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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_mpw.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 +1614,7 @@ Table of contents
</svg></a><span class="headline-id">emd.emw.mpw</span></h4>
<div class="outline-text-4" id="text-emd_emw_mpw">
<p>
A {\it monochromatic} wave is one having a single frequency in its temporal dependence. Say that the propagation direction is \(\hat{\boldsymbol z}\): we'd then have
A <b>monochromatic</b> wave is one having a single frequency in its temporal dependence. Say that the propagation direction is \(\hat{\boldsymbol z}\): we'd then have
\[
{\bf E} (z, t) = {\bf E}_0 e^{i(k z - \omega t)}, \hspace{1cm}
{\bf B} (z,t) = {\bf B}_0 e^{i(k z - \omega t)}
@@ -1632,7 +1624,7 @@ Maxwell's equations impose constraints. Since \({\boldsymbol \nabla} \cdot {\bf
(E_0)_z = 0 = (B_0)_z
\label{Gr(9.44)}
\]
so {\bf electromagnetic waves are transverse}.
so <b>electromagnetic waves are transverse</b>.
</p>
<p>
@@ -1643,9 +1635,26 @@ From Faraday: \({\boldsymbol \nabla} \times {\bf E} = -\partial {\bf B}/\partia
\label{Gr(9.46)}
\]
so \({\bf E}\) and \({\bf B}\) are mutually perpendicular, and
</p>
<div class="eqlabel" id="org4747373">
<p>
<a id="EBmpw"></a><a href="./emd_emw_mpw.html#EBmpw"><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="org2ea2f4c">
<ul class="org-ul">
<li>Gr (9.47)</li>
</ul>
</div>
</div>
<p>
\[
B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0.
\label{Gr(9.47)}
\tag{EBmpw}\label{EBmpw}
\]
</p>
@@ -1653,19 +1662,37 @@ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0.
Generalizing to propagation in the direction of an arbitrary wavevector
\({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
</p>
<div class="core div" id="org53e84bf">
<div class="core div" id="orga666428">
<p>
<b>E and B fields for a monochromatic EM plane wave</b>
</p>
<div class="eqlabel" id="orge12acff">
<p>
<a id="mpw"></a><a href="./emd_emw_mpw.html#mpw"><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="orgd889027">
<ul class="org-ul">
<li>Gr (9.49)</li>
</ul>
</div>
</div>
<p>
{\bf E and B fields for a monochromatic EM plane wave}
\[
{\boldsymbol E} ({\boldsymbol r},t ) = E_0 e^{i ({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol n},
\hspace{10mm}
{\boldsymbol B} ({\boldsymbol r}, t) = \frac{E_0}{c} e^{i({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol k} \times \hat{\boldsymbol n}
= \frac{1}{c} ~\hat{\boldsymbol k} \times {\boldsymbol E} ({\boldsymbol r}, t)
\]
with the transversality condition
{\boldsymbol E} ({\boldsymbol r},t ) = E_0 e^{i ({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol n},
\hspace{10mm}
{\boldsymbol B} ({\boldsymbol r}, t) = \frac{E_0}{c} e^{i({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol k} \times \hat{\boldsymbol n}
= \frac{1}{c} ~\hat{\boldsymbol k} \times {\boldsymbol E} ({\boldsymbol r}, t)
\tag{mpw}\label{mpw}
\]
with the <b>transversality condition</b>
\[
\hat{\boldsymbol k} \cdot \hat{\boldsymbol n} = 0
\]
\hat{\boldsymbol k} \cdot \hat{\boldsymbol n} = 0
\]
</p>
</div>
@@ -1697,7 +1724,7 @@ 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>