12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667686970717273747576777879808182838485868788899091929394959697989910010110210310410510610710810911011111211311411511611711811912012112212312412512612712812913013113213313413513613713813914014114214314414514614714814915015115215315415515615715815916016116216316416516616716816917017117217317417517617717817918018118218318418518618718818919019119219319419519619719819920020120220320420520620720820921021121221321421521621721821922022122222322422522622722822923023123223323423523623723823924024124224324424524624724824925025125225325425525625725825926026126226326426526626726826927027127227327427527627727827928028128228328428528628728828929029129229329429529629729829930030130230330430530630730830931031131231331431531631731831932032132232332432532632732832933033133233333433533633733833934034134234334434534634734834935035135235335435535635735835936036136236336436536636736836937037137237337437537637737837938038138238338438538638738838939039139239339439539639739839940040140240340440540640740840941041141241341441541641741841942042142242342442542642742842943043143243343443543643743843944044144244344444544644744844945045145245345445545645745845946046146246346446546646746846947047147247347447547647747847948048148248348448548648748848949049149249349449549649749849950050150250350450550650750850951051151251351451551651751851952052152252352452552652752852953053153253353453553653753853954054154254354454554654754854955055155255355455555655755855956056156256356456556656756856957057157257357457557657757857958058158258358458558658758858959059159259359459559659759859960060160260360460560660760860961061161261361461561661761861962062162262362462562662762862963063163263363463563663763863964064164264364464564664764864965065165265365465565665765865966066166266366466566666766866967067167267367467567667767867968068168268368468568668768868969069169269369469569669769869970070170270370470570670770870971071171271371471571671771871972072172272372472572672772872973073173273373473573673773873974074174274374474574674774874975075175275375475575675775875976076176276376476576676776876977077177277377477577677777877978078178278378478578678778878979079179279379479579679779879980080180280380480580680780880981081181281381481581681781881982082182282382482582682782882983083183283383483583683783883984084184284384484584684784884985085185285385485585685785885986086186286386486586686786886987087187287387487587687787887988088188288388488588688788888989089189289389489589689789889990090190290390490590690790890991091191291391491591691791891992092192292392492592692792892993093193293393493593693793893994094194294394494594694794894995095195295395495595695795895996096196296396496596696796896997097197297397497597697797897998098198298398498598698798898999099199299399499599699799899910001001100210031004100510061007100810091010101110121013101410151016101710181019102010211022102310241025102610271028102910301031103210331034103510361037103810391040104110421043104410451046104710481049105010511052105310541055105610571058105910601061106210631064106510661067106810691070107110721073107410751076107710781079108010811082108310841085108610871088108910901091109210931094109510961097109810991100110111021103110411051106110711081109111011111112111311141115111611171118111911201121112211231124112511261127112811291130113111321133113411351136113711381139114011411142114311441145114611471148114911501151115211531154115511561157115811591160116111621163116411651166116711681169117011711172117311741175117611771178117911801181118211831184118511861187118811891190119111921193119411951196119711981199120012011202120312041205120612071208120912101211121212131214121512161217121812191220122112221223122412251226122712281229123012311232123312341235123612371238123912401241124212431244124512461247124812491250125112521253125412551256125712581259126012611262126312641265126612671268126912701271127212731274127512761277127812791280128112821283128412851286128712881289129012911292129312941295129612971298129913001301130213031304130513061307130813091310131113121313131413151316131713181319132013211322132313241325132613271328132913301331133213331334133513361337133813391340134113421343134413451346134713481349135013511352135313541355135613571358135913601361136213631364136513661367136813691370137113721373137413751376137713781379138013811382138313841385138613871388138913901391139213931394139513961397139813991400140114021403140414051406140714081409141014111412141314141415141614171418141914201421142214231424142514261427142814291430143114321433143414351436143714381439144014411442144314441445144614471448144914501451145214531454145514561457145814591460146114621463146414651466146714681469147014711472147314741475147614771478147914801481148214831484148514861487148814891490149114921493149414951496149714981499150015011502150315041505150615071508150915101511151215131514151515161517151815191520152115221523152415251526152715281529153015311532153315341535153615371538153915401541154215431544154515461547154815491550155115521553155415551556155715581559156015611562156315641565156615671568156915701571157215731574157515761577157815791580158115821583158415851586158715881589159015911592159315941595159615971598159916001601160216031604160516061607160816091610161116121613161416151616161716181619162016211622162316241625162616271628162916301631163216331634163516361637163816391640164116421643164416451646164716481649165016511652165316541655165616571658165916601661166216631664166516661667166816691670167116721673167416751676167716781679168016811682168316841685168616871688168916901691169216931694169516961697169816991700170117021703170417051706170717081709171017111712171317141715171617171718171917201721172217231724 |
- <!DOCTYPE html>
- <html lang="en">
- <head>
- <!-- 2022-03-24 Thu 08:42 -->
- <meta charset="utf-8">
- <meta name="viewport" content="width=device-width, initial-scale=1">
- <title>Pre-Quantum Electrodynamics</title>
- <meta name="generator" content="Org mode">
