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Preface

Classical Electrodynamics is one of the most beautiful things in the world. Four simple vector equations (or one tensor equation and an asssociated dual) describe the unified electromagnetic field and more or less directly imply the theory of relativity. The discovery and proof that light is an electromagnetic wave and unifies two fields stands to this day as one of the greatest moments in the history of science.

These four equations even contain within them the seeds of their own destruction as a classical theory. Once Maxwell's equations were known in their entirety, it rapidly became clear that their predictions - however beautifully verified they were for freely propagating fields and the connection of those fields with macroscopic charge/current distributions - were inconsistent with virtually all observations at the atomic or nuclear level. This forced the classicists of the day, many of them metaphorically kicking or screaming, to invent quantum mechanics and quantum electrodynamics to explain physics at this scale.

Indeed, once the single fact that an accelerated charged particle necessarily radiates electromagnetic energy was known, it became virtually impossible to conceptually explain the persistence of structure at the microscopic level (since the forces associated with binding objects together out of discrete charged parts inevitably produce an oscillation of charge due to small perturbations of position, with an associated acceleration). The few hypotheses that were advanced to account for it ``without'' an overtly oscillatory model were rapidly and decisively shot down by (now famous) experiments by Rutherford, Millikan, and others.

Even though the Universe proves to be quantum mechanical at the microscopic level, classical electrodynamics is nevertheless extremely relevant and useful in the real world today at the macroscopic level. It describes extremely precisely nearly all the mundane aspects of ordinary electrical engineering and electromagnetic radiation from the static limit through optical frequencies. Even at the molecular level or photonic level where it breaks down and a quantum theory must be used it is first necessary to understand the classical theory before exploring the quantum theory, as the quantum theory is built on top of the entire relativistic electrodynamic conceptual framework already established.

This set of lecture notes is designed to be used to teach graduate students (and possibly advanced and motivated undergraduates) classical electrodynamics. In particular, it supports the second (more difficult) semester of a two semester course in electrodynamics that covers pretty much ``all'' of the theory itself (omitting, of course, many topics or specific areas where it can be applied) out to the points where the theory itself breaks down as noted above. At that point, to make further progress a student needs to learn about more fields, quantum (field) theory, advanced (general) relativity - topics generally beyond the scope of these notes.

The requirements for this course include a thorough understanding of electricity and magnetism at the level of at least one, ideally two, undergraduate courses. At Duke, for example, physics majors are first exposed first to an introductory course that covers the integral formulation of Maxwell's equations and light that uses no multivariate differential calculus, then a second course that develops the vector differential formulation of Maxwell's equations and their consequences) as does this course) but with considerably less mathematical rigor and completeness of the treatment as students taking it have likely still not had a course in e.g. contour integration. Students using these notes will find it useful to be at least somewhat comfortable with vector differential and integral calculus, to have had exposure to the theory and solution methodology of ordinary and partial differential equations, to be familiar with the mathematics of complex variables and analytic functions, contour integration, and it would be simply lovely if they at least knew what a ``tensor'' was.

However, even more so than is the case for most physics texts, this book will endeavor to provide internal support for students that are weak in one or more of these required mathematical skills. This support will come in one of several forms. At the very least, considerable effort has been made to hunt down on behalf of the student and explicitly recommend useful textbooks and online resources on various mathematical and physical topics that may be of use to them. Many of these resources are freely available on the web. Some mathematical methods are completely developed in the context of the discussion, either because it makes sense to do so or because there simply are no references a student is likely to be able to find. Finally, selected topics will be covered in e.g. appendices or as insertions in the text where they are short enough to be coverable in this way and important enough that students are likely to be highly confused without this sort of support.

A very brief review of the electrodynamics topics covered includes: Maxwell's equations themselves (skipping the usual coverage of electrostatics and magnetostatics that often makes up the first semester of a two-semester course), then plane waves, dispersion, penetration of waves at a boundary (skin depth), wave guides and cavities and the various (TE, TM, TEM) modes associated with them, and radiation in the more general case beginning with sources.

In the course of studying radiation from sources we develop multipolar radiation in detail. This text includes a fairly thorough exposition of the underlying PDEs, the properties of the Green's functions used to generate multipoles both approximate and exact, and formally precise solutions that extend inside the source charge-current density (as indeed they must for this formalism to be of use in e.g. self-consistent field theories treating extended charge density distributions). In addition to the vector spherical harmonics, it defines and derives the properties of the Hansen multipoles (which are otherwise very nearly a lost art) demonstrating their practical utility with example problems involving antennae. It concludes this part of the exposition with a short description of optical scattering as waves interact with ``media'', e.g. small spheres intended to model atoms or molecules.

