2022-02-07 14:11:58 +00:00
<!DOCTYPE html>
< html lang = "en" >
< head >
2022-02-09 21:41:42 +00:00
<!-- 2022 - 02 - 09 Wed 22:40 -->
2022-02-07 14:11:58 +00:00
< 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 >
2022-02-08 16:21:33 +00:00
< a href = "./in_t_l.html#in_t_l" > Section and equation labelling< / a > < span class = "headline-id" > in.t.l< / span >
< / li >
< li >
2022-02-07 14:11:58 +00:00
< 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 >
< summary >
< 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 >
2022-02-08 06:07:41 +00:00
< a href = "./ems_es_ep.html#ems_es_ep" > The Electrostatic Potential< / a > < span class = "headline-id" > ems.es.ep< / span >
2022-02-07 14:11:58 +00:00
< / 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 >
2022-02-09 21:41:42 +00:00
< 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 >
2022-02-07 14:11:58 +00:00
< / 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 >
2022-02-08 06:07:41 +00:00
< a href = "./ems_es_e.html#ems_es_e" > Electrostatic Energy from the Potential< / a > < span class = "headline-id" > ems.es.e< / span >
2022-02-07 14:11:58 +00:00
< / 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 >
< summary >
< 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 >
< summary >
< 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 >
< 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 Charge< / a > < span class = "headline-id" > ems.ms.lf.pc< / span >
< / li >
< li >
< a href = "./ems_ms_lf_c.html#ems_ms_lf_c" > Currents< / a > < span class = "headline-id" > ems.ms.lf.c< / span >
< / li >
< / ul >
< / details >
< / li >
< li >
< details >
< summary >
< a href = "./ems_ms_BS.html#ems_ms_BS" > Steady Currents: the Biot-Savart Law< / a > < span class = "headline-id" > ems.ms.BS< / span >
< / summary >
< ul >
< li >
< a href = "./ems_ms_BS_sc.html#ems_ms_BS_sc" > The Magnetic Field issuing from a Steady Current< / a > < span class = "headline-id" > ems.ms.BS.sc< / span >
< / li >
< / ul >
< / details >
< / 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_sc.html#ems_ms_dcB_sc" > Straight-line Currents< / a > < span class = "headline-id" > ems.ms.dcB.sc< / span >
< / li >
< li >
< a href = "./ems_ms_dcB_BS.html#ems_ms_dcB_BS" > Divergence and Curl of \({\bf B}\) from Biot-Savart< / a > < span class = "headline-id" > ems.ms.dcB.BS< / 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_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 open = "" >
< summary class = "toc-open" >
< a href = "./emsm.html#emsm" > Electromagnetostatics in matter< / a > < span class = "headline-id" > emsm< / span >
< / summary >
< ul >
< li >
< a href = "./emsm_esm_s.html#emsm_esm_s" > A proper definition of "statics"< / a > < span class = "headline-id" > emsm.esm.s< / span >
< / li >
< li >
< details open = "" >
< summary class = "toc-open" >
< a href = "./emsm_esm.html#emsm_esm" > Electrostatics in matter< / a > < span class = "headline-id" > emsm.esm< / span >
< / summary >
< ul >
< li >
< a href = "./emsm_esm_p.html#emsm_esm_p" > Polarization< / a > < span class = "headline-id" > emsm.esm.p< / span >
< / li >
< li >
2022-02-08 16:21:33 +00:00
< a href = "./emsm_esm_di.html#emsm_esm_di" > Dielectrics< / a > < span class = "headline-id" > emsm.esm.di< / span >
2022-02-07 14:11:58 +00:00
< / li >
< li >
< details >
< summary >
< a href = "./emsm_esm_fpo.html#emsm_esm_fpo" > The Field of a Polarized Object< / a > < span class = "headline-id" > emsm.esm.fpo< / span >
< / summary >
< ul >
< li >