- <meta name="author" content="Jean-Sébastien Caux">
- <style >
- <!--/*--><![CDATA[/*><!--*/
- .title { text-align: center;
- margin-bottom: .2em; }
- .subtitle { text-align: center;
- font-size: medium;
- font-weight: bold;
- margin-top:0; }
- .todo { font-family: monospace; color: red; }
- .done { font-family: monospace; color: green; }
- .priority { font-family: monospace; color: orange; }
- .tag { background-color: #eee; font-family: monospace;
- padding: 2px; font-size: 80%; font-weight: normal; }
- .timestamp { color: #bebebe; }
- .timestamp-kwd { color: #5f9ea0; }
- .org-right { margin-left: auto; margin-right: 0px; text-align: right; }
- .org-left { margin-left: 0px; margin-right: auto; text-align: left; }
- .org-center { margin-left: auto; margin-right: auto; text-align: center; }
- .underline { text-decoration: underline; }
- #postamble p, #preamble p { font-size: 90%; margin: .2em; }
- p.verse { margin-left: 3%; }
- pre {
- border: 1px solid #ccc;
- box-shadow: 3px 3px 3px #eee;
- padding: 8pt;
- font-family: monospace;
- overflow: auto;
- margin: 1.2em;
- }
- pre.src {
- position: relative;
- overflow: auto;
- padding-top: 1.2em;
- }
- pre.src:before {
- display: none;
- position: absolute;
- background-color: white;
- top: -10px;
- right: 10px;
- padding: 3px;
- border: 1px solid black;
- }
- pre.src:hover:before { display: inline; margin-top: 14px;}
- /* Languages per Org manual */
- pre.src-asymptote:before { content: 'Asymptote'; }
- pre.src-awk:before { content: 'Awk'; }
- pre.src-C:before { content: 'C'; }
- /* pre.src-C++ doesn't work in CSS */
- pre.src-clojure:before { content: 'Clojure'; }
- pre.src-css:before { content: 'CSS'; }
- pre.src-D:before { content: 'D'; }
- pre.src-ditaa:before { content: 'ditaa'; }
- pre.src-dot:before { content: 'Graphviz'; }
- pre.src-calc:before { content: 'Emacs Calc'; }
- pre.src-emacs-lisp:before { content: 'Emacs Lisp'; }
- pre.src-fortran:before { content: 'Fortran'; }
- pre.src-gnuplot:before { content: 'gnuplot'; }
- pre.src-haskell:before { content: 'Haskell'; }
- pre.src-hledger:before { content: 'hledger'; }
- pre.src-java:before { content: 'Java'; }
- pre.src-js:before { content: 'Javascript'; }
- pre.src-latex:before { content: 'LaTeX'; }
- pre.src-ledger:before { content: 'Ledger'; }
- pre.src-lisp:before { content: 'Lisp'; }
- pre.src-lilypond:before { content: 'Lilypond'; }
- pre.src-lua:before { content: 'Lua'; }
- pre.src-matlab:before { content: 'MATLAB'; }
- pre.src-mscgen:before { content: 'Mscgen'; }
- pre.src-ocaml:before { content: 'Objective Caml'; }
- pre.src-octave:before { content: 'Octave'; }
- pre.src-org:before { content: 'Org mode'; }
- pre.src-oz:before { content: 'OZ'; }
- pre.src-plantuml:before { content: 'Plantuml'; }
- pre.src-processing:before { content: 'Processing.js'; }
- pre.src-python:before { content: 'Python'; }
- pre.src-R:before { content: 'R'; }
- pre.src-ruby:before { content: 'Ruby'; }
- pre.src-sass:before { content: 'Sass'; }
- pre.src-scheme:before { content: 'Scheme'; }
- pre.src-screen:before { content: 'Gnu Screen'; }
- pre.src-sed:before { content: 'Sed'; }
- pre.src-sh:before { content: 'shell'; }
- pre.src-sql:before { content: 'SQL'; }
- pre.src-sqlite:before { content: 'SQLite'; }
- /* additional languages in org.el's org-babel-load-languages alist */
- pre.src-forth:before { content: 'Forth'; }
- pre.src-io:before { content: 'IO'; }
- pre.src-J:before { content: 'J'; }
- pre.src-makefile:before { content: 'Makefile'; }
- pre.src-maxima:before { content: 'Maxima'; }
- pre.src-perl:before { content: 'Perl'; }
- pre.src-picolisp:before { content: 'Pico Lisp'; }
- pre.src-scala:before { content: 'Scala'; }
- pre.src-shell:before { content: 'Shell Script'; }
- pre.src-ebnf2ps:before { content: 'ebfn2ps'; }
- /* additional language identifiers per "defun org-babel-execute"
- in ob-*.el */
- pre.src-cpp:before { content: 'C++'; }
- pre.src-abc:before { content: 'ABC'; }
- pre.src-coq:before { content: 'Coq'; }
- pre.src-groovy:before { content: 'Groovy'; }
- /* additional language identifiers from org-babel-shell-names in
- ob-shell.el: ob-shell is the only babel language using a lambda to put
- the execution function name together. */
- pre.src-bash:before { content: 'bash'; }
- pre.src-csh:before { content: 'csh'; }
- pre.src-ash:before { content: 'ash'; }
- pre.src-dash:before { content: 'dash'; }
- pre.src-ksh:before { content: 'ksh'; }
- pre.src-mksh:before { content: 'mksh'; }
- pre.src-posh:before { content: 'posh'; }
- /* Additional Emacs modes also supported by the LaTeX listings package */
- pre.src-ada:before { content: 'Ada'; }
- pre.src-asm:before { content: 'Assembler'; }
- pre.src-caml:before { content: 'Caml'; }
- pre.src-delphi:before { content: 'Delphi'; }
- pre.src-html:before { content: 'HTML'; }
- pre.src-idl:before { content: 'IDL'; }
- pre.src-mercury:before { content: 'Mercury'; }
- pre.src-metapost:before { content: 'MetaPost'; }
- pre.src-modula-2:before { content: 'Modula-2'; }
- pre.src-pascal:before { content: 'Pascal'; }
- pre.src-ps:before { content: 'PostScript'; }
- pre.src-prolog:before { content: 'Prolog'; }
- pre.src-simula:before { content: 'Simula'; }
- pre.src-tcl:before { content: 'tcl'; }
- pre.src-tex:before { content: 'TeX'; }
- pre.src-plain-tex:before { content: 'Plain TeX'; }
- pre.src-verilog:before { content: 'Verilog'; }
- pre.src-vhdl:before { content: 'VHDL'; }
- pre.src-xml:before { content: 'XML'; }
- pre.src-nxml:before { content: 'XML'; }
- /* add a generic configuration mode; LaTeX export needs an additional
- (add-to-list 'org-latex-listings-langs '(conf " ")) in .emacs */
- pre.src-conf:before { content: 'Configuration File'; }
-
- table { border-collapse:collapse; }
- caption.t-above { caption-side: top; }
- caption.t-bottom { caption-side: bottom; }
- td, th { vertical-align:top; }
- th.org-right { text-align: center; }
- th.org-left { text-align: center; }
