Lulu s book previewer (where one can also purchase an inexpensive clean download of the book PDFin Crown Quarto size inch pages that can be read using anyPDF browser or locally printed).In this way the text can be used by students all over the world,where eachstudent can pay (or not) according to their means. Nevertheless, I am hopingthat students who truly find this work useful will purchase either the PDFdownload or the current paper snapshot, if only to help subsidize me while Icontinue to write more inexpensive textbooks in physics or other are real lecture notes , and they therefore have errorsgreat and small,missing figures (that I usually draw from memory in class), and they cover andomit topics according to my own view of what is or isn t important to coverin a one-semester course.

3 Expect them to change without warning as I addcontent or correct errors. Purchasers of a paper version should be aware of itsimperfection and be prepared to either live with it or mark uptheirowncopieswith corrections or additions as need be in the lecture note spirit, as I do text has generous margins, is widely spaced, and contains scattered blankpages for students or instructors own use to facilitate cherish good-hearted communication from students or other instructorspointing out errors or suggesting new content (and have in the past done mybest to implement many such corrections or suggestions). The Interplay of Physics andMathematics .. Personal Contact Information .. Useful Texts and Web References .. 3I Mathematical Physics51 Mathematical Prelude72 Real Numbers .. Complex Numbers .. 103 Vectors and Vector Scalars and Vectors .. The Scalar, or Dot Product .. The Law of Cosines.

4 The Vector, or Cross Product .. Triple Products of Vectors .. ijand ijk.. The Kronecker Delta Function and the Einstein Summa-tion Convention .. The Levi-Civita Tensor .. The Epsilon-Delta Identity .. 224 The Dyad andN-adic Forms .. Coordinate Transformations .. 28i5 Group Subgroups .. Abelian (Commutative) Groups .. Lie (Continuous) Groups .. Coordinate Transformation Groups .. The Translation Group .. The Rotation Group .. The Inversion Group .. 376 Scalar and Vector Scalar Differentiation .. Vector Differentiation .. The Partial Derivative .. The Gradient .. Vector Derivatives .. The Sum Rules .. The Product Rules .. Second Derivatives .. Scalar Integration .. The Fundamental Theorem of Calculus .. Vector Integration .. The Fundamental Theorem(s) of Vector Calculus .. A Scalar Function of Vector Coordinates.

5 The Divergence Theorem .. Stokes Theorem .. Integration by Parts .. Scalar Integration by Parts .. Vector Integration by Parts .. Integration By Parts in Electrodynamics .. 517 Coordinate Cartesian .. Spherical Polar .. Cylindrical .. 608 The Dirac -Function639 Math References67II Non-Relativistic Electrodynamics6910 Maxwell s The Maxwell Displacement Current .. Potentials .. Gauge Transformations .. The Lorenz Gauge .. The Coulomb or Transverse Gauge .. Poynting s Theorem, Work and Energy .. Magnetic Monopoles .. Dirac Monopoles .. 8911 Plane The Free Space Wave Equation .. Maxwell s Equations .. The Wave Equation .. Plane Waves .. Polarization of Plane Waves .. Reflection and Refraction at aPlane Interface .. Kinematics and Snell s Law .. Dynamics and Reflection/Refraction .. Dispersion .. Static Case .. Dynamic Case .. Things to Note.

6 Anomalous Dispersion, and Resonant Absorption .. Attenuation by a complex .. Low Frequency Behavior .. High Frequency Limit; Plasma Frequency .. Penetration of Waves Into a Conductor Skin Depth .. Wave Attenuation in Two Limits .. Kramers-Kronig Relations .. Plane Waves Assignment .. 12312 Wave Boundary Conditions at a Conducting Surface: Skin Depth .. Mutilated Maxwell s Equations (MMEs) .. TEM Waves .. TE and TM Waves .. TM Waves .. Summary of TE/TM waves .. Rectangular Waveguides .. Resonant Cavities .. Wave Guides Assignment .. 14113 Maxwell s Equations, Yet Again .. Quickie Review of Chapter 6 .. Green s Functions for the Wave Equation .. Poisson Equation .. Green s Function for the Helmholtz Equation .. Green s Function for the Wave Equation .. Simple Radiating Systems .. The Zones .. The Near Zone .. The Far Zone .. The Homogeneous Helmholtz Equation .. Properties of Spherical Bessel Functions.

