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Update 2022-03-15 10:07

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Jean-Sébastien
2022-03-15 10:07:27 +01:00
parent 4808df71e6
commit 55f0de8197
193 changed files with 2416 additions and 2082 deletions
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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>
@@ -1663,8 +1663,8 @@ density, we get
Going back to our setup with plates in the \(xz\) plane which we started from,
in the moving frame, there is now a magnetic field due to surface currents:
\[
{\boldsymbol K}_{\mbox{\tiny top}} = \sigma v_0 \hat{\boldsymbol x}
= -{\boldsymbol K}_{\mbox{\tiny bot}}.
{\boldsymbol K}_{\mbox{top}} = \sigma v_0 \hat{\boldsymbol x}
= -{\boldsymbol K}_{\mbox{bot}}.
\]
This magnetic field between the plates is thus
\[
@@ -1736,10 +1736,22 @@ These factors cancel so \(\bar{B}_x = B_x\).
<p>
We thus obtain the
</p>
<div class="core div" id="orgb2cea03">
<div class="core div" id="org24ecb59">
<p>
{\bf EM field transformation laws (motion along \(x\) with velocity \(v\))}
<b>EM field transformation laws</b> <i>(motion along \(x\) with velocity \(v\))</i>
</p>
<div class="eqlabel" id="orgd271f80">
<p>
<a id="EMtr"></a><a href="./red_rem_Ltf.html#EMtr"><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="orgec94183">
</div>
</div>
\begin{align}
\bar{E}_x &amp;= E_x, \hspace{10mm} &amp;
\bar{E}_y &amp;= \gamma (E_y - v B_z), \hspace{10mm} &amp;
@@ -1747,7 +1759,7 @@ We thus obtain the
\bar{B}_x &amp;= B_x, &amp;
\bar{B}_y &amp;= \gamma \left( B_y + \frac{v}{c^2} E_z \right), &amp;
\bar{B}_z &amp;= \gamma \left( B_z - \frac{v}{c^2} E_y \right)
\label{eq:EMFieldsLorentzTransfo}
\tag{EMtr}\label{EMtr}
\end{align}
</div>
@@ -1757,7 +1769,7 @@ Two special cases can be mentioned:
</p>
<p>
\paragraph{If \({\boldsymbol B} = 0\) in \({\cal S}\):}
<b>If</b> \({\boldsymbol B} = 0\) <b>in</b> \({\cal S}\):
Then, \(\bar{\boldsymbol B} = \gamma \frac{v}{c^2} (E_z \hat{\boldsymbol y} - E_y \hat{\boldsymbol z}) = \frac{v}{c^2} (\bar{E}_z \hat{\boldsymbol y} - \bar{E}_y \hat{\boldsymbol z})\) so
\[
\bar{\boldsymbol B} = -\frac{1}{c^2} {\boldsymbol v} \times \bar{\boldsymbol E}.
@@ -1765,7 +1777,7 @@ Then, \(\bar{\boldsymbol B} = \gamma \frac{v}{c^2} (E_z \hat{\boldsymbol y} - E_
</p>
<p>
\paragraph{If \({\boldsymbol E} = 0\) in \({\cal S}\):}
<b>If</b> \({\boldsymbol E} = 0\) <b>in</b> \({\cal S}\):
Then, \(\hat{\boldsymbol E} = -\gamma v (B_z \hat{\boldsymbol y} - B_y \hat{\boldsymbol z}) = -v (\bar{B}_z \hat{\boldsymbol y} - \bar{B}_y \hat{\boldsymbol z})\)
so
\[
@@ -1793,7 +1805,7 @@ 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>