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Update 2022-02-15 10:32

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Jean-Sébastien
2022-02-15 10:32:42 +01:00
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-14 Mon 20:35 -->
<!-- 2022-02-15 Tue 10:14 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1604,29 +1604,40 @@ Table of contents
<p>
The fact that conductors are equipotentials means that we can fomally solve
loads of electrostatic problems involving conductors of various shapes, by
looking at combinations of point charges.
This section could almost be called "guessing the potential" or "pulling the
potential out of a hat". We will rely on two things:
</p>
<ul class="org-ul">
<li>our ability to assemble potentials by combining simple potentials we already know (invoking the principle of superposition)</li>
<li>our certainty that if the potentials we assemble fulfill the desired boundary conditions, we have found the one and true potential we were originally seeking.</li>
</ul>
<p>
One type of physical system which gives us particularly simple conditions at boundaries
are conductors: the fact that conductors are equipotentials means that we can fomally solve
loads of electrostatic problems involving conductors of various shapes.
</p>
<p>
Let's consider the simplest electrostatic problem above a single point charge:
a system of two point charges \(\pm q\) separated by distance \(d\).
For definiteness, we put a charge \(q\) at coordinate
Let's consider one of the simplest electrostatic setups we can think of which
goes beyond a point charge, namely a system of two point charges \(\pm q\) separated
by distance \(d\). For definiteness, we put a charge \(q\) at coordinate
\(\frac{d}{2} ~\hat{z}\), and a charge \(-q\) at \(-\frac{d}{2} ~\hat{z}\).
</p>
<p>
By superposition, we have that
</p>
<div class="eqlabel" id="org2ce446c">
<div class="eqlabel" id="org71ec126">
<p>
<a id="p_dip_z"></a><a href="./ems_ca_mi.html#p_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">
<a id="p_di_z"></a><a href="./ems_ca_mi.html#p_di_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="orgb1af7bf">
<div class="alteqlabels" id="orgef817d5">
<ul class="org-ul">
<li>FLS II (6.8)</li>
<li>Gr (3.9)</li>
@@ -1637,7 +1648,7 @@ By superposition, we have that
</div>
\begin{align}
\phi(x,y,z) = \frac{1}{4\pi \varepsilon_0} \left[ \frac{q}{[x^2 + y^2 + (z-d/2)^2]^{1/2}} - \frac{q}{[x^2 + y^2 + (z+d/2)^2]^{1/2}} \right]
\tag{p_dip_z}\label{p_dip_z}
\tag{p_di_z}\label{p_di_z}
\end{align}
<p>
@@ -1684,7 +1695,7 @@ 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: 2022-02-14 Mon 20:35</p>
<p class="date">Created: 2022-02-15 Tue 10:14</p>
<p class="validation"></p>
</div>