Update 2022-03-15 10:07
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<title>Pre-Quantum Electrodynamics</title>
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</summary>
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<ul>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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</li>
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<li>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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<li>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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</li>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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@ -1637,7 +1637,7 @@ 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: 2022-03-07 Mon 20:38</p>
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<title>Pre-Quantum Electrodynamics</title>
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</summary>
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<ul>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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</li>
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<li>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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</li>
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<li>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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</li>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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</li>
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@ -1720,7 +1720,7 @@ 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: 2022-03-07 Mon 20:38</p>
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<p class="validation"></p>
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<meta charset="utf-8">
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<title>Pre-Quantum Electrodynamics</title>
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</summary>
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<ul>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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</li>
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<li>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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</li>
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@ -1352,6 +1348,10 @@ Table of contents
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<li>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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</li>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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</li>
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</ul>
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@ -1637,7 +1637,7 @@ 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: 2022-03-07 Mon 20:38</p>
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<p class="validation"></p>
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<meta charset="utf-8">
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<title>Pre-Quantum Electrodynamics</title>
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</summary>
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<ul>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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</li>
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<li>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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</li>
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@ -1352,6 +1348,10 @@ Table of contents
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<li>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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</li>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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</li>
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</ul>
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@ -1643,7 +1643,7 @@ 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: 2022-03-07 Mon 20:38</p>
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<p class="validation"></p>
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</div>
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<!DOCTYPE html>
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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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@ -1310,10 +1310,6 @@ Table of contents
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</summary>
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<ul>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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</li>
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<li>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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</li>
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@ -1352,6 +1348,10 @@ Table of contents
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<li>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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</li>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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</li>
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</ul>
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@ -1639,7 +1639,7 @@ 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: 2022-03-07 Mon 20:38</p>
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<p class="validation"></p>
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<!DOCTYPE html>
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<head>
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<!-- 2022-03-07 Mon 20:38 -->
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<!-- 2022-03-15 Tue 08:10 -->
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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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@ -1310,10 +1310,6 @@ Table of contents
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</summary>
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<ul>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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</li>
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||||
<li>
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||||
<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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</li>
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@ -1352,6 +1348,10 @@ Table of contents
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<li>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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</li>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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</li>
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@ -1659,14 +1659,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
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<div id="outline-container-c_m_cs_cyl_grad" class="outline-6">
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<h6 id="c_m_cs_cyl_grad"><a href="#c_m_cs_cyl_grad">Gradient</a></h6>
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<div class="outline-text-6" id="text-c_m_cs_cyl_grad">
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<div class="eqlabel" id="orgfd5fb6e">
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<div class="eqlabel" id="org9ded6d5">
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<p>
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<a id="cylgrad"></a><a href="./c_m_cs_cyl.html#cylgrad"><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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<div class="alteqlabels" id="org16d29dc">
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<div class="alteqlabels" id="org3066755">
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<ul class="org-ul">
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<li>Gr4(1.79)</li>
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<div id="outline-container-c_m_cs_cyl_div" class="outline-6">
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<h6 id="c_m_cs_cyl_div"><a href="#c_m_cs_cyl_div">Divergence</a></h6>
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<div class="outline-text-6" id="text-c_m_cs_cyl_div">
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<div class="eqlabel" id="org1dfbb35">
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<p>
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<a id="cyl_div"></a><a href="./c_m_cs_cyl.html#cyl_div"><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="org035dafc">
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<div class="alteqlabels" id="orge14058b">
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<ul class="org-ul">
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<li>Gr4(2.21)</li>
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</ul>
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<div id="outline-container-c_m_cs_cyl_curl" class="outline-6">
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<h6 id="c_m_cs_cyl_curl"><a href="#c_m_cs_cyl_curl">Curl</a></h6>
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<div class="outline-text-6" id="text-c_m_cs_cyl_curl">
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<div class="eqlabel" id="org747dbea">
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<div class="eqlabel" id="org34649e5">
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<p>
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<a id="cyl_curl"></a><a href="./c_m_cs_cyl.html#cyl_curl"><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="orgcd134fe">
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<div class="alteqlabels" id="orgeec607c">
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<ul class="org-ul">
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||||
<li>Gr4(2.21)</li>
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</ul>
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||||
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@ -1771,7 +1771,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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|||
</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>
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||||
<p class="validation"></p>
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</div>
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<!DOCTYPE html>
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<html lang="en">
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<head>
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||||
<!-- 2022-03-07 Mon 20:38 -->
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<!-- 2022-03-15 Tue 08:10 -->
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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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@ -1310,10 +1310,6 @@ Table of contents
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</summary>
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||||
<ul>
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||||
<li>
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||||
<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
|
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||||
</li>
|
||||
<li>
|
||||
<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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||||
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||||
</li>
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@ -1352,6 +1348,10 @@ Table of contents
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<li>
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||||
<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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||||
</li>
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||||
<li>
|
||||
<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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||||
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||||
</li>
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||||
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||||
</ul>
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@ -1632,7 +1632,7 @@ 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>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
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||||
<p class="date">Created: 2022-03-15 Tue 08:10</p>
|
||||
<p class="validation"></p>
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</div>
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-03-07 Mon 20:38 -->
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<!-- 2022-03-15 Tue 08:10 -->
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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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@ -1310,10 +1310,6 @@ Table of contents
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</summary>
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||||
<ul>
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||||
<li>
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||||
<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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||||
</li>
|
||||
<li>
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||||
<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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||||
</li>
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||||
|
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@ -1352,6 +1348,10 @@ Table of contents
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|||
<li>
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||||
<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
|
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||||
</li>
|
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<li>
|
||||
<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
|
||||
|
||||
</li>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1623,14 +1623,14 @@ which \(r\) is the distance from the chosen origin,
|
|||
The usual Cartesian coordinates relate to spherical coordinates
|
||||
according to
|
||||
</p>
|
||||
<div class="eqlabel" id="org348038f">
|
||||
<div class="eqlabel" id="org309b2b3">
|
||||
<p>
|
||||
<a id="sph_xyz"></a><a href="./c_m_cs_sph.html#sph_xyz"><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="org5f5a1a3">
|
||||
<div class="alteqlabels" id="org9ad4ca2">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1655,14 +1655,14 @@ A generic vector can be expressed as
|
|||
where the explicit relation between spherical and
|
||||
Cartesian unit vectors is
|
||||
</p>
|
||||
<div class="eqlabel" id="orgbdd14bc">
|
||||
<div class="eqlabel" id="orgbf81903">
|
||||
<p>
|
||||
<a id="sph_uv"></a><a href="./c_m_cs_sph.html#sph_uv"><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="orga250579">
|
||||
<div class="alteqlabels" id="org437eb02">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1685,14 +1685,14 @@ and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).
|
|||
<p>
|
||||
An infinitesimal displacement \(d{\bf l}\) can be written as
|
||||
</p>
|
||||
<div class="eqlabel" id="org6494f4b">
|
||||
<div class="eqlabel" id="orgda909cf">
|
||||
<p>
|
||||
<a id="sph_dl"></a><a href="./c_m_cs_sph.html#sph_dl"><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="orgaafdc77">
|
||||
<div class="alteqlabels" id="org02f05bf">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1708,14 +1708,14 @@ d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin
|
|||
<p>
|
||||
Infinitesimal volume element:
|
||||
</p>
|
||||
<div class="eqlabel" id="orga40a4ab">
|
||||
<div class="eqlabel" id="orgd2cfc4a">
|
||||
<p>
|
||||
<a id="sph_dtau"></a><a href="./c_m_cs_sph.html#sph_dtau"><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="orgc7eb689">
|
||||
<div class="alteqlabels" id="org0a4c199">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1736,14 +1736,14 @@ Infinitesimal surface element: depends on situation.
|
|||
<div id="outline-container-c_m_cs_sph_grad" class="outline-6">
|
||||
<h6 id="c_m_cs_sph_grad"><a href="#c_m_cs_sph_grad">Gradient</a></h6>
|
||||
<div class="outline-text-6" id="text-c_m_cs_sph_grad">
|
||||
<div class="eqlabel" id="orgb45a655">
|
||||
<div class="eqlabel" id="orge44fa24">
|
||||
<p>
|
||||
<a id="sph_grad"></a><a href="./c_m_cs_sph.html#sph_grad"><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="org5dfccd6">
|
||||
<div class="alteqlabels" id="org6b20d40">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1760,14 +1760,14 @@ Infinitesimal surface element: depends on situation.
|
|||
<div id="outline-container-c_m_cs_sph_div" class="outline-6">
|
||||
<h6 id="c_m_cs_sph_div"><a href="#c_m_cs_sph_div">Divergence</a></h6>
|
||||
<div class="outline-text-6" id="text-c_m_cs_sph_div">
|
||||
<div class="eqlabel" id="orgdca9c80">
|
||||
<div class="eqlabel" id="org3d72303">
|
||||
<p>
|
||||
<a id="sph_div"></a><a href="./c_m_cs_sph.html#sph_div"><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="org52febbe">
|
||||
<div class="alteqlabels" id="org9b1dfae">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1784,14 +1784,14 @@ Infinitesimal surface element: depends on situation.
|
|||
<div id="outline-container-c_m_cs_sph_curl" class="outline-6">
|
||||
<h6 id="c_m_cs_sph_curl"><a href="#c_m_cs_sph_curl">Curl</a></h6>
|
||||
<div class="outline-text-6" id="text-c_m_cs_sph_curl">
|
||||
<div class="eqlabel" id="org26cc7dd">
|
||||
<div class="eqlabel" id="orge960708">
|
||||
<p>
|
||||
<a id="sph_curl"></a><a href="./c_m_cs_sph.html#sph_curl"><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="orgeafd92e">
|
||||
<div class="alteqlabels" id="org0f49079">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1810,14 +1810,14 @@ Infinitesimal surface element: depends on situation.
|
|||
<div id="outline-container-c_m_cs_sph_lap" class="outline-6">
|
||||
<h6 id="c_m_cs_sph_lap"><a href="#c_m_cs_sph_lap">Laplacian</a></h6>
|
||||
<div class="outline-text-6" id="text-c_m_cs_sph_lap">
|
||||
<div class="eqlabel" id="org9c1e636">
|
||||
<div class="eqlabel" id="org1d2bef1">
|
||||
<p>
|
||||
<a id="sph_Lap"></a><a href="./c_m_cs_sph.html#sph_Lap"><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="org83d937d">
|
||||
<div class="alteqlabels" id="org6ba97dc">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1851,7 +1851,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1642,7 +1642,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1644,7 +1644,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1615,9 +1615,9 @@ Table of contents
|
|||
<div class="outline-text-5" id="text-c_m_dc_d2">
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc12bd11" class="outline-6">
|
||||
<h6 id="orgc12bd11"><a href="#orgc12bd11">Divergence of gradient</a></h6>
|
||||
<div class="outline-text-6" id="text-orgc12bd11">
|
||||
<div id="outline-container-orgc162c65" class="outline-6">
|
||||
<h6 id="orgc162c65"><a href="#orgc162c65">Divergence of gradient</a></h6>
|
||||
<div class="outline-text-6" id="text-orgc162c65">
|
||||
<p>
|
||||
\({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
|
||||
The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
|
||||
|
|
@ -1626,44 +1626,44 @@ given by the Laplacian of the corresponding vector elements.
