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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-13 Sun 21:20 -->
<!-- 2022-02-14 Mon 20:35 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1598,110 +1598,227 @@ Table of contents
</svg></a><span class="headline-id">ems.ca.me.h</span></h5>
<div class="outline-text-5" id="text-ems_ca_me_h">
<p>
The next terms in the expansion are obtained similarly: the {\bf quadrupole term} is
The next terms in the expansion are obtained similarly: the <b>quadrupole</b> term is
\[
V_{\mbox{\tiny quad}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^3} \int_{\cal V} d\tau_s r_s^2 P_2 (\hat{\bf r} \cdot \hat{\bf r}_s) \rho({\bf r}_s)
\phi_q ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^3} \int_{\cal V} d\tau_s r_s^2 P_2 (\hat{\bf r} \cdot \hat{\bf r}_s) \rho({\bf r}_s)
= \sum_{a,b = x,y,z} \frac{r_a r_b}{r^5} \int_{\cal V} d\tau_s \frac{1}{2} (3 r_{s,a} r_{s,b} - r_s^2 \delta_{a,b}) \rho ({\bf r}_s)
\]
and can be rewritten as
</p>
<div class="main div" id="org2ba72a1">
<div class="eqlabel" id="org9cab05e">
<p>
<a id="p_quad"></a><a href="./ems_ca_me_h.html#p_quad"><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="orgbca6f1d">
</div>
</div>
<div class="main div" id="org5faa07e">
<p>
\[
V_{\mbox{\tiny quad}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab}
\]
\phi_q ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab}
\tag{p_quad}\label{p_quad}
\]
</p>
</div>
<p>
in terms of the {\bf quadrupole moment}
in terms of the <b>quadrupole moment</b>
</p>
<div class="main div" id="org9a117e2">
<div class="eqlabel" id="org78a53c2">
<p>
<a id="quadmom"></a><a href="./ems_ca_me_h.html#quadmom"><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="org23355b9">
</div>
</div>
<div class="main div" id="orga508802">
<p>
\[
Q_{ab} = \int_{\cal V} d\tau_s (3 r_{s,a} r_{s,b} - r_s^2 \delta_{a,b}) \rho ({\bf r}_s).
\]
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).
\tag{quadmom}\label{quadmom}
\]
</p>
</div>
<p>
This is a symmetric rank \(2\) tensor, \(Q_{ab} = Q_{ba}\).
Moreover, it is traceless, \(\sum_a Q_{aa} = 0\). It therefore has \(5\) independent components.
This is a symmetric, traceless rank \(2\) tensor: \(Q_{ab} = Q_{ba}\) and
\(\sum_a Q_{aa} = 0\). It therefore has \(5\) independent components.
</p>
<p>
Our expansion for the potential thus looks like
Our expansion for the potential, with terms up to quadrupole ones, thus looks like
</p>
<div class="main div" id="org807951f">
<div class="eqlabel" id="orgd3632c3">
<p>
<a id="p_mdq"></a><a href="./ems_ca_me_h.html#p_mdq"><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="orgebae3fc">
</div>
</div>
<div class="main div" id="orge9b9790">
<p>
\[
V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \left( \frac{Q}{r} + \sum_a \frac{r_a}{r^3} p_a + \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab} + ... \right)
\]
\phi({\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)
\tag{p_mdq}\label{p_mdq}
\]
</p>
</div>
<p>
This can be carried further if we feel like it, with the {\bf octopole}, {\bf hexadecapole}, {\bf triacontadipole}, {\bf hexecontatetrapole}, … terms (see info box).
This can be carried further if we feel like it, with the <b>octopole</b>, <b>hexadecapole</b>,
<b>triacontadipole</b>, <b>hexecontatetrapole</b>, … terms (see info box).
</p>
<p>
\paragraph{Important property:}
<b>Important property</b>:
the leading nonvanishing multipole moment is independent of the chosen location for
the origin of the coordinate system (see Jackson Prob. 4.4).
</p>
<div class="info div" id="org3a15b33">
<div class="info div" id="org897641d">
<p>
{\bf A consistent nomenclature for the multipole expansion?}\\\\
You all know the terms {\bf monopole}, {\bf dipole} and {\bf quadrupole},
and perhaps also the less frequently used {\bf octupole}, {\bf hexadecapole} [16],
{\bf triacontadipole} (or {\bf dotriacontapole}) [32] and {\bf tetrahexacontapole}
(or {\bf hexacontatetrapole}) [64].
Physicists are clearly insufficiently educated in the humanities:
these terms sound very fancy and their choice seems to make sense, but it doesn't.
