Update 2022-03-24 08:43
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Jean-Sébastien
+21
-21
@@ -1,7 +1,7 @@
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-03-22 Tue 10:52 -->
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<!-- 2022-03-24 Thu 08:42 -->
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<title>Pre-Quantum Electrodynamics</title>
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@@ -1293,7 +1293,7 @@ Table of contents
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</summary>
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<ul>
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<li>
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<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
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<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
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</li>
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@@ -1648,18 +1648,18 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
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\]
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We therefore have the
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</p>
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<div class="core div" id="org64396a0">
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<div class="core div" id="orgc00904a">
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<p>
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<b>Polarization current density</b>
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</p>
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<div class="eqlabel" id="orgf37fc4f">
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<div class="eqlabel" id="org3742df8">
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<p>
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<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">
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<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"/>
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<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"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org12af1ad">
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<div class="alteqlabels" id="org6073635">
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<ul class="org-ul">
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<li>Gr (7.48)</li>
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</ul>
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@@ -1683,7 +1683,7 @@ the polarization current is the result of linear motion of charge when
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polarization changes). We can check consistency with the continuity equation
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associated to the conservation of bound charges:
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</p>
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<aside id="org0322793">
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<aside id="orgff2c75f">
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<p>
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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.
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</p>
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@@ -1705,15 +1705,15 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
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In view of this: total charge density can be separated into 2 parts,
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<b>free</b> and <b>bound</b>:
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</p>
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<div class="main div" id="org56027a6">
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<div class="eqlabel" id="org6fd2aed">
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<div class="main div" id="org7866a36">
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<div class="eqlabel" id="orge94a27e">
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<p>
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<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">
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<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"/>
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<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"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org9e03f13">
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<div class="alteqlabels" id="orgbb91da6">
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<ul class="org-ul">
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<li>Gr (7.49)</li>
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</ul>
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@@ -1733,15 +1733,15 @@ In view of this: total charge density can be separated into 2 parts,
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and current can be separated into three parts, <b>free</b>, <b>bound</b> and
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<b>polarization</b>:
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</p>
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<div class="main div" id="org96443a3">
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<div class="eqlabel" id="orga6b8e4a">
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<div class="main div" id="org765eeee">
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<div class="eqlabel" id="org49803b2">
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<p>
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<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">
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<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"/>
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<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"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="orgddcfaf9">
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<div class="alteqlabels" id="orgc87ffc7">
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<ul class="org-ul">
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<li>Gr (7.50)</li>
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</ul>
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@@ -1767,7 +1767,7 @@ Gauss's law: can be rewritten
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\]
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where (as in static case)
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</p>
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<div class="core div" id="orgdbd92f1">
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<div class="core div" id="org91a1e38">
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<p>
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\[
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{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
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@@ -1788,7 +1788,7 @@ or
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\]
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where as before
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</p>
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<div class="core div" id="org7839f4c">
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<div class="core div" id="org0c2e84f">
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<p>
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\[
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{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
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@@ -1806,18 +1806,18 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
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<p>
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In terms of free charges and currents, we thus get
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</p>
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<div class="core div" id="org6597738">
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<div class="core div" id="orgeeb0b35">
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<p>
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<b>Maxwell's equations</b> <i>(in matter)</i>
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</p>
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<div class="eqlabel" id="orgffcbbe9">
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<div class="eqlabel" id="orgb9e6675">
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<p>
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<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">
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<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"/>
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<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"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org5c6c62d">
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<div class="alteqlabels" id="org6367cef">
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<ul class="org-ul">
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<li>Gr (7.55)</li>
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</ul>
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@@ -1847,15 +1847,15 @@ This must all be complemented by the <b>constitutive relations</b> giving \({\bf
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in terms of \({\bf E}\) and \({\bf B}\).
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For the restricted case of linear media:
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</p>
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<div class="main div" id="org2d4f8dd">
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<div class="eqlabel" id="org03c06a4">
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<div class="main div" id="orgd78e0b5">
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<div class="eqlabel" id="org86e1a29">
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<p>
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<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">
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<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"/>
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<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"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org45b4fff">
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<div class="alteqlabels" id="org7ea2c8c">
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<ul class="org-ul">
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<li>Gr (7.56,7.57)</li>
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</ul>
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@@ -1898,7 +1898,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-03-22 Tue 10:52</p>
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<p class="date">Created: 2022-03-24 Thu 08:42</p>
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<p class="validation"></p>
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</div>
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