- th.org-center { text-align: center; }
- td.org-right { text-align: right; }
- td.org-left { text-align: left; }
- td.org-center { text-align: center; }
- dt { font-weight: bold; }
- .footpara { display: inline; }
- .footdef { margin-bottom: 1em; }
- .figure { padding: 1em; }
- .figure p { text-align: center; }
- .equation-container {
- display: table;
- text-align: center;
- width: 100%;
- }
- .equation {
- vertical-align: middle;
- }
- .equation-label {
- display: table-cell;
- text-align: right;
- vertical-align: middle;
- }
- .inlinetask {
- padding: 10px;
- border: 2px solid gray;
- margin: 10px;
- background: #ffffcc;
- }
- #org-div-home-and-up
- { text-align: right; font-size: 70%; white-space: nowrap; }
- textarea { overflow-x: auto; }
- .linenr { font-size: smaller }
- .code-highlighted { background-color: #ffff00; }
- .org-info-js_info-navigation { border-style: none; }
- #org-info-js_console-label
- { font-size: 10px; font-weight: bold; white-space: nowrap; }
- .org-info-js_search-highlight
- { background-color: #ffff00; color: #000000; font-weight: bold; }
- .org-svg { width: 90%; }
- /*]]>*/-->
- </style>
- <link rel="stylesheet" type="text/css" href="style.css">
- <script >
- // @license magnet:?xt=urn:btih:e95b018ef3580986a04669f1b5879592219e2a7a&dn=public-domain.txt Public Domain
- <!--/*--><![CDATA[/*><!--*/
- function CodeHighlightOn(elem, id)
- {
- var target = document.getElementById(id);
- if(null != target) {
- elem.classList.add("code-highlighted");
- target.classList.add("code-highlighted");
- }
- }
- function CodeHighlightOff(elem, id)
- {
- var target = document.getElementById(id);
- if(null != target) {
- elem.classList.remove("code-highlighted");
- target.classList.remove("code-highlighted");
- }
- }
- /*]]>*///-->
- // @license-end
- </script>
- <script type="text/x-mathjax-config">
- MathJax.Hub.Config({
- displayAlign: "center",
- displayIndent: "0em",
-
- "HTML-CSS": { scale: 100,
- linebreaks: { automatic: "false" },
- webFont: "TeX"
- },
- SVG: {scale: 100,
- linebreaks: { automatic: "false" },
- font: "TeX"},
- NativeMML: {scale: 100},
- TeX: { equationNumbers: {autoNumber: "AMS"},
- MultLineWidth: "85%",
- TagSide: "right",
- TagIndent: ".8em"
- }
- });
- </script>
- <script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_HTML"></script>
- </head>
-
- <div id="content">
- <header>
- <h1 class="title">
- <a href="./index.html" class="homepage-link">Pre-Quantum Electrodynamics</a>
- </h1>
- </header>
- <nav id="collapsed-table-of-contents">
- <details>
- <summary>
- Table of contents
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./in.html#in">Introduction</a><span class="headline-id">in</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./in_p.html#in_p">Preface</a><span class="headline-id">in.p</span>
-
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./in_t.html#in_t">Tips for the reader</a><span class="headline-id">in.t</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
-
- </li>
- <li>
- <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details open="">
- <summary class="toc-open">
- <a href="./ems.html#ems">Electromagnetostatics</a><span class="headline-id">ems</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./ems_es.html#ems_es">Electrostatics</a><span class="headline-id">ems.es</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./ems_es_ec.html#ems_es_ec">Electric Charge</a><span class="headline-id">ems.es.ec</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_es_ec_b.html#ems_es_ec_b">Basics</a><span class="headline-id">ems.es.ec.b</span>
-
- </li>
- <li>
- <a href="./ems_es_ec_c.html#ems_es_ec_c">Conservation</a><span class="headline-id">ems.es.ec.c</span>
-
- </li>
- <li>
- <a href="./ems_es_ec_q.html#ems_es_ec_q">Quantization</a><span class="headline-id">ems.es.ec.q</span>
-
- </li>
- <li>
- <a href="./ems_es_ec_s.html#ems_es_ec_s">Structure</a><span class="headline-id">ems.es.ec.s</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_es_efo.html#ems_es_efo">Electric Force and Energy</a><span class="headline-id">ems.es.efo</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_es_efo_cl.html#ems_es_efo_cl">Coulomb's Law</a><span class="headline-id">ems.es.efo.cl</span>
-
- </li>
- <li>
- <a href="./ems_es_efo_ps.html#ems_es_efo_ps">Principle of Superposition</a><span class="headline-id">ems.es.efo.ps</span>
-
- </li>
- <li>
- <a href="./ems_es_efo_exp.html#ems_es_efo_exp">Experimental Investigations</a><span class="headline-id">ems.es.efo.exp</span>
-
- </li>
- <li>
- <a href="./ems_es_efo_e.html#ems_es_efo_e">Energy in Systems of Point Charges</a><span class="headline-id">ems.es.efo.e</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_es_ef.html#ems_es_ef">Electrostatic Fields</a><span class="headline-id">ems.es.ef</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_es_ef_pc.html#ems_es_ef_pc">Electrostatic Field of Point Charges</a><span class="headline-id">ems.es.ef.pc</span>
-
- </li>
- <li>
- <a href="./ems_es_ef_ccd.html#ems_es_ef_ccd">Electrostatic Field of Continuous Charge Distributions</a><span class="headline-id">ems.es.ef.ccd</span>
-
- </li>
- <li>
- <a href="./ems_es_ef_cE.html#ems_es_ef_cE">The Curl of \({\bf E}\)</a><span class="headline-id">ems.es.ef.cE</span>
-
- </li>
- <li>
- <a href="./ems_es_ef_Gl.html#ems_es_ef_Gl">Gauss's Law: the divergence of \({\bf E}\)</a><span class="headline-id">ems.es.ef.Gl</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_es_ep.html#ems_es_ep">The Electrostatic Potential</a><span class="headline-id">ems.es.ep</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_es_ep_d.html#ems_es_ep_d">Definition</a><span class="headline-id">ems.es.ep.d</span>
-
- </li>
- <li>
- <a href="./ems_es_ep_fp.html#ems_es_ep_fp">Field in terms of the potential</a><span class="headline-id">ems.es.ep.fp</span>
-
- </li>
- <li>
- <a href="./ems_es_ep_ex.html#ems_es_ep_ex">Example calculations for the potential</a><span class="headline-id">ems.es.ep.ex</span>