|
|||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0ea7ae4" class="outline-6">
|
||||
<h6 id="org0ea7ae4"><a href="#org0ea7ae4">Curl of a gradient</a></h6>
|
||||
<div class="outline-text-6" id="text-org0ea7ae4">
|
||||
<div id="outline-container-org2f7ef92" class="outline-6">
|
||||
<h6 id="org2f7ef92"><a href="#org2f7ef92">Curl of a gradient</a></h6>
|
||||
<div class="outline-text-6" id="text-org2f7ef92">
|
||||
<p>
|
||||
This always vanishes.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org78d85e1" class="outline-6">
|
||||
<h6 id="org78d85e1"><a href="#org78d85e1">Gradient of the divergence</a></h6>
|
||||
<div class="outline-text-6" id="text-org78d85e1">
|
||||
<div id="outline-container-orgceccef1" class="outline-6">
|
||||
<h6 id="orgceccef1"><a href="#orgceccef1">Gradient of the divergence</a></h6>
|
||||
<div class="outline-text-6" id="text-orgceccef1">
|
||||
<p>
|
||||
\({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge7f86bf" class="outline-6">
|
||||
<h6 id="orge7f86bf"><a href="#orge7f86bf">Divergence of a curl</a></h6>
|
||||
<div class="outline-text-6" id="text-orge7f86bf">
|
||||
<div id="outline-container-orgc07e1dc" class="outline-6">
|
||||
<h6 id="orgc07e1dc"><a href="#orgc07e1dc">Divergence of a curl</a></h6>
|
||||
<div class="outline-text-6" id="text-orgc07e1dc">
|
||||
<p>
|
||||
This always vanishes.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbb8199b" class="outline-6">
|
||||
<h6 id="orgbb8199b"><a href="#orgbb8199b">Curl of curl</a></h6>
|
||||
<div class="outline-text-6" id="text-orgbb8199b">
|
||||
<div class="eqlabel" id="orga59fc73">
|
||||
<div id="outline-container-org1a67d52" class="outline-6">
|
||||
<h6 id="org1a67d52"><a href="#org1a67d52">Curl of curl</a></h6>
|
||||
<div class="outline-text-6" id="text-org1a67d52">
|
||||
<div class="eqlabel" id="org5d4ff40">
|
||||
<p>
|
||||
<a id="curlcurl"></a><a href="./c_m_dc_d2.html#curlcurl"><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="org384a104">
|
||||
<div class="alteqlabels" id="orgd2518b8">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1694,7 +1694,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1643,7 +1643,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1640,7 +1640,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1664,7 +1664,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1622,14 +1622,14 @@ explicited as follows:
|
|||
<p>
|
||||
<b>Gradient of a product</b>:
|
||||
</p>
|
||||
<div class="eqlabel" id="orged10256">
|
||||
<div class="eqlabel" id="org824336d">
|
||||
<p>
|
||||
<a id="grad_prod"></a><a href="./c_m_dc_pr.html#grad_prod"><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="orgfc8793a">
|
||||
<div class="alteqlabels" id="org2bf23af">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (3)</li>
|
||||
<li>W (1-111)</li>
|
||||
|
|
@ -1649,14 +1649,14 @@ explicited as follows:
|
|||
<p>
|
||||
<b>Gradient of a scalar product</b>:
|
||||
</p>
|
||||
<div class="eqlabel" id="org4b4bdf3">
|
||||
<div class="eqlabel" id="org41d1a68">
|
||||
<p>
|
||||
<a id="grad_sprod"></a><a href="./c_m_dc_pr.html#grad_sprod"><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="org02913cc">
|
||||
<div class="alteqlabels" id="org466aa7e">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (4)</li>
|
||||
<li>W (1-112)</li>
|
||||
|
|
@ -1676,14 +1676,14 @@ explicited as follows:
|
|||
<p>
|
||||
<b>Divergence of a product</b>:
|
||||
</p>
|
||||
<div class="eqlabel" id="org1df402a">
|
||||
<div class="eqlabel" id="org1ad95d1">
|
||||
<p>
|
||||
<a id="div_prod"></a><a href="./c_m_dc_pr.html#div_prod"><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="orgfa5050d">
|
||||
<div class="alteqlabels" id="org93785a5">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (5)</li>
|
||||
<li>W (1-115)</li>
|
||||
|
|
@ -1703,14 +1703,14 @@ explicited as follows:
|
|||
<p>
|
||||
<b>Divergence of a cross product</b>:
|
||||
</p>
|
||||
<div class="eqlabel" id="org5048e1e">
|
||||
<div class="eqlabel" id="org4333997">
|
||||
<p>
|
||||
<a id="div_xprod"></a><a href="./c_m_dc_pr.html#div_xprod"><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="orgd39dff6">
|
||||
<div class="alteqlabels" id="org2626fe4">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (6)</li>
|
||||
<li>W (1-116)</li>
|
||||
|
|
@ -1730,14 +1730,14 @@ explicited as follows:
|
|||
<p>
|
||||
<b>Curl of a product</b>:
|
||||
</p>
|
||||
<div class="eqlabel" id="orgbd31452">
|
||||
<div class="eqlabel" id="orgfa1c2f8">
|
||||
<p>
|
||||
<a id="curl_prod"></a><a href="./c_m_dc_pr.html#curl_prod"><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="org825aae4">
|
||||
<div class="alteqlabels" id="orgb82ff5c">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7)</li>
|
||||
<li>W (1-118)</li>
|
||||
|
|
@ -1757,14 +1757,14 @@ explicited as follows:
|
|||
<p>
|
||||
<b>Curl of a cross product</b>:
|
||||
</p>
|
||||
<div class="eqlabel" id="org9c6b6a4">
|
||||
<div class="eqlabel" id="org9d0ffb0">
|
||||
<p>
|
||||
<a id="curl_xprod"></a><a href="./c_m_dc_pr.html#curl_xprod"><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="orgb030476">
|
||||
<div class="alteqlabels" id="org6beb5fe">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (8)</li>
|
||||
<li>W (1-119)</li>
|
||||
|
|
@ -1805,7 +1805,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1639,7 +1639,7 @@ 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>
|
||||
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|
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|
|||
</summary>
|
||||
<ul>
|
||||
<li>
|
||||
<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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|
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</li>
|
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<li>
|
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
|
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|
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</li>
|
||||
|
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@ -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>
|
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</li>
|
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</ul>
|
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|
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@ -1651,7 +1651,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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|
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<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>
|
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<p class="validation"></p>
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<!DOCTYPE html>
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@ -1310,10 +1310,6 @@ Table of contents
|
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</summary>
|
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<ul>
|
||||
<li>
|
||||
<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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</li>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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|
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@ -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>
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</li>
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<li>
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||||
<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
|
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|
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</li>
|
||||
|
||||
</ul>
|
||||
|
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@ -1636,14 +1636,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
|
|||
More generally,
|
||||
</p>
|
||||
|
||||
<div class="eqlabel" id="org492d0c0">
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<div class="eqlabel" id="org501b97b">
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<p>
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||||
<a id="divdel"></a><a href="./c_m_dd_3d.html#divdel"><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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|
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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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||||
<div class="alteqlabels" id="org9b8bed8">
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||||
<div class="alteqlabels" id="org2eebb69">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (1.100)</li>
|
||||
</ul>
|
||||
|
|
@ -1662,14 +1662,14 @@ More generally,
|
|||
Since
|
||||
</p>
|
||||
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||||
<div class="eqlabel" id="orga3b7a91">
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<p>
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<a id="div1or"></a><a href="./c_m_dd_3d.html#div1or"><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="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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||||
</p>
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||||
<div class="alteqlabels" id="org98ac626">
|
||||
<div class="alteqlabels" id="org17b7d37">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (1.101)</li>
|
||||
</ul>
|
||||
|
|
@ -1685,14 +1685,14 @@ Since
|
|||
<p>
|
||||
we have that
|
||||
</p>
|
||||
<div class="eqlabel" id="org596ba00">
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||||
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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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||||
</p>
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||||
<div class="alteqlabels" id="org620c397">
|
||||
<div class="alteqlabels" id="orgcc41bc0">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (1.102)</li>
|
||||
</ul>
|
||||
|
|
@ -1724,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
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@ -1310,10 +1310,6 @@ Table of contents
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</summary>
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<ul>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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@ -1352,6 +1348,10 @@ Table of contents
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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@ -1657,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
|
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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@ -1310,10 +1310,6 @@ Table of contents
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</summary>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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@ -1352,6 +1348,10 @@ Table of contents
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|||
<li>
|
||||
<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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@ -1642,7 +1642,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
|
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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@ -1310,10 +1310,6 @@ Table of contents
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</summary>
|
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<ul>
|
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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@ -1352,6 +1348,10 @@ Table of contents
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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@ -1310,10 +1310,6 @@ Table of contents
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</summary>
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<ul>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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@ -1352,6 +1348,10 @@ Table of contents
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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@ -1649,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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@ -1310,10 +1310,6 @@ Table of contents
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</summary>
|
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<ul>
|
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<li>
|
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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@ -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>
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@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
|
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
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</summary>
|
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<ul>
|
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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@ -1352,6 +1348,10 @@ Table of contents
|
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<li>
|
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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@ -1650,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
|
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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<div id="outline-container-org7fab499" class="outline-6">
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<h6 id="org7fab499"><a href="#org7fab499">Line Integrals</a></h6>
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<div class="outline-text-6" id="text-org7fab499">
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<div id="outline-container-org6c52cc6" class="outline-6">
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<h6 id="org6c52cc6"><a href="#org6c52cc6">Line Integrals</a></h6>
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<div class="outline-text-6" id="text-org6c52cc6">
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<p>
|
||||
\[
|
||||
{\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l}
|
||||
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@ -1646,9 +1646,9 @@ Integral over a closed loop:
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<div id="outline-container-org543b5ed" class="outline-6">
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<h6 id="org543b5ed"><a href="#org543b5ed">Surface Integrals</a></h6>
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<div class="outline-text-6" id="text-org543b5ed">
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<div id="outline-container-org7c42ba4" class="outline-6">
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<h6 id="org7c42ba4"><a href="#org7c42ba4">Surface Integrals</a></h6>
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<div class="outline-text-6" id="text-org7c42ba4">
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|
||||
\[
|
||||
\int_{\cal S} {\bf v} \cdot d{\bf a}
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|
||||
<div id="outline-container-org015d5b1" class="outline-6">
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||||
<h6 id="org015d5b1"><a href="#org015d5b1">Volume Integrals</a></h6>
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<div class="outline-text-6" id="text-org015d5b1">
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<div id="outline-container-org25bec25" class="outline-6">
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<h6 id="org25bec25"><a href="#org25bec25">Volume Integrals</a></h6>
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<div class="outline-text-6" id="text-org25bec25">
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\[
|
||||
\int_{\cal V} T d\tau
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|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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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><span class="headline-id">c.m.ic.stokes</span></h5>
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@ -1665,7 +1665,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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@ -1652,7 +1652,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
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</summary>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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@ -1652,7 +1652,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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|
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|
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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@ -1641,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
|
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</summary>
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|
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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|
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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<p class="author">Author: Jean-Sébastien Caux</p>
|
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<p class="author">Author: Jean-Sébastien Caux</p>
|
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<p class="author">Author: Jean-Sébastien Caux</p>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<p class="author">Author: Jean-Sébastien Caux</p>
|
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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<p class="author">Author: Jean-Sébastien Caux</p>
|
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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<p class="author">Author: Jean-Sébastien Caux</p>
|
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<li><a href="d_m.html">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span></li>
|
||||
<li><a href="d_ems.html">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span></li>
|
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<li><a href="d_ems_ca.html">Diagnostics: Calculating or Approximating the Electostatic Potential</a><span class="headline-id">d.ems.ca</span></li>
|
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<li><a href="d_emsm.html">Diagnostics: Electromagnetostatics in Matter</a><span class="headline-id">d.emsm</span></li>
|
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@ -1662,9 +1661,10 @@ I have little doubt you'll pass the course.
