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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>
@@ -1598,31 +1598,72 @@ 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>
Put \({\bf d}\) along \(\hat{\bf z}\). Then, (\ref{eq:electric_dipole}) becomes
To compute the electric field of a dipole, we will apply relation <a href="./ems_es_ep_fp.html#Emgp">Empg</a>
on <a href="./ems_ca_me_md.html#p_di">p_di</a>.
</p>
<p>
Let us first proceed simplistically, by putting
\({\bf d}\) along \(\hat{\bf z}\). Then, <a href="./ems_ca_me_md.html#p_di">p_di</a> becomes
\[
V_{\mbox{\tiny di}}({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{p \cos \theta}{r^2}
\label{Gr(3.102)}
\]
Taking the gradient,
\[
E_r = -\frac{\partial V}{\partial r} = \frac{1}{4\pi \varepsilon_0} \frac{2p\cos \theta}{r^3}, \hspace{1cm}
E_\theta = -\frac{1}{r} \frac{\partial V}{\partial \theta} = \frac{1}{4\pi \varepsilon_0} \frac{p \sin \theta}{r^3}, \hspace{1cm}
E_\phi = -\frac{1}{r \sin \theta} \frac{\partial V}{\partial \phi} = 0,
\phi_d ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{p \cos \theta}{r^2}
\]
Taking the gradient (in spherical coordinates, c.f. <a href="./c_m_cs_sph.html#sph_grad">sph_grad</a>),
</p>
\begin{align*}
E_r &amp;= -\frac{\partial \phi}{\partial r} = \frac{1}{4\pi \varepsilon_0} \frac{2p\cos \theta}{r^3}, \nonumber \\
E_\theta &amp;= -\frac{1}{r} \frac{\partial \phi}{\partial \theta} = \frac{1}{4\pi \varepsilon_0} \frac{p \sin \theta}{r^3}, \nonumber \\
E_\phi &amp;= -\frac{1}{r \sin \theta} \frac{\partial \phi}{\partial \varphi} = 0,
\end{align*}
<p>
we get
</p>
<div class="eqlabel" id="org1080e24">
<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="orgaea3bf0">
<ul class="org-ul">
<li>Gr (3.103)</li>
</ul>
</div>
</div>
<p>
\[
{\bf E}_{\mbox{\tiny di}} (r, \theta) = \frac{1}{4\pi \varepsilon_0} \frac{p}{r^3} (2\cos \theta \hat{\bf r} + \sin \theta \hat{\bf \theta})
\label{Gr(3.103)}
{\bf E}_d (r, \theta) = \frac{1}{4\pi \varepsilon_0} \frac{p}{r^3} (2\cos \theta ~\hat{\bf r} + \sin \theta ~\hat{\bf \theta})
\tag{E_di_1}\label{E_di_1}
\]
or in a better coordinate-free form
</p>
<div class="eqlabel" id="org1f76dd3">
<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="org8e1c28a">
<ul class="org-ul">
<li>Gr (3.104)</li>
</ul>
</div>
</div>
<p>
\[
{\bf E}_{\mbox{\tiny di}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^3} \left[3 ({\bf p} \cdot \hat{\bf r}) \hat{\bf r} - {\bf p}\right]
\label{eq:dipole_field}
{\bf E}_{\mbox{\tiny di}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^3} \left[3 ({\bf p} \cdot \hat{\bf r}) \hat{\bf r} - {\bf p}\right]
\tag{E_di}\label{E_di}
\]
</p>
<p>
<b>Dipole energy</b>
<b>Dipole energy</b> (tbd)
</p>
</div>
</div>
@@ -1644,7 +1685,7 @@ 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>