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Update 2022-03-07 20:40

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Jean-Sébastien
2022-03-07 20:40:36 +01:00
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commit 4808df71e6
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build/emd_emw_we.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-02 Wed 15:45 -->
<!-- 2022-03-07 Mon 20:38 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1098,14 +1098,6 @@ Table of contents
<li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li>
</ul>
@@ -1622,7 +1614,6 @@ Table of contents
</svg></a><span class="headline-id">emd.emw.we</span></h4>
<div class="outline-text-4" id="text-emd_emw_we">
<p>
\subsubsection*{The wave equation for \({\bf E}\) and \({\bf B}\)}
Take Maxwell's equations in vacuum:
</p>
\begin{align}
@@ -1650,15 +1641,32 @@ These take the form of coupled first-order partial differential equations for \(
Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
we get the
</p>
<div class="core div" id="orgb3cd985">
<div class="core div" id="orgc9c5d44">
<p>
<b>Wave equations for electric and magnetic fields in vacuum</b>
</p>
<div class="eqlabel" id="orgb29f135">
<p>
<a id="WaveEq"></a><a href="./emd_emw_we.html#WaveEq"><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="orge5c16c5">
<ul class="org-ul">
<li>Gr (9.41)</li>
</ul>
</div>
</div>
<p>
{\bf Wave equations for electric and magnetic fields in vacuum}
\[
{\boldsymbol \nabla}^2 {\bf E} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf E}}{\partial t^2},
\hspace{1cm}
{\boldsymbol \nabla}^2 {\bf B} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf B}}{\partial t^2}.
\label{Gr(9.41)}
\]
{\boldsymbol \nabla}^2 {\bf E} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf E}}{\partial t^2},
\hspace{1cm}
{\boldsymbol \nabla}^2 {\bf B} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf B}}{\partial t^2}.
\tag{WaveEq}\label{WaveEq}
\]
</p>
</div>
@@ -1681,7 +1689,7 @@ That is, a form
{\bf E} ({\bf r},t) = {\bf E}_0 e^{i ({\bf k} \cdot {\bf r} - \omega t)}, \hspace{1cm}
{\bf B} ({\bf r},t) = {\bf B}_0 e^{i ({\bf k} \cdot {\bf r} - \omega t)},
\]
solves (\ref{Gr(9.41)}) for \(\omega = c |{\bf k}|\).
solves <a href="./emd_emw_we.html#WaveEq">WaveEq</a> for \(\omega = c |{\bf k}|\).
Here and under, we use complex exponentials for convenience, remembering that
the actual electric and magnetic fields are given by the real part.
</p>
@@ -1706,7 +1714,7 @@ 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: 2022-03-02 Wed 15:45</p>
<p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p>
</div>