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Update 2022-02-10 08:34

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4
build/a.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1619,7 +1619,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

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build/a_l.html
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<!DOCTYPE html>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1702,7 +1702,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

4
build/c.html
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<!DOCTYPE html>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1619,7 +1619,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

4
build/c_m.html
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<!DOCTYPE html>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1625,7 +1625,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

4
build/c_m_cs.html
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<!DOCTYPE html>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1621,7 +1621,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

16
build/c_m_cs_cyl.html
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<!DOCTYPE html>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1643,14 +1643,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
<div id="outline-container-c_m_cs_cyl_grad" class="outline-6">
<h6 id="c_m_cs_cyl_grad"><a href="#c_m_cs_cyl_grad">Gradient</a></h6>
<div class="outline-text-6" id="text-c_m_cs_cyl_grad">
<div class="eqlabel" id="org5eb0619">
<div class="eqlabel" id="orgbec1efd">
<p>
<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">
<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="orgfcf9b20">
<div class="alteqlabels" id="org21e7bea">
<ul class="org-ul">
<li>Gr4(1.79)</li>
</ul>
@ -1671,14 +1671,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
<div id="outline-container-c_m_cs_cyl_div" class="outline-6">
<h6 id="c_m_cs_cyl_div"><a href="#c_m_cs_cyl_div">Divergence</a></h6>
<div class="outline-text-6" id="text-c_m_cs_cyl_div">
<div class="eqlabel" id="org0f08fb2">
<div class="eqlabel" id="org59ba5e0">
<p>
<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">
<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="org483e629">
<div class="alteqlabels" id="orgd4f5fe9">
<ul class="org-ul">
<li>Gr4(2.21)</li>
</ul>
@ -1699,14 +1699,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
<div id="outline-container-c_m_cs_cyl_curl" class="outline-6">
<h6 id="c_m_cs_cyl_curl"><a href="#c_m_cs_cyl_curl">Curl</a></h6>
<div class="outline-text-6" id="text-c_m_cs_cyl_curl">
<div class="eqlabel" id="org805726b">
<div class="eqlabel" id="orgc77e287">
<p>
<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">
<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="orgd6e48a1">
<div class="alteqlabels" id="org808a8dd">
<ul class="org-ul">
<li>Gr4(2.21)</li>
</ul>
@ -1753,7 +1753,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

4
build/c_m_cs_hyp.html
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<!DOCTYPE html>
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<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1614,7 +1614,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

4
build/c_m_cs_sph.html
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<!DOCTYPE html>
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<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1727,7 +1727,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

4
build/c_m_dc.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1624,7 +1624,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

4
build/c_m_dc_curl.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1626,7 +1626,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

