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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
209 changed files with 3663 additions and 1367 deletions
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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_ca_me_md.html">Monopole and Dipole Terms&emsp;<small>[ems.ca.me.md]</small></a></li><li>Next:&nbsp;<a href="ems_ca_me_Ed.html">The Electric Field of a Dipole&emsp;<small>[ems.ca.me.Ed]</small></a></li><li>Up:&nbsp;<a href="ems_ca_me.html">The Multipole Expansion&emsp;<small>[ems.ca.me]</small></a></li></ul><div id="outline-container-ems_ca_me_h" class="outline-5">
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="ems.html">Electromagnetostatics</a></li><li><a class="breadcrumb-link"href="ems_ca.html">Calculating or Approximating the Electrostatic Potential</a></li><li><a class="breadcrumb-link"href="ems_ca_me.html">The Multipole Expansion</a></li><li>Higher Moments</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca_me_md.html">Monopole and Dipole Terms&emsp;<small>[ems.ca.me.md]</small></a></li><li>Next:&nbsp;<a href="ems_ca_me_Ed.html">The Electric Field of a Dipole&emsp;<small>[ems.ca.me.Ed]</small></a></li><li>Up:&nbsp;<a href="ems_ca_me.html">The Multipole Expansion&emsp;<small>[ems.ca.me]</small></a></li></ul><div id="outline-container-ems_ca_me_h" class="outline-5">
<h5 id="ems_ca_me_h">Higher Moments<a class="headline-permalink" href="./ems_ca_me_h.html#ems_ca_me_h"><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"/>
@@ -1630,7 +1630,7 @@ V_{\mbox{\tiny quad}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^3} \in
\]
and can be rewritten as
</p>
<div class="main div" id="org7b16036">
<div class="main div" id="org33b3bb2">
<p>
\[
V_{\mbox{\tiny quad}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab}
@@ -1641,7 +1641,7 @@ and can be rewritten as
<p>
in terms of the {\bf quadrupole moment}
</p>
<div class="main div" id="orga1ea430">
<div class="main div" id="orgf43684b">
<p>
\[
Q_{ab} = \int_{\cal V} d\tau_s (3 r_{s,a} r_{s,b} - r_s^2 \delta_{a,b}) \rho ({\bf r}_s).
@@ -1659,7 +1659,7 @@ Moreover, it is traceless, \(\sum_a Q_{aa} = 0\). It therefore has \(5\) indepe
<p>
Our expansion for the potential thus looks like
</p>
<div class="main div" id="org1059aa3">
<div class="main div" id="orgbba2343">
<p>
\[
V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \left( \frac{Q}{r} + \sum_a \frac{r_a}{r^3} p_a + \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab} + ... \right)
@@ -1678,7 +1678,7 @@ the leading nonvanishing multipole moment is independent of the chosen location
the origin of the coordinate system (see Jackson Prob. 4.4).
</p>
<div class="info div" id="org778fe8a">
<div class="info div" id="orgd3efe3a">
<p>
{\bf A consistent nomenclature for the multipole expansion?}\\\\
You all know the terms {\bf monopole}, {\bf dipole} and {\bf quadrupole},
@@ -1735,10 +1735,21 @@ more terms than you might ever (hopefully) need:<br>
<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>