< a href = "./emsm_esm_fpo_pibc.html#emsm_esm_fpo_pibc" > Physical Interpretation of Bound Charges< / a > < span class = "headline-id" > emsm.esm.fpo.pibc< / span >
< / li >
< li >
< a href = "./emsm_esm_fpo_fid.html#emsm_esm_fpo_fid" > The Field Inside a Dielectric< / a > < span class = "headline-id" > emsm.esm.fpo.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 >
< details open = "" >
< summary class = "toc-open" >
< a href = "./emsm_esm_di.html#emsm_esm_di" > Dielectrics< / a > < span class = "headline-id" > emsm.esm.di< / span >
< / summary >
< ul >
2022-02-08 06:07:41 +00:00
< li class = "toc-currentpage" >
2022-02-07 14:11:58 +00:00
< a href = "./emsm_esm_di_ld.html#emsm_esm_di_ld" > Linear Dielectrics< / a > < span class = "headline-id" > emsm.esm.di.ld< / 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 >
< li >
< a href = "./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe" > Fresnel's Equations< / a > < span class = "headline-id" > emdm.emwm.refl.Fe< / span >
< / li >
< li >
< a href = "./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba" > Brewster's Angle< / a > < span class = "headline-id" > emdm.emwm.refl.Ba< / 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_t.html#qed_t" > QED today< / a > < span class = "headline-id" > qed.t< / 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_m.html#d_m" > Diagnostics: Mathematical Preliminaries< / a > < span class = "headline-id" > d.m< / span >
< / li >
< 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 >
< / 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 Rules< / 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 >
2022-02-09 06:44:58 +00:00
< ul class = "breadcrumbs" > < li > < a class = "breadcrumb-link" href = "emsm.html" > Electromagnetostatics in matter< / a > < / li > < li > < a class = "breadcrumb-link" href = "emsm_esm.html" > Electrostatics in matter< / a > < / li > < li > < a class = "breadcrumb-link" href = "emsm_esm_di.html" > Dielectrics< / a > < / li > < li > Linear Dielectrics< / li > < / ul > < ul class = "navigation-links" > < li > Prev: < a href = "emsm_esm_di.html" > Dielectrics  < small > [emsm.esm.di]< / small > < / a > < / li > < li > Next: < a href = "emsm_msm.html" > Magnetostatics in matter  < small > [emsm.msm]< / small > < / a > < / li > < li > Up: < a href = "emsm_esm_di.html" > Dielectrics  < small > [emsm.esm.di]< / small > < / a > < / li > < / ul > < div id = "outline-container-emsm_esm_di_ld" class = "outline-5" >
2022-02-07 14:11:58 +00:00
< h5 id = "emsm_esm_di_ld" > Linear Dielectrics< a class = "headline-permalink" href = "./emsm_esm_di_ld.html#emsm_esm_di_ld" > < 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" > emsm.esm.di.ld< / span > < / h5 >
< div class = "outline-text-5" id = "text-emsm_esm_di_ld" >
< / div >
2022-02-08 16:21:33 +00:00
< div id = "outline-container-emsm_esm_di_ld_sp" class = "outline-6" >
< h6 id = "emsm_esm_di_ld_sp" > < a href = "#emsm_esm_di_ld_sp" > Susceptibility, Permittivity, Dielectric Constant< / a > < / h6 >
< div class = "outline-text-6" id = "text-emsm_esm_di_ld_sp" >
2022-02-07 14:11:58 +00:00
< p >
For many substances: polarization is proportional to field, if the latter isn't too strong:
< / p >
2022-02-09 21:41:42 +00:00
< div class = "main div" id = "orga3278d6" >
2022-02-07 14:11:58 +00:00
< p >
\[
{\bf P} = \varepsilon_0 \chi_e {\bf E}
\label{Gr(4.30)}
\]
< / p >
< / div >
< p >
Constant \(\chi_e\): called the {\bf electric susceptibility} of the medium. Since \(\varepsilon_0\) is
there, \(\chi_e\) is dimensionless. Materials that obey \ref{Gr(4.30)} are called
{\bf linear dielectrics}.
