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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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<!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>
@@ -1598,29 +1598,56 @@ Table of contents
</svg></a><span class="headline-id">ems.ca.mi.fe</span></h5>
<div class="outline-text-5" id="text-ems_ca_mi_fe">
<p>
Force is the same as case of two point charges:
The force between the point charge and the plane
is the same as in the case of two point charges:
</p>
<div class="eqlabel" id="org59bf877">
<p>
<a id="F_dip_z"></a><a href="./ems_ca_mi_fe.html#F_dip_z"><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="orgc3f251d">
<ul class="org-ul">
<li>Gr (3.12)</li>
</ul>
</div>
</div>
<p>
\[
{\bf F} = -\frac{1}{4\pi \varepsilon_0} \frac{q^2}{(2d)^2} \hat{\bf z}
\label{Gr(3.12)}
\]
Energy is however not the same: for two point charges,
(\ref{Gr(2.42)}) gives
</p>
<p>
The energy is however not exactly the same: for two point charges,
<a href="./ems_es_e.html#W_pcd">W_pcd</a> gives
\[
W = -\frac{1}{4\pi \varepsilon_0} \frac{q^2}{2d}
\label{Gr(3.13)}
\]
whereas for the single charge and conducting plane,
</p>
<p>
\[
W = -\frac{1}{4\pi \varepsilon_0} \frac{q^2}{4d}
\label{Gr(3.14)}
\]
Half: \(E^2\) integrated over half the space in (\ref{eq:Energy_as_int_E2}).
The simple reason for the \(1/2\) factor between these two results
is that in the latter case, we are integrating \(E^2\)
in <a href="./ems_es_e.html#W_intEsq">W_intEsq</a> but over only half the space as in the
pure dipole case.
</p>
</div>
</div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca_mi_isc.html">Induced Surface Charges&emsp;<small>[ems.ca.mi.isc]</small></a></li><li>Next:&nbsp;<a href="ems_ca_mi_o.html">Other Image Problems&emsp;<small>[ems.ca.mi.o]</small></a></li><li>Up:&nbsp;<a href="ems_ca_mi.html">The Method of Images&emsp;<small>[ems.ca.mi]</small></a></li></ul>
<br>
<hr>
<div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@@ -1634,7 +1661,7 @@ 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>