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Update 2022-02-14 20:42

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Jean-Sébastien
2022-02-14 20:42:37 +01:00
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build/emsm_msm_fmo_bc.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-13 Sun 21:20 -->
<!-- 2022-02-14 Mon 20:35 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1608,7 +1608,7 @@ For a single dipole: refer to \ref{Gr(5.83)} (vector potential of single dipole
For a chunk of material with local magnetization \({\bf M} ({\bf r})\),
by the principle of superposition we thus have:
</p>
<div class="main div" id="org86bd6f7">
<div class="main div" id="org362b749">
<p>
\[
{\bf A} ({\bf r}) = \frac{\mu_0}{4\pi} \int_{\cal V} d\tau' ~\frac{{\bf M} ({\bf r}') \times ({\bf r} - {\bf r}')}{|{\bf r} - {\bf r}'|^3}
@@ -1650,7 +1650,7 @@ Problem 1.61 b) (p.56): leads to
\]
Reinterpretation: first term: potential from volume current,
</p>
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<div class="main div" id="orgfe22f72">
<p>
\[
{\bf J}_b = {\boldsymbol \nabla} \times {\bf M}
@@ -1662,7 +1662,7 @@ Reinterpretation: first term: potential from volume current,
<p>
second term: potential from surface current,
</p>
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<div class="main div" id="org2d7aeed">
<p>
\[
{\bf K}_b = {\bf M} \times \hat{\bf n}
@@ -1674,7 +1674,7 @@ second term: potential from surface current,
<p>
With these definitions,
</p>
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<div class="main div" id="org19936af">
<p>
\[
{\bf A} ({\bf r}) = \frac{\mu_0}{4\pi} ∫_{\cal V} dτ' \frac{{\bf J}_b ({\bf r}')}{|{\bf r} - {\bf r}'|}
@@ -1695,18 +1695,18 @@ in the volume and surface of the material.
<div class="example div" id="org6d7ec15">
<div class="example div" id="org9fe198f">
<p>
\paragraph{Example 6.1:} find field of uniformly magnetized sphere.
\paragraph{Solution:} put z axis along \({\bf M}\).
\[
{\bf J}_b = {\boldsymbol \nabla} \times {\bf M} = 0,
\hspace{1cm}
{\bf K}_b = {\bf M} \times \hat{\bf n} = M \sin \theta \hat{\boldsymbol \phi}.
{\bf K}_b = {\bf M} \times \hat{\bf n} = M \sin \theta \hat{\boldsymbol \varphi}.
\]
Rotating spherical shell of uniform surface charge \(\sigma\): surface current density
\[
{\bf K} = \sigma {\bf v} = \sigma \omega R \sin \theta \hat{\boldsymbol \phi}.
{\bf K} = \sigma {\bf v} = \sigma \omega R \sin \theta \hat{\boldsymbol \varphi}.
\]
Same if \(\sigma R {\boldsymbol \omega} = {\bf M}\). Refer to Example 5.11,% ({\it not done in class !}),
\[
@@ -1741,7 +1741,7 @@ 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: 2022-02-13 Sun 21:20</p>
<p class="date">Created: 2022-02-14 Mon 20:35</p>
<p class="validation"></p>
</div>