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Update 2022-02-17 08:44

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4
build/a.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-15 Tue 10:14 -->
<!-- 2022-02-17 Thu 08:42 -->
<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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>

4
build/a_l.html
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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-15 Tue 10:14 -->
<!-- 2022-02-17 Thu 08:42 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1704,7 +1704,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>

4
build/c.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-15 Tue 10:14 -->
<!-- 2022-02-17 Thu 08:42 -->
<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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>

4
build/c_m.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-15 Tue 10:14 -->
<!-- 2022-02-17 Thu 08:42 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1627,7 +1627,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>

4
build/c_m_cs.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-15 Tue 10:14 -->
<!-- 2022-02-17 Thu 08:42 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>

16
build/c_m_cs_cyl.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-15 Tue 10:14 -->
<!-- 2022-02-17 Thu 08:42 -->
<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: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
<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="orgd10bfd1">
<div class="eqlabel" id="org7b9a99f">
<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="orgdbb0a24">
<div class="alteqlabels" id="org2c751cd">
<ul class="org-ul">
<li>Gr4(1.79)</li>
</ul>
@ -1671,14 +1671,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
<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="orgc1d1989">
<div class="eqlabel" id="orga3e29d1">
<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="org6cb8be3">
<div class="alteqlabels" id="org3d0b2fa">
<ul class="org-ul">
<li>Gr4(2.21)</li>
</ul>
@ -1699,14 +1699,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
<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="orgb5f05c6">
<div class="eqlabel" id="org204f3ee">
<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="org2b16116">
<div class="alteqlabels" id="org42e15eb">
<ul class="org-ul">
<li>Gr4(2.21)</li>
</ul>
@ -1755,7 +1755,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>

4
build/c_m_cs_hyp.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-15 Tue 10:14 -->
<!-- 2022-02-17 Thu 08:42 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1616,7 +1616,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>

36
build/c_m_cs_sph.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-15 Tue 10:14 -->
<!-- 2022-02-17 Thu 08:42 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1607,14 +1607,14 @@ which \(r\) is the distance from the chosen origin,
The usual Cartesian coordinates relate to spherical coordinates
according to
</p>
<div class="eqlabel" id="orga151b97">
<div class="eqlabel" id="orgbcfbea1">
<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="orgb53bb4c">
<div class="alteqlabels" id="org71b8aed">
</div>
@ -1639,14 +1639,14 @@ A generic vector can be expressed as
where the explicit relation between spherical and
Cartesian unit vectors is
</p>
<div class="eqlabel" id="orgbbd22fd">
<div class="eqlabel" id="org6e45fd1">
<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="org83278a4">
<div class="alteqlabels" id="org5b78c82">
</div>
@ -1669,14 +1669,14 @@ and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).
<p>
An infinitesimal displacement \(d{\bf l}\) can be written as
</p>
<div class="eqlabel" id="org4e012db">
<div class="eqlabel" id="org673b085">
<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="orgbe0d2b4">
<div class="alteqlabels" id="org77c531c">
</div>
@ -1692,14 +1692,14 @@ d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin
<p>
Infinitesimal volume element:
</p>
<div class="eqlabel" id="org899678f">
<div class="eqlabel" id="org1dbbdd5">
<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="orgefd25f6">
<div class="alteqlabels" id="org00dd06e">
</div>
@ -1720,14 +1720,14 @@ 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="org7d9f0e0">
<div class="eqlabel" id="org80c4f55">
<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="org5efd485">
<div class="alteqlabels" id="org0ea0f03">
</div>
@ -1744,14 +1744,14 @@ 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="orgd0fb802">
<div class="eqlabel" id="orgfe29a0f">
<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="orgb9c869a">
<div class="alteqlabels" id="org5b16e24">
</div>
@ -1768,14 +1768,14 @@ 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="orgefcd74f">
<div class="eqlabel" id="orga4479e8">
<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="orge79eaf4">
<div class="alteqlabels" id="org557f5ed">
</div>
@ -1794,14 +1794,14 @@ 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="org533eddc">
<div class="eqlabel" id="orgecfce1d">
<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="org5ad2288">
<div class="alteqlabels" id="org01d6eb3">
</div>
@ -1835,7 +1835,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>

4
build/c_m_dc.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-15 Tue 10:14 -->
<!-- 2022-02-17 Thu 08:42 -->
<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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>

4
build/c_m_dc_curl.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-15 Tue 10:14 -->
<!-- 2022-02-17 Thu 08:42 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>