-
- </li>
- <li>
- <a href="./ems_es_ep_PL.html#ems_es_ep_PL">Poisson's and Laplace's Equations</a><span class="headline-id">ems.es.ep.PL</span>
-
- </li>
- <li>
- <a href="./ems_es_ep_bc.html#ems_es_ep_bc">Electrostatic Boundary Conditions</a><span class="headline-id">ems.es.ep.bc</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
- <a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>
-
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_es_c.html#ems_es_c">Conductors</a><span class="headline-id">ems.es.c</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_es_c_p.html#ems_es_c_p">Properties</a><span class="headline-id">ems.es.c.p</span>
-
- </li>
- <li>
- <a href="./ems_es_c_ic.html#ems_es_c_ic">Induced Charges</a><span class="headline-id">ems.es.c.ic</span>
-
- </li>
- <li>
- <a href="./ems_es_c_sc.html#ems_es_c_sc">Surface Charge and the Force on a Conductor</a><span class="headline-id">ems.es.c.sc</span>
-
- </li>
- <li>
- <a href="./ems_es_c_cap.html#ems_es_c_cap">Capacitors</a><span class="headline-id">ems.es.c.cap</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details open="">
- <summary class="toc-open">
- <a href="./ems_ca.html#ems_ca">Calculating or Approximating the Electrostatic Potential</a><span class="headline-id">ems.ca</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ca_fe.html#ems_ca_fe">Fundamental Equations for the Electrostatic Potential</a><span class="headline-id">ems.ca.fe</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_ca_fe_L.html#ems_ca_fe_L">The Laplace Equation</a><span class="headline-id">ems.ca.fe.L</span>
-
- </li>
- <li>
- <a href="./ems_ca_fe_g.html#ems_ca_fe_g">Green's Identities</a><span class="headline-id">ems.ca.fe.g</span>
-
- </li>
- <li>
- <a href="./ems_ca_fe_uP.html#ems_ca_fe_uP">Uniqueness of Solution to Poisson's Equation</a><span class="headline-id">ems.ca.fe.uP</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ca_mi.html#ems_ca_mi">The Method of Images</a><span class="headline-id">ems.ca.mi</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_ca_mi_isc.html#ems_ca_mi_isc">Induced Surface Charges</a><span class="headline-id">ems.ca.mi.isc</span>
-
- </li>
- <li>
- <a href="./ems_ca_mi_fe.html#ems_ca_mi_fe">Force and Energy</a><span class="headline-id">ems.ca.mi.fe</span>
-
- </li>
- <li>
- <a href="./ems_ca_mi_o.html#ems_ca_mi_o">Other Image Problems</a><span class="headline-id">ems.ca.mi.o</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ca_sv.html#ems_ca_sv">Separation of Variables</a><span class="headline-id">ems.ca.sv</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_ca_sv_car.html#ems_ca_sv_car">Cartesian Coordinates</a><span class="headline-id">ems.ca.sv.car</span>
-
- </li>
- <li>
- <a href="./ems_ca_sv_cyl.html#ems_ca_sv_cyl">Cylindrical Coordinates</a><span class="headline-id">ems.ca.sv.cyl</span>
-
- </li>
- <li>
- <a href="./ems_ca_sv_sph.html#ems_ca_sv_sph">Spherical Coordinates</a><span class="headline-id">ems.ca.sv.sph</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details open="">
- <summary class="toc-open">
- <a href="./ems_ca_me.html#ems_ca_me">The Multipole Expansion</a><span class="headline-id">ems.ca.me</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_ca_me_a.html#ems_ca_me_a">Approximate Potential at Large Distance</a><span class="headline-id">ems.ca.me.a</span>
-
- </li>
- <li class="toc-currentpage">
- <a href="./ems_ca_me_md.html#ems_ca_me_md">Monopole and Dipole Terms</a><span class="headline-id">ems.ca.me.md</span>
-
- </li>
- <li>
- <a href="./ems_ca_me_h.html#ems_ca_me_h">Higher Moments</a><span class="headline-id">ems.ca.me.h</span>
-
- </li>
- <li>
- <a href="./ems_ca_me_Ed.html#ems_ca_me_Ed">The Electric Field of a Dipole</a><span class="headline-id">ems.ca.me.Ed</span>
-
- </li>
- <li>
- <a href="./ems_ca_me_Eq.html#ems_ca_me_Eq">The Electric Field of a Quadrupole</a><span class="headline-id">ems.ca.me.Eq</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ms.html#ems_ms">Magnetostatics</a><span class="headline-id">ems.ms</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ms_lf.html#ems_ms_lf">Charges in Motion: the Lorentz Force Law</a><span class="headline-id">ems.ms.lf</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charges</a><span class="headline-id">ems.ms.lf.pc</span>
-
- </li>
- <li>
- <a href="./ems_ms_lf_sc.html#ems_ms_lf_sc">Steady Currents</a><span class="headline-id">ems.ms.lf.sc</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
- <a href="./ems_ms_ce.html#ems_ms_ce">Charge Conservation and the Continuity Equation</a><span class="headline-id">ems.ms.ce</span>
-
- </li>
- <li>
- <a href="./ems_ms_BS.html#ems_ms_BS">Steady Currents: the Biot-Savart Law</a><span class="headline-id">ems.ms.BS</span>
-
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ms_dcB.html#ems_ms_dcB">Divergence and Curl of \({\bf B}\)</a><span class="headline-id">ems.ms.dcB</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_ms_dcB_iw.html#ems_ms_dcB_iw">Simplistic case: infinite wire</a><span class="headline-id">ems.ms.dcB.iw</span>
-
- </li>
- <li>
- <a href="./ems_ms_dcB_d.html#ems_ms_dcB_d">Divergence of \({\bf B}\) from Biot-Savart</a><span class="headline-id">ems.ms.dcB.d</span>
-
- </li>
- <li>
- <a href="./ems_ms_dcB_c.html#ems_ms_dcB_c">Curl of \({\bf B}\) from Biot-Savart; Ampère's Law</a><span class="headline-id">ems.ms.dcB.c</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ms_vp.html#ems_ms_vp">The Vector Potential</a><span class="headline-id">ems.ms.vp</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_ms_vp_A.html#ems_ms_vp_A">Definition; Gauge Choices</a><span class="headline-id">ems.ms.vp.A</span>
-
- </li>
- <li>
- <a href="./ems_ms_vp_mbc.html#ems_ms_vp_mbc">Magnetic Boundary Conditions</a><span class="headline-id">ems.ms.vp.mbc</span>
-
- </li>
- <li>
- <a href="./ems_ms_vp_me.html#ems_ms_vp_me">Multipole Expansion of the Vector Potential</a><span class="headline-id">ems.ms.vp.me</span>
