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Jean-Sébastien
2022-03-15 10:07:27 +01:00
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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>
@@ -1616,13 +1616,12 @@ Table of contents
<p>
Maxwell's equations in vacuum:
</p>
\begin{align}
\begin{align*}
(i)~~ &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0}, \hspace{1cm} &amp;\mbox{Gauss}, \nonumber \\
(ii)~~ &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \hspace{1cm} &amp;\mbox{anonymous} \nonumber \\
(iii)~~ &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \hspace{1cm} &amp;\mbox{Faraday}, \nonumber \\
(iv)~~ &amp;{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, \hspace{1cm} &amp;\mbox{Ampère + Maxwell}.
\label{Gr(7.39)}
\end{align}
\end{align*}
<p>
are complete as they stand. In presence of matter: more convenient to write sources in terms of
free charges and currents.
@@ -1642,25 +1641,25 @@ and magnetization \({\bf M}\) produces bound current density <a href="./emsm_msm
One new feature in nonstatic case: change in electric polarization involves flow of bound charge
(call it \({\bf J}_p\)) which must be included in total current.
Consider a small chunk of polarized material.
Polarization induces charge density \(\sigma_b = P\) at one end and \(-\sigma_b\) at other. If \(P\)
Polarization induces charge density \(\sigma_b = P\) at one end and \(-\sigma_b\) at other (refer to <a href="./emsm_esm_po.html#sigmab">sigmab</a>. If \(P\)
increases a bit, charge increases, giving net current
\[
dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partial t} da_{\perp}.
\]
We therefore have the
</p>
<div class="core div" id="orgf1835ae">
<div class="core div" id="orge0518b1">
<p>
<b>Polarization current density</b>
</p>
<div class="eqlabel" id="orgfad2da8">
<div class="eqlabel" id="org62db796">
<p>
<a id="Jp"></a><a href="./emdm_Me_Mem.html#Jp"><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="orgf15b4cb">
<div class="alteqlabels" id="orgebf1aab">
<ul class="org-ul">
<li>Gr (7.48)</li>
</ul>
@@ -1677,14 +1676,14 @@ We therefore have the
</div>
<p>
otherwise simply called the {\bf polarization current}.
otherwise simply called the <b>polarization current</b>.
This has nothing to do with the bound current \({\bf J}_b\)
(the latter is associated to magnetization;
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="orgadb90c7">
<aside id="orgfd82638">
<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 common convention.
</p>
@@ -1706,15 +1705,15 @@ 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,
<b>free</b> and <b>bound</b>:
</p>
<div class="main div" id="orgbe78924">
<div class="eqlabel" id="org4679dd8">
<div class="main div" id="org5bd748a">
<div class="eqlabel" id="org94dfd02">
<p>
<a id="rhofb"></a><a href="./emdm_Me_Mem.html#rhofb"><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="orgb7e7ed8">
<div class="alteqlabels" id="orgd2d4d1d">
<ul class="org-ul">
<li>Gr (7.49)</li>
</ul>
@@ -1734,15 +1733,15 @@ In view of this: total charge density can be separated into 2 parts,
and current can be separated into three parts, <b>free</b>, <b>bound</b> and
<b>polarization</b>:
</p>
<div class="main div" id="org1c506a7">
<div class="eqlabel" id="orgbf0c17d">
<div class="main div" id="org7ad8d3b">
<div class="eqlabel" id="org19a6c97">
<p>
<a id="Jfbp"></a><a href="./emdm_Me_Mem.html#Jfbp"><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="org33aea7e">
<div class="alteqlabels" id="orgac73457">
<ul class="org-ul">
<li>Gr (7.50)</li>
</ul>
@@ -1768,7 +1767,7 @@ Gauss's law: can be rewritten
\]
where (as in static case)
</p>
<div class="core div" id="org501f375">
<div class="core div" id="org65ac5cc">
<p>
\[
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
@@ -1789,7 +1788,7 @@ or
\]
where as before
</p>
<div class="core div" id="orgfaac9ca">
<div class="core div" id="org7e8e1b5">
<p>
\[
{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
@@ -1807,18 +1806,18 @@ 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="org4a1df55">
<div class="core div" id="org089bb13">
<p>
<b>Maxwell's equations</b> <i>(in matter)</i>
</p>
<div class="eqlabel" id="orga6eb31a">
<div class="eqlabel" id="orgfabaa36">
<p>
<a id="Max_mat"></a><a href="./emdm_Me_Mem.html#Max_mat"><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="org154a0ec">
<div class="alteqlabels" id="orgaba63f2">
<ul class="org-ul">
<li>Gr (7.55)</li>
</ul>
@@ -1848,15 +1847,15 @@ This must all be complemented by the <b>constitutive relations</b> giving \({\bf
in terms of \({\bf E}\) and \({\bf B}\).
For the restricted case of linear media:
</p>
<div class="main div" id="orge60b08f">
<div class="eqlabel" id="org88284f8">
<div class="main div" id="org435d93d">
<div class="eqlabel" id="orgb5db623">
<p>
<a id="consrel"></a><a href="./emdm_Me_Mem.html#consrel"><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="org21479eb">
<div class="alteqlabels" id="org6a8a0ba">
<ul class="org-ul">
<li>Gr (7.56,7.57)</li>
</ul>
@@ -1899,7 +1898,7 @@ 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>