34
build/c_m_dc_d2.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-15 Tue 10:14 -->
<!-- 2022-02-17 Thu 08:42 -->
<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-orga412b41" class="outline-6">
<h6 id="orga412b41"><a href="#orga412b41">Divergence of gradient</a></h6>
<div class="outline-text-6" id="text-orga412b41">
<div id="outline-container-org48d9aa2" class="outline-6">
<h6 id="org48d9aa2"><a href="#org48d9aa2">Divergence of gradient</a></h6>
<div class="outline-text-6" id="text-org48d9aa2">
<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-orgf64a0dc" class="outline-6">
<h6 id="orgf64a0dc"><a href="#orgf64a0dc">Curl of a gradient</a></h6>
<div class="outline-text-6" id="text-orgf64a0dc">
<div id="outline-container-org527fbc8" class="outline-6">
<h6 id="org527fbc8"><a href="#org527fbc8">Curl of a gradient</a></h6>
<div class="outline-text-6" id="text-org527fbc8">
<p>
This always vanishes.
</p>
</div>
</div>
<div id="outline-container-org51f1f70" class="outline-6">
<h6 id="org51f1f70"><a href="#org51f1f70">Gradient of the divergence</a></h6>
<div class="outline-text-6" id="text-org51f1f70">
<div id="outline-container-org9758e01" class="outline-6">
<h6 id="org9758e01"><a href="#org9758e01">Gradient of the divergence</a></h6>
<div class="outline-text-6" id="text-org9758e01">
<p>
\({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.
</p>
</div>
</div>
<div id="outline-container-orge147e0c" class="outline-6">
<h6 id="orge147e0c"><a href="#orge147e0c">Divergence of a curl</a></h6>
<div class="outline-text-6" id="text-orge147e0c">
<div id="outline-container-org5e919b4" class="outline-6">
<h6 id="org5e919b4"><a href="#org5e919b4">Divergence of a curl</a></h6>
<div class="outline-text-6" id="text-org5e919b4">
<p>
This always vanishes.
</p>
</div>
</div>
<div id="outline-container-org2ce8322" class="outline-6">
<h6 id="org2ce8322"><a href="#org2ce8322">Curl of curl</a></h6>
<div class="outline-text-6" id="text-org2ce8322">
<div id="outline-container-org0eb22e6" class="outline-6">
<h6 id="org0eb22e6"><a href="#org0eb22e6">Curl of curl</a></h6>
<div class="outline-text-6" id="text-org0eb22e6">
<p>
\[
{\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v}
@ -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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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@ -1648,7 +1648,7 @@ 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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@ -1620,14 +1620,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
More generally,
</p>
<div class="eqlabel" id="org1e8620f">
<div class="eqlabel" id="org53bf72e">
<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="org811a9d7">
<div class="alteqlabels" id="orgf5950bf">
<ul class="org-ul">
<li>Gr (1.100)</li>
</ul>
@ -1646,14 +1646,14 @@ More generally,
Since
</p>
<div class="eqlabel" id="org23b7fff">
<div class="eqlabel" id="org95ea3a2">
<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="orgabc977b">
<div class="alteqlabels" id="org173d151">
<ul class="org-ul">
<li>Gr (1.101)</li>
</ul>
@ -1669,14 +1669,14 @@ Since
<p>
we have that
</p>
<div class="eqlabel" id="orgbf46b11">
<div class="eqlabel" id="orgf8803cf">
<p>
<a id="Lap1or"></a><a href="./c_m_dd_3d.html#Lap1or"><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="org436deb8">
<div class="alteqlabels" id="orgaa458a9">
<ul class="org-ul">
<li>Gr (1.102)</li>
</ul>
@ -1708,7 +1708,7 @@ 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: 2022-02-15 Tue 10:14</p>
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@ -1641,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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@ -1599,9 +1599,9 @@ Table of contents
<div class="outline-text-5" id="text-c_m_ic_lsv">
</div>
<div id="outline-container-org16d704a" class="outline-6">
<h6 id="org16d704a"><a href="#org16d704a">Line Integrals</a></h6>
<div class="outline-text-6" id="text-org16d704a">
<div id="outline-container-org524bc35" class="outline-6">
<h6 id="org524bc35"><a href="#org524bc35">Line Integrals</a></h6>
<div class="outline-text-6" id="text-org524bc35">
<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-orgeef7716" class="outline-6">
<h6 id="orgeef7716"><a href="#orgeef7716">Surface Integrals</a></h6>
<div class="outline-text-6" id="text-orgeef7716">
<div id="outline-container-org0cc38e3" class="outline-6">
<h6 id="org0cc38e3"><a href="#org0cc38e3">Surface Integrals</a></h6>
<div class="outline-text-6" id="text-org0cc38e3">
<p>
\[
\int_{\cal S} {\bf v} \cdot d{\bf a}
@ -1652,9 +1652,9 @@ Over a closed surface:
</div>
</div>
<div id="outline-container-org999b522" class="outline-6">
<h6 id="org999b522"><a href="#org999b522">Volume Integrals</a></h6>
<div class="outline-text-6" id="text-org999b522">
<div id="outline-container-org9857c59" class="outline-6">
<h6 id="org9857c59"><a href="#org9857c59">Volume Integrals</a></h6>
<div class="outline-text-6" id="text-org9857c59">
<p>
\[
\int_{\cal V} T d\tau
@ -1695,7 +1695,7 @@ 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: 2022-02-15 Tue 10:14</p>
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@ -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="orgdf16255">
<div class="eqlabel" id="orgc6d532c">
<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">
<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="org13ed8c2">
<div class="alteqlabels" id="org6c9bf8e">
<ul class="org-ul">
<li>Gr (1.57)</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: 2022-02-15 Tue 10:14</p>
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@ -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="org81af346">
<summary id="orgad161fb">
<details class="objectives" id="org060fffc">
<summary id="org324a296">
Objectives
</summary>
@ -1663,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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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="org8d271e4">
<summary id="org0e0a7b5">
<details class="prereq" id="org9fd257c">
<summary id="org03c921a">
Prerequisites
</summary>
<ul class="org-ul">
@ -1608,8 +1608,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="org6301fe6">
<summary id="orgd491539">
<details class="objectives" id="orgde2512c">
<summary id="org1721848">
Objectives
</summary>
<ul class="org-ul">
@ -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>
<p class="date">Created: 2022-02-15 Tue 10:14</p>
