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Update 2022-03-15 10:07

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Jean-Sébastien
2022-03-15 10:07:27 +01:00
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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-07 Mon 20:38 -->
<!-- 2022-03-15 Tue 08:10 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1310,10 +1310,6 @@ Table of contents
</summary>
<ul>
<li>
<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
</li>
<li>
<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
</li>
@@ -1352,6 +1348,10 @@ Table of contents
<li>
<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
</li>
<li>
<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
</li>
</ul>
@@ -1614,25 +1614,25 @@ Table of contents
</svg></a><span class="headline-id">emf.g.Cg</span></h4>
<div class="outline-text-4" id="text-emf_g_Cg">
<p>
The {\bf Coulomb Gauge} is specified by taking
The <b>Coulomb Gauge</b> is specified by taking
\[
{\boldsymbol \nabla} \cdot {\boldsymbol A} = 0
\]
in which case (\ref{eq:LaplacianV}) becomes simply
in which case <a href="./emf_svp.html#Lapphi">Lapphi</a> becomes simply
\[
{\boldsymbol \nabla}^2 V = -\frac{\rho}{\varepsilon_0}
{\boldsymbol \nabla}^2 \phi = -\frac{\rho}{\varepsilon_0}
\]
{\it i.e.} Poisson's equation, whose solution we already know:
<i>i.e.</i> Poisson's equation, whose solution we already know:
\[
V({\boldsymbol r}, t) = \frac{1}{4\pi \varepsilon_0} \int d\tau' \frac{\rho({\boldsymbol r}^\prime, t)}{|{\boldsymbol r} - {\boldsymbol r}^\prime|}
\phi({\boldsymbol r}, t) = \frac{1}{4\pi \varepsilon_0} \int d\tau' \frac{\rho({\boldsymbol r}^\prime, t)}{|{\boldsymbol r} - {\boldsymbol r}^\prime|}
\]
Note that this is an equal-time relationship (it does not mean instantaneous action at a distance, since \(V\) by itself is not physically measurable).
Note that this is an equal-time relationship (it does not mean instantaneous action at a distance, since \(\phi\) by itself is not physically measurable).
</p>
<p>
Although Gauss's law looks nice in the Coulomb gauge, Amp{\`e}re-Maxwell does not:
Although Gauss's law looks nice in the Coulomb gauge, Ampère-Maxwell does not:
\[
{\boldsymbol \nabla}^2 {\boldsymbol A} - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} = -\mu_0 {\boldsymbol J} + \mu_0 \varepsilon_0 {\boldsymbol \nabla} \left( \frac{\partial V}{\partial t} \right).
{\boldsymbol \nabla}^2 {\boldsymbol A} - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} = -\mu_0 {\boldsymbol J} + \mu_0 \varepsilon_0 {\boldsymbol \nabla} \left( \frac{\partial \phi}{\partial t} \right).
\]
</p>
</div>
@@ -1656,7 +1656,7 @@ 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: 2022-03-07 Mon 20:38</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p>
<p class="validation"></p>
</div>