|
|||
<li><a href="d_emd_emw.html">Diagnostics: Electromagnetic Waves</a><span class="headline-id">d.emd.emw</span></li>
|
||||
<li><a href="d_emf.html">Diagnostics: Potentials, Gauges and Fields</a><span class="headline-id">d.emf</span></li>
|
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<li><a href="d_red.html">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span></li>
|
||||
<li><a href="d_m.html">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span></li>
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<br><ul class="navigation-links"><li>Next: <a href="d_m.html">Diagnostics: Mathematical Preliminaries <small>[d.m]</small></a></li></ul>
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<br><ul class="navigation-links"><li>Next: <a href="d_ems.html">Diagnostics: Electromagnetostatics <small>[d.ems]</small></a></li></ul>
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|
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<p class="author">Author: Jean-Sébastien Caux</p>
|
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<ul>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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|
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|
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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|
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|
|
@ -1352,6 +1348,10 @@ Table of contents
|
|||
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|
||||
<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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@ -1647,12 +1647,7 @@ Things you should be able to do (ideally: from scratch, on a blank sheet of pape
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@ -1667,7 +1662,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
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@ -1,7 +1,7 @@
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@ -1310,10 +1310,6 @@ Table of contents
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</summary>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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@ -1352,6 +1348,10 @@ Table of contents
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<li class="toc-currentpage">
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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@ -1607,7 +1607,7 @@ Table of contents
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<h3 id="d_red">Diagnostics: Relativistic Electrodynamics<a class="headline-permalink" href="./d_red.html#d_red"><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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|
||||
|
|
@ -1646,7 +1646,9 @@ As a strict minimum, you should be able to:
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</div>
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|
||||
<br><ul class="navigation-links"><li>Prev: <a href="d_emf.html">Diagnostics: Potentials, Gauges and Fields <small>[d.emf]</small></a></li><li>Next: <a href="a.html">Appendices <small>[a]</small></a></li><li>Up: <a href="d.html">Diagnostics <small>[d]</small></a></li></ul>
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<br><ul class="navigation-links"><li>Prev: <a href="d_emf.html">Diagnostics: Potentials, Gauges and Fields <small>[d.emf]</small></a></li><li>Next: <a href="d_m.html">Diagnostics: Compendium - Mathematics <small>[d.m]</small></a></li><li>Up: <a href="d.html">Diagnostics <small>[d]</small></a></li></ul>
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@ -1661,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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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: 2022-03-07 Mon 20:38</p>
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
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@ -1,7 +1,7 @@
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@ -1310,10 +1310,6 @@ Table of contents
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</summary>
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<ul>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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@ -1352,6 +1348,10 @@ Table of contents
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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</ul>
|
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@ -1614,8 +1614,8 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emd</span></h2>
|
||||
|
||||
<div class="outline-text-2" id="text-emd">
|
||||
<details class="prereq" id="org417b126">
|
||||
<summary id="org6bf52b1">
|
||||
<details class="prereq" id="orgeb31271">
|
||||
<summary id="org69f893c">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1624,8 +1624,8 @@ Prerequisites
|
|||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="org93939b2">
|
||||
<summary id="org78ba2fd">
|
||||
<details class="objectives" id="orgd436fef">
|
||||
<summary id="orgff05dfe">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1666,7 +1666,7 @@ 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>
|
||||
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|
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|
||||
|
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|
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|
|
@ -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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<meta name="viewport" content="width=device-width, initial-scale=1">
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<title>Pre-Quantum Electrodynamics</title>
|
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|
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@ -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>
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|
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</li>
|
||||
<li>
|
||||
<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
|
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|
||||
</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>
|
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|
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</li>
|
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<li>
|
||||
<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
|
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|
||||
</li>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1640,7 +1640,7 @@ 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>
|
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</div>
|
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|
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|
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|
|
@ -1,7 +1,7 @@
|
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<!DOCTYPE html>
|
||||
<html lang="en">
|
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<head>
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<title>Pre-Quantum Electrodynamics</title>
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@ -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>
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|
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</li>
|
||||
<li>
|
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
|
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|
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</li>
|
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|
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@ -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>
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|
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</li>
|
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<li>
|
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
|
||||
|
||||
</li>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1644,18 +1644,18 @@ Empirically: the changing magnetic field induces an electric current around
|
|||
the circuit. This current is really driven by an electric field having a component
|
||||
along the wire. The line integral of this field is called the
|
||||
</p>
|
||||
<div class="core div" id="org9cc8d48">
|
||||
<div class="core div" id="orgfd2df2d">
|
||||
<p>
|
||||
<b>Electromotive force (or electromotance)</b>,
|
||||
</p>
|
||||
<div class="eqlabel" id="orgb9f75a5">
|
||||
<div class="eqlabel" id="org510ac9c">
|
||||
<p>
|
||||
<a id="elmofo"></a><a href="./emd_Fl_Fl.html#elmofo"><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="orgcb4fab4">
|
||||
<div class="alteqlabels" id="org85afd05">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.9)</li>
|
||||
</ul>
|
||||
|
|
@ -1680,14 +1680,14 @@ The precise statement associated to Faraday's observations
|
|||
is that the electromotive force is proportional
|
||||
to the rate of change of the magnetic flux,
|
||||
</p>
|
||||
<div class="eqlabel" id="orgb7faf93">
|
||||
<div class="eqlabel" id="org4496ec0">
|
||||
<p>
|
||||
<a id="Fl_flux"></a><a href="./emd_Fl_Fl.html#Fl_flux"><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="orgffe5f68">
|
||||
<div class="alteqlabels" id="orga00ad3a">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.14)</li>
|
||||
</ul>
|
||||
|
|
@ -1702,18 +1702,18 @@ to the rate of change of the magnetic flux,
|
|||
\]
|
||||
so we obtain
|
||||
</p>
|
||||
<div class="core div" id="org80fe686">
|
||||
<div class="core div" id="org35a9412">
|
||||
<p>
|
||||
<b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
|
||||
</p>
|
||||
<div class="eqlabel" id="orgfa5af5c">
|
||||
<div class="eqlabel" id="org6646a96">
|
||||
<p>
|
||||
<a id="Fl_int"></a><a href="./emd_Fl_Fl.html#Fl_int"><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="orga84de5d">
|
||||
<div class="alteqlabels" id="org4c06533">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.15)</li>
|
||||
</ul>
|
||||
|
|
@ -1737,15 +1737,15 @@ for any loop (on a wire or not). Using Stokes' theorem,
|
|||
\]
|
||||
we obtain
|
||||
</p>
|
||||
<div class="core div" id="org2c503e3">
|
||||
<div class="eqlabel" id="org9023002">
|
||||
<div class="core div" id="orgfef0799">
|
||||
<div class="eqlabel" id="org07be4b3">
|
||||
<p>
|
||||
<a id="Fl"></a><a href="./emd_Fl_Fl.html#Fl"><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="orgb7f3817">
|
||||
<div class="alteqlabels" id="orgd458eb7">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.16)</li>
|
||||
</ul>
|
||||
|
|
@ -1790,7 +1790,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1639,14 +1639,14 @@ W = \frac{1}{2} I \oint {\bf A} \cdot d{\bf l} = \frac{1}{2} \oint ({\bf A} \cdo
|
|||
\]
|
||||
Generalization to volume currents:
|
||||
</p>
|
||||
<div class="eqlabel" id="org78f90c5">
|
||||
<div class="eqlabel" id="org9c43b0a">
|
||||
<p>
|
||||
<a id="W_intAJ"></a><a href="./emd_Fl_e.html#W_intAJ"><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="org0277370">
|
||||
<div class="alteqlabels" id="org1b4fa12">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.31)</li>
|
||||
</ul>
|
||||
|
|
@ -1681,15 +1681,15 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold
|
|||
\]
|
||||
We can integrate over all space: after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with
|
||||
</p>
|
||||
<div class="core div" id="org9994672">
|
||||
<div class="eqlabel" id="org2cf22c6">
|
||||
<div class="core div" id="orgc92d34b">
|
||||
<div class="eqlabel" id="org7ec4181">
|
||||
<p>
|
||||
<a id="W_intBsq"></a><a href="./emd_Fl_e.html#W_intBsq"><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="orgde5daac">
|
||||
<div class="alteqlabels" id="org073f8ff">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.34)</li>
|
||||
</ul>
|
||||
|
|
@ -1718,7 +1718,7 @@ W_{mag} &= \frac{1}{2} \int d\tau ~({\bf A} \cdot {\bf J}) &= \frac{1}{2
|
|||
which are equations <a href="./ems_es_e.html#W_vcd">W_vcd</a>, <a href="./ems_es_e.html#W_intEsq">W_intEsq</a>, <a href="./emd_Fl_e.html#W_intAJ">W_intAJ</a> and <a href="./emd_Fl_e.html#W_intBsq">W_intBsq</a>.
|
||||
</p>
|
||||
|
||||
<div class="example div" id="orgd2854e0">
|
||||
<div class="example div" id="org60417d4">
|
||||
<p>
|
||||
<b>Example: energy in coaxial cable</b>
|
||||
</p>
|
||||
|
|
@ -1768,7 +1768,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1641,14 +1641,14 @@ so
|
|||
\]
|
||||
and we can write the mutual inductance as the <b>Neumann formula</b>,
|
||||
</p>
|
||||
<div class="eqlabel" id="org766e684">
|
||||
<div class="eqlabel" id="org554c244">
|
||||
<p>
|
||||
<a id="Newmann_M"></a><a href="./emd_Fl_i.html#Newmann_M"><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="orga92dd51">
|
||||
<div class="alteqlabels" id="orgf48cffa">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.22)</li>
|
||||
</ul>
|
||||
|
|
@ -1665,14 +1665,14 @@ M_{21} = \frac{\mu_0}{4\pi} \oint_{{\cal P}_1} \oint_{{\cal P}_2} \frac{d{\bf l}
|
|||
Two things:
|
||||
first, \(M_{21}\) is purely geometrical. Second,
|
||||
</p>
|
||||
<div class="eqlabel" id="org577027c">
|
||||
<div class="eqlabel" id="org12f955a">
|
||||
<p>
|
||||
<a id="Msym"></a><a href="./emd_Fl_i.html#Msym"><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="org21aff84">
|
||||
<div class="alteqlabels" id="orga59d4b0">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.23)</li>
|
||||
</ul>
|
||||
|
|
@ -1687,7 +1687,7 @@ M_{12} = M_{21}
|
|||
\]
|
||||
</p>
|
||||
|
||||
<div class="example div" id="org9334fe0">
|
||||
<div class="example div" id="orgdd5f18b">
|
||||
<p>
|
||||
<b>Example: solenoid in solenoid</b>
|
||||
</p>
|
||||
|
|
@ -1735,14 +1735,14 @@ What if we vary current in loop 1? Flux in 2 will vary. Induces EMF in loop 2:
|
|||
\]
|
||||
Changing current also induces EMF in the source loop itself:
|
||||
</p>
|
||||
<div class="eqlabel" id="org4807c27">
|
||||
<div class="eqlabel" id="org06772cd">
|
||||
<p>
|
||||
<a id="PLI"></a><a href="./emd_Fl_i.html#PLI"><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="orga19e4d1">
|
||||
<div class="alteqlabels" id="org5ce4f36">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.25)</li>
|
||||
</ul>
|
||||
|
|
@ -1765,7 +1765,7 @@ Inductance: measured in <b>henries</b> (\(H\)). \(H = V s/A\).
|
|||
</p>
|
||||
|
||||
|
||||
<div class="example div" id="org940f5ab">
|
||||
<div class="example div" id="org678dce1">
|
||||
<p>
|
||||
<b>Example: self-inductance of toroidal coil</b>
|
||||
</p>
|
||||
|
|
@ -1804,7 +1804,7 @@ Inductance (like capacitance) is intrinsically positive. Use Lenz law.
|
|||
Think of <i>back EMF</i>.
|
||||
</p>
|
||||
|
||||
<div class="example div" id="orgbeea66c">
|
||||
<div class="example div" id="org42aa8b9">
|
||||
<p>
|
||||
<b>Example: circuit</b>
|
||||
</p>
|
||||
|
|
@ -1853,7 +1853,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1637,7 +1637,7 @@ law in integral form:
|
|||
|
||||
|
||||
|
||||
<div class="example div" id="org56facb7">
|
||||
<div class="example div" id="orgcfd6427">
|
||||
<p>
|
||||
<b>Example: loop with time-dependent flux</b>
|
||||
</p>
|
||||
|
|
@ -1664,7 +1664,7 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
|
|||
</div>
|
||||
|
||||
|
||||
<div class="example div" id="org367fcfc">
|
||||
<div class="example div" id="orgfc45720">
|
||||
<p>
|
||||
<b>Example: wheel with charged rim traversed by flux</b>
|
||||
</p>
|
||||
|
|
@ -1705,7 +1705,7 @@ called the <b>quasistatic</b> approximation, and works provided we deal with
|
|||
<i>slow enough</i> phenomena.
|
||||
</p>
|
||||
|
||||
<div class="example div" id="orgb2a276c">
|
||||
<div class="example div" id="org1d9bce6">
|
||||
<p>
|
||||
<b>Example: field from wire with time-dependent current</b>
|
||||
</p>
|
||||
|
|
@ -1764,7 +1764,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1640,7 +1640,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1616,18 +1616,18 @@ Table of contents
|
|||
<p>
|
||||
Full set of equations for the electromagnetic field:
|
||||
</p>
|
||||
<div class="core div" id="orgce899b3">
|
||||
<div class="core div" id="orgee05fb0">
|
||||
<p>
|
||||
<b>Maxwell's equations</b> <i>(in vacuum)</i>
|
||||
</p>
|
||||
<div class="eqlabel" id="org6517c73">
|
||||
<div class="eqlabel" id="orgf090588">
|
||||
<p>
|
||||
<a id="Max_vac"></a><a href="./emd_Me_Me.html#Max_vac"><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="orgfa86c68">
|
||||
<div class="alteqlabels" id="org538c235">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1644,7 +1644,7 @@ Full set of equations for the electromagnetic field:
|
|||
<p>
|
||||
Complement:
|
||||
</p>
|
||||
<div class="core div" id="orgba4da4a">
|
||||
<div class="core div" id="org14efa58">
|
||||
<p>
|
||||
Force law <a href="./ems_ms_lf_pc.html#LorFo">LorFo</a>
|
||||
\[
|
||||
|
|
@ -1667,15 +1667,15 @@ take divergence of \((iv)\) and use \((i)\).