{\bf Mono-} is derived from the Greek
{\it monos} ('alone'); {\bf di-} is derived from the Greek {\it dis} ('twice');
{\bf quadru-} is a fake Latin prefix ({\it quadri-} would be genuine)
<b>A consistent nomenclature for the multipole expansion?</b>
</p>
<p>
You all know the terms <b>monopole</b>, <b>dipole</b> and <b>quadrupole</b>,
and perhaps also the less frequently used <b>octupole</b>, <b>hexadecapole</b> [16],
<b>triacontadipole</b> (or <b>dotriacontapole</b>) [32] and <b>tetrahexacontapole</b>
(or <b>hexacontatetrapole</b>) [64].
</p>
<p>
Well I'm afraid physicists are clearly insufficiently educated in the humanities,
and have badly screwed up with this nomeclature. Although these terms sound very
fancy and their choice seems to make sense, it doesn't.
</p>
<p>
<i>Mono</i>- is derived from the Greek <i>monos</i> ('alone');
<i>di</i>- is derived from the Greek <i>dis</i> ('twice');
<i>quadru</i>- is a fake Latin prefix (<i>quadri</i>- would be genuine)
meaning 'something to do with the number 4', and
{\bf octu-} is another (fake) Latin prefix (both Greek and Latin have {\it octo/okto} for 8,
but Greek makes compounds with {\it octo-} or {\it octa-}, never {\it octu-}).
<i>octu</i>- is another (fake) Latin prefix (both Greek and Latin have
<i>octo/okto</i> for 8, but Greek makes compounds with <i>octo</i>- or <i>octa</i>-,
never <i>octu</i>-).
</p>
<p>
A more consistent nomenclature would be to go either fully Greek {\it or} Latin,
yielding:<br>
A more consistent nomenclature would be to go either fully Greek <i>or</i> Latin,
yielding:
</p>
<p>
\begin{tabular}{r|ll}
&amp; Greek-inspired &amp; Latin-inspired <br>
\hline
1 &amp; monopole &amp; unipole <br>
2 &amp; dipole &amp; duopole <br>
4 &amp; tetrapole &amp; quadrupole <br>
8 &amp; octopole &amp; octopole <br>
\end{tabular}<br>
</p>
<table>
<colgroup>
<col class="org-right">
<col class="org-left">
<col class="org-left">
</colgroup>
<thead>
<tr>
<th scope="col" class="org-right"> </th>
<th scope="col" class="org-left">Greek-inspired</th>
<th scope="col" class="org-left">Latin-inspired</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-right">1</td>
<td class="org-left">monopole</td>
<td class="org-left">unipole</td>
</tr>
<tr>
<td class="org-right">2</td>
<td class="org-left">dipole</td>
<td class="org-left">duopole</td>
</tr>
<tr>
<td class="org-right">4</td>
<td class="org-left">tetrapole</td>
<td class="org-left">quadrupole</td>
</tr>
<tr>
<td class="org-right">8</td>
<td class="org-left">octopole</td>
<td class="org-left">octopole</td>
</tr>
</tbody>
</table>
<p>
Irrespective of whether you have a predilection for the Greek or Latin version,
you can go wild and ask how this could generalize. The way to do this is not
uniquely defined; here is a set of possibilities for
more terms than you might ever (hopefully) need:<br>
more terms than you might ever (hopefully) need:
</p>
<p>
\begin{tabular}{r|ll}
16 &amp; hexadecapole &amp; sexdecapole<br>
32 &amp; triacontadipole &amp; trigentiduopole<br>
64 &amp; hexecontatetrapole &amp; sexagintiquadrupole<br>
128 &amp; hecatonikosioctopole &amp; viginticentioctopole<br>
256 &amp; diacosipentecontahexapole &amp; ducentiquinquagintisexapole
\end{tabular}<br>
</p>
<table>
<colgroup>
<col class="org-right">
<col class="org-left">
<col class="org-left">
</colgroup>
<tbody>
<tr>
<td class="org-right">16</td>
<td class="org-left">hexadecapole</td>
<td class="org-left">sexdecapole</td>
</tr>
<tr>
<td class="org-right">32</td>
<td class="org-left">triacontadipole</td>
<td class="org-left">trigentiduopole</td>
</tr>
<tr>
<td class="org-right">64</td>
<td class="org-left">hexecontatetrapole</td>
<td class="org-left">sexagintiquadrupole</td>
</tr>
<tr>
<td class="org-right">128</td>
<td class="org-left">hecatonikosioctopole</td>
<td class="org-left">viginticentioctopole</td>
</tr>
<tr>
<td class="org-right">256</td>
<td class="org-left">diacosipentecontahexapole</td>
<td class="org-left">ducentiquinquagintisexapole</td>
</tr>
</tbody>
</table>
</div>
</div>
@@ -1725,7 +1842,7 @@ 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: 2022-02-13 Sun 21:20</p>
<p class="date">Created: 2022-02-14 Mon 20:35</p>
<p class="validation"></p>
</div>