jscaux/pqed_html
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Update 2022-02-14 20:42

Browse Source
This commit is contained in:
Jean-Sébastien
2022-02-14 20:42:37 +01:00
parent 4cfe8cef59
commit 09a8ba5fb6
204 changed files with 1968 additions and 1206 deletions
Download Patch File Download Diff File Expand all files Collapse all files
build/ems_ca_me_md.html
+52 -19
View File
@@ -1,7 +1,7 @@
<!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>
@@ -1598,16 +1598,31 @@ Table of contents
</svg></a><span class="headline-id">ems.ca.me.md</span></h5>
<div class="outline-text-5" id="text-ems_ca_me_md">
<p>
The series (\ref{Gr(3.95)}) is organized in increasing powers of inverse distance.
The leading term is called the {\bf monopole} term, and is
The series <a href="./ems_ca_me_a.html#p_Leg">p_Leg</a> is organized in increasing powers of inverse distance.
The leading term is called the *monopole term, and is
</p>
<div class="core div" id="org0dea0f3">
<div class="eqlabel" id="org4619ff2">
<p>
<a id="p_mono"></a><a href="./ems_ca_me_md.html#p_mono"><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="org5a3bf3c">
<ul class="org-ul">
<li>Gr (3.97)</li>
</ul>
</div>
</div>
<div class="core div" id="org333d379">
<p>
\[
V_{\mbox{\tiny mono}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{Q}{r}, \hspace{1cm}
Q = \int_{\cal V} d\tau_s \rho({\bf r}_s).
\label{Gr(3.97)}
\]
\phi_m ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{Q}{r}, \hspace{1cm}
Q = \int_{\cal \phi} d\tau_s \rho({\bf r}_s).
\tag{p_mono}\label{p_mono}
\]
</p>
</div>
@@ -1616,25 +1631,38 @@ For a point charge, the monopole term gives the exact potential.
</p>
<p>
The next term is the {\bf dipole} term: by using \(P_1 (x) = x\), we have
The next term is the <b>dipole</b> term: by using \(P_1 (x) = x\), we have
\[
V_{\mbox{\tiny di}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^2} \int_{\cal V} d\tau_s \hat{\bf r} \cdot {\bf r}_s \rho({\bf r}_s)
\phi_d ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^2} \int_{\cal V} d\tau_s \hat{\bf r} \cdot {\bf r}_s \rho({\bf r}_s)
\]
This can be written
</p>
<div class="core div" id="orga21233c">
<div class="eqlabel" id="org82b714c">
<p>
<a id="p_di"></a><a href="./ems_ca_me_md.html#p_di"><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="org0d44254">
</div>
</div>
<div class="core div" id="org0634724">
<p>
\[
V_{\mbox{\tiny di}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{\hat{\bf r} \cdot {\bf p}}{r^2},
\hspace{10mm}{\bf p} \equiv \int_{\cal V} d\tau_s ~{\bf r}_s ~\rho({\bf r}_s).
\label{eq:electric_dipole}
\]
\phi_d ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{\hat{\bf r} \cdot {\bf p}}{r^2},
\hspace{10mm}{\bf p} \equiv \int_{\cal V} d\tau_s ~{\bf r}_s ~\rho({\bf r}_s).
\tag{p_di}\label{p_di}
\]
</p>
</div>
<p>
in terms of the {\bf dipole moment} \({\bf p}\).
Note that the dipole moment is an internal property of the source charges, and that it
in terms of the <b>dipole moment</b> \({\bf p}\).
Note that the dipole moment is a property related to the internal distribution
of the source charges, and that it
in general depends on the chosen point of origin (more on this later).
</p>
@@ -1643,7 +1671,11 @@ Since dipole moments are vectors, they are summed following vector addition rule
</p>
<p>
{\bf Pure dipole:} two charges closer and closer together, but charges higher and higher such that \({\bf p}\) remains finite.
In the literature, one comes across the notion of a <b>mathematical dipole</b> or
synonymously a <b>pure dipole</b>: this is simply an abstraction in which the charges
are moved infinitesimally close together, while their charges are proportionally
increased such that the dipole moment remains the same. In such cases, the
multipole expansion terminates at the dipole term.
</p>
</div>
</div>
@@ -1651,6 +1683,7 @@ Since dipole moments are vectors, they are summed following vector addition rule
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca_me_a.html">Approximate Potential at Large Distance&emsp;<small>[ems.ca.me.a]</small></a></li><li>Next:&nbsp;<a href="ems_ca_me_h.html">Higher Moments&emsp;<small>[ems.ca.me.h]</small></a></li><li>Up:&nbsp;<a href="ems_ca_me.html">The Multipole Expansion&emsp;<small>[ems.ca.me]</small></a></li></ul>
<br>
<hr>
@@ -1666,7 +1699,7 @@ 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>