|
|||
<p>
|
||||
Better way of writing: all fields on left, all sources on right,
|
||||
</p>
|
||||
<div class="core div" id="org8fce2a6">
|
||||
<div class="eqlabel" id="org384b902">
|
||||
<div class="core div" id="orge7a8009">
|
||||
<div class="eqlabel" id="orgaf03f6a">
|
||||
<p>
|
||||
<a id="Max_vac_s"></a><a href="./emd_Me_Me.html#Max_vac_s"><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="orgb8b5d0a">
|
||||
<div class="alteqlabels" id="orgc1dd627">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.42)</li>
|
||||
</ul>
|
||||
|
|
@ -1713,7 +1713,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1622,15 +1622,15 @@ The term which should be zero (but isn't) in <a href="./emd_Me_ebM.html#divcurlB
|
|||
\]
|
||||
The extra term would thus be eliminated if we were to put
|
||||
</p>
|
||||
<div class="core div" id="org76622f8">
|
||||
<div class="eqlabel" id="org47af899">
|
||||
<div class="core div" id="org9029eb1">
|
||||
<div class="eqlabel" id="org7937f05">
|
||||
<p>
|
||||
<a id="AmpMax"></a><a href="./emd_Me_dc.html#AmpMax"><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="org4b2df9a">
|
||||
<div class="alteqlabels" id="orgc4ec53c">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.36)</li>
|
||||
</ul>
|
||||
|
|
@ -1659,18 +1659,18 @@ Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation
|
|||
<p>
|
||||
Maxwell baptized this term the
|
||||
</p>
|
||||
<div class="core div" id="org1ac2c47">
|
||||
<div class="core div" id="orgf5b87fa">
|
||||
<p>
|
||||
<b>Displacement current</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="orgd74a8cb">
|
||||
<div class="eqlabel" id="orgff1f2c8">
|
||||
<p>
|
||||
<a id="Jd"></a><a href="./emd_Me_dc.html#Jd"><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="org44de33e">
|
||||
<div class="alteqlabels" id="org87a5cc7">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.37)</li>
|
||||
</ul>
|
||||
|
|
@ -1724,7 +1724,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1630,14 +1630,14 @@ Fatal inconsistency: div of curl must always vanish. Check on \((iii)\):
|
|||
\]
|
||||
But: try same with \((iv)\):
|
||||
</p>
|
||||
<div class="eqlabel" id="org83e3419">
|
||||
<div class="eqlabel" id="org76390cc">
|
||||
<p>
|
||||
<a id="divcurlB"></a><a href="./emd_Me_ebM.html#divcurlB"><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="org20c79ee">
|
||||
<div class="alteqlabels" id="org64ca711">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.35)</li>
|
||||
</ul>
|
||||
|
|
@ -1682,7 +1682,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1663,7 +1663,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1614,8 +1614,8 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emd.ce</span></h3>
|
||||
|
||||
<div class="outline-text-3" id="text-emd_ce">
|
||||
<details class="prereq" id="org36d2595">
|
||||
<summary id="orgac28d59">
|
||||
<details class="prereq" id="orgd007ccd">
|
||||
<summary id="org4b65898">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1623,8 +1623,8 @@ Prerequisites
|
|||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="org84c31d2">
|
||||
<summary id="org5891649">
|
||||
<details class="objectives" id="org1ca55b0">
|
||||
<summary id="org22a98ae">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1662,7 +1662,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1616,18 +1616,18 @@ Table of contents
|
|||
<p>
|
||||
The angular momentum of EM fields is directly given by
|
||||
</p>
|
||||
<div class="main div" id="org7d69a07">
|
||||
<div class="main div" id="org9f96756">
|
||||
<p>
|
||||
<b>Angular momentum of EM fields</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org0855fcd">
|
||||
<div class="eqlabel" id="org2a11d69">
|
||||
<p>
|
||||
<a id="lrxg"></a><a href="./emd_ce_amom.html#lrxg"><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="org434669a">
|
||||
<div class="alteqlabels" id="orgfb3f8f7">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1661,7 +1661,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1635,7 +1635,7 @@ This means that
|
|||
\]
|
||||
Since this is true for any volume, we have (re)derived the
|
||||
</p>
|
||||
<div class="core div" id="org75f48b5">
|
||||
<div class="core div" id="org3f0dec4">
|
||||
<p>
|
||||
<b>Continuity equation</b> <a href="./ems_ms_ce.html#conteq">conteq</a>
|
||||
\[
|
||||
|
|
@ -1675,7 +1675,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1628,18 +1628,18 @@ in which the first integral can be interpreted as the momentum stored in the EM
|
|||
<p>
|
||||
This is thus simply a conservation law for momentum, with
|
||||
</p>
|
||||
<div class="main div" id="org60dbc44">
|
||||
<div class="main div" id="orgc119c5d">
|
||||
<p>
|
||||
<b>Momentum density in the EM fields</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org7808fcd">
|
||||
<div class="eqlabel" id="org4c00e17">
|
||||
<p>
|
||||
<a id="gExB"></a><a href="./emd_ce_mom.html#gExB"><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="org1473c1c">
|
||||
<div class="alteqlabels" id="org62e68b8">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1655,18 +1655,18 @@ This is thus simply a conservation law for momentum, with
|
|||
<p>
|
||||
In a region in which the mechanical momentum is not changing due to external influences, we then have the
|
||||
</p>
|
||||
<div class="main div" id="orgf7ee289">
|
||||
<div class="main div" id="org55398a2">
|
||||
<p>
|
||||
<b>Continuity equation for EM momentum</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="orgd7db956">
|
||||
<div class="eqlabel" id="org02bb382">
|
||||
<p>
|
||||
<a id="contg"></a><a href="./emd_ce_mom.html#contg"><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="orge8fbc29">
|
||||
<div class="alteqlabels" id="orgf040d5c">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1699,7 +1699,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1667,18 +1667,18 @@ and similarly for \({\boldsymbol B}\). We thus get
|
|||
<p>
|
||||
This expression can be greatly simplified by introducing the
|
||||
</p>
|
||||
<div class="main div" id="orgaa8145a">
|
||||
<div class="main div" id="org6a32dc6">
|
||||
<p>
|
||||
<b>Maxwell stress tensor</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org0e3a6cc">
|
||||
<div class="eqlabel" id="org46015d8">
|
||||
<p>
|
||||
<a id="MaxST"></a><a href="./emd_ce_mst.html#MaxST"><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="org4826577">
|
||||
<div class="alteqlabels" id="org9675080">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1699,18 +1699,18 @@ The element \(T_{ij}\) represents the force per unit area in the \(i\) direction
|
|||
<p>
|
||||
We then obtain the
|
||||
</p>
|
||||
<div class="main div" id="org66e8efe">
|
||||
<div class="main div" id="org0171e33">
|
||||
<p>
|
||||
<b>EM force per unit volume</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org49902c2">
|
||||
<div class="eqlabel" id="org5306def">
|
||||
<p>
|
||||
<a id="fT"></a><a href="./emd_ce_mst.html#fT"><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="orge0c52b1">
|
||||
<div class="alteqlabels" id="orgc0f4f98">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1726,18 +1726,18 @@ We then obtain the
|
|||
<p>
|
||||
where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
|
||||
</p>
|
||||
<div class="main div" id="org887bc5d">
|
||||
<div class="main div" id="org28ab853">
|
||||
<p>
|
||||
<b>Total force on charges in volume</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="orgd12201e">
|
||||
<div class="eqlabel" id="orgb821a87">
|
||||
<p>
|
||||
<a id="totFo"></a><a href="./emd_ce_mst.html#totFo"><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="org0dc7e2f">
|
||||
<div class="alteqlabels" id="orgdcbb5c3">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1770,7 +1770,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1617,7 +1617,6 @@ Table of contents
|
|||
Earlier: work done to assemble a static charge distribution:
|
||||
\[
|
||||
W_e = \frac{\varepsilon_0}{2} \int d\tau ~E^2
|
||||
\tag{\ref{eq:Energy_as_int_E2}}
|
||||
\]
|
||||
Work necessary to get currents going:
|
||||
\[
|
||||
|
|
@ -1637,14 +1636,14 @@ done by EM forces? From Lorentz force law:
|
|||
Really, we're looking at a small volume element \(d\tau\) carrying charge \(\rho d\tau\), moving
|
||||
at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\). Thus,
|
||||
</p>
|
||||
<div class="eqlabel" id="orgc9958b2">
|
||||
<div class="eqlabel" id="org935dce9">
|
||||
<p>
|
||||
<a id="dWdt_intEJ"></a><a href="./emd_ce_poy.html#dWdt_intEJ"><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="orgf9bbccb">
|
||||
<div class="alteqlabels" id="orgad7d22c">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (8.6)</li>
|
||||
</ul>
|
||||
|
|
@ -1684,18 +1683,18 @@ so we get
|
|||
Substituting this in <a href="./emd_ce_poy.html#dWdt_intEJ">dWdt_intEJ</a> and using the divergence theorem,
|
||||
we obtain
|
||||
</p>
|
||||
<div class="main div" id="orgead023b">
|
||||
<div class="main div" id="orgdbc3e5c">
|
||||
<p>
|
||||
<b>Poynting's theorem</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org1b1ef48">
|
||||
<div class="eqlabel" id="orgc64c555">
|
||||
<p>
|
||||
<a id="👉Thm"></a><a href="./emd_ce_poy.html#👉Thm"><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="org9c51283">
|
||||
<div class="alteqlabels" id="orgf1ac980">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (8.9)</li>
|
||||
</ul>
|
||||
|
|
@ -1721,18 +1720,18 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
|
|||
<p>
|
||||
Energy per unit time, per unit area carried by EM fields: given by the
|
||||
</p>
|
||||
<div class="core div" id="orgafa4bdd">
|
||||
<div class="core div" id="orga8e7c87">
|
||||
<p>
|
||||
<b>Poynting vector</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org0aaf227">
|
||||
<div class="eqlabel" id="org7fb23da">
|
||||
<p>
|
||||
<a id="PoyntingVec"></a><a href="./emd_ce_poy.html#PoyntingVec"><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="org05edf23">
|
||||
<div class="alteqlabels" id="org8150e29">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (8.10)</li>
|
||||
</ul>
|
||||
|
|
@ -1751,18 +1750,18 @@ Energy per unit time, per unit area carried by EM fields: given by the
|
|||
<p>
|
||||
We can thus express Poynting's theorem more compactly:
|
||||
</p>
|
||||
<div class="core div" id="orgf7b3c73">
|
||||
<div class="core div" id="org41e1182">
|
||||
<p>
|
||||
<b>Poynting's theorem</b> (integral form)
|
||||
</p>
|
||||
<div class="eqlabel" id="org610ce5e">
|
||||
<div class="eqlabel" id="orgfa25d4a">
|
||||
<p>
|
||||
<a id="PoyntingThm_int"></a><a href="./emd_ce_poy.html#PoyntingThm_int"><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="org462a7a9">
|
||||
<div class="alteqlabels" id="org264bb7a">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (8.11)</li>
|
||||
</ul>
|
||||
|
|
@ -1781,18 +1780,18 @@ We can thus express Poynting's theorem more compactly:
|
|||
<p>
|
||||
where we have defined the total
|
||||
</p>
|
||||
<div class="core div" id="org49394fc">
|
||||
<div class="core div" id="orga6dad30">
|
||||
<p>
|
||||
<b>Energy in electromagnetic fields</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org2fd18f3">
|
||||
<div class="eqlabel" id="org16b17ef">
|
||||
<p>
|
||||
<a id="Uem"></a><a href="./emd_ce_poy.html#Uem"><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="orge0e6bd0">
|
||||
<div class="alteqlabels" id="orgdf4c40f">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (8.5)</li>
|
||||
</ul>
|
||||
|
|
@ -1821,18 +1820,18 @@ Then,
|
|||
\]
|
||||
so we get the
|
||||
</p>
|
||||
<div class="core div" id="orgb7f6aa8">
|
||||
<div class="core div" id="orgfd64b94">
|
||||
<p>
|
||||
<b>Poynting theorem</b> (differential form)
|
||||
</p>
|
||||
<div class="eqlabel" id="orgc43f9ba">
|
||||
<div class="eqlabel" id="org47e48c8">
|
||||
<p>
|
||||
<a id="PoyntingThm"></a><a href="./emd_ce_poy.html#PoyntingThm"><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="org50e67b6">
|
||||
<div class="alteqlabels" id="orge4a2af6">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (8.14)</li>
|
||||
</ul>
|
||||
|
|
@ -1855,7 +1854,7 @@ and has a similar for to the continuity equation
|
|||
|
||||
|
||||
|
||||
<div class="example div" id="org76ab6ea">
|
||||