-
- </li>
- <li>
- <a href="./ems_ms_vp_comp.html#ems_ms_vp_comp">Comparison of Electrostatics and Magnetostatics</a><span class="headline-id">ems.ms.vp.comp</span>
-
- </li>
- <li>
- <a href="./ems_ms_vp_LC.html#ems_ms_vp_LC">The Levi-Civita Symbol</a><span class="headline-id">ems.ms.vp.LC</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emsm.html#emsm">Electromagnetostatics in matter</a><span class="headline-id">emsm</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_esm.html#emsm_esm">Electrostatics in matter</a><span class="headline-id">emsm.esm</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_esm_mE.html#emsm_esm_mE">Matter Bathed in E Fields; Polarization</a><span class="headline-id">emsm.esm.mE</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emsm_esm_mE_o.html#emsm_esm_mE_o">Overview</a><span class="headline-id">emsm.esm.mE.o</span>
-
- </li>
- <li>
- <a href="./emsm_esm_mE_P.html#emsm_esm_mE_P">Polarization</a><span class="headline-id">emsm.esm.mE.P</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_esm_po.html#emsm_esm_po">Polarized Objects; Bound Charges</a><span class="headline-id">emsm.esm.po</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emsm_esm_po_pibc.html#emsm_esm_po_pibc">Physical Interpretation of Bound Charges</a><span class="headline-id">emsm.esm.po.pibc</span>
-
- </li>
- <li>
- <a href="./emsm_esm_po_fid.html#emsm_esm_po_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.po.fid</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_esm_D.html#emsm_esm_D">The Electric Displacement</a><span class="headline-id">emsm.esm.D</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emsm_esm_D_bc.html#emsm_esm_D_bc">Boundary Conditions</a><span class="headline-id">emsm.esm.D.bc</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
- <a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
-
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_esm_ld.html#emsm_esm_ld">Linear Dielectrics</a><span class="headline-id">emsm.esm.ld</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emsm_esm_ld_sp.html#emsm_esm_ld_sp">Susceptibility, Permittivity, Dielectric Constant</a><span class="headline-id">emsm.esm.ld.sp</span>
-
- </li>
- <li>
- <a href="./emsm_esm_ld_bvp.html#emsm_esm_ld_bvp">Boundary Value Problems with Linear Dielectrics</a><span class="headline-id">emsm.esm.ld.bvp</span>
-
- </li>
- <li>
- <a href="./emsm_esm_ld_e.html#emsm_esm_ld_e">Energy in Dielectric Systems</a><span class="headline-id">emsm.esm.ld.e</span>
-
- </li>
- <li>
- <a href="./emsm_esm_ld_f.html#emsm_esm_ld_f">Forces on Dielectrics</a><span class="headline-id">emsm.esm.ld.f</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_msm.html#emsm_msm">Magnetostatics in matter</a><span class="headline-id">emsm.msm</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_msm_m.html#emsm_msm_m">Magnetization</a><span class="headline-id">emsm.msm.m</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emsm_msm_m_dpf.html#emsm_msm_m_dpf">Diamagnetism, Paramagnetism, Ferromagnetism</a><span class="headline-id">emsm.msm.m.dpf</span>
-
- </li>
- <li>
- <a href="./emsm_msm_m_fdi.html#emsm_msm_m_fdi">Torques and Forces on Magnetic Dipoles</a><span class="headline-id">emsm.msm.m.fdi</span>
-
- </li>
- <li>
- <a href="./emsm_msm_a.html#emsm_msm_a">Effect of Magnetic Field on Atomic Orbits</a><span class="headline-id">emsm.msm.a</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_msm_fmo.html#emsm_msm_fmo">The Field of a Magnetized Object</a><span class="headline-id">emsm.msm.fmo</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emsm_msm_fmo_bc.html#emsm_msm_fmo_bc">Bound Currents</a><span class="headline-id">emsm.msm.fmo.bc</span>
-
- </li>
- <li>
- <a href="./emsm_msm_fmo_pibc.html#emsm_msm_fmo_pibc">Physical Interpretation of Bound Currents</a><span class="headline-id">emsm.msm.fmo.pibc</span>
-
- </li>
- <li>
- <a href="./emsm_msm_fmo_fim.html#emsm_msm_fmo_fim">The Magnetic Field Inside Matter</a><span class="headline-id">emsm.msm.fmo.fim</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_msm_H.html#emsm_msm_H">The H Field</a><span class="headline-id">emsm.msm.H</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emsm_msm_H_A.html#emsm_msm_H_A">Ampère's Law in Magnetized Materials</a><span class="headline-id">emsm.msm.H.A</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_msm_lnlm.html#emsm_msm_lnlm">Linear and Nonlinear Media</a><span class="headline-id">emsm.msm.lnlm</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emsm_msm_lnlm_sp.html#emsm_msm_lnlm_sp">Magnetic Susceptibility and Permeability</a><span class="headline-id">emsm.msm.lnlm.sp</span>
-
- </li>
- <li>
- <a href="./emsm_msm_lnlm_fm.html#emsm_msm_lnlm_fm">Ferromagnetism</a><span class="headline-id">emsm.msm.lnlm.fm</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emd.html#emd">Electromagnetodynamics</a><span class="headline-id">emd</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./emd_Fl.html#emd_Fl">Induction: Faraday's Law</a><span class="headline-id">emd.Fl</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emd_Fl_Fl.html#emd_Fl_Fl">Faraday's Law</a><span class="headline-id">emd.Fl.Fl</span>
-
- </li>
- <li>
- <a href="./emd_Fl_ief.html#emd_Fl_ief">The Induced Electric Field</a><span class="headline-id">emd.Fl.ief</span>
-
- </li>
- <li>
- <a href="./emd_Fl_i.html#emd_Fl_i">Inductance</a><span class="headline-id">emd.Fl.i</span>
-
- </li>
- <li>
- <a href="./emd_Fl_e.html#emd_Fl_e">Energy in Magnetic Fields</a><span class="headline-id">emd.Fl.e</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emd_Me.html#emd_Me">Maxwell's Equations</a><span class="headline-id">emd.Me</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emd_Me_ebM.html#emd_Me_ebM">Electrodynamics Before Maxwell</a><span class="headline-id">emd.Me.ebM</span>
-
- </li>
- <li>
- <a href="./emd_Me_dc.html#emd_Me_dc">Maxwell's Correction to Ampère's Law; the Displacement Current</a><span class="headline-id">emd.Me.dc</span>
