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Jean-Sébastien
2022-03-02 15:47:54 +01:00
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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1624,53 +1624,79 @@ Table of contents
<p>
Full set of equations for the electromagnetic field:
</p>
<div class="core div" id="org883ebf1">
<div class="core div" id="org40e4789">
<p>
{\bf Maxwell's equations} {\it (in vacuum)}
<b>Maxwell's equations</b> <i>(in vacuum)</i>
</p>
<div class="eqlabel" id="org3e8ad9b">
<p>
<a id="Max_vac"></a><a href="./emd_Me_Me.html#Max_vac"><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="orgc606bec">
</div>
</div>
\begin{align}
(i) {\boldsymbol \nabla} \cdot {\bf E} &amp;= \frac{\rho}{\varepsilon_0}, \hspace{1cm} &amp;\mbox{Gauss}, \nonumber \\
(ii) {\boldsymbol \nabla} \cdot {\bf B} &amp;= 0, \hspace{1cm} &amp;\mbox{anonymous} \nonumber \\
(iii) {\boldsymbol \nabla} \times {\bf E} &amp;= -\frac{\partial {\bf B}}{\partial t}, \hspace{1cm} &amp;\mbox{Faraday}, \nonumber \\
(iv) {\boldsymbol \nabla} \times {\bf B} &amp;= \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, \hspace{1cm} &amp;\mbox{Ampère + Maxwell}.
\label{Gr(7.39)}
(i)~ {\boldsymbol \nabla} \cdot {\bf E} &amp;= \frac{\rho}{\varepsilon_0}, \hspace{1cm} &amp;\mbox{Gauss}, \nonumber \\
(ii)~ {\boldsymbol \nabla} \cdot {\bf B} &amp;= 0, &amp;\mbox{anonymous} \nonumber \\
(iii)~ {\boldsymbol \nabla} \times {\bf E} &amp;= -\frac{\partial {\bf B}}{\partial t}, &amp;\mbox{Faraday}, \nonumber \\
(iv)~ {\boldsymbol \nabla} \times {\bf B} &amp;= \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, &amp;\mbox{Ampère + Maxwell}.
\tag{Max_vac}\label{Max_vac}
\end{align}
</div>
<p>
Complement:
</p>
<div class="core div" id="org2481694">
<div class="core div" id="org28daec0">
<p>
{\bf Force law}
Force law <a href="./ems_ms_lf_pc.html#LorFo">LorFo</a>
\[
{\bf F} = q ({\bf E} + {\bf v} \times {\bf B}).
\label{Gr(7.40)}
\]
{\bf F} = q ({\bf E} + {\bf v} \times {\bf B}).
\label{Gr(7.40)}
\]
</p>
</div>
<p>
These equations summarize the {\bf entire content of classical electrodynamics}.
These equations contain the <b>entirety of pre-quantum electrodynamics</b>.
</p>
<p>
\paragraph{Note:} even the continuity equation can be derived from Maxwell's equations:
take divergence of \((iv)\).
<b>Note</b>: even the continuity equation can be derived from Maxwell's equations:
take divergence of \((iv)\) and use \((i)\).
</p>
<p>
Better way of writing: all fields on left, all sources on right,
</p>
<div class="core div" id="org47f05c2">
<div class="core div" id="org4bb3e78">
<div class="eqlabel" id="org05437e5">
<p>
<a id="Max_vac_s"></a><a href="./emd_Me_Me.html#Max_vac_s"><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="org61742b2">
<ul class="org-ul">
<li>Gr (7.42)</li>
</ul>
</div>
</div>
\begin{align}
(i) &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0},
&amp;(iii) {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} = 0, \\
(ii) &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0,
&amp;(iv) {\boldsymbol \nabla} \times {\bf B} - \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} = \mu_0 {\bf J},
\label{Gr(7.42)}
(i)~ {\boldsymbol \nabla} \cdot {\bf E} &amp;= \frac{\rho}{\varepsilon_0}, \nonumber \\
(ii)~{\boldsymbol \nabla} \cdot {\bf B} &amp;= 0, \nonumber \\
(iii)~ {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} &amp;= 0, \nonumber \\
(iv)~ {\boldsymbol \nabla} \times {\bf B} - \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} &amp;= \mu_0 {\bf J},
\tag{Max_vac_s}\label{Max_vac_s}
\end{align}
</div>
@@ -1679,6 +1705,7 @@ Better way of writing: all fields on left, all sources on right,
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Me_dc.html">Maxwell's Correction to Ampère's Law; the Displacement Current&emsp;<small>[emd.Me.dc]</small></a></li><li>Next:&nbsp;<a href="emd_Me_mc.html">Magnetic Charge&emsp;<small>[emd.Me.mc]</small></a></li><li>Up:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li></ul>
<br>
<hr>
@@ -1694,7 +1721,7 @@ 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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>