<div class="example div" id="org5f795de">
|
||||
<p>
|
||||
<b>Example: Joule heating</b>
|
||||
</p>
|
||||
|
|
@ -1881,7 +1880,7 @@ wire of radius \(a\),
|
|||
\]
|
||||
Poynting:
|
||||
\[
|
||||
{\boldsymbol S} = \frac{1}{\mu_0} \frac{V}{L} \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol x} \times \hat{\boldsymbol \varphi} = -\frac{VI}{2\pi a L} \hat{\boldsymbol s}
|
||||
{\boldsymbol S} = \frac{1}{\mu_0} \frac{V}{L} \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol x} \times \hat{\boldsymbol \varphi} = -\frac{VI}{2\pi a L} \hat{\boldsymbol r}
|
||||
\]
|
||||
and points radially inwards. Energy per unit time passing surface of wire:
|
||||
\[
|
||||
|
|
@ -1912,7 +1911,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1614,8 +1614,8 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emd.emw</span></h3>
|
||||
|
||||
<div class="outline-text-3" id="text-emd_emw">
|
||||
<details class="prereq" id="org42aca70">
|
||||
<summary id="org3fdd75c">
|
||||
<details class="prereq" id="org81c4de7">
|
||||
<summary id="org94e2b41">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1624,21 +1624,14 @@ Prerequisites
|
|||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="org8aeb437">
|
||||
<summary id="org16d834d">
|
||||
<details class="objectives" id="org04214f1">
|
||||
<summary id="org840c379">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
<li>Understand how to obtain the wave equation from Maxwell's equations</li>
|
||||
<li>Understand monochromatic plane waves in vacuum</li>
|
||||
<li>Understand their energy and momentum</li>
|
||||
<li>Understand EM waves in linear media</li>
|
||||
<li>Understand reflection and transmission of EM waves at an interface</li>
|
||||
<li>Know the laws of reflection and refraction</li>
|
||||
<li>Know Fresnel's equations (parallel polarization)</li>
|
||||
<li>Know Brewster's angle</li>
|
||||
<li>Understand EM waves in conductors</li>
|
||||
<li>Understand simple waveguides and coaxial cables</li>
|
||||
</ul>
|
||||
</details>
|
||||
</div>
|
||||
|
|
@ -1666,7 +1659,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1644,18 +1644,18 @@ so for a monochromatic EM plan wave,
|
|||
\]
|
||||
or more succinctly:
|
||||
</p>
|
||||
<div class="main div" id="orgee78283">
|
||||
<div class="main div" id="orge6fd2ac">
|
||||
<p>
|
||||
<b>Poynting vector of a monochromatic EM wave</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="orgdedd25b">
|
||||
<div class="eqlabel" id="orgc1c96f7">
|
||||
<p>
|
||||
<a id="Poynting_mpw"></a><a href="./emd_emw_ep.html#Poynting_mpw"><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="orgdfa9b5e">
|
||||
<div class="alteqlabels" id="org907a483">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (9.57)</li>
|
||||
</ul>
|
||||
|
|
@ -1678,7 +1678,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
|
|||
<p>
|
||||
Similary, we get the
|
||||
</p>
|
||||
<div class="main div" id="org108cc57">
|
||||
<div class="main div" id="orga2199bc">
|
||||
<p>
|
||||
<b>Momentum density of a monochromatic EM wave</b>
|
||||
\[
|
||||
|
|
@ -1729,7 +1729,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1636,14 +1636,14 @@ From Faraday: \({\boldsymbol \nabla} \times {\bf E} = -\partial {\bf B}/\partia
|
|||
\]
|
||||
so \({\bf E}\) and \({\bf B}\) are mutually perpendicular, and
|
||||
</p>
|
||||
<div class="eqlabel" id="org4747373">
|
||||
<div class="eqlabel" id="org4864134">
|
||||
<p>
|
||||
<a id="EBmpw"></a><a href="./emd_emw_mpw.html#EBmpw"><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="org2ea2f4c">
|
||||
<div class="alteqlabels" id="org6d00653">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (9.47)</li>
|
||||
</ul>
|
||||
|
|
@ -1662,18 +1662,18 @@ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0.
|
|||
Generalizing to propagation in the direction of an arbitrary wavevector
|
||||
\({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
|
||||
</p>
|
||||
<div class="core div" id="orga666428">
|
||||
<div class="core div" id="org9581019">
|
||||
<p>
|
||||
<b>E and B fields for a monochromatic EM plane wave</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="orge12acff">
|
||||
<div class="eqlabel" id="org98b7671">
|
||||
<p>
|
||||
<a id="mpw"></a><a href="./emd_emw_mpw.html#mpw"><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="orgd889027">
|
||||
<div class="alteqlabels" id="orge381054">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (9.49)</li>
|
||||
</ul>
|
||||
|
|
@ -1724,7 +1724,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1641,18 +1641,18 @@ These take the form of coupled first-order partial differential equations for \(
|
|||
Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
|
||||
we get the
|
||||
</p>
|
||||
<div class="core div" id="orgc9c5d44">
|
||||
<div class="core div" id="org672468f">
|
||||
<p>
|
||||
<b>Wave equations for electric and magnetic fields in vacuum</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="orgb29f135">
|
||||
<div class="eqlabel" id="orgf656d76">
|
||||
<p>
|
||||
<a id="WaveEq"></a><a href="./emd_emw_we.html#WaveEq"><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="orge5c16c5">
|
||||
<div class="alteqlabels" id="org1ac2ec3">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (9.41)</li>
|
||||
</ul>
|
||||
|
|
@ -1714,7 +1714,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1638,7 +1638,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1638,7 +1638,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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)~~ &{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0}, \hspace{1cm} &\mbox{Gauss}, \nonumber \\
|
||||
(ii)~~ &{\boldsymbol \nabla} \cdot {\bf B} = 0, \hspace{1cm} &\mbox{anonymous} \nonumber \\
|
||||
(iii)~~ &{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \hspace{1cm} &\mbox{Faraday}, \nonumber \\
|
||||
(iv)~~ &{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, \hspace{1cm} &\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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1616,18 +1616,18 @@ Table of contents
|
|||
<p>
|
||||
Discontinuities between different media, deduced from
|
||||
</p>
|
||||
<div class="core div" id="org2152f07">
|
||||
<div class="core div" id="org31953cb">
|
||||
<p>
|
||||
<b>Maxwell's equations</b> <i>(in matter)</i>, <i>integral form</i>
|
||||
</p>
|
||||
<div class="eqlabel" id="org896f0bf">
|
||||
<div class="eqlabel" id="org853b142">
|
||||
<p>
|
||||
<a id="Max_mat_int"></a><a href="./emdm_Me_bc.html#Max_mat_int"><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="org37cc7e4">
|
||||
<div class="alteqlabels" id="org734ccfd">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1645,15 +1645,15 @@ Discontinuities between different media, deduced from
|
|||
Applying \((i)\) to wafer-thin Gaussian pillbox straddling boundary between 2 materials:
|
||||
\({\bf D}_1 \cdot {\bf a} - {\bf D}_2 \cdot {\bf a} = \sigma_f a\) so
|
||||
</p>
|
||||
<div class="main div" id="orgaee2676">
|
||||
<div class="eqlabel" id="org35b4f05">
|
||||
<div class="main div" id="org74cc983">
|
||||
<div class="eqlabel" id="org49821b6">
|
||||
<p>
|
||||
<a id="Ddisc"></a><a href="./emdm_Me_bc.html#Ddisc"><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="orgc9c66ee">
|
||||
<div class="alteqlabels" id="orgb4f898e">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.60)</li>
|
||||
</ul>
|
||||
|
|
@ -1672,9 +1672,9 @@ D^{\perp}_1 - D^{\perp}_2 = \sigma_f
|
|||
<p>
|
||||
Same reasoning applied to \((ii)\) gives <a href="./ems_ms_vp_mbc.html#Bdisc">Bdisc</a>
|
||||
</p>
|
||||
<div class="main div" id="org0c24b54">
|
||||
<div class="eqlabel" id="org262505f">
|
||||
<div class="alteqlabels" id="org560c237">
|
||||
<div class="main div" id="org167c3b0">
|
||||
<div class="eqlabel" id="orgd15716f">
|
||||
<div class="alteqlabels" id="orgc56e074">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1690,7 +1690,7 @@ B^{\perp}_1 - B^{\perp}_2 = 0
|
|||
For \((iii)\): Amperian loop straddling surface: \({\bf E}_1 \cdot {\bf l} - {\bf E}_2 \cdot {\bf l} =
|
||||
-\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}\). Limit of small loop: flux vanishes, therefore
|
||||
</p>
|
||||
<div class="main div" id="org2c8267d">
|
||||
<div class="main div" id="org944c540">
|
||||
<p>
|
||||
\[
|
||||
{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0
|
||||
|
|
@ -1702,27 +1702,9 @@ For \((iii)\): Amperian loop straddling surface: \({\bf E}_1 \cdot {\bf l} - {
|
|||
Similarly, \((iv)\) implies \({\bf H}_1 \cdot {\bf l} - {\bf H}_2 \cdot {\bf l} = I_{f_{enc}}\).
|
||||
No volume current can contribute, but a surface current can. Can write
|
||||
\(I_{f_{enc}} = {\bf K}_f \cdot (\hat{\bf n} \times {\bf l}) = ({\bf K}_f \times \hat{\bf n}) \cdot {\bf l}\)
|
||||
and thus
|
||||
and thus (as we got before in <a href="./emsm_msm_H_A.html#Hdisc">Hdisc</a>)
|
||||
</p>
|
||||
<div class="main div" id="orge9d0c89">
|
||||
<div class="eqlabel" id="org15b0fbc">
|
||||
<p>
|
||||
<a id="Hdisc"></a><a href="./emsm_msm_H_A.html#Hdisc"><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><a href="./emdm_Me_bc.html#Hdisc"><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="orgc1f1499">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.63)</li>
|
||||
</ul>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="main div" id="org44aa4fb">
|
||||
<p>
|
||||
\[
|
||||
{\bf H}_1^{\parallel} - {\bf H}_2^{\parallel} = {\bf K}_f \times \hat{\bf n}
|
||||
|
|
@ -1738,15 +1720,15 @@ These are the general boundary conditions for electrodynamics.
|
|||
<p>
|
||||
In case of linear media: can be expressed in terms of \({\bf E}\) and \({\bf B}\) alone:
|
||||
</p>
|
||||
<div class="main div" id="org8b0c892">
|
||||
<div class="eqlabel" id="org2d4783e">
|
||||
<div class="main div" id="org74a1c76">
|
||||
<div class="eqlabel" id="orgbb2af38">
|
||||
<p>
|
||||
<a id="disc_lm"></a><a href="./emdm_Me_bc.html#disc_lm"><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="orga65ee45">
|
||||
<div class="alteqlabels" id="org98cdce0">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.64)</li>
|
||||
</ul>
|
||||
|
|
@ -1766,14 +1748,14 @@ In case of linear media: can be expressed in terms of \({\bf E}\) and \({\bf B}
|
|||
<p>
|
||||
If there is no free charge and no free current at boundary:
|
||||
</p>
|
||||
<div class="eqlabel" id="org656d25d">
|
||||
<div class="eqlabel" id="orga2465cb">
|
||||
<p>
|
||||
<a id="disc_nfc"></a><a href="./emdm_Me_bc.html#disc_nfc"><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="org856682c">
|
||||
<div class="alteqlabels" id="org06ac001">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.64)</li>
|
||||
</ul>
|
||||
|
|
@ -1810,7 +1792,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1614,6 +1614,31 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emdm.emwm</span></h3>
|
||||
|
||||
<div class="outline-text-3" id="text-emdm_emwm">
|
||||
<details class="prereq" id="orgf2f0eda">
|
||||
<summary id="orged680b6">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
<li>Electrodynamics</li>
|
||||
<li>Convervation Laws</li>
|
||||
<li>Electromagnetic waves in vacuum</li>
|
||||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="orge99509d">
|
||||
<summary id="orge43e5c1">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
<li>Understand EM waves in linear media</li>
|
||||
<li>Understand reflection and transmission of EM waves at an interface</li>
|
||||
<li>Know the laws of reflection and refraction</li>
|
||||
<li>Know Fresnel's equations (parallel polarization)</li>
|
||||
<li>Know Brewster's angle</li>
|
||||
<li>Understand EM waves in conductors</li>
|
||||
<li>Understand simple waveguides and coaxial cables</li>
|
||||
</ul>
|
||||
</details>
|
||||
</div>
|
||||
|
||||
|
||||
|
|
@ -1641,7 +1666,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1637,7 +1637,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1661,7 +1661,7 @@ with plane-wave solutions
|
|||
\[
|
||||
{\boldsymbol E} ({\boldsymbol r}, t) = {\boldsymbol E}_0 e^{i \tilde{\boldsymbol k} \cdot {\boldsymbol r} - \omega t}
|
||||
\]
|
||||
with
|
||||
with complex-valued wavevector
|
||||
\[
|
||||
\tilde{k}^2 \equiv \tilde{\boldsymbol k} \cdot \tilde{\boldsymbol k} = \mu \varepsilon \omega^2 + i\mu \sigma \omega.