-
- </li>
- <li>
- <a href="./emd_Me_Me.html#emd_Me_Me">Maxwell's Equations</a><span class="headline-id">emd.Me.Me</span>
-
- </li>
- <li>
- <a href="./emd_Me_mc.html#emd_Me_mc">Magnetic Charge</a><span class="headline-id">emd.Me.mc</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emd_ce.html#emd_ce">Charge and Energy Flows</a><span class="headline-id">emd.ce</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emd_ce_ce.html#emd_ce_ce">The Continuity Equation</a><span class="headline-id">emd.ce.ce</span>
-
- </li>
- <li>
- <a href="./emd_ce_poy.html#emd_ce_poy">Poynting's Theorem; the Poynting Vector</a><span class="headline-id">emd.ce.poy</span>
-
- </li>
- <li>
- <a href="./emd_ce_mst.html#emd_ce_mst">Maxwell's Stress Tensor</a><span class="headline-id">emd.ce.mst</span>
-
- </li>
- <li>
- <a href="./emd_ce_mom.html#emd_ce_mom">Momentum</a><span class="headline-id">emd.ce.mom</span>
-
- </li>
- <li>
- <a href="./emd_ce_amom.html#emd_ce_amom">Angular Momentum</a><span class="headline-id">emd.ce.amom</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emd_emw.html#emd_emw">Electromagnetic waves in vacuum</a><span class="headline-id">emd.emw</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emd_emw_we.html#emd_emw_we">The Wave Equation</a><span class="headline-id">emd.emw.we</span>
-
- </li>
- <li>
- <a href="./emd_emw_mpw.html#emd_emw_mpw">Monochromatic Plane Waves</a><span class="headline-id">emd.emw.mpw</span>
-
- </li>
- <li>
- <a href="./emd_emw_ep.html#emd_emw_ep">Energy and Momentum</a><span class="headline-id">emd.emw.ep</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emdm.html#emdm">Electromagnetodynamics in Matter</a><span class="headline-id">emdm</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./emdm_Me.html#emdm_Me">Maxwell's Equations in Matter</a><span class="headline-id">emdm.Me</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emdm_Me_Mem.html#emdm_Me_Mem">Maxwell's Equations in Matter</a><span class="headline-id">emdm.Me.Mem</span>
-
- </li>
- <li>
- <a href="./emdm_Me_bc.html#emdm_Me_bc">Boundary Conditions</a><span class="headline-id">emdm.Me.bc</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emdm_emwm.html#emdm_emwm">Electromagnetic Waves in Matter</a><span class="headline-id">emdm.emwm</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emdm_emwm_plm.html#emdm_emwm_plm">Propagation in Linear Media</a><span class="headline-id">emdm.emwm.plm</span>
-
- </li>
- <li>
- <a href="./emdm_emwm_refr.html#emdm_emwm_refr">Refraction</a><span class="headline-id">emdm.emwm.refr</span>
-
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emdm_emwm_refl.html#emdm_emwm_refl">Reflection and Transmission</a><span class="headline-id">emdm.emwm.refl</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emdm_emwm_refl_ni.html#emdm_emwm_refl_ni">Normal Incidence</a><span class="headline-id">emdm.emwm.refl.ni</span>
-
- </li>
- <li>
- <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emdm_emwm_ad.html#emdm_emwm_ad">Absorption and Dispersion</a><span class="headline-id">emdm.emwm.ad</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emdm_emwm_ad_c.html#emdm_emwm_ad_c">EM Waves in Conductors</a><span class="headline-id">emdm.emwm.ad.c</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emdm_emwm_wg.html#emdm_emwm_wg">Waveguides</a><span class="headline-id">emdm.emwm.wg</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emdm_emwm_wg_gw.html#emdm_emwm_wg_gw">Guided waves</a><span class="headline-id">emdm.emwm.wg.gw</span>
-
- </li>
- <li>
- <a href="./emdm_emwm_wg_r.html#emdm_emwm_wg_r">Rectangular Waveguides</a><span class="headline-id">emdm.emwm.wg.r</span>
-
- </li>
- <li>
- <a href="./emdm_emwm_wg_c.html#emdm_emwm_wg_c">Coaxial Lines</a><span class="headline-id">emdm.emwm.wg.c</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emf.html#emf">Electromagnetic Fields</a><span class="headline-id">emf</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emf_svp.html#emf_svp">Scalar and Vector Potentials</a><span class="headline-id">emf.svp</span>
-
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emf_g.html#emf_g">Gauge Freedom and Choices</a><span class="headline-id">emf.g</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emf_g_Cg.html#emf_g_Cg">Coulomb Gauge</a><span class="headline-id">emf.g.Cg</span>
-
- </li>
- <li>
- <a href="./emf_g_Lg.html#emf_g_Lg">Lorenz Gauge; d'Alembertian; Inhomogeneous Maxwell Equations</a><span class="headline-id">emf.g.Lg</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./red.html#red">Relativistic Electrodynamics</a><span class="headline-id">red</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./red_sr.html#red_sr">Special Relativity</a><span class="headline-id">red.sr</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./red_sr_p.html#red_sr_p">Postulates and their consequences</a><span class="headline-id">red.sr.p</span>
-
- </li>
- <li>
- <a href="./red_sr_Lt.html#red_sr_Lt">Lorentz Transformations</a><span class="headline-id">red.sr.Lt</span>
-
- </li>
- <li>
- <a href="./red_sr_4v.html#red_sr_4v">Covariant and Contravariant Four-Vectors</a><span class="headline-id">red.sr.4v</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./red_rm.html#red_rm">Relativistic Mechanics</a><span class="headline-id">red.rm</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./red_rm_pt.html#red_rm_pt">Proper Time and Proper Velocity</a><span class="headline-id">red.rm.pt</span>
-
- </li>
- <li>
- <a href="./red_rm_rme.html#red_rm_rme">Relativistic Momentum and Energy</a><span class="headline-id">red.rm.rme</span>
-
- </li>
- <li>
- <a href="./red_rm_Mf.html#red_rm_Mf">Relativistic version of Newton's Laws; the Minkowski Force</a><span class="headline-id">red.rm.Mf</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./red_rem.html#red_rem">Relativistic Electromagnetism</a><span class="headline-id">red.rem</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./red_rem_mre.html#red_rem_mre">Magnetism as a Relativistic Effect</a><span class="headline-id">red.rem.mre</span>
