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Update 2022-02-09 07:44

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Jean-Sébastien
2022-02-09 07:44:58 +01:00
parent 84531d125a
commit 3c40f5bfe8
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build/ems_es_ef_Gl.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-08 Tue 17:21 -->
<!-- 2022-02-09 Wed 07:31 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1616,7 +1616,7 @@ Table of contents
</ul>
</details>
</nav>
<ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_ef_cE.html">The Curl of \({\bf E}\)&emsp;<small>[ems.es.ef.cE]</small></a></li><li>Next:&nbsp;<a href="ems_es_ep.html">The Electrostatic Potential&emsp;<small>[ems.es.ep]</small></a></li><li>Up:&nbsp;<a href="ems_es_ef.html">Electrostatic Fields&emsp;<small>[ems.es.ef]</small></a></li></ul><div id="outline-container-ems_es_ef_Gl" class="outline-5">
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="ems.html">Electromagnetostatics</a></li><li><a class="breadcrumb-link"href="ems_es.html">Electrostatics</a></li><li><a class="breadcrumb-link"href="ems_es_ef.html">Electrostatic Fields</a></li><li>Gauss's Law: the divergence of \({\bf E}\)</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_ef_cE.html">The Curl of \({\bf E}\)&emsp;<small>[ems.es.ef.cE]</small></a></li><li>Next:&nbsp;<a href="ems_es_ep.html">The Electrostatic Potential&emsp;<small>[ems.es.ep]</small></a></li><li>Up:&nbsp;<a href="ems_es_ef.html">Electrostatic Fields&emsp;<small>[ems.es.ef]</small></a></li></ul><div id="outline-container-ems_es_ef_Gl" class="outline-5">
<h5 id="ems_es_ef_Gl">Gauss's Law: the divergence of \({\bf E}\)<a class="headline-permalink" href="./ems_es_ef_Gl.html#ems_es_ef_Gl"><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"/>
@@ -1667,14 +1667,14 @@ sphere of radius \(r\) around the charge,
<p>
so by superposition, we obtain
</p>
<div class="eqlabel" id="org804902c">
<div class="eqlabel" id="org552cd00">
<p>
<a id="Gl_i"></a><a href="./ems_es_ef_Gl.html#Gl_i"><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="org04a2212">
<div class="alteqlabels" id="org6dc5103">
<ul class="org-ul">
<li>Gr (2.13)</li>
</ul>
@@ -1682,7 +1682,7 @@ so by superposition, we obtain
</div>
</div>
<div class="core div" id="orgbbc9bc1">
<div class="core div" id="orgebfc03c">
<p>
<b>Gauss' law (in integral form)</b>
</p>
@@ -1714,14 +1714,14 @@ By applying the divergence theorem,
and using \(Q_{\mbox{enc}} = \int_{\cal V} \rho d\tau\), and using the fact the the choice of volume
is arbitrary, we get
</p>
<div class="eqlabel" id="org06ff080">
<div class="eqlabel" id="orga1e9bf0">
<p>
<a id="Gl_d"></a><a href="./ems_es_ef_Gl.html#Gl_d"><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="orgcf50b9d">
<div class="alteqlabels" id="org79262b5">
<ul class="org-ul">
<li>Gr (2.14)</li>
</ul>
@@ -1729,7 +1729,7 @@ is arbitrary, we get
</div>
</div>
<div class="core div" id="orgf131b44">
<div class="core div" id="org1c01bba">
<p>
<b>Gauss' law in differential form</b>
</p>
@@ -1788,7 +1788,7 @@ cylindrical or plane symmetry.
Gaussian surfaces: respectively, concentric sphere, coaxial cylinder, pillbox.
</p>
<div class="example div" id="org57bb4a4">
<div class="example div" id="org139e93f">
<p>
<b>Example 2.2</b>: Field outside a uniformly charged sphere of radius \(R\) and total charge \(q\).
</p>
@@ -1819,7 +1819,7 @@ Same as point charge at origin!
</div>
<div class="example div" id="orgc83260f">
<div class="example div" id="org9001b7b">
<p>
<b>Example 2.3</b>: infinitely long cylinder carrying charge density \(\rho = k s\) for some constant \(k\). Find \({\bf E}\) within the cylinder.
</p>
@@ -1858,7 +1858,7 @@ Therefore,
</div>
<div class="example div" id="orgabb478b">
<div class="example div" id="orgb2c877e">
<p>
<b>Example 2.4</b>: infinite plane (defined by \(z = 0\)) with uniform surface charge density \(\sigma\). Find \({\bf E}\).
</p>
@@ -1884,7 +1884,7 @@ where \(\hat{\bf n}\) is a unit vector extending away from the plane. Independe
</div>
<div class="example div" id="orgd7a4c2f">
<div class="example div" id="orga697386">
<p>
<b>Example 2.5</b>: two infinite planes (put them vertical) carrying equal but opposite uniform surface charge densities \(\pm \sigma\).
</p>
@@ -1899,10 +1899,21 @@ where \(\hat{\bf n}\) is a unit vector extending away from the plane. Independe
</div>
<hr><div id="postamble" class="status">
<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: 2022-02-08 Tue 17:21</p>
<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
<p class="date">Created: 2022-02-09 Wed 07:31</p>
<p class="validation"></p>
</div>