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Update 2022-02-17 08:44

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Jean-Sébastien
2022-02-17 08:44:22 +01:00
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commit ec8a4ca406
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build/ems_ca_me_a.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-15 Tue 10:14 -->
<!-- 2022-02-17 Thu 08:42 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1612,14 +1612,14 @@ 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="org3027486">
<div class="eqlabel" id="orgbcad955">
<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="org77ac74d">
<div class="alteqlabels" id="org0e29a64">
</div>
@@ -1637,14 +1637,14 @@ 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="org9cc1631">
<div class="eqlabel" id="org3dde779">
<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="org47be670">
<div class="alteqlabels" id="org2eb9674">
<ul class="org-ul">
<li>Gr(3.94)</li>
</ul>
@@ -1652,11 +1652,13 @@ Reading the parameters, we get \(A_l = 0\), \(B_l = r_s^l\). Putting back a gen
</div>
</div>
\begin{equation}
\frac{1}{|{\bf r} - {\bf r}_s|} = \sum_{l=0}^{\infty} \frac{r_s^l}{r^{l+1}} P_l (\cos \theta),
\hspace{1cm} \cos \theta \equiv \hat{\bf r} \cdot \hat{\bf r}_s, \hspace{1cm} r_s &lt; r.
\tag{1or_Leg\label{1or_Leg}
\tag{1or_Leg}\label{1or_Leg}
\end{equation}
<p>
We thus see that our beloved Legendre polynomials are quite handy beasts indeed.
Considering an arbitrary charge distribution over a volume \({\cal V}\),
@@ -1664,14 +1666,14 @@ 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="org2d8ac35">
<div class="eqlabel" id="org2f77251">
<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="orgefd2a26">
<div class="alteqlabels" id="org802f8b6">
<ul class="org-ul">
<li>Gr (3.95)</li>
</ul>
@@ -1703,27 +1705,45 @@ two equal and opposite charges \(\pm q\) separated by a distance \(d\).
<p>
To be more general, we here put \(q\) at position \({\bf d}/2\) and \(-q\) at \(-{\bf d}/2\).
The potential is then
</p>
<p>
\[
\phi({\bf r}) = \frac{q}{4\pi \varepsilon_0} \left( \frac{1}{|{\bf r} - {\bf d}/2|} - \frac{1}{|{\bf r} + {\bf d}/2|} \right).
\]
We can write \(|{\bf r} \pm {\bf d}/2|^2 = r^2 \mp {\bf r} \cdot {\bf d} + (d/2)^2
= r^2 \left( 1 \mp \frac{{\bf d} \cdot \hat{\bf r}}{r} + \frac{d^2}{4r^2}\right)\).
</p>
<p>
We can write
</p>
\begin{equation*}
|{\bf r} \pm {\bf d}/2|^2 = r^2 \pm {\bf r} \cdot {\bf d} + (d/2)^2 = r^2 \left( 1 \pm \frac{{\bf d} \cdot \hat{\bf r}}{r} + \frac{d^2}{4r^2}\right)
\end{equation*}
<p>
For \(r \gg d\), we can expand (immediately dropping terms of order \(d^2/r^2\))
</p>
<p>
\[
\frac{1}{|{\bf r} \mp {\bf d}/2|} \simeq \frac{1}{r} \left( 1 \pm \frac{{\bf d} \cdot \hat{\bf r}}{r} \right)^{1/2}
\simeq \frac{1}{r} \left( 1 \pm \frac{{\bf d} \cdot \hat{\bf r}}{2r} \right).
\frac{1}{|{\bf r} \pm {\bf d}/2|} \simeq \frac{1}{r} \left( 1 \pm \frac{{\bf d} \cdot \hat{\bf r}}{r} \right)^{-1/2}
\simeq \frac{1}{r} \left( 1 \mp \frac{{\bf d} \cdot \hat{\bf r}}{2r} \right).
\]
</p>
<p>
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="org7e345bb">
<div class="eqlabel" id="orga1d3321">
<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="org8f0806f">
<div class="alteqlabels" id="org7b58d38">
<ul class="org-ul">
<li>Gr (3.90)</li>
</ul>
@@ -1770,7 +1790,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>