34
build/c_m_dc_d2.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
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<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1599,9 +1599,9 @@ Table of contents
<div class="outline-text-5" id="text-c_m_dc_d2">
</div>
<div id="outline-container-orgddf1a80" class="outline-6">
<h6 id="orgddf1a80"><a href="#orgddf1a80">Divergence of gradient</a></h6>
<div class="outline-text-6" id="text-orgddf1a80">
<div id="outline-container-orgd5309ec" class="outline-6">
<h6 id="orgd5309ec"><a href="#orgd5309ec">Divergence of gradient</a></h6>
<div class="outline-text-6" id="text-orgd5309ec">
<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
@ -1610,36 +1610,36 @@ given by the Laplacian of the corresponding vector elements.
</div>
</div>
<div id="outline-container-org1c6301d" class="outline-6">
<h6 id="org1c6301d"><a href="#org1c6301d">Curl of a gradient</a></h6>
<div class="outline-text-6" id="text-org1c6301d">
<div id="outline-container-org3f8f01e" class="outline-6">
<h6 id="org3f8f01e"><a href="#org3f8f01e">Curl of a gradient</a></h6>
<div class="outline-text-6" id="text-org3f8f01e">
<p>
This always vanishes.
</p>
</div>
</div>
<div id="outline-container-orgce82ec3" class="outline-6">
<h6 id="orgce82ec3"><a href="#orgce82ec3">Gradient of the divergence</a></h6>
<div class="outline-text-6" id="text-orgce82ec3">
<div id="outline-container-org86a1e1f" class="outline-6">
<h6 id="org86a1e1f"><a href="#org86a1e1f">Gradient of the divergence</a></h6>
<div class="outline-text-6" id="text-org86a1e1f">
<p>
\({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.
</p>
</div>
</div>
<div id="outline-container-org820bb5e" class="outline-6">
<h6 id="org820bb5e"><a href="#org820bb5e">Divergence of a curl</a></h6>
<div class="outline-text-6" id="text-org820bb5e">
<div id="outline-container-orgdca9d9a" class="outline-6">
<h6 id="orgdca9d9a"><a href="#orgdca9d9a">Divergence of a curl</a></h6>
<div class="outline-text-6" id="text-orgdca9d9a">
<p>
This always vanishes.
</p>
</div>
</div>
<div id="outline-container-org75d969a" class="outline-6">
<h6 id="org75d969a"><a href="#org75d969a">Curl of curl</a></h6>
<div class="outline-text-6" id="text-org75d969a">
<div id="outline-container-org613cb02" class="outline-6">
<h6 id="org613cb02"><a href="#org613cb02">Curl of curl</a></h6>
<div class="outline-text-6" id="text-org613cb02">
<p>
\[
{\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v}
@ -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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

4
build/c_m_dc_del.html
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@ -1,7 +1,7 @@
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<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1625,7 +1625,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

4
build/c_m_dc_div.html
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<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1622,7 +1622,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

4
build/c_m_dc_g.html
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<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1646,7 +1646,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

4
build/c_m_dc_pr.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
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<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1619,7 +1619,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

4
build/c_m_dd.html
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@ -1,7 +1,7 @@
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<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1621,7 +1621,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

4
build/c_m_dd_1d.html
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@ -1,7 +1,7 @@
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1633,7 +1633,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

12
build/c_m_dd_3d.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
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<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1620,14 +1620,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
More generally,
</p>
<div class="eqlabel" id="orga2fc830">
<div class="eqlabel" id="org16a4f22">
<p>
<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">
<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="org6add17c">
<div class="alteqlabels" id="orgf96ad98">
<ul class="org-ul">
<li>Gr (1.100)</li>
</ul>
@ -1646,14 +1646,14 @@ More generally,
Since
</p>
<div class="eqlabel" id="org321d3a4">
<div class="eqlabel" id="org71f92c6">
<p>
<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">
<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="orgb03949d">
<div class="alteqlabels" id="org39f00df">
<ul class="org-ul">
<li>Gr (1.101)</li>
</ul>
@ -1691,7 +1691,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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</div>

