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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>
@@ -1622,7 +1622,7 @@ implies the existence of magnetism, and vice-versa.
<p>
To illustrate this, we take a simple example
</p>
<aside id="org404a58c">
<aside id="orgbe19a87">
<p>
See Purcell \&amp; Morin, section 5.9.
</p>
@@ -1630,7 +1630,7 @@ See Purcell \&amp; Morin, section 5.9.
<p>
Consider a wire at rest in the lab frame. This wire carries a current made of a (positive) line charge density \(\lambda\) moving towards the right at velocity \(v\), and a (negative) line charge density \(-\lambda\) moving towards the left at velocity \(-v\). In the lab frame, the current is thus
\[
I_{\mbox{\tiny lab}} = 2 \lambda v
I_{\mbox{lab}} = 2 \lambda v
\]
Additionally, in the lab frame, there is a charge \(q\) situated
at a distance \(s\) from the cable and moving with a velocity \(u &lt; v\) parallel to the wire. In the lab frame, there is no electrical force between the wire and the charge, since the wire carries no net charge.
@@ -1668,7 +1668,7 @@ The dilation factors in the test particle's frame are thus
<p>
so the resultant line charge in the test frame is
\[
\lambda_{\mbox{\tiny test}} = \lambda_+ + \lambda_-
\lambda_{\mbox{test}} = \lambda_+ + \lambda_-
= \lambda_0 ( \gamma_+ - \gamma_-)
= -\frac{2\lambda u v}{c^2 \sqrt{1 - u^2/c^2}}.
\]
@@ -1680,16 +1680,16 @@ neutral in one reference frame, can appear to be charged in another.
In the frame of the test particle, there is an electric field
equal to that of a uniformly charged wire:
\[
E = \frac{\lambda_{\mbox{\tiny test}}}{2\pi \epsilon_0 s}
E = \frac{\lambda_{\mbox{test}}}{2\pi \epsilon_0 s}
\]
so the force (in the test frame) is
\[
F_{\mbox{\tiny test}} = q E = -\frac{\lambda v}{\pi \epsilon_0 c^2 s} \frac{q u}{\sqrt{1 - u^2/c^2}}.
F_{\mbox{test}} = q E = -\frac{\lambda v}{\pi \epsilon_0 c^2 s} \frac{q u}{\sqrt{1 - u^2/c^2}}.
\]
If there is a force on our test charge in this test frame, there must
also be one in the lab frame. Using equation (\ref{eq:ForceTransfoSimple}),
also be one in the lab frame. Using equation <a href="./red_rm_Mf.html#Ftr0">Ftr0</a>,
\[
F = \sqrt{1 - u^2/c^2} ~F_{\mbox{\tiny test}}
F = \sqrt{1 - u^2/c^2} ~F_{\mbox{test}}
= -\frac{\lambda v}{\pi \varepsilon_0 c^2} \frac{qu}{s}
\]
which upon recognizing \(c^2 = \frac{1}{\varepsilon_0 \mu_0}\)
@@ -1722,7 +1722,7 @@ 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>