< / p >
< p >
\paragraph{Note:} \({\bf E}\) on the RHS of \ref{Gr(4.30)} is the {\bf total} electric field,
due to free charges and to the polarization itself. Putting dielectric in field \({\bf E}_0\),
we can't compute \({\bf P}\) directly from \ref{Gr(4.30)}. Better way: compute \({\bf D}\).
< / p >
< p >
In linear dielectrics:
\[
{\bf D} = \varepsilon_0 {\bf E} + {\bf P} = \varepsilon_0 {\bf E} + \varepsilon_0 \chi_e {\bf E}
= \varepsilon_0 (1 + \chi_e) {\bf E}
\label{Gr(4.31)}
\]
so
< / p >
2022-02-09 21:41:42 +00:00
< div class = "main div" id = "org78784fd" >
2022-02-07 14:11:58 +00:00
< p >
\[
{\bf D} = \varepsilon {\bf E}
\label{Gr(4.32)}
\]
< / p >
< / div >
< p >
where \(\varepsilon\) is called the {\bf permittivity} of the material. In vacuum, susceptibility is zero,
permittivity is \(\varepsilon_0\). Also,
\[
\varepsilon_r \equiv 1 + \chi_e = \frac{\varepsilon}{\varepsilon_0}
\label{Gr(4.34)}
\]
is called the {\bf relative permittivity} or {\bf dielectric constant} of the material.
This is all just nomenclature, everything is already in \ref{Gr(4.30)}.
< / p >
2022-02-09 21:41:42 +00:00
< div class = "example div" id = "org85c03d5" >
2022-02-07 14:11:58 +00:00
< p >
\paragraph{Example 4.5:} metal sphere of radius \(a\) carrying charge \(Q\), surrounded out to radius \(b\) by
a linear dielectric material of permittivity \(\varepsilon\). Find potential at center (relative to infinity).
\paragraph{Solution:} need to know \({\bf E}\). Could try to locate bound charge: but we don't know \({\bf P}\) !
What we do know: free charge, situation is spherically symmetric, so can calculate \({\bf D}\)
using \ref{Gr(4.23)}:
\[
{\bf D} = \frac{Q}{4\pi r^2} \hat{\bf r}, \hspace{1cm} r > a.
\]
Inside sphere, \({\bf E} = {\bf P} = {\bf D} = 0\). Find \({\bf E}\) using \ref{Gr(4.32)}:
\[
{\bf E} = \left\{ \begin{array}{cc}
\frac{Q}{4\pi \varepsilon r^2} \hat{\bf r}, & a < r < b, \\
\frac{Q}{4\pi \varepsilon_0 r^2} \hat{\bf r}, & r > b. \end{array} \right.
\]
The potential is thus
\[
V = -\int_\infty^0 d{\bf l} \cdot {\bf E} = -\int_\infty^b dr \frac{Q}{4\pi\varepsilon_0 r^2} - \int_b^a dr \frac{Q}{4\pi\varepsilon r^2}
= \frac{Q}{4\pi} \left( \frac{1}{\varepsilon_0 b} + \frac{1}{\varepsilon a} - \frac{1}{\varepsilon b} \right).
\]
< / p >
< p >
It was thus not necessary to compute the polarization or the bound charge explicitly. This can be done:
\[
{\bf P} = \varepsilon_0 \chi_e {\bf E} = \frac{\varepsilon_0 \chi_e Q}{4\pi \varepsilon r^2} \hat{\bf r},
\]
so
\[
\rho_b = -{\boldsymbol \nabla} \cdot {\bf P} = 0, \hspace{1cm}
\sigma_b = {\bf P} \cdot \hat{\bf n} = \left\{ \begin{array}{cc}
\frac{\varepsilon_0 \chi_e Q}{4\pi \varepsilon b^2}, & \mbox{outer surface} \\
-\frac{\varepsilon_0 \chi_e Q}{4\pi \varepsilon a^2}, & \mbox{inner surface} \end{array} \right.
