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@@ -1,7 +1,7 @@
<!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>
@@ -1614,49 +1614,121 @@ Table of contents
</svg></a><span class="headline-id">emf.g.Lg</span></h4>
<div class="outline-text-4" id="text-emf_g_Lg">
<p>
A more aesthetic choice is the {\bf Lorenz gauge}:
A more aesthetic choice is the <b>Lorenz gauge</b>:
</p>
<div class="eqlabel" id="orgb11b2c1">
<p>
<a id="LorenzG"></a><a href="./emf_g_Lg.html#LorenzG"><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="org9cfa6f0">
</div>
</div>
<p>
\[
{\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial V}{\partial t} = 0
\label{eq:LorenzGauge}
{\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial \phi}{\partial t} = 0
\tag{LorenzG}\label{LorenzG}
\]
which is chosen to put the second term in the left-hand side of (\ref{eq:LaplacianA}) to zero. What remains is then
which is chosen to put the second term in the left-hand side of <a href="./emf_svp.html#LapA">LapA</a> to zero. What remains is then
</p>
<div class="eqlabel" id="org8bdd983">
<p>
<a id="ALorenzG"></a><a href="./emf_g_Lg.html#ALorenzG"><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="orgda10464">
</div>
</div>
<p>
\[
{\boldsymbol \nabla}^2 {\boldsymbol A} - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} = -\mu_0 {\boldsymbol J}
\tag{ALorenzG}\label{ALorenzG}
\]
while the equation for \(V\) becomes
while the equation for \(\phi\) becomes
</p>
<div class="eqlabel" id="org92ed269">
<p>
<a id="phiLorenzG"></a><a href="./emf_g_Lg.html#phiLorenzG"><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="orgf853d29">
</div>
</div>
<p>
\[
{\boldsymbol \nabla}^2 V - \mu_0 \varepsilon_0 \frac{\partial^2 V}{\partial t^2} = -\frac{\rho}{\varepsilon_0}.
{\boldsymbol \nabla}^2 \phi - \mu_0 \varepsilon_0 \frac{\partial^2 \phi}{\partial t^2} = -\frac{\rho}{\varepsilon_0}.
\tag{phiLorenzG}\label{phiLorenzG}
\]
These can be written compactly upon introducing a new operator: the
</p>
<div class="core div" id="org0aaab77">
<div class="core div" id="orgd77f9a6">
<p>
<b>d'Alembertian operator</b>
</p>
<div class="eqlabel" id="orgc902ec7">
<p>
<a id="dAl"></a><a href="./emf_g_Lg.html#dAl"><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="orgac35877">
</div>
</div>
<p>
{\bf d'Alembertian operator}
\[
\square^2 \equiv {\boldsymbol \nabla}^2 - \mu_0 \varepsilon_0 \frac{\partial^2}{\partial t^2}
\label{eq:dAlembertian}
\]
\square^2 \equiv {\boldsymbol \nabla}^2 - \mu_0 \varepsilon_0 \frac{\partial^2}{\partial t^2}
\tag{dAl}\label{dAl}
\]
</p>
</div>
<p>
so we get the
</p>
<div class="core div" id="orgc8e8b42">
<div class="core div" id="org12b2ff9">
<p>
<b>Inhomogeneous Maxwell equations</b> <i>(Lorenz gauge)</i>
</p>
<div class="eqlabel" id="orge48b8c0">
<p>
<a id="MaxLor"></a><a href="./emf_g_Lg.html#MaxLor"><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="org140821b">
</div>
</div>
<p>
{\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
\[
\square^2 V = -\frac{\rho}{\varepsilon_0}, \hspace{10mm}
\square^2 {\boldsymbol A} = -\mu_0 {\boldsymbol J}
\label{eq:InhomogeneousMaxwellLorenzGauge}
\]
\square^2 \phi = -\frac{\rho}{\varepsilon_0}, \hspace{10mm}
\square^2 {\boldsymbol A} = -\mu_0 {\boldsymbol J}
\tag{MaxLor}\label{MaxLor}
\]
</p>
</div>
<p>
This gauge is especially nice in the context of special relativity.
The whole of electrodynamics has thus reduced to solving the inhomogeneous
wave equations (\ref{eq:InhomogeneousMaxwellLorenzGauge})
wave equations <a href="./emf_g_Lg.html#MaxLor">MaxLor</a>
in terms of specified sources.
</p>
@@ -1665,11 +1737,11 @@ in terms of specified sources.
Without choosing the Lorenz gauge, we can still write the inhomogeneous
Maxwell equations in a simpler form. Defining
\[
L \equiv {\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial V}{\partial t},
L \equiv {\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial \phi}{\partial t},
\]
we have by direct inspection
\[
\square^2 V + \frac{\partial L}{\partial t} = -\frac{\rho}{\varepsilon_0}, \hspace{10mm}
\square^2 \phi + \frac{\partial L}{\partial t} = -\frac{\rho}{\varepsilon_0}, \hspace{10mm}
\square^2 {\boldsymbol A} - {\boldsymbol \nabla} L = -\mu_0 {\boldsymbol J}.
\]
</p>
@@ -1692,7 +1764,7 @@ 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>