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<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-02-09 Wed 22:40</p>
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@ -1624,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
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@ -1631,7 +1631,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
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@ -1632,7 +1632,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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build/c_m_ic_lsv.html
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<meta charset="utf-8">
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<title>Pre-Quantum Electrodynamics</title>
@ -1599,9 +1599,9 @@ Table of contents
<div class="outline-text-5" id="text-c_m_ic_lsv">
</div>
<div id="outline-container-org4fa4b6a" class="outline-6">
<h6 id="org4fa4b6a"><a href="#org4fa4b6a">Line Integrals</a></h6>
<div class="outline-text-6" id="text-org4fa4b6a">
<div id="outline-container-org3f3c8c7" class="outline-6">
<h6 id="org3f3c8c7"><a href="#org3f3c8c7">Line Integrals</a></h6>
<div class="outline-text-6" id="text-org3f3c8c7">
<p>
\[
{\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l}
@ -1630,9 +1630,9 @@ Integral over a closed loop:
</div>
</div>
<div id="outline-container-org3cf12c2" class="outline-6">
<h6 id="org3cf12c2"><a href="#org3cf12c2">Surface Integrals</a></h6>
<div class="outline-text-6" id="text-org3cf12c2">
<div id="outline-container-orgbda984c" class="outline-6">
<h6 id="orgbda984c"><a href="#orgbda984c">Surface Integrals</a></h6>
<div class="outline-text-6" id="text-orgbda984c">
<p>
\[
\int_{\cal S} {\bf v} \cdot d{\bf a}
@ -1652,9 +1652,9 @@ Over a closed surface:
</div>
</div>
<div id="outline-container-orgd48f6eb" class="outline-6">
<h6 id="orgd48f6eb"><a href="#orgd48f6eb">Volume Integrals</a></h6>
<div class="outline-text-6" id="text-orgd48f6eb">
<div id="outline-container-orgc3891db" class="outline-6">
<h6 id="orgc3891db"><a href="#orgc3891db">Volume Integrals</a></h6>
<div class="outline-text-6" id="text-orgc3891db">
<p>
\[
\int_{\cal V} T d\tau
@ -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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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</div>

