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Update 2022-02-09 07:44

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Jean-Sébastien
2022-02-09 07:44:58 +01:00
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-08 Tue 17:21 -->
<!-- 2022-02-09 Wed 07:31 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1616,7 +1616,7 @@ Table of contents
</ul>
</details>
</nav>
<ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_Me.html">Maxwell's Equations in Matter&emsp;<small>[emdm.Me]</small></a></li><li>Next:&nbsp;<a href="emdm_Me_bc.html">Boundary Conditions&emsp;<small>[emdm.Me.bc]</small></a></li><li>Up:&nbsp;<a href="emdm_Me.html">Maxwell's Equations in Matter&emsp;<small>[emdm.Me]</small></a></li></ul><div id="outline-container-emdm_Me_Mem" class="outline-4">
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_Me.html">Maxwell's Equations in Matter</a></li><li>Maxwell's Equations in Matter</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_Me.html">Maxwell's Equations in Matter&emsp;<small>[emdm.Me]</small></a></li><li>Next:&nbsp;<a href="emdm_Me_bc.html">Boundary Conditions&emsp;<small>[emdm.Me.bc]</small></a></li><li>Up:&nbsp;<a href="emdm_Me.html">Maxwell's Equations in Matter&emsp;<small>[emdm.Me]</small></a></li></ul><div id="outline-container-emdm_Me_Mem" class="outline-4">
<h4 id="emdm_Me_Mem">Maxwell's Equations in Matter<a class="headline-permalink" href="./emdm_Me_Mem.html#emdm_Me_Mem"><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"/>
@@ -1658,7 +1658,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
\]
We therefore have the
</p>
<div class="core div" id="orgff46cb4">
<div class="core div" id="org15d7510">
<p>
{\bf Polarization current density}
\[
@@ -1676,7 +1676,7 @@ the polarization current is the result of linear motion of charge when
polarization changes). We can check consistency with the continuity equation
associated to the conservation of bound charges:
</p>
<aside id="orgbce7d73">
<aside id="org9c1ef52">
<p>
Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current
\({\boldsymbol J}_b\) but this is not the convention used here.
@@ -1699,7 +1699,7 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
In view of this: total charge density can be separated into 2 parts,
{\it free} and {\it bound}:
</p>
<div class="main div" id="org4df7865">
<div class="main div" id="org2c2e80a">
<p>
\[
\rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
@@ -1712,7 +1712,7 @@ In view of this: total charge density can be separated into 2 parts,
and current can be separated into three parts, {\it free}, {\it bound} and
{\it polarization}:
</p>
<div class="main div" id="orgcc32502">
<div class="main div" id="orgafd7f00">
<p>
\[
{\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
@@ -1736,7 +1736,7 @@ Gauss's law: can be rewritten
\]
where (as in static case)
</p>
<div class="core div" id="org7600f00">
<div class="core div" id="orgbd7306c">
<p>
\[
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
@@ -1762,7 +1762,7 @@ or
\]
where as before
</p>
<div class="core div" id="orgb942ad8">
<div class="core div" id="orgde049dc">
<p>
\[
{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
@@ -1780,7 +1780,7 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
<p>
In terms of free charges and currents, we thus get
</p>
<div class="core div" id="orgd9d30d5">
<div class="core div" id="orgf901bae">
<p>
{\bf Maxwell's equations {\it (in matter)}}
</p>
@@ -1806,7 +1806,7 @@ Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and
in terms of \({\bf E}\) and \({\bf B}\).
For the restricted case of linear media:
</p>
<div class="main div" id="orge1df28e">
<div class="main div" id="org5288a81">
<p>
\[
{\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
@@ -1828,10 +1828,21 @@ where \(\varepsilon \equiv \varepsilon_0(1 + \chi_e)\) and \(\mu \equiv \mu_0 (1
<hr><div id="postamble" class="status">
<hr>
<div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
target="_blank" class="m-2">
<img alt="Creative Commons License" style="border-width:0"
src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
</a>
Except where otherwise noted, all content is licensed under a
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
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: 2022-02-08 Tue 17:21</p>
<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
<p class="date">Created: 2022-02-09 Wed 07:31</p>
<p class="validation"></p>
</div>