jscaux/pqed_html
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Update 2022-02-10 08:34

Browse Source
This commit is contained in:
Jean-Sébastien
2022-02-10 08:34:34 +01:00
parent f3c2446d19
commit f8446c1405
204 changed files with 803 additions and 790 deletions
Download Patch File Download Diff File Expand all files Collapse all files
build/ems_es_e.html
+26 -21
View File
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-09 Wed 22:40 -->
<!-- 2022-02-10 Thu 08:32 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1615,17 +1615,17 @@ W_u = -\int_{{\bf a}}^{{\bf b}} {\bf E} \cdot d{\bf l}
</p>
<p>
For a point charge distribution, this
can also be rewritten in terms of the potential,
For a point charge distribution, the total work required to assemble it
can also be rewritten in terms of the potential by using <a href="./ems_es_ep_d.html#p">p</a>,
</p>
<div class="eqlabel" id="orgfd9aa20">
<div class="eqlabel" id="org4987cb4">
<p>
<a id="W_pcd"></a><a href="./ems_es_e.html#W_pcd"><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="org87b466f">
<div class="alteqlabels" id="orgae0f38c">
<ul class="org-ul">
<li>Gr (2.42)</li>
</ul>
@@ -1644,14 +1644,14 @@ W = \frac{1}{2} \sum_{i=1}^m q_i \phi({\bf r}_i)
For a volume charge distribution, this becomes
</p>
<div class="eqlabel" id="orgc4a08c9">
<div class="eqlabel" id="orgaae8f35">
<p>
<a id="W_vcd"></a><a href="./ems_es_e.html#W_vcd"><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="org46ddad6">
<div class="alteqlabels" id="org9e1426e">
<ul class="org-ul">
<li>Gr (2.43)</li>
</ul>
@@ -1661,7 +1661,7 @@ For a volume charge distribution, this becomes
</div>
<p>
\[
W = \frac{1}{2} \int \rho({\bf r}) V({\bf r}) ~d\tau
W = \frac{1}{2} \int \rho({\bf r}) \phi({\bf r}) ~d\tau
\tag{W_vcd}\label{W_vcd}
\]
</p>
@@ -1684,22 +1684,23 @@ Integrating by parts and using \({\boldsymbol \nabla} \phi = -{\bf E}\),
<p>
\[
W = \frac{\varepsilon_0}{2} \left( \int_{\cal V} E^2 d\tau + \oint_{\cal S} \phi {\bf E} \cdot d{\bf a} \right)
W = \frac{\varepsilon_0}{2} \left( \int_{\cal V} E^2 d\tau + \oint_{\cal S} \phi ~{\bf E} \cdot d{\bf a} \right)
\label{Gr(2.44)}
\]
</p>
<p>
Integrating over all space,
Integrating over all space (and thus dropping the surface integrale whose integrand is assumed
to vanish at infinity), we get
</p>
<div class="eqlabel" id="org23414f1">
<div class="eqlabel" id="org7354478">
<p>
<a id="W_intEsq"></a><a href="./ems_es_e.html#W_intEsq"><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="orgba1f377">
<div class="alteqlabels" id="org72489d5">
<ul class="org-ul">
<li>Gr (2.45)</li>
</ul>
@@ -1707,7 +1708,7 @@ Integrating over all space,
</div>
</div>
<div class="core div" id="org93e1ca1">
<div class="core div" id="orged969c0">
<p>
</p>
@@ -1721,7 +1722,7 @@ W = \frac{\varepsilon_0}{2} \int E^2 d\tau
</div>
<div class="example div" id="org0020c97">
<div class="example div" id="org362282e">
<p>
<b>Energy of spherical shell</b>
</p>
@@ -1732,10 +1733,13 @@ We wish to compute its electrostatic energy.
</p>
<ul class="org-ul">
<li>
<b>First way</b>: we use <a href="./ems_es_e.html#W_vcd">W_vcd</a> adapted for surface charges,</li>
</ul>
<p>
<b>first way</b>: we use <a href="./ems_es_e.html#W_vcd">W_vcd</a> adapted for surface charges,
\[
W = \frac{1}{2} \int_S da \sigma \phi.
W = \frac{1}{2} \int_S da ~\sigma \phi.
\]
</p>
@@ -1746,13 +1750,14 @@ The potential at the surface of the sphere is constant,
<p>
\[
W = \frac{1}{8\pi \varepsilon_0} \frac{q}{R} \int da \sigma = \frac{1}{8\pi \varepsilon_0} \frac{q^2}{R}.
W = \frac{1}{8\pi \varepsilon_0} \frac{q}{R} \int da ~\sigma = \frac{1}{8\pi \varepsilon_0} \frac{q^2}{R}.
\]
</p>
<p>
<b>second way</b> we use <a href="./ems_es_e.html#W_intEsq">W_intEsq</a>. Inside the sphere, \({\bf E} = 0\); outside,
</p>
<ul class="org-ul">
<li>
<b>Second way</b>: we use <a href="./ems_es_e.html#W_intEsq">W_intEsq</a>. Inside the sphere, \({\bf E} = 0\); outside,</li>
</ul>
<p>
\[
@@ -1788,7 +1793,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>