|
||||
\]
|
||||
|
|
@ -1733,7 +1733,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1632,15 +1632,15 @@ For linear medium:
|
|||
\]
|
||||
If the medium is homogeneous (no spatial dependence of \(\varepsilon\) or \(\mu\)),
|
||||
</p>
|
||||
<div class="main div" id="orga20c833">
|
||||
<div class="eqlabel" id="org28a8d44">
|
||||
<div class="main div" id="org7ad5a88">
|
||||
<div class="eqlabel" id="orgde6c1b4">
|
||||
<p>
|
||||
<a id="Max_lh"></a><a href="./emdm_emwm_plm.html#Max_lh"><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="org79eef6d">
|
||||
<div class="alteqlabels" id="org8b72259">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (9.65)</li>
|
||||
</ul>
|
||||
|
|
@ -1669,18 +1669,18 @@ v = \frac{1}{\sqrt{\mu \varepsilon}} = \frac{c}{n}
|
|||
\]
|
||||
where the index of refraction of the material is defined as
|
||||
</p>
|
||||
<div class="main div" id="orga25a79e">
|
||||
<div class="main div" id="org56849ee">
|
||||
<p>
|
||||
<b>Index of refraction</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="orgc9df98d">
|
||||
<div class="eqlabel" id="orgff8c108">
|
||||
<p>
|
||||
<a id="n"></a><a href="./emdm_emwm_plm.html#n"><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="org574afc6">
|
||||
<div class="alteqlabels" id="orgf8eaa9d">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (9.69)</li>
|
||||
</ul>
|
||||
|
|
@ -1743,7 +1743,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1649,7 +1649,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1621,7 +1621,8 @@ in \(x\) direction approaches surface:
|
|||
{\boldsymbol B}_I (z,t) = \frac{1}{v_1} E_{0_I} e^{i (k_1 z - \omega t)} \hat{\boldsymbol y}
|
||||
\label{Gr(9.75)}
|
||||
\]
|
||||
(in which we have used \(B_0 = \frac{1}{v} E_0\), <a href="./emd_emw_mpw.html#EBmpw">EBmpw</a>.
|
||||
(in which we have used \(B_0 = \frac{1}{v} E_0\), <a href="./emd_emw_mpw.html#EBmpw">EBmpw</a>,
|
||||
adapted for the light velocity in this medium).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
|
|
@ -1670,14 +1671,14 @@ E_{0_I} - E_{0_R} = \beta E_{0_T}, \hspace{1cm}
|
|||
Solving these coupled equations, we can write
|
||||
outgoing amplitudes in terms of incident ones:
|
||||
</p>
|
||||
<div class="eqlabel" id="org653a14a">
|
||||
<div class="eqlabel" id="org1b7d37e">
|
||||
<p>
|
||||
<a id="ERT"></a><a href="./emdm_emwm_refl_ni.html#ERT"><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="org5690572">
|
||||
<div class="alteqlabels" id="org2907578">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (9.82)</li>
|
||||
</ul>
|
||||
|
|
@ -1743,7 +1744,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1635,14 +1635,14 @@ Transmitted wave:
|
|||
\]
|
||||
All waves have the same frequency \(\omega\). Since \(\omega = k v\), the three wavevectors are related by
|
||||
</p>
|
||||
<div class="eqlabel" id="org862531f">
|
||||
<div class="eqlabel" id="org792ed2b">
|
||||
<p>
|
||||
<a id="RTobliquek"></a><a href="./emdm_emwm_refl_oi.html#RTobliquek"><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="org4435847">
|
||||
<div class="alteqlabels" id="org1997374">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1665,7 +1665,7 @@ These forms for incident, reflected and transmitted wave can be substituted in t
|
|||
<p>
|
||||
From now on we will orient the axes so that \({\boldsymbol k}_I\) lies in the \(xz\) plane. This means that \({\boldsymbol k}_R\) and \({\boldsymbol k}_T\) also lie in that plane. This is the
|
||||
</p>
|
||||
<div class="core div" id="org6f173e3">
|
||||
<div class="core div" id="orgbf226b1">
|
||||
<p>
|
||||
<b>First law of reflection:</b>
|
||||
the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
|
||||
|
|
@ -1680,7 +1680,7 @@ Specializing <a href="./emdm_emwm_refl_oi.html#RTobliquek">RTobliquek</a> to our
|
|||
with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
|
||||
and the angle of refraction (\(\theta_T\)) obey the following laws:
|
||||
</p>
|
||||
<div class="core div" id="orgbf9459c">
|
||||
<div class="core div" id="org000f37c">
|
||||
<p>
|
||||
<b>Law of reflection</b>
|
||||
\[
|
||||
|
|
@ -1697,14 +1697,14 @@ and the angle of refraction (\(\theta_T\)) obey the following laws:
|
|||
<p>
|
||||
This takes care of the spatially-dependent exponential factors in the boundary conditions. The coefficients must further obey
|
||||
</p>
|
||||
<div class="eqlabel" id="org8d76e61">
|
||||
<div class="eqlabel" id="orgdc1db4c">
|
||||
<p>
|
||||
<a id="EBRT"></a><a href="./emdm_emwm_refl_oi.html#EBRT"><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="org2c4d03c">
|
||||
<div class="alteqlabels" id="orgf69a0d1">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1750,18 +1750,18 @@ while the third equation becomes
|
|||
\]
|
||||
Writing everything in terms of the incident amplitude, we get
|
||||
</p>
|
||||
<div class="main div" id="org3866053">
|
||||
<div class="main div" id="org7e347a7">
|
||||
<p>
|
||||
<b>Fresnel's equations for reflection and transmission amplitudes (parallel case)</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org31e8ee3">
|
||||
<div class="eqlabel" id="org5c049e2">
|
||||
<p>
|
||||
<a id="Fresnel"></a><a href="./emdm_emwm_refl_oi.html#Fresnel"><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="org1f91c78">
|
||||
<div class="alteqlabels" id="org50e3086">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1785,15 +1785,15 @@ Amplitudes for transmitted and reflected wave: depend on angle of incidence:
|
|||
Behaviour: for \(\theta_I = 0\) we recover <a href="./emdm_emwm_refl_ni.html#ERT">ERT</a>.
|
||||
For grazing waves \(\theta_I \rightarrow \pi/2\) we have that \(\alpha \rightarrow \infty\) and the wave is totally reflected. The most interesting angle is the one at which \(\alpha = \beta\) and the reflected wave has zero amplitude. This is known as
|
||||
</p>
|
||||
<div class="main div" id="org328f2a5">
|
||||
<div class="eqlabel" id="org3b9cec2">
|
||||
<div class="main div" id="org73df67c">
|
||||
<div class="eqlabel" id="org8295eee">
|
||||
<p>
|
||||
<a id="Brewster"></a><a href="./emdm_emwm_refl_oi.html#Brewster"><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"/>
|
||||
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||||
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||||
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||||
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||||
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@ -1846,7 +1846,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
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|
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<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
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|
|||
</summary>
|
||||
<ul>
|
||||
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|
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
|
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|
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</li>
|
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
|
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|
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|
||||
|
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@ -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>
|
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|
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</li>
|
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<li>
|
||||
<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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</ul>
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@ -1633,7 +1633,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
|
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
|
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<p class="validation"></p>
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<!DOCTYPE html>
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@ -1310,10 +1310,6 @@ Table of contents
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</summary>
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<ul>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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</li>
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<li>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
|
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|
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</li>
|
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|
|
@ -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>
|
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|
||||
</li>
|
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<li>
|
||||
<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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</li>
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</ul>
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@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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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: 2022-03-07 Mon 20:38</p>
|
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
|
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<p class="validation"></p>
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<!DOCTYPE html>
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</summary>
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<ul>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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</li>
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@ -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>
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</li>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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</ul>
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@ -1659,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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</div>
|
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<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
|
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<p class="validation"></p>
|
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|
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<!DOCTYPE html>
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<html lang="en">
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|
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</summary>
|
||||
<ul>
|
||||
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|
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
|
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</li>
|
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@ -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>
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</li>
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<li>
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||||
<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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</li>
|
||||
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|
||||
|
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@ -1629,8 +1629,8 @@ We are interested in monochromatic waves propagating down the waveguide and will
|
|||
\]
|
||||
These fields must obey Maxwell's equations. Unlike waves in vacuum, it will turn out that EM waves in waveguides are not purely transverse. We will thus start with a general ansatz for the fields:
|
||||
\[
|
||||
{\boldsymbol E}_0 = \sum_i E_i \hat{\boldsymbol i}, \hspace{10mm}
|
||||
{\boldsymbol B}_0 = \sum_i B_i \hat{\boldsymbol i}, \hspace{10mm}
|
||||
{\boldsymbol E}_0 = \sum_i E_i ~\hat{\boldsymbol i}, \hspace{10mm}
|
||||
{\boldsymbol B}_0 = \sum_i B_i ~\hat{\boldsymbol i}, \hspace{10mm}
|
||||
i = x, y, z.
|
||||
\]
|
||||
Maxwell's equations (iii) and (iv) then give
|
||||
|
|
@ -1683,7 +1683,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
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<head>
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|
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@ -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>
|
||||
|
|
@ -1636,9 +1636,9 @@ The general solution for \(X\) is
|
|||
\[
|
||||
X(x) = A \sin (k_x x) + B \cos (k_x x)
|
||||
\]
|
||||
and to fit the boundary conditions we must have that \(B_x = 0\) so
|
||||
and to fit the boundary conditions we must have that
|
||||
\(\partial B_x/\partial x\) must vanish at \(x=0, a\),
|
||||
so \(dX/dx\) must also vanish at \(x = 0, a\) and thus \(A = 0\) and
|
||||
so \(dX/dx\) must vanish at \(x = 0, a\) and thus \(A = 0\) and
|
||||
\[
|
||||
k_x = m \pi/a, \hspace{5mm} m = 0, 1, \cdots
|
||||
\]
|
||||
|
|
@ -1693,7 +1693,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
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|
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|
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@ -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>
|
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|
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|
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
|
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|
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|
||||
|
|
@ -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>
|
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|
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|
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|
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|
||||
|
|
@ -1614,8 +1614,8 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emf</span></h2>
|
||||
|
||||
<div class="outline-text-2" id="text-emf">
|
||||
<details class="prereq" id="org8e30f8b">
|
||||
<summary id="org2c7b815">
|
||||
<details class="prereq" id="org101ede2">
|
||||
<summary id="org9fe7d9c">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1623,8 +1623,8 @@ Prerequisites
|
|||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="org5818573">
|
||||
<summary id="orgda754dd">
|
||||
<details class="objectives" id="org01ce394">
|
||||
<summary id="org119d5b0">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1660,7 +1660,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
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<head>
|
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|
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<title>Pre-Quantum Electrodynamics</title>
|
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|
|
@ -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>
|
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|
||||
</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>
|
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|
||||
</li>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1621,10 +1621,10 @@ When rewriting our fields into potentials, we passed from six unknowns (the thre
|
|||
<p>
|
||||
There is however a certain ambiguity left when representing fields in terms of potentials. Namely, it is easy to verify that new fields
|
||||
\[
|
||||
V^\prime = V - \frac{\partial \lambda}{\partial t}, \hspace{10mm}
|
||||
\phi^\prime = \phi - \frac{\partial \lambda}{\partial t}, \hspace{10mm}
|
||||
{\boldsymbol A}^\prime = {\boldsymbol A} + {\boldsymbol \nabla} \lambda
|
||||
\]
|
||||
give the same \({\boldsymbol E}\) and \({\boldsymbol B}\) fields as \(V, {\boldsymbol A}\) for any scalar function \(\lambda ({\boldsymbol r}, t)\).
