Update 2022-02-09 22:41
This commit is contained in:
Jean-Sébastien
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-36
@@ -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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<!-- 2022-02-09 Wed 07:31 -->
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<!-- 2022-02-09 Wed 22:40 -->
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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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@@ -408,17 +408,13 @@ Table of contents
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<li>
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<a href="./ems_es_ep_fp.html#ems_es_ep_fp">Field in terms of the potential</a><span class="headline-id">ems.es.ep.fp</span>
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</li>
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<li>
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<a href="./ems_es_ep_c.html#ems_es_ep_c">Comments on the Electrostatic Potential</a><span class="headline-id">ems.es.ep.c</span>
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</li>
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<li>
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<a href="./ems_es_ep_ex.html#ems_es_ep_ex">Example calculations for the potential</a><span class="headline-id">ems.es.ep.ex</span>
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</li>
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<li>
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<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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<a href="./ems_es_ep_PL.html#ems_es_ep_PL">Poisson's and Laplace's Equations</a><span class="headline-id">ems.es.ep.PL</span>
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</li>
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<li>
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@@ -430,29 +426,8 @@ Table of contents
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</details>
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</li>
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<li>
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<details>
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<summary>
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<a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>
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</summary>
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<ul>
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<li>
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<a href="./ems_es_e_pcd.html#ems_es_e_pcd">The Energy of a Point Charge Distribution</a><span class="headline-id">ems.es.e.pcd</span>
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</li>
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<li>
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<a href="./ems_es_e_ccd.html#ems_es_e_ccd">The Energy of a Continuous Charge Distribution</a><span class="headline-id">ems.es.e.ccd</span>
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</li>
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<li>
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<a href="./ems_es_e_c.html#ems_es_e_c">Comments on Electrostatic Energy</a><span class="headline-id">ems.es.e.c</span>
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</li>
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</ul>
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</details>
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</li>
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<li>
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@@ -1658,7 +1633,7 @@ 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="org15d7510">
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<div class="core div" id="org643b4aa">
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<p>
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{\bf Polarization current density}
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\[
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@@ -1676,7 +1651,7 @@ 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="org9c1ef52">
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<aside id="orgf5df19a">
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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
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\({\boldsymbol J}_b\) but this is not the convention used here.
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@@ -1699,7 +1674,7 @@ 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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{\it free} and {\it bound}:
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</p>
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<div class="main div" id="org2c2e80a">
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<div class="main div" id="orgfd1ad39">
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<p>
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\[
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\rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
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@@ -1712,7 +1687,7 @@ 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, {\it free}, {\it bound} and
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{\it polarization}:
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</p>
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<div class="main div" id="orgafd7f00">
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<div class="main div" id="org09b6579">
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<p>
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\[
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{\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
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@@ -1736,7 +1711,7 @@ Gauss's law: can be rewritten
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\]
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where (as in static case)
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</p>
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<div class="core div" id="orgbd7306c">
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<div class="core div" id="orgcb69874">
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<p>
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\[
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{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
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@@ -1762,7 +1737,7 @@ or
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\]
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where as before
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</p>
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<div class="core div" id="orgde049dc">
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<div class="core div" id="orga76243f">
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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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@@ -1780,7 +1755,7 @@ 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="orgf901bae">
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<div class="core div" id="org5d9c2e0">
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<p>
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{\bf Maxwell's equations {\it (in matter)}}
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</p>
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@@ -1806,7 +1781,7 @@ Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and
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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="org5288a81">
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<div class="main div" id="orgca90a26">
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<p>
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\[
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{\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
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@@ -1841,7 +1816,7 @@ 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: 2022-02-09 Wed 07:31</p>
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<p class="date">Created: 2022-02-09 Wed 22:40</p>
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<p class="validation"></p>
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</div>
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