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build/c_m_ic_stokes.html
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<title>Pre-Quantum Electrodynamics</title>
@ -1597,14 +1597,14 @@ 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">c.m.ic.stokes</span></h5>
<div class="outline-text-5" id="text-c_m_ic_stokes">
<div class="eqlabel" id="org54da3f4">
<div class="eqlabel" id="org9333ada">
<p>
<a id="Stokes"></a><a href="./c_m_ic_stokes.html#Stokes"><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"/>
</svg></a>
</p>
<div class="alteqlabels" id="org3a4e967">
<div class="alteqlabels" id="orgf5a9283">
<ul class="org-ul">
<li>Gr (1.57)</li>
</ul>
@ -1647,7 +1647,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
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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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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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build/d.html
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<!DOCTYPE html>
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<meta charset="utf-8">
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<title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">d</span></h2>
<div class="outline-text-2" id="text-d">
<details class="objectives" id="org40c4648">
<summary id="orgce477d7">
<details class="objectives" id="orge321671">
<summary id="orgc02d8c9">
Objectives
</summary>
@ -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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1641,7 +1641,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1636,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1636,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -1658,7 +1658,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="date">Created: 2022-02-09 Wed 22:40</p>
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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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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
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<p class="author">Author: Jean-Sébastien Caux</p>
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@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">emd</span></h2>
<div class="outline-text-2" id="text-emd">
<details class="prereq" id="orgc5008fa">
<summary id="org88d0f1c">
<details class="prereq" id="org09895e9">
<summary id="org4a0bf93">
Prerequisites
</summary>
<ul class="org-ul">
@ -1608,8 +1608,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="org036b173">
<summary id="org9627cee">
<details class="objectives" id="orgef3f98e">
<summary id="org1810cca">
Objectives
</summary>
<ul class="org-ul">
@ -1648,7 +1648,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -1622,7 +1622,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1638,7 +1638,7 @@ 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="orge05960f">
<div class="core div" id="orgee21897">
<p>
<b>Electromotive force (or electromotance)</b>,
\[
@ -1660,7 +1660,7 @@ to the rate of change of the magnetic flux,
\]
so we obtain
</p>
<div class="core div" id="org726542f">
<div class="core div" id="orgd1f18df">
<p>
<b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
\[
@ -1678,7 +1678,7 @@ for any loop (on a wire or not). Using Stokes' theorem,
\]
we obtain
</p>
<div class="core div" id="orge08eeaf">
<div class="core div" id="org04bf1f1">
<p>
<b>Faraday's law</b> (differential form)
\[
@ -1713,7 +1713,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1657,7 +1657,7 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold
\]
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="org430dd21">
<div class="core div" id="org183e158">
<p>
\[
W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2
@ -1678,7 +1678,7 @@ W_{mag} = \frac{1}{2} \int d\tau ({\bf A} \cdot {\bf J}) = \frac{1}{2\mu_0} \int
\hspace{2cm} \mbox{(7.31 and 7.34)}
\end{align}
<div class="example div" id="org2441b5d">
<div class="example div" id="org5a2dc32">
<p>
\paragraph{Example 7.13:} coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\).
Find energy stored in section of length \(l\).
@ -1715,7 +1715,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1637,7 +1637,7 @@ M_{12} = M_{21}
\]
</p>
<div class="example div" id="org1839798">
<div class="example div" id="orgb4beaba">
<p>
\paragraph{Example 7.10:}
short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) lies concentrically inside
@ -1687,7 +1687,7 @@ Inductance: measured in {\bf henries} (\(H\)). \(H = V s/A\).
</p>
<div class="example div" id="org435f19e">
<div class="example div" id="org24edf9c">
<p>
\paragraph{Example 7.11:} find self-inductance of toroidal coil with
rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\))
@ -1714,7 +1714,7 @@ Total flux: \(N\) times this, so self-inductance is
Inductance (like capacitance) is intrinsically positive. Use Lenz law. Think of {\bf back EMF}.
</p>
<div class="example div" id="orgadb84c5">
<div class="example div" id="org0b14b9f">
<p>
\paragraph{Example 7.12:} circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\).
What is the current ?
@ -1751,7 +1751,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -1621,7 +1621,7 @@ law in integral form:
<div class="example div" id="org0e50d64">
<div class="example div" id="org7f7b579">
<p>
{\bf Example 7.7:}
\({\bf B}(t)\) points up in circular region of radius \(R\). What is the induced \({\bf E}(t)\) ?
@ -1637,7 +1637,7 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
</div>
<div class="example div" id="orgda9093e">
<div class="example div" id="orgd8d7c07">
<p>
{\bf Example 7.8:} wheel or radius \(b\) with line charge \(\lambda\) on the rim.
Uniform magnetic field \({\bf B}_0\) in central region up to \(a &lt; b\),
@ -1671,7 +1671,7 @@ called the {\bf quasistatic} approximation, and works provided we deal with
'slow enough' phenomena.
</p>
<div class="example div" id="orgcb11b45">
<div class="example div" id="org1188389">
<p>
{\bf Example 7.9:} infinitely long straight wire carries \(I(t)\). Find
induced \({\bf E}\) field as a function of distance \(s\) from wire.
@ -1717,7 +1717,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -1622,7 +1622,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -1600,7 +1600,7 @@ Table of contents
<p>
Full set of equations for the electromagnetic field:
</p>
<div class="core div" id="orga5920b3">
<div class="core div" id="org8953ea9">
<p>
{\bf Maxwell's equations} {\it (in vacuum)}
</p>
@ -1616,7 +1616,7 @@ Full set of equations for the electromagnetic field:
<p>
Complement:
</p>
<div class="core div" id="org9ee0da3">
<div class="core div" id="org04de767">
<p>
{\bf Force law}
\[
@ -1640,7 +1640,7 @@ take divergence of \((iv)\).
<p>
Better way of writing: all fields on left, all sources on right,
</p>
<div class="core div" id="orgdda2f9a">
<div class="core div" id="org8800b57">
\begin{align}
(i) &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0},
&amp;(iii) {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} = 0, \\
@ -1668,7 +1668,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
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@ -1607,7 +1607,7 @@ the continuity equation as
\]
The extra term would thus be eliminated if we were to put
</p>
<div class="core div" id="org45d3060">
<div class="core div" id="orgba15335">
<p>
\[
{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}
@ -1631,7 +1631,7 @@ Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation
<p>
Maxwell baptized this term the
</p>
<div class="core div" id="org90773a4">
<div class="core div" id="org79a35ce">
<p>
{\bf Displacement current}
\[
@ -1677,7 +1677,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -1647,7 +1647,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -1,7 +1,7 @@
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<title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,8 @@ 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="org9908421">
<summary id="orgdbb6aa4">
<details class="prereq" id="orga7642d4">
<summary id="org89bea54">
Prerequisites
</summary>
<ul class="org-ul">
@ -1607,8 +1607,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="org1b20ea0">
<summary id="orgc447c59">
<details class="objectives" id="orgf85a819">
<summary id="orgb7f774b">
Objectives
</summary>
<ul class="org-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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