|
||||
give the same \({\boldsymbol E}\) and \({\boldsymbol B}\) fields as \(\phi, {\boldsymbol A}\) for any scalar function \(\lambda ({\boldsymbol r}, t)\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
|
|
@ -1657,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
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</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>
|
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|
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|
|
|
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|
|
@ -1,7 +1,7 @@
|
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<!DOCTYPE html>
|
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<html lang="en">
|
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@ -1310,10 +1310,6 @@ Table of contents
|
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</summary>
|
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<ul>
|
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<li>
|
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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|
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</li>
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|
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
|
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|
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</li>
|
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|
|
@ -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>
|
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|
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</li>
|
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<li>
|
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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|
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|
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@ -1614,25 +1614,25 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emf.g.Cg</span></h4>
|
||||
<div class="outline-text-4" id="text-emf_g_Cg">
|
||||
<p>
|
||||
The {\bf Coulomb Gauge} is specified by taking
|
||||
The <b>Coulomb Gauge</b> is specified by taking
|
||||
\[
|
||||
{\boldsymbol \nabla} \cdot {\boldsymbol A} = 0
|
||||
\]
|
||||
in which case (\ref{eq:LaplacianV}) becomes simply
|
||||
in which case <a href="./emf_svp.html#Lapphi">Lapphi</a> becomes simply
|
||||
\[
|
||||
{\boldsymbol \nabla}^2 V = -\frac{\rho}{\varepsilon_0}
|
||||
{\boldsymbol \nabla}^2 \phi = -\frac{\rho}{\varepsilon_0}
|
||||
\]
|
||||
{\it i.e.} Poisson's equation, whose solution we already know:
|
||||
<i>i.e.</i> Poisson's equation, whose solution we already know:
|
||||
\[
|
||||
V({\boldsymbol r}, t) = \frac{1}{4\pi \varepsilon_0} \int d\tau' \frac{\rho({\boldsymbol r}^\prime, t)}{|{\boldsymbol r} - {\boldsymbol r}^\prime|}
|
||||
\phi({\boldsymbol r}, t) = \frac{1}{4\pi \varepsilon_0} \int d\tau' \frac{\rho({\boldsymbol r}^\prime, t)}{|{\boldsymbol r} - {\boldsymbol r}^\prime|}
|
||||
\]
|
||||
Note that this is an equal-time relationship (it does not mean instantaneous action at a distance, since \(V\) by itself is not physically measurable).
|
||||
Note that this is an equal-time relationship (it does not mean instantaneous action at a distance, since \(\phi\) by itself is not physically measurable).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Although Gauss's law looks nice in the Coulomb gauge, Amp{\`e}re-Maxwell does not:
|
||||
Although Gauss's law looks nice in the Coulomb gauge, Ampère-Maxwell does not:
|
||||
\[
|
||||
{\boldsymbol \nabla}^2 {\boldsymbol A} - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} = -\mu_0 {\boldsymbol J} + \mu_0 \varepsilon_0 {\boldsymbol \nabla} \left( \frac{\partial V}{\partial t} \right).
|
||||
{\boldsymbol \nabla}^2 {\boldsymbol A} - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} = -\mu_0 {\boldsymbol J} + \mu_0 \varepsilon_0 {\boldsymbol \nabla} \left( \frac{\partial \phi}{\partial t} \right).
|
||||
\]
|
||||
</p>
|
||||
</div>
|
||||
|
|
@ -1656,7 +1656,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1614,49 +1614,121 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emf.g.Lg</span></h4>
|
||||
<div class="outline-text-4" id="text-emf_g_Lg">
|
||||
<p>
|
||||
A more aesthetic choice is the {\bf Lorenz gauge}:
|
||||
A more aesthetic choice is the <b>Lorenz gauge</b>:
|
||||
</p>
|
||||
<div class="eqlabel" id="orgb11b2c1">
|
||||
<p>
|
||||
<a id="LorenzG"></a><a href="./emf_g_Lg.html#LorenzG"><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="org9cfa6f0">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
\[
|
||||
{\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial V}{\partial t} = 0
|
||||
\label{eq:LorenzGauge}
|
||||
{\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial \phi}{\partial t} = 0
|
||||
\tag{LorenzG}\label{LorenzG}
|
||||
\]
|
||||
which is chosen to put the second term in the left-hand side of (\ref{eq:LaplacianA}) to zero. What remains is then
|
||||
which is chosen to put the second term in the left-hand side of <a href="./emf_svp.html#LapA">LapA</a> to zero. What remains is then
|
||||
</p>
|
||||
<div class="eqlabel" id="org8bdd983">
|
||||
<p>
|
||||
<a id="ALorenzG"></a><a href="./emf_g_Lg.html#ALorenzG"><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="orgda10464">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
\[
|
||||
{\boldsymbol \nabla}^2 {\boldsymbol A} - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} = -\mu_0 {\boldsymbol J}
|
||||
\tag{ALorenzG}\label{ALorenzG}
|
||||
\]
|
||||
while the equation for \(V\) becomes
|
||||
while the equation for \(\phi\) becomes
|
||||
</p>
|
||||
<div class="eqlabel" id="org92ed269">
|
||||
<p>
|
||||
<a id="phiLorenzG"></a><a href="./emf_g_Lg.html#phiLorenzG"><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="orgf853d29">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
\[
|
||||
{\boldsymbol \nabla}^2 V - \mu_0 \varepsilon_0 \frac{\partial^2 V}{\partial t^2} = -\frac{\rho}{\varepsilon_0}.
|
||||
{\boldsymbol \nabla}^2 \phi - \mu_0 \varepsilon_0 \frac{\partial^2 \phi}{\partial t^2} = -\frac{\rho}{\varepsilon_0}.
|
||||
\tag{phiLorenzG}\label{phiLorenzG}
|
||||
\]
|
||||
These can be written compactly upon introducing a new operator: the
|
||||
</p>
|
||||
<div class="core div" id="org0aaab77">
|
||||
<div class="core div" id="orgd77f9a6">
|
||||
<p>
|
||||
<b>d'Alembertian operator</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="orgc902ec7">
|
||||
<p>
|
||||
<a id="dAl"></a><a href="./emf_g_Lg.html#dAl"><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="orgac35877">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
{\bf d'Alembertian operator}
|
||||
\[
|
||||
\square^2 \equiv {\boldsymbol \nabla}^2 - \mu_0 \varepsilon_0 \frac{\partial^2}{\partial t^2}
|
||||
\label{eq:dAlembertian}
|
||||
\]
|
||||
\square^2 \equiv {\boldsymbol \nabla}^2 - \mu_0 \varepsilon_0 \frac{\partial^2}{\partial t^2}
|
||||
\tag{dAl}\label{dAl}
|
||||
\]
|
||||
</p>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
so we get the
|
||||
</p>
|
||||
<div class="core div" id="orgc8e8b42">
|
||||
<div class="core div" id="org12b2ff9">
|
||||
<p>
|
||||
<b>Inhomogeneous Maxwell equations</b> <i>(Lorenz gauge)</i>
|
||||
</p>
|
||||
<div class="eqlabel" id="orge48b8c0">
|
||||
<p>
|
||||
<a id="MaxLor"></a><a href="./emf_g_Lg.html#MaxLor"><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="org140821b">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
{\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
|
||||
\[
|
||||
\square^2 V = -\frac{\rho}{\varepsilon_0}, \hspace{10mm}
|
||||
\square^2 {\boldsymbol A} = -\mu_0 {\boldsymbol J}
|
||||
\label{eq:InhomogeneousMaxwellLorenzGauge}
|
||||
\]
|
||||
\square^2 \phi = -\frac{\rho}{\varepsilon_0}, \hspace{10mm}
|
||||
\square^2 {\boldsymbol A} = -\mu_0 {\boldsymbol J}
|
||||
\tag{MaxLor}\label{MaxLor}
|
||||
\]
|
||||
</p>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
This gauge is especially nice in the context of special relativity.
|
||||
The whole of electrodynamics has thus reduced to solving the inhomogeneous
|
||||
wave equations (\ref{eq:InhomogeneousMaxwellLorenzGauge})
|
||||
wave equations <a href="./emf_g_Lg.html#MaxLor">MaxLor</a>
|
||||
in terms of specified sources.
|
||||
</p>
|
||||
|
||||
|
|
@ -1665,11 +1737,11 @@ in terms of specified sources.
|
|||
Without choosing the Lorenz gauge, we can still write the inhomogeneous
|
||||
Maxwell equations in a simpler form. Defining
|
||||
\[
|
||||
L \equiv {\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial V}{\partial t},
|
||||
L \equiv {\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial \phi}{\partial t},
|
||||
\]
|
||||
we have by direct inspection
|
||||
\[
|
||||
\square^2 V + \frac{\partial L}{\partial t} = -\frac{\rho}{\varepsilon_0}, \hspace{10mm}
|
||||
\square^2 \phi + \frac{\partial L}{\partial t} = -\frac{\rho}{\varepsilon_0}, \hspace{10mm}
|
||||
\square^2 {\boldsymbol A} - {\boldsymbol \nabla} L = -\mu_0 {\boldsymbol J}.
|
||||
\]
|
||||
</p>
|
||||
|
|
@ -1692,7 +1764,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1628,63 +1628,96 @@ Useful strategy: represent fields in terms of potentials.
|
|||
</p>
|
||||
|
||||
<p>
|
||||
Easiest:
|
||||
Easiest: as we already saw (<a href="./ems_ms_vp_A.html#BcurlA">BcurlA</a>), we can write the magnetic
|
||||
field as a pure curl (since its divergence always vanishes):
|
||||
</p>
|
||||
<div class="core div" id="org5bf4193">
|
||||
<div class="core div" id="orgb63eb69">
|
||||
<p>
|
||||
\[
|
||||
{\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
|
||||
\]
|
||||
{\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
|
||||
\]
|
||||
</p>
|
||||
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Putting this into Faraday's law gives
|
||||
Putting this into Faraday's law <a href="./emd_Fl_Fl.html#Fl">Fl</a> gives
|
||||
\[
|
||||
{\boldsymbol \nabla} \times \left({\boldsymbol E} + \frac{\partial {\boldsymbol A}}{\partial t} \right) = 0
|
||||
\]
|
||||
so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
|
||||
so this can be written as the gradient of a scalar.