The text then procedes to develop relativity theory, first reviewing the elementary theory presumably already familiar to students, then developing the full Lorentz Group. As students tend to not be familiar with tensors, the notes contain a special appendix on tensors and tensor notation as a supplement. It also contains a bit of supplemental support on at least those aspects of contour integration relevant to the course for similar reasons. With relativity in hand, relativistic electrodynamics is developed, including the properties of radiation emitted from a point charge as it is accelerated.

Finally, the text concludes with a nice overview of radiation reaction (exploring the work of Lorentz, Dirac, and Wheeler and Feynman) and the puzzles therein - self-interaction versus action at a distance, the need for a classical renormalization in a theory based on self-interaction. This makes the text just a bit too long to present in a single semester (at least to my own experience); instructors that begin with Maxwell's equations in detail (including the treatment of monopoles) may not have time to get to radiation reaction, but instructors who begin with plane waves or waveguides likely will.

One note-worthy feature of this text in its online form (sorry, but I do like puns and you'll just have to get used to them:-) is that the electronic/online version of them includes several inventions of my own such as a wikinote1 , a reference to supporting wikipedia articles that appears as a URL and footnote in the text copy but which is an active link in a PDF or HTML (online) copy. Similarly, there are google links and ordinary web links presented in the same way.

This text a set of real lecture notes and is therefore likely to change as they are used, semester by semester. In some cases the changes are quite important, for example when a kind reader gently points out a bone-headed mistake I made that makes some aspect of the physics or presentation quite incorrect. In others they are smaller improvements: a new link, a slightly improved discussion, fixing clumsy language, a new figure (or putting in one of the missing old ones), more or better problems.

For all of these reasons, students who are using this textbook may wish to have both a bound paper copy (homemade or purchased for a fairly nominal sum through Lulu or Amazon) - that will inevitably contain omissions and mistakes or material I don't actually cover in this year's class - and the current electronic copy. I generally maintain the current snapshot of the electronic copy that I'm actually using to teach from where it is available, for free to all comers, on my personal/class website at:

http://www.phy.duke.edu/$\sim$rgb/Class/Electrodynamics.php

(which cleverly and self-consistently demonstrates an active link in action, as did the wikilink above). In this way a student or instructor can have the convenience of a slightly-out-of-date paper copy to browse or study or follow and mark up during lecture as well as an electronic copy that is up to date and which contains useful active links.

Let it be noted that I'm as greedy and needy as the next human, and can always use extra money. As I've worked quite hard on this text (and from observation they go quite beyond what e.g. most of my colleagues in the physics world make available as online notes for their own courses) and I have done the work required to transform them into an actual bound book that students can elect to purchase all at once instead of downloading the free PDF, printing it out as two-sided pages, punching it, and inserting it into a three ring binder that anonymously joins the rest of their notes and ultimately is thrown away or lost.

This printed book is remarkably inexpensive by the standards of modern textbooks (where e.g Wyld, which I once purchased now at $16 a copy, is not available new for $70 a copy). At the same site, students can find the actual PDF from which the book is generated available for a very low cost and are at liberty to purchase and keep that on their personal laptops or PDF-capable e-book readers, or for that matter to have it printed and bound by a local printer. In both cases I make a small royalty (on the order of $5) from their sale, which is both fair and helps support me so that I can write more texts such as this.

However, students around the world have very different means. Purchasing a $7.50 download in the United States means (for most students) that a student has to give up a few Latte Enormes from Starbucks. Purchasing that same download could be a real hardship for students from many countries around the world including some from the United States. For this reason students will always have the option of using the online notes directly from the class website for free or printing their own copy on paper at cost. All that I ask of students who elect to use them for free is that they ``pay it forward'' - that one day they help others who are less fortunate in some way for free so that we can all keep the world moving along in a positive direction.

The one restriction I have, and I think it is entirely fair, is that instructors who elect to use these notes to help support the teaching of their own classes (either building them with or without modifications from the sources or using any of the free prebuilt images) may not resell these notes to their own students for a profit or otherwise without my explicit permission, nor may they alter this preface, the authorship or copyright notice (basically all the front-matter) or the license. Instructors are free to add to or edit the content to support their own class, however, and the notes should easily build on any e.g. linux system.

Anyway, good luck and remember that I do cherish feedback of all sorts, corrections, additions (especially in ready-to-build latex with EPS figures:-), suggestions, criticisms, and or course money. You can always send me money...



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Robert G. Brown 2014-08-19