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Update 2022-03-07 20:40

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Jean-Sébastien
2022-03-07 20:40:36 +01:00
parent 21bf9fdba5
commit 4808df71e6
194 changed files with 1487 additions and 5980 deletions
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build/emd_Fl_ief.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>
@@ -1645,7 +1637,7 @@ law in integral form:
<div class="example div" id="org045463b">
<div class="example div" id="org56facb7">
<p>
<b>Example: loop with time-dependent flux</b>
</p>
@@ -1661,10 +1653,10 @@ through a horizontal circular region of radius \(R\).
<p>
<b>Solution</b>:
amperian loop of radius \(s\), apply Faraday:
amperian loop of radius \(r\), apply Faraday:
\[
\oint {\bf E} \cdot d{\bf l} = E (2\pi s) = -\frac{d\Phi}{dt} = -\pi s^2 \frac{dB}{dt}
\Rightarrow {\bf E} = -\frac{s}{2} \frac{dB}{dt} \hat{\boldsymbol \varphi}.
\oint {\bf E} \cdot d{\bf l} = E (2\pi r) = -\frac{d\Phi}{dt} = -\pi r^2 \frac{dB}{dt}
\Rightarrow {\bf E} = -\frac{r}{2} \frac{dB}{dt} \hat{\boldsymbol \varphi}.
\]
Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
</p>
@@ -1672,7 +1664,7 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
</div>
<div class="example div" id="org047ea10">
<div class="example div" id="org367fcfc">
<p>
<b>Example: wheel with charged rim traversed by flux</b>
</p>
@@ -1713,7 +1705,7 @@ called the <b>quasistatic</b> approximation, and works provided we deal with
<i>slow enough</i> phenomena.
</p>
<div class="example div" id="org733cbdd">
<div class="example div" id="orgb2a276c">
<p>
<b>Example: field from wire with time-dependent current</b>
</p>
@@ -1723,31 +1715,31 @@ Consider an infinitely long straight wire which carries current \(I(t)\).
</p>
<p>
<b>Task</b>: find the induced \({\bf E}\) field as a function of distance \(s\) from wire.
<b>Task</b>: find the induced \({\bf E}\) field as a function of distance \(r\) from wire.
</p>
<p>
<b>Solution</b>: assuming we can use the quasistatic approximation, the
magnetic field is \(B = \frac{\mu_0 I}{2\pi s}\)
magnetic field is \(B = \frac{\mu_0 I}{2\pi r}\)
and circles the wire. Like \({\bf B}\) field of solenoid, \({\bf E}\) runs parallel
to wire. Amperian loop with sides at distances \(s_0\) and \(s\):
to wire. Amperian loop with sides at distances \(r_0\) and \(r\):
\[
\oint {\bf E} \cdot d{\bf l} = E(s_0)l - E(s)l = -\frac{d}{dt} \int {\bf B} \cdot d{\bf a}
= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \int_{s_0}^s \frac{ds'}{s'}
= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \ln(s/s_0).
\oint {\bf E} \cdot d{\bf l} = E(r_0)l - E(r)l = -\frac{d}{dt} \int {\bf B} \cdot d{\bf a}
= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \int_{r_0}^s \frac{dr'}{r'}
= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \ln(r/r_0).
\]
So:
\[
{\bf E} (s) = \left[ \frac{\mu_0}{2\pi} \frac{dI}{dt} \ln s + K \right] \hat{\bf x}
{\bf E} (r) = \left[ \frac{\mu_0}{2\pi} \frac{dI}{dt} \ln r + K \right] \hat{\bf x}
\label{Gr(7.19)}
\]
where \(K\) is a constant (depends on the history of \(I(t)\)).
</p>
<p>
<b>N.B.</b>: this can't be true always, since it blows up as \(s \rightarrow \infty\).
<b>N.B.</b>: this can't be true always, since it blows up as \(r \rightarrow \infty\).
Reason: in this case, we've overstepped the quasistatic limit. We need
\(s \ll c\tau\) where \(\tau\) is a typical time scale for change of \(I(t)\).
\(r \ll c\tau\) where \(\tau\) is a typical time scale for change of \(I(t)\).
</p>
</div>
@@ -1772,7 +1764,7 @@ 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>