7 (r),NL(r), andH L(r) .. General Solutions to the HHE .. Green s Functions and Free Spherical Waves .. Electric Dipole Radiation .. Radiation outside the source .. Radiation.. Magnetic Dipole and Electric Quadrupole Radiation Fields .. Magnetic Dipole Radiation .. Electric Quadrupole Radiation .. Radiation Assignment .. 17114 Vector Angular momentum and spherical harmonics .. Magnetic and Electric Multipoles Revisited .. Vector Spherical Harmonics and Multipoles .. 17915 The Hansen The Hansen Multipoles .. The Basic Solutions .. Their Significant Properties .. Explicit Forms .. Green s Functions for the Vector Helmholtz Equation .. Multipolar Radiation, revisited .. A Linear Center-Fed Half-Wave Antenna .. Connection to Old (Approximate) Multipole Moments .. Angular Momentum Flux .. Concluding Remarks About Multipoles .. Table of Properties of Vector Harmonics .. 20616 Optical Radiation Reaction of a Polarizable Medium.

8 Scattering from a Small Dielectric Sphere .. Scattering from a Small Conducting Sphere .. Many Scatterers .. 220 III Relativistic Electrodynamics22517 Special Einstein s Postulates .. The Elementary Lorentz Transformation .. 4-Vectors .. Proper Time and Time Dilation .. Addition of Velocities .. Relativistic Energy and Momentum .. 24018 The Lorentz The Geometry of Space Time .. Tensors in 4 Dimensions .. The Metric Tensor .. Generators of the Lorentz Group .. Infinitesimal Transformations .. Thomas Precession .. Covariant Formulation of Electrodynamics .. The Transformation of Electromagnetic Fields .. 27219 Relativistic Covariant Field Theory .. The Brute Force Way .. The Elegant Way .. Motion of a Point Charge in a Static Magnetic Field .. Building a Relativistic Field Theory .. The Symmetric Stress Tensor .. Covariant Green s Functions .. 29020 Radiation from Point Larmor s Formula.

9 Thomson Scattering of Radiation .. 30521 Radiation The Death of Classical Physics .. Radiation Reaction and Energy Conservation .. Integrodifferential Equations of Motion .. Radiation Damping of an Oscillating Charge .. Dirac s Derivation of Radiation Reaction .. Wheeler and Feynman s Derivation of Radiation Reaction .. My Own Field-Free Derivation of Radiation Reaction .. 320 PrefaceClassical Electrodynamics is one of the most beautiful things in the simple vector equations (or one tensor equation and an asssociated dual)describe the unified electromagnetic field and more or less directly imply thetheory of relativity. The discovery and proof that light is an electromagneticwave and unifies two fields stands to this day as one of the greatest moments inthe history of four equations even contain within them the seeds of their own de-struction as a classical theory. Once Maxwell s equations were known in theirentirety, it rapidly became clear that their predictions however beautifullyverified they were for freely propagating fields and the connection of those fieldswith macroscopic charge/current distributions wereinconsistentwith virtu-ally all observations at the atomic or nuclear level.

10 This forced the classicists ofthe day, many of them metaphorically kicking or screaming, to inventquantummechanics andquantumelectrodynamics to explain physics at this , once the single fact that an accelerated charged particle necessarilyradiates electromagnetic energy was known, it became virtually impossible toconceptually explain the persistence of structure at the microscopic level (sincethe forces associated with binding objects together out of discrete charged partsinevitably produce an oscillation of charge due to small perturbations of posi-tion, with an associated acceleration). The few hypothesesthat were advancedto account for it without an overtly oscillatory model were rapidly and de-cisively shot down by (now famous) experiments by Rutherford, Millikan, though the Universe proves to be quantum mechanical at the micro-scopic level, classical Electrodynamics is nevertheless extremely relevant anduseful in the real world today at themacroscopiclevel.