\]
< / p >
< p >
Dielectric thus like an imperfect conductor: charge \(Q\) not fully screened.
< / p >
< / div >
< p >
In linear dielectrics, the parallel between \({\bf E}\) and \({\bf D}\) is also not perfect.
Remark: since \({\bf P}\) and \({\bf D}\) are both proportional to \({\bf E}\) inside the dielectric,
does it mean that their curl vanishes like for \({\bf E}\) ? {\bf No}: if there is a boundary
between two materials with different dielectric constants, then a closed loop integral
of {\it e.g.} \({\bf P}\) would not vanish.
< / p >
< p >
Only case where parallel works: space entirely filled with homogeneous linear dielectric.
< / p >
2022-02-09 21:41:42 +00:00
< div class = "example div" id = "orgc0d8c4c" >
2022-02-07 14:11:58 +00:00
< p >
\paragraph{Example 4.6:} parallel-plate capacitor filled with insulating material of
dielectric constant \(\varepsilon_r\). What is the effect on the capacitance ?
\paragraph{Solution:} field confined between plates, and reduced by factor \(1/\varepsilon_r\).
Potential difference \(V\) also reduced by same factor. Since \(Q = C/V\), capacitance
is increased by factor of \(\varepsilon_r\), so
\[
C = \varepsilon_r C_{vac}
\label{Gr(4.37)}
\]
< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-emsm_esm_di_ld_bvp" class = "outline-6" >
< h6 id = "emsm_esm_di_ld_bvp" > < a href = "#emsm_esm_di_ld_bvp" > Boundary Value Problems with Linear Dielectrics< / a > < / h6 >
< div class = "outline-text-6" id = "text-emsm_esm_di_ld_bvp" >
< p >
In homogeneous linear dielectric:
\[
\rho_b = -{\boldsymbol \nabla} \cdot {\bf P} = - {\boldsymbol \nabla} \cdot \left(\varepsilon_0 \frac{\chi_e}{\varepsilon} {\bf D}\right)
= -\left( \frac{\chi_e}{1 + \chi_e} \right) \rho_f
\label{Gr(4.39)}
\]
If \(\rho = 0\), any net charge is on surface, potential then obeys Laplace.
< / p >
< p >
Convenient to rewrite boundary conditions in terms of free charge: from \ref{Gr(4.26)},
< / p >
2022-02-09 21:41:42 +00:00
< div class = "main div" id = "org738acac" >
2022-02-07 14:11:58 +00:00
< p >
\[
\varepsilon_{above} E^{\perp}_{above} - \varepsilon_{below} E^{\perp}_{below} = \sigma_f
\label{Gr(4.40)}
\]
< / p >
< / div >
< p >
or in terms of the potential,
< / p >
2022-02-09 21:41:42 +00:00
< div class = "main div" id = "org6a5bb81" >
2022-02-07 14:11:58 +00:00
< p >
\[
\varepsilon_{above} \frac{\partial V_{above}}{\partial n} -
\varepsilon_{below} \frac{\partial V_{below}}{\partial n} = -\sigma_f
\label{Gr(4.41)}
\]
< / p >
< / div >
< p >
Potential itself is continuous,
< / p >
2022-02-09 21:41:42 +00:00
< div class = "main div" id = "org00bd1e4" >
2022-02-07 14:11:58 +00:00
< p >
\[
V_{above} = V_{below}
\label{Gr(4.42)}
\]
< / p >
< / div >
2022-02-09 21:41:42 +00:00
< div class = "example div" id = "orgf22ddc4" >
2022-02-07 14:11:58 +00:00
< p >
\paragraph{Example 4.7:} sphere of homogeneous dielectric material in uniform electric field \({\bf E}_0\).
Find electric field inside sphere.
\paragraph{Solution:} resembles Example 3.8 (conducting sphere), here cancellation is not total.
