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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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build/ems_ca_me_Ed.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>
@@ -1598,12 +1598,12 @@ Table of contents
</svg></a><span class="headline-id">ems.ca.me.Ed</span></h5>
<div class="outline-text-5" id="text-ems_ca_me_Ed">
<p>
To compute the electric field of a dipole, we will apply relation <a href="./ems_es_ep_fp.html#Emgp">Empg</a>
To compute the electric field of a dipole, we will apply relation <a href="./ems_es_ep_fp.html#Emgp">Emgp</a>
on <a href="./ems_ca_me_md.html#p_di">p_di</a>.
</p>
<p>
Let us first proceed simplistically, by putting
Let us first proceed in a slightly pedestrian manner, putting
\({\bf d}\) along \(\hat{\bf z}\). Then, <a href="./ems_ca_me_md.html#p_di">p_di</a> becomes
\[
\phi_d ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{p \cos \theta}{r^2}
@@ -1618,14 +1618,14 @@ E_\phi &amp;= -\frac{1}{r \sin \theta} \frac{\partial \phi}{\partial \varphi} =
<p>
we get
</p>
<div class="eqlabel" id="org3d8213b">
<div class="eqlabel" id="orgbe4b01d">
<p>
<a id="E_di_1"></a><a href="./ems_ca_me_Ed.html#E_di_1"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org9e4601a">
<div class="alteqlabels" id="org5aa2b75">
<ul class="org-ul">
<li>Gr (3.103)</li>
</ul>
@@ -1638,16 +1638,23 @@ we get
{\bf E}_d (r, \theta) = \frac{1}{4\pi \varepsilon_0} \frac{p}{r^3} (2\cos \theta ~\hat{\bf r} + \sin \theta ~\hat{\bf \theta})
\tag{E_di_1}\label{E_di_1}
\]
or in a better coordinate-free form
</p>
<div class="eqlabel" id="orgb9d9fc2">
<p>
A smarter derivation directly applies the gradient on <a href="./ems_ca_me_md.html#p_di">p_di</a>,
\[
E_d ({\bf r}) = - \frac{1}{4\pi \varepsilon_0} \nabla \frac{{\bf p} \cdot {\bf r}}{r^3}
\]
immediately yielding the coordinate-free expression
</p>
<div class="eqlabel" id="org9838564">
<p>
<a id="E_di"></a><a href="./ems_ca_me_Ed.html#E_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="orgb7c1eba">
<div class="alteqlabels" id="orga3ac4a7">
<ul class="org-ul">
<li>Gr (3.104)</li>
</ul>
@@ -1657,11 +1664,13 @@ or in a better coordinate-free form
</div>
<p>
\[
{\bf E}_{\mbox{\tiny di}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^3} \left[3 ({\bf p} \cdot \hat{\bf r}) \hat{\bf r} - {\bf p}\right]
{\bf E}_d ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^3} \left[3 ({\bf p} \cdot \hat{\bf r}) \hat{\bf r} - {\bf p}\right]
\tag{E_di}\label{E_di}
\]
</p>
<p>
<b>Dipole energy</b> (tbd)
</p>
@@ -1685,7 +1694,7 @@ 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>