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@ -1,7 +1,7 @@
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<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1600,7 +1600,7 @@ Table of contents
<p>
The angular momentum of EM fields is directly given by
</p>
<div class="main div" id="orgd94c7c7">
<div class="main div" id="orgbedab67">
<p>
{\bf Angular momentum of EM fields}
\[
@ -1628,7 +1628,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

6
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@ -1,7 +1,7 @@
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<title>Pre-Quantum Electrodynamics</title>
@ -1619,7 +1619,7 @@ This means that
\]
Since this is true for any volume, we have (re)derived the
</p>
<div class="core div" id="org17b3983">
<div class="core div" id="org8328070">
<p>
{\bf Continuity equation}
\[
@ -1658,7 +1658,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
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8
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@ -1,7 +1,7 @@
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<title>Pre-Quantum Electrodynamics</title>
@ -1612,7 +1612,7 @@ 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="org37cec17">
<div class="main div" id="orgfb9b180">
<p>
{\bf Momentum density in the EM fields}
\[
@ -1624,7 +1624,7 @@ 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="orgcd3036a">
<div class="main div" id="org03df6a3">
<p>
{\bf Continuity equation for EM momentum}
\[
@ -1651,7 +1651,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

10
build/emd_ce_mst.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1654,7 +1654,7 @@ and similarly for \({\boldsymbol B}\). We thus get
<p>
This expression can be greatly simplified by introducing the
</p>
<div class="main div" id="orge78bc78">
<div class="main div" id="org7f141f8">
<p>
{\bf Maxwell stress tensor}
\[
@ -1677,7 +1677,7 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
<p>
We then obtain
</p>
<div class="main div" id="org8f28e32">
<div class="main div" id="org6e156bd">
<p>
{\bf EM force per unit volume}
\[
@ -1689,7 +1689,7 @@ We then obtain
<p>
where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
</p>
<div class="main div" id="org942d34e">
<div class="main div" id="org7c98547">
<p>
{\bf Total force on charges in volume}
\[
@ -1716,7 +1716,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

16
build/emd_ce_poy.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<title>Pre-Quantum Electrodynamics</title>
@ -1666,7 +1666,7 @@ so we get
Substituting this in \ref{Gr(8.6)} and using the divergence theorem,
we obtain
</p>
<div class="main div" id="orgc3e7fad">
<div class="main div" id="orgab4b619">
<p>
{\bf Poynting's theorem}
\[
@ -1691,7 +1691,7 @@ 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:
</p>
<div class="core div" id="orgf0750fa">
<div class="core div" id="org9cfe7aa">
<p>
{\bf Poynting vector}
\[
@ -1704,7 +1704,7 @@ Energy per unit time, per unit area carried by EM fields:
<p>
We can thus express Poynting's theorem more compactly:
</p>
<div class="core div" id="orged7fe8f">
<div class="core div" id="orgdeee59b">
<p>
{\bf Poynting's theorem}
\[
@ -1717,7 +1717,7 @@ We can thus express Poynting's theorem more compactly:
<p>
where we have defined the total
</p>
<div class="core div" id="org27d8494">
<div class="core div" id="org57db6a8">
<p>
{\bf Energy in electromagnetic fields}
\[
@ -1740,7 +1740,7 @@ Then,
\]
so we get the
</p>
<div class="core div" id="org40d56f8">
<div class="core div" id="org65ae443">
<p>
{\bf Poynting theorem (differential form)}
\[
@ -1757,7 +1757,7 @@ and has a similar for to the continuity equation
<div class="example div" id="orge1eb64c">
<div class="example div" id="org0e3381a">
<p>
\paragraph{Example 8.1} Current in a wire: Joule heating. Energy per unit time delivered to wire: from Poynting.
Assuming that the field is uniform, the electric field parallel to the wire is
@ -1800,7 +1800,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