< / p >
< p >
Need to solve Laplace's equation for \(V(r, \theta)\) with boundary conditions
< / p >
\begin{align}
& (i)~~V_{in} (R,\theta) = V_{out} (R, \theta), \nonumber \\
& (ii)~~\varepsilon \frac{\partial V_{in} (R,\theta)}{\partial n}
= \varepsilon_0 \frac{\partial V_{out} (R,\theta)}{\partial n}, \nonumber \\
& (iii)~~V_{out} (r) \rightarrow -E_0 r \cos \theta, ~~r \gg R.
\end{align}
< p >
Inside and outside sphere:
\[
V_{in} (r,\theta) = \sum_{l=0}^\infty A_l r^l P_l (\cos \theta), \hspace{1cm}
V_{out} (r,\theta) = -E_0 r \cos \theta + \sum_{l=0}^\infty \frac{B_l}{r^{l+1}} P_l (\cos \theta).
\]
Boundary condition \((i)\) imposes
\[
\sum_{l=0}^\infty A_l R^l P_l(\cos \theta) = -E_0 R \cos \theta + \sum_{l=0}^\infty \frac{B_l}{R^{l+1}} P_l (\cos \theta)
\]
so
\[
A_l R^l = \frac{B_l}{R^{l+1}}, ~~ l \neq 1, \hspace{1cm}
A_1 R = -E_0 R + \frac{B_1}{R^2}.
\]
Boundary condition \((ii)\):
\[
\varepsilon_r \sum_{l=0}^\infty l A_l R^{l-1} P_l (\cos \theta) = - E_0 \cos \theta - \sum_{l=0}^\infty \frac{(l+1)B_l}{R^{l+2}} P_l (\cos \theta)
\]
so
\[
\varepsilon_r l A_l R^{l-1} = -\frac{(l+1)B_l}{R^{l+2}}, ~~ l \neq 1, \hspace{1cm}
\varepsilon_r A_1 = -E_0 - \frac{2B_1}{R^3}.
\]
We then have
\[
A_l = 0 = B_l, ~~ l \neq 1, \hspace{1cm}
A_l = -\frac{3}{\varepsilon_r + 2} E_0, ~~ B_1 = \frac{\varepsilon_r - 1}{\varepsilon_r + 2} R^3 E_0.
\]
Thus,
\[
V_{in} (r, \theta) = -\frac{3E_0}{\varepsilon_r + 2} r \cos \theta = -\frac{3E_0}{\varepsilon_r + 2} z,
\hspace{1cm}
{\bf E} = \frac{3}{\varepsilon_r + 2} {\bf E}_0.
\]
< / p >
< / div >
2022-02-09 21:41:42 +00:00
< div class = "example div" id = "org3d34c22" >
2022-02-07 14:11:58 +00:00
< p >
\paragraph{Example 4.8:} suppose region below \(z = 0\) is filled with uniform linear dielectric with susceptibility \(\chi_e\).
Calculate force on point charge \(q\) situated a distance \(d\) above origin.
\paragraph{Solution:} bound surface charge is of opposite sign, force is attractive. No bound volume charge
because of \ref{Gr(4.39)}. Using \ref{Gr(4.11)} and \ref{Gr(4.30)},
\[
\sigma_b = {\bf P} \cdot \hat{\bf n} = P_z = \varepsilon_0 \chi_e E_z
\]
where \(E_z\) is the z-component of total field just below surface of dielectric (due to \(q\) and to bound charge).
Contribution from charge \(q\) from Coulomb (careful: \(\theta\) is $π$-rotated as compared to spherical coord so \(\theta = 0\) represents \(-\hat{z}\))
\[
-\frac{1}{4\pi\varepsilon_0} \frac{q}{r^2 + d^2} \cos \theta = -\frac{1}{4\pi\varepsilon_0} \frac{qd}{(r^2 + d^2)^{3/2}}, \hspace{1cm}
r = \sqrt{x^2 + y^2}.
