Classical 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 me his(mostly handwritten or crudely typed) lecture notes when in the naturalcourse of events I came to teach Electrodynamics for the first time. Mostof the notes have been completely rewritten, typeset with latex, changedto emphasize the things that I think are important, but there are still im-portant fragments that are more or less pure Biedenharn, in particular thelovely exposition of vector spherical harmonics and Hansen solutions (whicha student will very likely be unable to find anywhere else).I d also like to acknowledge and thank my many colleagues at Duke andelsewhere who have contributed ideas, criticisms, or encouragement to meover the years, in particular Mikael Ciftan (my other advisor for my beyond), Richard Palmer and Ronen NoticeCopyright Robert G.

2 Brown 1993, 2007 NoticeThis set of lecture notes is designed to support my personal teach-ing activities at Duke University , in particular teaching itsPhysics 318/319series (graduate level Classical Electrodynamics ) using J. D. Jackson sClas-sical Electrodynamicsas a primary text. However, the notes may be usefulto students studying from other texts or even as a standalone text in its 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 bookPDF in Crown Quarto size inch pages that can be readusing any PDF browser or locally printed).In this way the text can be used by students all over the world, whereeach student can pay (or not) according to their means.

3 Nevertheless, I amhoping that students who truly find this work useful will purchase eitherthe PDF download or the current paper snapshot, if only to help subsidizeme while I continue to write more inexpensive textbooks in physics or are real lecture notes, and they therefore have errors great andsmall, missing figures (that I usually draw from memory in class), and theycover and omit topics according to my own view of what is or isn t importantto cover in a one-semester course. Expect them to change withoutwarningas I add content or correct errors. Purchasers of a paper versionshouldbe aware of its imperfection and be prepared to either live with it or markup theirowncopies with corrections or additions as need be in the lecturenote spirit, as I do mine.

4 The text has generous margins, is widely spaced,and contains scattered blank pages for students or instructors own use tofacilitate cherish good-hearted communication from students or other instructorspointing out errors or suggesting new content (and have in the past donemy best to implement many such corrections or suggestions).ContentsPrefaceiii1 Syllabus and Course Contact Information .. Useful Texts and Web References .. Course Description .. Basis of Grade .. Percentages .. Research Project: .. Course Rules .. The Interplay of Physics andMathematics .. 10I Mathematical Physics132 Mathematical Prelude153 Vector Calculus: Integration by Parts194 Real Numbers .. Complex Numbers.

5 Contour Integration .. Geometric Algebra .. 295 Partial Differential The Laplace Equation .. The Helmholtz Equation .. The Wave Equation .. 316 The Dyad andN-adic Forms .. Coordinate Transformations .. 377 Group Definition of a Group .. Groups of Transformation .. 438 Math References45II Non-Relativistic Electrodynamics499 Plane The Free Space Wave Equation .. Maxwell s Equations .. The Wave Equation .. Plane Waves .. Polarization of Plane Waves .. Reflection and Refraction at a Plane Interface .. Kinematics and Snell s Law .. Dynamics and Reflection/Refraction .. Dispersion .. Static Case .. Dynamic Case .. Things to Note .. Anomalous Dispersion, and Resonant Absorption.

6 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 .. 8410 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 .. 10611 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.

7 Simple Radiating Systems .. The Zones .. The Near Zone .. The Far Zone .. The Homogeneous Helmholtz Equation .. Properties of Spherical Bessel Functions .. (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 .. 14012 Vector Angular momentum and spherical harmonics .. Magnetic and Electric Multipoles Revisited .. Vector Spherical Harmonics and Multipoles .. 15013 The Hansen The Hansen Multipoles .. The Basic Solutions .. Their Significant Properties.

8 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 .. 18114 Optical Radiation Reaction of a Polarizable Medium .. Scattering from a Small Dielectric Sphere .. Scattering from a Small Conducting Sphere .. Many Scatterers .. 197 III Relativistic Electrodynamics20115 Special Einstein s Postulates .. The Elementary Lorentz Transformation .. 4-Vectors .. Proper Time and Time Dilation .. Addition of Velocities .. Relativistic Energy and Momentum.

9 21916 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 ..25917 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 .. 28018 Radiation from Point Larmor s Formula .. Thomson Scattering of Radiation .. 29719 Radiation The Death of Classical Physics .. Radiation Reaction and Energy Conservation.

10 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 .. 314 PrefaceClassical Electrodynamics is one of the most beautiful things in the simple vector equations (or one tensor equation and an asssociateddual) describe the unified electromagnetic field and more or less directlyimply the theory of relativity. The discovery and proof that light is anelectromagnetic wave stands to this day as one of the greatest moments inthe history of four equations even contain within them the seeds of their owndestruction as a Classical theory. Once Maxwell s equations were known,theinconsistencyof the Classical physics one could then easily derive fromthem with countless experimental results associated with electromagnetismforced the classicists of the day, many of them metaphorically kicking orscreaming, to inventquantummechanics andquantumelectrodynamics toexplain them.