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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-09 Wed 07:31 -->
<!-- 2022-02-09 Wed 22:40 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -408,17 +408,13 @@ Table of contents
<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_c.html#ems_es_ep_c">Comments on the Electrostatic Potential</a><span class="headline-id">ems.es.ep.c</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">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
<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>
@@ -429,30 +425,9 @@ Table of contents
</ul>
</details>
</li>
<li>
<details open="">
<summary class="toc-currentpage">
<li class="toc-currentpage">
<a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>
</summary>
<ul>
<li>
<a href="./ems_es_e_pcd.html#ems_es_e_pcd">The Energy of a Point Charge Distribution</a><span class="headline-id">ems.es.e.pcd</span>
</li>
<li>
<a href="./ems_es_e_ccd.html#ems_es_e_ccd">The Energy of a Continuous Charge Distribution</a><span class="headline-id">ems.es.e.ccd</span>
</li>
<li>
<a href="./ems_es_e_c.html#ems_es_e_c">Comments on Electrostatic Energy</a><span class="headline-id">ems.es.e.c</span>
</li>
</ul>
</details>
</li>
<li>
@@ -1616,12 +1591,11 @@ Table of contents
</ul>
</details>
</nav>
<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>Electrostatic Energy from the Potential</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_ep_bc.html">Electrostatic Boundary Conditions&emsp;<small>[ems.es.ep.bc]</small></a></li><li>Next:&nbsp;<a href="ems_es_e_pcd.html">The Energy of a Point Charge Distribution&emsp;<small>[ems.es.e.pcd]</small></a></li><li>Up:&nbsp;<a href="ems_es.html">Electrostatics&emsp;<small>[ems.es]</small></a></li></ul>
<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>Electrostatic Energy from the Potential</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_ep_bc.html">Electrostatic Boundary Conditions&emsp;<small>[ems.es.ep.bc]</small></a></li><li>Next:&nbsp;<a href="ems_es_c.html">Conductors&emsp;<small>[ems.es.c]</small></a></li><li>Up:&nbsp;<a href="ems_es.html">Electrostatics&emsp;<small>[ems.es]</small></a></li></ul><div id="outline-container-ems_es_e" class="outline-4">
<h4 id="ems_es_e">Electrostatic Energy from the Potential<a class="headline-permalink" href="./ems_es_e.html#ems_es_e"><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.es.e</span></h4>
<div class="outline-text-4" id="text-ems_es_e">
<ul class="org-ul altsecnrs">
<li>FLS II 8</li>
@@ -1630,12 +1604,8 @@ Table of contents
</ul>
<p>
We have already seen that the electrostatic potential is the potential energy of a unit charge brought from infinity.
</p>
<p>
Since the force is proportional to the charge we're moving,
we can calculate the work per unit charge/ \(W_u\) (so \(W = q W_u\)),
We have already seen that the electrostatic potential is the potential energy of a unit charge brought from infinity, and that since the force is proportional to the charge we're moving,
we can simply calculate the work per unit charge \(W_u\) (so \(W = q W_u\)):
</p>
<p>
@@ -1643,18 +1613,168 @@ we can calculate the work per unit charge/ \(W_u\) (so \(W = q W_u\)),
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,
</p>
<div class="eqlabel" id="orgfd9aa20">
<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">
<ul class="org-ul">
<li>Gr (2.42)</li>
</ul>
</div>
</div>
<p>
\[
W = \frac{1}{2} \sum_{i=1}^m q_i \phi({\bf r}_i)
\tag{W_pcd}\label{W_pcd}
\]
</p>
<p>
For a volume charge distribution, this becomes
</p>
<div class="eqlabel" id="orgc4a08c9">
<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">
<ul class="org-ul">
<li>Gr (2.43)</li>
</ul>
</div>
</div>
<p>
\[
W = \frac{1}{2} \int \rho({\bf r}) V({\bf r}) ~d\tau
\tag{W_vcd}\label{W_vcd}
\]
</p>
<p>
In this, we can eliminate \(\rho\) and \(\phi\) in favour of \({\bf E}\):
</p>
<p>
\[
\rho = \varepsilon_0 {\boldsymbol \nabla} \cdot {\bf E}
\longrightarrow
W = \frac{\varepsilon_0}{2} \int ({\boldsymbol \nabla} \cdot {\bf E}) ~\phi d\tau.
\]
</p>
<p>
Integrating by parts and using \({\boldsymbol \nabla} \phi = -{\bf E}\),
</p>
<p>
\[
W = \frac{\varepsilon_0}{2} \left( \int_{\cal V} E^2 d\tau + \oint_{\cal S} \phi {\bf E} \cdot d{\bf a} \right)
\label{Gr(2.44)}
\]
</p>
<p>
Integrating over all space,
</p>
<div class="eqlabel" id="org23414f1">
<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">
<ul class="org-ul">
<li>Gr (2.45)</li>
</ul>
</div>
</div>
<div class="core div" id="org93e1ca1">
<p>
</p>
<p>
\[
W = \frac{\varepsilon_0}{2} \int E^2 d\tau
\tag{W_intEsq}\label{W_intEsq}
\]
</p>
</div>
<div class="example div" id="org0020c97">
<p>
<b>Energy of spherical shell</b>
</p>
<p>
Consider a uniformly charged spherical shell of total charge \(q\) and radius \(R\).
We wish to compute its electrostatic energy.
</p>
<p>
<b>first way</b>: we use <a href="./ems_es_e.html#W_vcd">W_vcd</a> adapted for surface charges,
\[
W = \frac{1}{2} \int_S da \sigma \phi.
\]
</p>
<p>
The potential at the surface of the sphere is constant,
\(\phi = \frac{1}{4\pi \varepsilon_0} \frac{q}{R}\), so
</p>
<p>
\[
W = \frac{1}{8\pi \varepsilon_0} \frac{q}{R} \int da \sigma = \frac{1}{8\pi \varepsilon_0} \frac{q^2}{R}.
\]
</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>
<p>
\[
{\bf E} = \frac{1}{4\pi \varepsilon_0} q \frac{\hat{\bf r}}{r^2} \rightarrow E^2 = \frac{q^2}{(4\pi \varepsilon_0)^2 r^4}.
\]
</p>
<p>
so
</p>
\begin{align}
W &amp;= \frac{\varepsilon_0}{2} \int_{outside~sphere} \frac{q^2}{(4\pi \varepsilon_0)^2 r^4} r^2 \sin \theta dr d\theta d\phi \nonumber \\
&amp;= \frac{1}{32 \pi^2 \varepsilon_0} q^2 4\pi \int_R^{\infty} \frac{dr}{r^2} = \frac{1}{8\pi \varepsilon_0} \frac{q^2}{R}.
\end{align}
</div>
</div>
</div>
<h5>In this section:</h5>
<ul class="child-links-list">
<li><a href="ems_es_e_pcd.html">The Energy of a Point Charge Distribution</a><span class="headline-id">ems.es.e.pcd</span></li>
<li><a href="ems_es_e_ccd.html">The Energy of a Continuous Charge Distribution</a><span class="headline-id">ems.es.e.ccd</span></li>
<li><a href="ems_es_e_c.html">Comments on Electrostatic Energy</a><span class="headline-id">ems.es.e.c</span></li>
</ul>
<hr>
<div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@@ -1668,7 +1788,7 @@ 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 07:31</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
<p class="validation"></p>
</div>