12
build/emd_emw.html
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@ -1,7 +1,7 @@
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<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,8 @@ 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="org2757254">
<summary id="org0bb0a4d">
<details class="prereq" id="org3bd5d73">
<summary id="orgec57498">
Prerequisites
</summary>
<ul class="org-ul">
@ -1608,8 +1608,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="orga6e02f8">
<summary id="org0f51f09">
<details class="objectives" id="org7eaef42">
<summary id="org71ec3b5">
Objectives
</summary>
<ul class="org-ul">
@ -1648,7 +1648,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

8
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@ -1,7 +1,7 @@
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<title>Pre-Quantum Electrodynamics</title>
@ -1628,7 +1628,7 @@ so for a monochromatic EM plan wave,
\]
or more succinctly:
</p>
<div class="main div" id="org19f6b89">
<div class="main div" id="org7992b89">
<p>
{\bf Poynting vector of a monochromatic EM wave}
\[
@ -1644,7 +1644,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
<p>
Similary, we get the
</p>
<div class="main div" id="org8e7b46e">
<div class="main div" id="org0141dca">
<p>
{\bf Momentum density of a monochromatic EM wave}
\[
@ -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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

6
build/emd_emw_mpw.html
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@ -1,7 +1,7 @@
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<title>Pre-Quantum Electrodynamics</title>
@ -1629,7 +1629,7 @@ 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="org7add9b6">
<div class="core div" id="org0336f47">
<p>
{\bf E and B fields for a monochromatic EM plane wave}
\[
@ -1671,7 +1671,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

6
build/emd_emw_we.html
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@ -1,7 +1,7 @@
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<title>Pre-Quantum Electrodynamics</title>
@ -1626,7 +1626,7 @@ 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="org2e89450">
<div class="core div" id="org832615b">
<p>
{\bf Wave equations for electric and magnetic fields in vacuum}
\[
@ -1680,7 +1680,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

4
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@ -1,7 +1,7 @@
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<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1620,7 +1620,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
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4
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1620,7 +1620,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
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20
build/emdm_Me_Mem.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<title>Pre-Quantum Electrodynamics</title>
@ -1633,7 +1633,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
\]
We therefore have the
</p>
<div class="core div" id="org643b4aa">
<div class="core div" id="orga19bc43">
<p>
{\bf Polarization current density}
\[
@ -1651,7 +1651,7 @@ the polarization current is the result of linear motion of charge when
polarization changes). We can check consistency with the continuity equation
associated to the conservation of bound charges:
</p>
<aside id="orgf5df19a">
<aside id="org7492618">
<p>
Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current
\({\boldsymbol J}_b\) but this is not the convention used here.
@ -1674,7 +1674,7 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
In view of this: total charge density can be separated into 2 parts,
{\it free} and {\it bound}:
</p>
<div class="main div" id="orgfd1ad39">
<div class="main div" id="org6458291">
<p>
\[
\rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
@ -1687,7 +1687,7 @@ In view of this: total charge density can be separated into 2 parts,
and current can be separated into three parts, {\it free}, {\it bound} and
{\it polarization}:
</p>
<div class="main div" id="org09b6579">
<div class="main div" id="org79300fd">
<p>
\[
{\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
@ -1711,7 +1711,7 @@ Gauss's law: can be rewritten
\]
where (as in static case)
</p>
<div class="core div" id="orgcb69874">
<div class="core div" id="org0cf7843">
<p>
\[
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
@ -1737,7 +1737,7 @@ or
\]
where as before
</p>
<div class="core div" id="orga76243f">
<div class="core div" id="org4d58c35">
<p>
\[
{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
@ -1755,7 +1755,7 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
<p>
In terms of free charges and currents, we thus get
</p>
<div class="core div" id="org5d9c2e0">
<div class="core div" id="org4ca5e56">
<p>
{\bf Maxwell's equations {\it (in matter)}}
</p>
@ -1781,7 +1781,7 @@ Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and
in terms of \({\bf E}\) and \({\bf B}\).
For the restricted case of linear media:
</p>
<div class="main div" id="orgca90a26">
<div class="main div" id="orgac29d47">
<p>
\[
{\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
@ -1816,7 +1816,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