\]
$z$-component of field from bound charge: \(-\sigma_b/2\varepsilon_0\), so
\[
\sigma_b = \varepsilon_0 \chi_e \left[ -\frac{1}{4\pi\varepsilon_0} \frac{qd}{(r^2 + d^2)^{3/2}} - \frac{\sigma_b}{2\varepsilon_0} \right],
\]
so
\[
\sigma_b = -\frac{1}{2\pi} \left(\frac{\chi_e}{\chi_e + 2}\right) \frac{qd}{(r^2 + d^2)^{3/2}}.
\label{Gr(4.50)}
\]
As per conducting plane, except for factor \(\chi_e/(\chi_e + 2)\). Total bound charge:
\[
q_b = -\left(\frac{\chi_e}{\chi_e + 2}\right) q.
\]
Field: by direct integration, or more nicely by method of images: replace dielectric by single point charge
\(q_b\) at \((0,0,-d)\):
\[
V (x,y,z> 0) = \frac{1}{4\pi\varepsilon_0} \left[ \frac{q}{\sqrt{x^2 + y^2 + (z-d)^2}} + \frac{q_b}{\sqrt{x^2 + y^2 + (z+d)^2}}\right]
\]
A charge \(q + q_b\) at \((0,0,d)\) gives
\[
V (x,y,z< 0) = \frac{1}{4\pi\varepsilon_0} \left[ \frac{q + q_b}{\sqrt{x^2 + y^2 + (z-d)^2}}\right]
\]
Putting these two together yields a solution to Poisson going to zero at infinity, and is therefore the unique solution.
Correct discontinuity at \(z = 0\):
\[
-\varepsilon_0 \left( \frac{\partial V}{\partial z}|_{z = 0^+} - \frac{\partial V}{\partial z}|_{z = 0^-} \right)
= -\frac{1}{2\pi} \left(\frac{\chi_e}{\chi_e + 2}\right) \frac{qd}{(r^2 + d^2)^{3/2}}.
\]
Force on \(q\):
\[
{\bf F} = \frac{1}{4\pi\varepsilon_0} \frac{q q_b}{4d^2} \hat{\bf z} = -\frac{1}{4\pi\varepsilon_0} \left(
\frac{\chi_e}{\chi_e + 2} \right) \frac{q^2}{4d^2} \hat{\bf z}
\label{Gr(4.54)}
\]
< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-emsm_esm_di_ld_e" class = "outline-6" >
< h6 id = "emsm_esm_di_ld_e" > < a href = "#emsm_esm_di_ld_e" > Energy in Dielectric Systems< / a > < / h6 >
< div class = "outline-text-6" id = "text-emsm_esm_di_ld_e" >
< p >
To charge a capacitor:
\[
W = \frac{1}{2} C V^2
\]
If capacitor filled with linear isotropic dielectric (Ex. 4.6):
\[
C = \varepsilon_r C_{vac}
\]
From Chap. 2:
\[
W = \frac{\varepsilon_0}{2} \int d\tau E^2
\]
How is this changed ? Counting only energy in fields: should decrease by
factor \(1/\varepsilon_r^2\). However, this would neglect the {\bf strain energy}
associated to the distortion of the polarized constituents of the dielectric medium.
< / p >
< p >
Derivation from scratch: bring in free charge: \(\rho_f\) increased by \(\Delta \rho_f\), polarization changes
(also bound charge distribution). Work done on free charges (only that matters):
\[
\Delta W = \int d\tau (\Delta \rho_f ({\bf r})) V ({\bf r}).
\label{Gr(4.56)}
\]
But \(\rho_f = {\boldsymbol \nabla} \cdot {\bf D}\) so \(\Delta \rho_f = {\boldsymbol \nabla} \cdot (\Delta {\bf D})\), so
\[
\Delta W = \int d\tau ({\boldsymbol \nabla} \cdot (\Delta {\bf D})) V
= \int d\tau {\boldsymbol \nabla} \cdot ((\Delta {\bf D}) V) + \int d\tau (\Delta {\bf D}) \cdot {\bf E}
\]
First integral: divergence theorem changes it to a surface integral which vanishes when integrating over all space.