|
||||
Making the choice \(-{\boldsymbol \nabla} \phi\) for this, we get
|
||||
</p>
|
||||
<div class="core div" id="orgffce8dc">
|
||||
<div class="core div" id="org23c61e0">
|
||||
<div class="eqlabel" id="org51cbd1b">
|
||||
<p>
|
||||
\[
|
||||
{\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}
|
||||
\label{eq:E_from_Potentials}
|
||||
\]
|
||||
<a id="E_phiA"></a><a href="./emf_svp.html#E_phiA"><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="org25412c6">
|
||||
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting (\ref{eq:E_from_Potentials}) into Gauss's law gives
|
||||
</p>
|
||||
<div class="main div" id="org26743e8">
|
||||
<p>
|
||||
\[
|
||||
{\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
|
||||
\label{eq:LaplacianV}
|
||||
\]
|
||||
</p>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
whereas Amp{\`ere}-Maxwell becomes
|
||||
\[
|
||||
{\boldsymbol \nabla} \times \left({\boldsymbol \nabla} {\boldsymbol A}\right) = \mu_0 {\boldsymbol J} - \mu_0 \varepsilon_0 {\boldsymbol \nabla} \left(\frac{\partial V}{\partial t}\right) - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2}
|
||||
{\boldsymbol E} = -{\boldsymbol \nabla} \phi - \frac{\partial {\boldsymbol A}}{\partial t}
|
||||
\tag{E_phiA}\label{E_phiA}
|
||||
\]
|
||||
which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
|
||||
</p>
|
||||
<div class="main div" id="orga86c8c2">
|
||||
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting <a href="./emf_svp.html#E_phiA">E_phiA</a> into Gauss's law gives
|
||||
</p>
|
||||
<div class="main div" id="orgf26abb4">
|
||||
<div class="eqlabel" id="org986b9b3">
|
||||
<p>
|
||||
<a id="Lapphi"></a><a href="./emf_svp.html#Lapphi"><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="orgcd04ad9">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
\[
|
||||
\left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
|
||||
{\boldsymbol \nabla}^2 \phi + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
|
||||
\tag{Lapphi}\label{Lapphi}
|
||||
\]
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>{\boldsymbol ∇} \left({\boldsymbol ∇} ⋅ {\boldsymbol A} + μ_0 ε_0 \frac{\partial V}{\partial t} \right) = -μ_0 {\boldsymbol J}</li>
|
||||
</ul>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
\label{eq:LaplacianA}
|
||||
whereas Ampère-Maxwell becomes
|
||||
\[
|
||||
{\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = \mu_0 {\boldsymbol J} - \mu_0 \varepsilon_0 {\boldsymbol \nabla} \left(\frac{\partial \phi}{\partial t}\right) - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2}
|
||||
\]
|
||||
which becomes after simple rearrangement and use of the <a href="./c_m_dc_d2.html#curlcurl">curlcurl</a> identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
|
||||
</p>
|
||||
<div class="main div" id="org135124e">
|
||||
<div class="eqlabel" id="org73667ac">
|
||||
<p>
|
||||
<a id="LapA"></a><a href="./emf_svp.html#LapA"><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="org3f09a58">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
\[
|
||||
\left( {\boldsymbol \nabla}^2 {\boldsymbol A} - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} \right) - {\boldsymbol \nabla} \left({\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial \phi}{\partial t} \right) = -\mu_0 {\boldsymbol J}
|
||||
\tag{LapA}\label{LapA}
|
||||
\]
|
||||
</p>
|
||||
|
||||
|
|
@ -1711,7 +1744,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1640,7 +1640,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1618,8 +1618,8 @@ Table of contents
|
|||
<li>Gr 3</li>
|
||||
</ul>
|
||||
|
||||
<details class="prereq" id="org192cbf5">
|
||||
<summary id="org78425f6">
|
||||
<details class="prereq" id="org1512362">
|
||||
<summary id="orgf95a20b">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1627,8 +1627,8 @@ Prerequisites
|
|||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="orge457a7e">
|
||||
<summary id="orge350b5c">
|
||||
<details class="objectives" id="org60b3244">
|
||||
<summary id="org14c49b9">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1666,7 +1666,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1624,7 +1624,7 @@ A generic configuration of static charges coupled via the Coulomb interaction
|
|||
defines an electrostatic problem, whose solution is in principle obtained
|
||||
from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
|
||||
</p>
|
||||
<div class="main div" id="orgb661984">
|
||||
<div class="main div" id="org91ccc03">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
|
|
@ -1638,7 +1638,7 @@ from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_v
|
|||
or (often simpler) by calculating the electrostatic potential, using either the
|
||||
explicit construction <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
|
||||
</p>
|
||||
<div class="main div" id="org5d15b9f">
|
||||
<div class="main div" id="orge494253">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
|
|
@ -1658,7 +1658,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
|
|||
<a href="./ems_es_ep_PL.html#Poi">🐟</a>
|
||||
</p>
|
||||
|
||||
<div class="core div" id="org7aef84c">
|
||||
<div class="core div" id="org4db04a9">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
|
|
@ -1674,7 +1674,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
|
|||
<p>
|
||||
In the specific case where the charge density vanishes, we fall back onto the simpler Laplace equation <a href="./ems_es_ep_PL.html#Lap">Lap</a>
|
||||
</p>
|
||||
<div class="core div" id="org3f412f2">
|
||||
<div class="core div" id="orgd28c357">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
|
|
@ -1712,7 +1712,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1630,14 +1630,14 @@ In one dimension, the potential is a single-variable
|
|||
function \(\phi (x)\) and the Laplace equation reads
|
||||
</p>
|
||||
|
||||
<div class="eqlabel" id="org5c6cddc">
|
||||
<div class="eqlabel" id="orgc211f0a">
|
||||
<p>
|
||||
<a id="Lap_1d"></a><a href="./ems_ca_fe_L.html#Lap_1d"><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="org5f00f86">
|
||||
<div class="alteqlabels" id="org36bff82">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1652,14 +1652,14 @@ function \(\phi (x)\) and the Laplace equation reads
|
|||
<p>
|
||||
The solution to this is
|
||||
</p>
|
||||
<div class="eqlabel" id="orgc4fd451">
|
||||
<div class="eqlabel" id="orgf5f202e">
|
||||
<p>
|
||||
<a id="Lap_1d_sol"></a><a href="./ems_ca_fe_L.html#Lap_1d_sol"><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="org1f2c6cb">
|
||||
<div class="alteqlabels" id="org25a65ef">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (3.6)</li>
|
||||
</ul>
|
||||
|
|
@ -1718,14 +1718,14 @@ In two dimensions, the potential becomes a function
|
|||
of two variables (here: \(x\) and \(y\)), so Laplace's
|
||||
equation now reads
|
||||
</p>
|
||||
<div class="eqlabel" id="orge167f78">
|
||||
<div class="eqlabel" id="org4bb6b92">
|
||||
<p>
|
||||
<a id="Lap_2d"></a><a href="./ems_ca_fe_L.html#Lap_2d"><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="org9aae751">
|
||||
<div class="alteqlabels" id="org482a454">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1778,14 +1778,14 @@ a point equals its value averaged over a sphere
|
|||
\(S_R({\bf r})\) of any radius \(R\) centered on this point
|
||||
(and of course not containing any charges),
|
||||
</p>
|
||||
<div class="eqlabel" id="orge400754">
|
||||
<div class="eqlabel" id="org96c7b40">
|
||||
<p>
|
||||
<a id="p_ball_avg"></a><a href="./ems_ca_fe_L.html#p_ball_avg"><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="orgf37e10c">
|
||||
<div class="alteqlabels" id="org550802d">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1797,8 +1797,8 @@ a point equals its value averaged over a sphere
|
|||
\]
|
||||
</p>
|
||||
|
||||
<details id="org4bb0c5f">
|
||||
<summary id="orgaf332f4">
|
||||
<details id="org8644d79">
|
||||
<summary id="orgbe634df">
|
||||
<strong>Physicist's proof</strong>
|
||||
</summary>
|
||||
<p>
|
||||
|
|
@ -1860,8 +1860,8 @@ proving the theorem.
|
|||
</p>
|
||||
</details>
|
||||
|
||||
<details id="orgfce7dc3">
|
||||
<summary id="org452b0d4">
|
||||
<details id="orgb4be215">
|
||||
<summary id="orgb09f919">
|
||||
<strong>Formal proof</strong>
|
||||
</summary>
|
||||
|
||||
|
|
@ -1911,14 +1911,14 @@ we get the following general
|
|||
<p>
|
||||
<b>Theorem</b>:
|
||||
</p>
|
||||
<div class="eqlabel" id="orgfba2553">
|
||||
<div class="eqlabel" id="org51f491a">
|
||||
<p>
|
||||
<a id="dfdR_intLap"></a><a href="./ems_ca_fe_L.html#dfdR_intLap"><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="orgc5c6de6">
|
||||
<div class="alteqlabels" id="org9bebf97">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1971,19 +1971,19 @@ are necessarily positive, we thus require \(f_x > 0\), \(f_y > 0\) and \(f
|
|||
of the \(f_x + f_y + f_z = 0\) condition above.
|
||||
</p>
|
||||
|
||||
<div class="eqlabel" id="org8465450">
|
||||
<div class="eqlabel" id="org02999c2">
|
||||
<p>
|
||||
<a id="Earnshaw"></a><a href="./ems_ca_fe_L.html#Earnshaw"><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="org0fbdf78">
|
||||
<div class="alteqlabels" id="org3cd554f">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="info div" id="org9dd741d">
|
||||
<div class="info div" id="org8708715">
|
||||
<p>
|
||||
<b>Earnshaw's theorem (physical version)</b> <br>
|
||||
</p>
|
||||
|
|
@ -2102,7 +2102,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1613,11 +1613,11 @@ Table of contents
|
|||
<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><span class="headline-id">ems.ca.fe.g</span></h5>
|
||||
<div class="outline-text-5" id="text-ems_ca_fe_g">
|
||||
<div class="info div" id="orga6b382e">
|
||||
<div class="info div" id="orgce63324">
|
||||
<p>
|
||||
<b>George Green</b>
|
||||
</p>
|
||||
<aside id="org8e26888">
|
||||
<aside id="org3d8c3b4">
|
||||
<p>
|
||||
See a <a href="https://en.wikipedia.org/wiki/George%5C_Green%5C_(mathematician)">short bio on wikipedia</a>
|
||||
</p>
|
||||
|
|
@ -1650,14 +1650,14 @@ and
|
|||
\]
|
||||
Substituting this in the divergence theorem gives <b>Green's first identity</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="orgeb35436">
|
||||
<div class="eqlabel" id="org86176bc">
|
||||
<p>
|
||||
<a id="Green1"></a><a href="./ems_ca_fe_g.html#Green1"><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="org6a05402">
|
||||
<div class="alteqlabels" id="org8e9389d">
|
||||
<ul class="org-ul">
|
||||
<li>J (1.34)</li>
|
||||
</ul>
|
||||
|
|
@ -1675,14 +1675,14 @@ As an aside for now, for completeness, if we do the same thing again but with \(
|
|||
interchanged, and subtract the result, we obtain another useful result known as
|
||||
<b>Green's second identity</b> or <b>Green's theorem</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org3cd71e0">
|
||||
<div class="eqlabel" id="orgee2e293">
|
||||
<p>
|
||||
<a id="Green2"></a><a href="./ems_ca_fe_g.html#Green2"><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="org2a9c8de">
|
||||
<div class="alteqlabels" id="org20f4498">
|
||||
<ul class="org-ul">
|
||||
<li>J (1.35)</li>
|
||||
</ul>
|
||||
|
|
@ -1718,7 +1718,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1635,14 +1635,14 @@ This yields
|
|||
The first term on the left-hand side vanishes since \(\Phi\) satisfies Laplace.
|
||||
The right-hand side can be made to vanish if \(\Phi\) obeys either
|
||||
</p>
|
||||
<div class="eqlabel" id="orgdd9023d">
|
||||
<div class="eqlabel" id="org13e35fc">
|
||||
<p>
|
||||
<a id="Dirichlet"></a><a href="./ems_ca_fe_uP.html#Dirichlet"><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="org85a3f6c">
|
||||
<div class="alteqlabels" id="orgccd338e">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1654,14 +1654,14 @@ The right-hand side can be made to vanish if \(\Phi\) obeys either
|
|||
<p>
|
||||
or
|
||||
</p>
|
||||
<div class="eqlabel" id="orgf116b4b">
|
||||
<div class="eqlabel" id="org68968dd">
|
||||
<p>
|
||||
<a id="Neumann"></a><a href="./ems_ca_fe_uP.html#Neumann"><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="org018d4be">
|
||||
<div class="alteqlabels" id="org062fea2">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1689,19 +1689,19 @@ additive constant.
|
|||
<p>
|
||||
We can thus finally state the
|
||||
</p>
|
||||
<div class="eqlabel" id="org6b09b4c">
|
||||
<div class="eqlabel" id="org154b9ea">
|
||||
<p>
|
||||
<a id="uniq_thm"></a><a href="./ems_ca_fe_uP.html#uniq_thm"><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="org2d6da13">
|
||||
<div class="alteqlabels" id="org43b26fb">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="core div" id="org10be721">
|
||||
<div class="core div" id="orgc742981">
|
||||
<p>
|
||||
<b>Uniqueness Theorem</b>
|
||||
</p>
|
||||
|
|
@ -1740,7 +1740,7 @@ Reading other books, you might be misled into thinking that there are numerous c
|
|||
and corollaries and that things are
|
||||
</p>
|
||||
|
||||
<div class="info div" id="orgf42bc42">
|
||||
<div class="info div" id="org2adbb79">
|
||||
<p>
|
||||
<b>Comment/warning</b>: <b>uniqueness theorem on uniqueness theorems</b> <br>
|
||||
Do not be misled: there is a <i>unique</i> uniqueness theorem for the
|
||||
|
|
@ -1767,7 +1767,7 @@ 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>
|
||||
|
||||
|
|
|
|||
|
|
@ -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>
|
||||
|
|
@ -1645,7 +1645,7 @@ 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>
|
||||
|
||||
|
|
|
|||
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Reference in New Issue