6
build/emdm_Me_bc.html
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@ -1,7 +1,7 @@
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<title>Pre-Quantum Electrodynamics</title>
@ -1600,7 +1600,7 @@ Table of contents
<p>
Discontinuities between different media, deduced from
</p>
<div class="core div" id="org1ce4cee">
<div class="core div" id="org72eb54c">
<p>
{\bf Maxwell's equations {\it (in matter)}, integral form}
</p>
@ -1689,7 +1689,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
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4
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<title>Pre-Quantum Electrodynamics</title>
@ -1623,7 +1623,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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4
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@ -1619,7 +1619,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -1715,7 +1715,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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6
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<title>Pre-Quantum Electrodynamics</title>
@ -1635,7 +1635,7 @@ 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="org4373079">
<div class="main div" id="orgc7bc968">
<p>
{\bf Index of refraction}
\[
@ -1690,7 +1690,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
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4
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<title>Pre-Quantum Electrodynamics</title>
@ -1634,7 +1634,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -1614,7 +1614,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -1615,7 +1615,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -1706,7 +1706,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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12
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@ -1,7 +1,7 @@
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<title>Pre-Quantum Electrodynamics</title>
@ -1635,7 +1635,7 @@ 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="orga0e77f3">
<div class="core div" id="orgb0a9710">
<p>
{\bf First law of reflection:}
the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
@ -1650,7 +1650,7 @@ Specializing (\ref{eq:RTObliquek}) to our notations, we have
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="orgc1a2ec9">
<div class="core div" id="orga98e40b">
<p>
{\bf Law of reflection}
\[
@ -1708,7 +1708,7 @@ while the third equation becomes
\]
Writing everything in terms of the incident amplitude, we get
</p>
<div class="main div" id="org933ec71">
<div class="main div" id="org521f546">
<p>
{\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)}
\[
@ -1728,7 +1728,7 @@ Amplitudes for transmitted and reflected wave: depend on angle of incidence:
Behaviour: for \(\theta_I = 0\) we recover (\ref{Gr(9.82)}).
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="orga399926">
<div class="main div" id="org51d503c">
<p>
{\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}}
\[
@ -1776,7 +1776,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1615,7 +1615,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -1621,7 +1621,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -1641,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
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<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -1665,7 +1665,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1674,7 +1674,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">emf</span></h2>
<div class="outline-text-2" id="text-emf">
<details class="prereq" id="orge054fff">
<summary id="orgb667583">
<details class="prereq" id="orgc2e1e28">
<summary id="orge525c1b">
Prerequisites
</summary>
<ul class="org-ul">
@ -1607,8 +1607,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="org92751e6">
<summary id="org8a7509c">
<details class="objectives" id="orgcd01d6b">
<summary id="org510a3e2">
Objectives
</summary>
<ul class="org-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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