Therefore,
\[
\Delta W = \int d\tau (\Delta {\bf D}) \cdot {\bf E}
\label{Gr(4.57)}
\]
This applies to any material.
< / p >
< p >
Special case of linear isotropic dielectric: \({\bf D} = \varepsilon {\bf E}\), so
\[
\Delta W = \Delta \left( \frac{1}{2} \int d\tau {\bf D} \cdot {\bf E} \right)
\]
Total work done:
< / p >
2022-02-09 21:41:42 +00:00
< div class = "main div" id = "org4c9c303" >
2022-02-07 14:11:58 +00:00
< p >
\[
W = \frac{1}{2} \int d\tau {\bf D} \cdot {\bf E}
\label{Gr(4.58)}
\]
< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-emsm_esm_di_ld_f" class = "outline-6" >
< h6 id = "emsm_esm_di_ld_f" > < a href = "#emsm_esm_di_ld_f" > Forces on Dielectrics< / a > < / h6 >
< div class = "outline-text-6" id = "text-emsm_esm_di_ld_f" >
< p >
As for a conductor: a dielectric is attracted into an electric field. Calculations can be
complicated: parallel plate capacitor with partially inserted dielectric: force comes
from {\bf fringing field} around edges.
< / p >
< p >
Better: reason from energy. Pull dielectric out by \(dx\). Energy change equal to work done:
\[
dW = F_{me} dx
\label{Gr(4.59)}
\]
where \(F_{me}\) is mechanical force exerted by external agent. \(F_{me} = -F\), where \(F\) is
electrical force on dielectric. Electrical force on slab:
\[
F = -\frac{dW}{dx}
\label{Gr(4.60)}
\]
Energy stored in capacitor:
\[
W = \frac{1}{2} C V^2
\label{Gr(4.61)}
\]
Capacitance in configuration considered:
\[
C = \frac{\varepsilon_0 w}{d} (\varepsilon_r l - \chi_e x)
\label{Gr(4.62)}
\]
where \(l\) is the length of the plates, and \(w\) is their width. Assume total charge \(Q\) on each
plate is held constant as \(x\) changes. In terms of \(Q\),
\[
W = \frac{1}{2} \frac{Q^2}{C}
\label{Gr(4.63)}
\]
so
\[
F = -\frac{dW}{dx} = \frac{1}{2} \frac{Q^2}{C^2} \frac{dC}{dx} = \frac{1}{2} V^2 \frac{dC}{dx}.
\label{Gr(4.64)}
\]
But
\[
\frac{dC}{dx} = -\frac{\varepsilon_0 \chi_e w}{d}
\]
so
\[
F = -\frac{\varepsilon_0 \chi_e w}{2d} V^2.
\label{Gr(4.65)}
\]
< / p >
< p >
Common mistake: to use \ref{Gr(4.61)} (for \(V\) constant) instead of \ref{Gr(4.63)}
(for \(Q\) constant) in computing the force. In this case, sign is reversed,
\[
F = -\frac{1}{2} V^2 \frac{dC}{dx}.
\]
Here, the battery also does work, so
\[
dW = F_{me} dx + V dQ
\label{Gr(4.66)}
\]
and
\[
F = -\frac{dW}{dx} + V \frac{dQ}{dx} = -\frac{1}{2} V^2 \frac{dC}{dx} + V^2 \frac{dC}{dx} = \frac{1}{2} V^2 \frac{dC}{dx}
\label{Gr(4.67)}
\]
so like before but with the correct sign.
< / p >
< / div >
< / div >
< / div >
2022-02-09 06:44:58 +00:00
< 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" >
2022-02-07 14:11:58 +00:00
< p class = "author" > Author: Jean-Sébastien Caux< / p >
2022-02-09 21:41:42 +00:00
< p class = "date" > Created: 2022-02-09 Wed 22:40< / p >
2022-02-09 06:44:58 +00:00
< p class = "validation" > < / p >
2022-02-07 14:11:58 +00:00
< / div >
< / div >
< / html >