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Update 2022-02-14 06:33

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Jean-Sébastien
2022-02-14 06:33:37 +01:00
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commit 4cfe8cef59
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build/ems_es_c_sc.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-10 Thu 08:32 -->
<!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1601,20 +1601,67 @@ Table of contents
For a conductor, we can exploit the fact that electrical fields vanish on the inside to
get a proper boundary condition for the potential. Namely, here, boundary condition
<a href="./ems_es_ep_bc.html#Edisc">Edisc</a> yields
</p>
<div class="eqlabel" id="org14f3dda">
<p>
<a id="E_sur_cond"></a><a href="./ems_es_c_sc.html#E_sur_cond"><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="org284af32">
<ul class="org-ul">
<li>Gr (2.48)</li>
</ul>
</div>
</div>
<p>
\[
{\bf E} = \frac{\sigma}{\varepsilon_0} \hat{\bf n}
\label{Gr(2.48)}
\tag{E_sur_cond}\label{E_sur_cond}
\]
</p>
<p>
which in terms of potential reads
</p>
<div class="eqlabel" id="orge65a3f8">
<p>
<a id="scd_cond"></a><a href="./ems_es_c_sc.html#scd_cond"><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="orgda137bd">
<ul class="org-ul">
<li>Gr (2.49)</li>
</ul>
</div>
</div>
<p>
\[
\sigma = -\varepsilon_0 \frac{\partial \phi}{\partial n}.
\label{Gr(2.49)}
\tag{scd_cond}\label{scd_cond}
\]
</p>
<p>
The force per unit area on the surface of an object is
</p>
<p>
\[
{\bf f} = \sigma {\bf E}_{\mbox{average}} = \frac{\sigma}{2} ({\bf E}_{\mbox{above}} + {\bf E}_{\mbox{below}})
\label{Gr(2.50)}
\]
</p>
<p>
For a conductor, \({\bf E}_{\mbox{below}} = 0\), \({\bf E}_{\mbox{above}} = \frac{\sigma}{\varepsilon_0} \hat{\bf n}\), so
\[
{\bf f} = \frac{\sigma^2}{2\varepsilon_0} \hat{\bf n}
@@ -1634,6 +1681,8 @@ change rate of energy with respect to volume).
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_c_ic.html">Induced Charges&emsp;<small>[ems.es.c.ic]</small></a></li><li>Next:&nbsp;<a href="ems_es_c_cap.html">Capacitors&emsp;<small>[ems.es.c.cap]</small></a></li><li>Up:&nbsp;<a href="ems_es_c.html">Conductors&emsp;<small>[ems.es.c]</small></a></li></ul>
<br>
<hr>
<div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@@ -1647,7 +1696,7 @@ 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: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p>
</div>