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<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
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8
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<title>Pre-Quantum Electrodynamics</title>
@ -1613,7 +1613,7 @@ while the equation for \(V\) becomes
\]
These can be written compactly upon introducing a new operator: the
</p>
<div class="core div" id="org0a132cf">
<div class="core div" id="org9685743">
<p>
{\bf d'Alembertian operator}
\[
@ -1626,7 +1626,7 @@ These can be written compactly upon introducing a new operator: the
<p>
so we get the
</p>
<div class="core div" id="org6df9df9">
<div class="core div" id="org8c1d592">
<p>
{\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
\[
@ -1674,7 +1674,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
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@ -1,7 +1,7 @@
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<title>Pre-Quantum Electrodynamics</title>
@ -1614,7 +1614,7 @@ Useful strategy: represent fields in terms of potentials.
<p>
Easiest:
</p>
<div class="core div" id="org043dbf2">
<div class="core div" id="org449388e">
<p>
\[
{\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
@ -1630,7 +1630,7 @@ Putting this into Faraday's law gives
\]
so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
</p>
<div class="core div" id="org8471d76">
<div class="core div" id="org8d2221f">
<p>
\[
{\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}
@ -1643,7 +1643,7 @@ so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \
<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="orgb6c325d">
<div class="main div" id="orgf979da5">
<p>
\[
{\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
@ -1659,7 +1659,7 @@ whereas Amp{\`ere}-Maxwell becomes
\]
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="org4caa993">
<div class="main div" id="orgc475199">
<p>
\[
\left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
@ -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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
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@ -1622,7 +1622,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">ems.ca</span></h3>
<div class="outline-text-3" id="text-ems_ca">
<details class="prereq" id="orgd1b9be9">
<summary id="orgc5d1771">
<details class="prereq" id="orga055812">
<summary id="org2cb689b">
Prerequisites
</summary>
<ul class="org-ul">
@ -1607,8 +1607,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="org547b0d5">
<summary id="org5765814">
<details class="objectives" id="orgb006f6c">
<summary id="org037b124">
Objectives
</summary>
<ul class="org-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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
</div>

12
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@ -1,7 +1,7 @@
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<title>Pre-Quantum Electrodynamics</title>
@ -1603,7 +1603,7 @@ 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="org6cf413c">
<div class="main div" id="org27034ee">
\begin{equation*}
{\bf E} ({\bf r}) = \frac{1}{4\pi\varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') \frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3}
\end{equation*}
@ -1613,7 +1613,7 @@ 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 (\ref{eq:V_from_rho})
</p>
<div class="main div" id="orgde2ce45">
<div class="main div" id="org1620b08">
<p>
\[
V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|}.
@ -1629,7 +1629,7 @@ condition (\ref{Gr(2.20)}) can be expressed as the single
'local' (differential) condition (Poisson's equation) (\ref{eq:Poisson})
</p>
<div class="core div" id="org10253c2">
<div class="core div" id="org1361b41">
<p>
\[
{\boldsymbol \nabla}^2 V = -\frac{\rho}{\varepsilon_0}.
@ -1643,7 +1643,7 @@ condition (\ref{Gr(2.20)}) can be expressed as the single
In the specific case where the charge density vanishes, we fall back onto the simpler
Laplace equation
</p>
<div class="core div" id="orgd876622">
<div class="core div" id="org2305746">
<p>
\[
{\boldsymbol \nabla}^2 V = 0
@ -1676,7 +1676,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
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@ -1677,7 +1677,7 @@ are necessarily positive, we thus require \(f_x &gt; 0\), \(f_y &gt; 0\) and \(f
of the \(f_x + f_y + f_z = 0\) condition above.
</p>
<div class="info div" id="org5574d7b">
<div class="info div" id="orge499be0">
<p>
<b>Earnshaw's theorem</b> <br>
Since solutions to Laplace's equation have no local minimum,
@ -1778,7 +1778,7 @@ 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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1597,11 +1597,11 @@ 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="org487295f">
<div class="info div" id="org5bc5392">
<p>
<b>George Green</b>
</p>
<aside id="orgf55ecae">
<aside id="orgb80e217">
<p>
See a \href{short biography of George Green on Wikipedia}{https://en.wikipedia.org/wiki/George\_Green\_(mathematician)}.
</p>
@ -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: 2022-02-09 Wed 22:40</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="validation"></p>
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