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Update 2022-03-02 15:47

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Jean-Sébastien
2022-03-02 15:47:54 +01:00
parent ac1e628013
commit 21bf9fdba5
194 changed files with 1653 additions and 1216 deletions
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build/ems_ca_me_a.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1626,7 +1626,7 @@ Let's consider the spatial function in the potential for a single point source c
\[
\frac{1}{|{\bf r} - {\bf r}_s|}
\]
How does this look when we're at large distances \(|{\bf r}| \gg |{\bf r}_s|\) ?
How does this look when we're at large distances \(|{\bf r}| \gg |{\bf r}_s|\)?
We can formally expand this in powers of \(|{\bf r}_s|/|{\bf r}|\). For simplicity, let's start by
putting \({\bf r} = r ~\hat{\bf z}, r &gt; 0\) and \({\bf r}_s = r_s \hat{\bf z}\), with \(|r_s| &lt; r\).
By Taylor expanding, we get
@@ -1636,14 +1636,14 @@ By Taylor expanding, we get
Formally, we could do this for any vector \({\bf r}_s\) such that \(|{\bf r}_s| &lt; |{\bf r}|\) by
Taylor expanding with the \({\boldsymbol \nabla}\) operator,
</p>
<div class="eqlabel" id="org522710d">
<div class="eqlabel" id="orgbba2c7c">
<p>
<a id="1or_grad"></a><a href="./ems_ca_me_a.html#1or_grad"><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="org3aaf125">
<div class="alteqlabels" id="orgba8e23d">
</div>
@@ -1661,14 +1661,14 @@ the potential takes the form of the general solution of Laplace's equation <a hr
Reading the parameters, we get \(A_l = 0\), \(B_l = r_s^l\). Putting back a generic angle
(the coefficients remain the same), we thus get
</p>
<div class="eqlabel" id="org22f2df1">
<div class="eqlabel" id="org50a8092">
<p>
<a id="1or_Leg"></a><a href="./ems_ca_me_a.html#1or_Leg"><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="org555af0c">
<div class="alteqlabels" id="org6a1a9bb">
<ul class="org-ul">
<li>Gr(3.94)</li>
</ul>
@@ -1690,14 +1690,14 @@ we can expand the potential at a point \({\bf r}\) outside \({\cal V}\) accordin
(here, we put the origin of our coordinate system closer to all points in \({\cal V}\) than to \({\bf r}\)
to ensure convergence)
</p>
<div class="eqlabel" id="org64fa160">
<div class="eqlabel" id="orgcf8de8f">
<p>
<a id="p_Leg"></a><a href="./ems_ca_me_a.html#p_Leg"><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="org33b5b72">
<div class="alteqlabels" id="org09dcc77">
<ul class="org-ul">
<li>Gr (3.95)</li>
</ul>
@@ -1760,14 +1760,14 @@ For \(r \gg d\), we can expand (immediately dropping terms of order \(d^2/r^2\))
Putting things together, the leading term in the expansion <a href="./ems_ca_me_a.html#p_Leg">p_Leg</a> for the
potential of the physical dipole is given by
</p>
<div class="eqlabel" id="org99e07b3">
<div class="eqlabel" id="org5950916">
<p>
<a id="p_physdi"></a><a href="./ems_ca_me_a.html#p_physdi"><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="org0f12ac7">
<div class="alteqlabels" id="org1ac0511">
<ul class="org-ul">
<li>Gr (3.90)</li>
</ul>
@@ -1814,7 +1814,7 @@ 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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>