-
- </li>
- <li>
- <a href="./red_rem_Ltf.html#red_rem_Ltf">Lorentz Transformation of Electromagnetic Fields</a><span class="headline-id">red.rem.Ltf</span>
-
- </li>
- <li>
- <a href="./red_rem_Fmunu.html#red_rem_Fmunu">The Field Tensor</a><span class="headline-id">red.rem.Fmunu</span>
-
- </li>
- <li>
- <a href="./red_rem_Me.html#red_rem_Me">Maxwell's Equations in Relativistic Notation</a><span class="headline-id">red.rem.Me</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./qed.html#qed">Quantum Electrodynamics</a><span class="headline-id">qed</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./d.html#d">Diagnostics</a><span class="headline-id">d</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
-
- </li>
- <li>
- <a href="./d_ems_ca.html#d_ems_ca">Diagnostics: Calculating or Approximating the Electostatic Potential</a><span class="headline-id">d.ems.ca</span>
-
- </li>
- <li>
- <a href="./d_emsm.html#d_emsm">Diagnostics: Electromagnetostatics in Matter</a><span class="headline-id">d.emsm</span>
-
- </li>
- <li>
- <a href="./d_ems_ms.html#d_ems_ms">Diagnostics: Magnetostatics</a><span class="headline-id">d.ems.ms</span>
-
- </li>
- <li>
- <a href="./d_emsm_msm.html#d_emsm_msm">Diagnostics: Magnetostatics in Matter</a><span class="headline-id">d.emsm.msm</span>
-
- </li>
- <li>
- <a href="./d_emd.html#d_emd">Diagnostics: Electromagnetodynamics</a><span class="headline-id">d.emd</span>
-
- </li>
- <li>
- <a href="./d_emd_ce.html#d_emd_ce">Diagnostics: Conservation Laws</a><span class="headline-id">d.emd.ce</span>
-
- </li>
- <li>
- <a href="./d_emd_emw.html#d_emd_emw">Diagnostics: Electromagnetic Waves</a><span class="headline-id">d.emd.emw</span>
-
- </li>
- <li>
- <a href="./d_emf.html#d_emf">Diagnostics: Potentials, Gauges and Fields</a><span class="headline-id">d.emf</span>
-
- </li>
- <li>
- <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
-
- </li>
- <li>
- <a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./a.html#a">Appendices</a><span class="headline-id">a</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./a_l.html#a_l">Literature</a><span class="headline-id">a.l</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./c.html#c">Compendium</a><span class="headline-id">c</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./c_m.html#c_m">Mathematics</a><span class="headline-id">c.m</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./c_m_va.html#c_m_va">Vector Analysis</a><span class="headline-id">c.m.va</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./c_m_va_n.html#c_m_va_n">Notation and algebraic properties</a><span class="headline-id">c.m.va.n</span>
-
- </li>
- <li>
- <a href="./c_m_va_sp.html#c_m_va_sp">Scalar product</a><span class="headline-id">c.m.va.sp</span>
-
- </li>
- <li>
- <a href="./c_m_va_cp.html#c_m_va_cp">Cross product</a><span class="headline-id">c.m.va.cp</span>
-
- </li>
- <li>
- <a href="./c_m_va_tp.html#c_m_va_tp">Triple Products</a><span class="headline-id">c.m.va.tp</span>
-
- </li>
- <li>
- <a href="./c_m_va_pds.html#c_m_va_pds">Position, Displacement and Separation Vectors</a><span class="headline-id">c.m.va.pds</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./c_m_dc.html#c_m_dc">Differential Calculus</a><span class="headline-id">c.m.dc</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./c_m_dc_g.html#c_m_dc_g">Gradient</a><span class="headline-id">c.m.dc.g</span>
-
- </li>
- <li>
- <a href="./c_m_dc_del.html#c_m_dc_del">The \({\boldsymbol \nabla}\) Operator</a><span class="headline-id">c.m.dc.del</span>
-
- </li>
- <li>
- <a href="./c_m_dc_div.html#c_m_dc_div">The Divergence</a><span class="headline-id">c.m.dc.div</span>
-
- </li>
- <li>
- <a href="./c_m_dc_curl.html#c_m_dc_curl">The Curl</a><span class="headline-id">c.m.dc.curl</span>
-
- </li>
- <li>
- <a href="./c_m_dc_pr.html#c_m_dc_pr">Product arguments</a><span class="headline-id">c.m.dc.pr</span>
-
- </li>
- <li>
- <a href="./c_m_dc_d2.html#c_m_dc_d2">Second Derivatives</a><span class="headline-id">c.m.dc.d2</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./c_m_ic.html#c_m_ic">Integral Calculus</a><span class="headline-id">c.m.ic</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./c_m_ic_lsv.html#c_m_ic_lsv">Line, Surface and Volume Integrals</a><span class="headline-id">c.m.ic.lsv</span>
-
- </li>
- <li>
- <a href="./c_m_ic_ftc.html#c_m_ic_ftc">The Fundamental Theorem of Calculus</a><span class="headline-id">c.m.ic.ftc</span>
-
- </li>
- <li>
- <a href="./c_m_ic_ftg.html#c_m_ic_ftg">The Fundamental Theorem for Gradients</a><span class="headline-id">c.m.ic.ftg</span>
-
- </li>
- <li>
- <a href="./c_m_ic_gauss.html#c_m_ic_gauss">Gauss' Theorem</a><span class="headline-id">c.m.ic.gauss</span>
-
- </li>
- <li>
- <a href="./c_m_ic_stokes.html#c_m_ic_stokes">Stokes' Theorem</a><span class="headline-id">c.m.ic.stokes</span>
-
- </li>
- <li>
- <a href="./c_m_ic_ip.html#c_m_ic_ip">Integration by Parts</a><span class="headline-id">c.m.ic.ip</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./c_m_cs.html#c_m_cs">Coordinate Systems</a><span class="headline-id">c.m.cs</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./c_m_cs_sph.html#c_m_cs_sph">Spherical Coordinates</a><span class="headline-id">c.m.cs.sph</span>
-
- </li>
- <li>
- <a href="./c_m_cs_cyl.html#c_m_cs_cyl">Cylindrical Coordinates</a><span class="headline-id">c.m.cs.cyl</span>
-
- </li>
- <li>
- <a href="./c_m_cs_hyp.html#c_m_cs_hyp">Hyperbolic Coordinates</a><span class="headline-id">c.m.cs.hyp</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./c_m_dd.html#c_m_dd">Dirac delta Distribution</a><span class="headline-id">c.m.dd</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./c_m_dd_div.html#c_m_dd_div">The Divergence of \(\hat{\bf r}/r^2\)</a><span class="headline-id">c.m.dd.div</span>
-
- </li>
- <li>
- <a href="./c_m_dd_1d.html#c_m_dd_1d">The One-Dimensional Dirac Delta Function</a><span class="headline-id">c.m.dd.1d</span>
-
- </li>
- <li>
- <a href="./c_m_dd_3d.html#c_m_dd_3d">The Three-Dimensional Delta Function</a><span class="headline-id">c.m.dd.3d</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./c_m_vf.html#c_m_vf">Vector Fields</a><span class="headline-id">c.m.vf</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./c_m_vf_helm.html#c_m_vf_helm">The Helmholtz Theorem</a><span class="headline-id">c.m.vf.helm</span>
-
- </li>
- <li>
- <a href="./c_m_vf_pot.html#c_m_vf_pot">Potentials</a><span class="headline-id">c.m.vf.pot</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./c_m_uf.html#c_m_uf">Useful Formulas</a><span class="headline-id">c.m.uf</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./c_m_uf_cyl.html#c_m_uf_cyl">Cylindrical coordinates</a><span class="headline-id">c.m.uf.cyl</span>
-
- </li>
- <li>
- <a href="./c_m_uf_sph.html#c_m_uf_sph">Spherical coordinates</a><span class="headline-id">c.m.uf.sph</span>
-
- </li>
- <li>
- <a href="./c_m_uf_vi.html#c_m_uf_vi">Vector identities</a><span class="headline-id">c.m.uf.vi</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </nav>
- <ul class="breadcrumbs"><li><a class="breadcrumb-link"href="ems.html">Electromagnetostatics</a></li><li><a class="breadcrumb-link"href="ems_ca.html">Calculating or Approximating the Electrostatic Potential</a></li><li><a class="breadcrumb-link"href="ems_ca_me.html">The Multipole Expansion</a></li><li>Monopole and Dipole Terms</li></ul><ul class="navigation-links"><li>Prev: <a href="ems_ca_me_a.html">Approximate Potential at Large Distance <small>[ems.ca.me.a]</small></a></li><li>Next: <a href="ems_ca_me_h.html">Higher Moments <small>[ems.ca.me.h]</small></a></li><li>Up: <a href="ems_ca_me.html">The Multipole Expansion <small>[ems.ca.me]</small></a></li></ul><div id="outline-container-ems_ca_me_md" class="outline-5">
- <h5 id="ems_ca_me_md">Monopole and Dipole Terms<a class="headline-permalink" href="./ems_ca_me_md.html#ems_ca_me_md"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
- <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
- <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
- </svg></a><span class="headline-id">ems.ca.me.md</span></h5>
- <div class="outline-text-5" id="text-ems_ca_me_md">
- <p>
- The series <a href="./ems_ca_me_a.html#p_Leg">p_Leg</a> is organized in increasing powers of inverse distance.
- The leading term is called the *monopole term, and is
- </p>
- <div class="eqlabel" id="orga2bb8f0">
- <p>
- <a id="p_mono"></a><a href="./ems_ca_me_md.html#p_mono"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
- <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
- <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
- </svg></a>
- </p>
- <div class="alteqlabels" id="org80cce4b">
- <ul class="org-ul">
- <li>Gr (3.97)</li>
- </ul>
-
- </div>
-
- </div>
- <div class="core div" id="org93f1445">
- <p>
- \[
- \phi_m ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{Q}{r}, \hspace{1cm}
- Q = \int_{\cal \phi} d\tau_s \rho({\bf r}_s).
- \tag{p_mono}\label{p_mono}
- \]
- </p>
-
- </div>
- <p>
- For a point charge, the monopole term gives the exact potential.
- </p>
-
- <p>
- The next term is the <b>dipole</b> term: by using \(P_1 (x) = x\), we have
- \[
- \phi_d ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^2} \int_{\cal V} d\tau_s \hat{\bf r} \cdot {\bf r}_s \rho({\bf r}_s)
- \]
- This can be written
- </p>
- <div class="eqlabel" id="org000288f">
- <p>
- <a id="p_di"></a><a href="./ems_ca_me_md.html#p_di"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
- <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
- <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
- </svg></a>
- </p>
- <div class="alteqlabels" id="org6eaa92c">
-
- </div>
-
- </div>
- <div class="core div" id="org964cce5">
- <p>
- \[
- \phi_d ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{\hat{\bf r} \cdot {\bf p}}{r^2},
- \hspace{10mm}{\bf p} \equiv \int_{\cal V} d\tau_s ~{\bf r}_s ~\rho({\bf r}_s).
- \tag{p_di}\label{p_di}
- \]
- </p>
-
- </div>
- <p>
- in terms of the <b>dipole moment</b> \({\bf p}\).
- Note that the dipole moment is a property related to the internal distribution
- of the source charges, and that it
- in general depends on the chosen point of origin (more on this later).
- </p>
-
- <p>
- Since dipole moments are vectors, they are summed following vector addition rules.
- </p>
-
- <p>
- In the literature, one comes across the notion of a <b>mathematical dipole</b> or
- synonymously a <b>pure dipole</b>: this is simply an abstraction in which the charges
- are moved infinitesimally close together, while their charges are proportionally
- increased such that the dipole moment remains the same. In such cases, the
- multipole expansion terminates at the dipole term.
- </p>
- </div>
- </div>
-
-
-
-
-
- <br><ul class="navigation-links"><li>Prev: <a href="ems_ca_me_a.html">Approximate Potential at Large Distance <small>[ems.ca.me.a]</small></a></li><li>Next: <a href="ems_ca_me_h.html">Higher Moments <small>[ems.ca.me.h]</small></a></li><li>Up: <a href="ems_ca_me.html">The Multipole Expansion <small>[ems.ca.me]</small></a></li></ul>
- <br>
- <hr>
- <div class="license">
- <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
- target="_blank" class="m-2">
- <img alt="Creative Commons License" style="border-width:0"
- src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
- </a>
- Except where otherwise noted, all content is licensed under a
- <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
- target="_blank">Creative Commons Attribution 4.0 International License</a>.
- </div>
- <div id="postamble" class="status">
- <p class="author">Author: Jean-Sébastien Caux</p>
- <p class="date">Created: 2022-03-24 Thu 08:42</p>
- <p class="validation"></p>
- </div>
-
-
- </div>
- </html>
|