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@ -1624,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="date">Created: 2022-02-15 Tue 10:14</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="org866e4fd">
<div class="core div" id="org48bd452">
<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="org9e4ada0">
<div class="core div" id="orgb48b333">
<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="org2c677eb">
<div class="core div" id="org7ffdde2">
<p>
<b>Faraday's law</b> (differential form)
\[
@ -1715,7 +1715,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
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<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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@ -1657,7 +1657,7 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold
\]
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="orge56d812">
<div class="core div" id="org05103f7">
<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="org9d858d6">
<div class="example div" id="orgfe6530a">
<p>
\paragraph{Example 7.13:} coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\).
Find energy stored in section of length \(l\).
@ -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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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@ -1637,7 +1637,7 @@ M_{12} = M_{21}
\]
</p>
<div class="example div" id="org25a0125">
<div class="example div" id="org8c11522">
<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="org86fe21b">
<div class="example div" id="org9c04d6d">
<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="org8db34bd">
<div class="example div" id="org856ea2f">
<p>
\paragraph{Example 7.12:} circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\).
What is the current ?
@ -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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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@ -1621,7 +1621,7 @@ law in integral form:
<div class="example div" id="orgdba2a6f">
<div class="example div" id="org9995269">
<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="org27b2361">
<div class="example div" id="org10525d2">
<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="orgc05de8a">
<div class="example div" id="orga3b651f">
<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.
@ -1719,7 +1719,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
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@ -1624,7 +1624,7 @@ 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: 2022-02-15 Tue 10:14</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="orgc22744b">
<div class="core div" id="orgb65b34c">
<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="orgb9c1836">
<div class="core div" id="orga966eba">
<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="orge08490a">
<div class="core div" id="org660ec71">
\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, \\
@ -1670,7 +1670,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</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="org3047d46">
<div class="core div" id="orgf32f14b">
<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="org8e0a58e">
<div class="core div" id="orgf353887">
<p>
{\bf Displacement current}
\[
@ -1679,7 +1679,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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@ -1649,7 +1649,7 @@ 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: 2022-02-15 Tue 10:14</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: 2022-02-15 Tue 10:14</p>
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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="org9a6ec8b">
<summary id="org41aa432">
<details class="prereq" id="orgd0bc04d">
<summary id="org26554fe">
Prerequisites
</summary>
<ul class="org-ul">
@ -1607,8 +1607,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="orgb318458">
<summary id="org762126b">
<details class="objectives" id="org41b8c54">
<summary id="orgeccfbac">
Objectives
</summary>
<ul class="org-ul">
@ -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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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<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="orgda2b7ea">
<div class="main div" id="org5125785">
<p>
{\bf Angular momentum of EM fields}
\[
@ -1630,7 +1630,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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6
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@ -1619,7 +1619,7 @@ This means that
\]
Since this is true for any volume, we have (re)derived the
</p>
<div class="core div" id="org8f4f0df">
<div class="core div" id="org3a26ba8">
<p>
{\bf Continuity equation}
\[
@ -1660,7 +1660,7 @@ 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: 2022-02-15 Tue 10:14</p>
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@ -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="org9f1f09d">
<div class="main div" id="org908691c">
<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="org81b4ed1">
<div class="main div" id="org1908c28">
<p>
{\bf Continuity equation for EM momentum}
\[
@ -1653,7 +1653,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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@ -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="org66f1141">
<div class="main div" id="orgf914cc8">
<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="org7f831b7">
<div class="main div" id="orgb679cf4">
<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="orgfd6d79f">
<div class="main div" id="org4a6eed2">
<p>
{\bf Total force on charges in volume}
\[
@ -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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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</div>

16
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@ -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="org4f11200">
<div class="main div" id="orgc46594c">
<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="org3a27803">
<div class="core div" id="org499e389">
<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="orgd91ad4e">
<div class="core div" id="org47309b1">
<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="org2526f44">
<div class="core div" id="org06b3424">
<p>
{\bf Energy in electromagnetic fields}
\[
@ -1740,7 +1740,7 @@ Then,
\]
so we get the
</p>
<div class="core div" id="orgd77ae2e">
<div class="core div" id="org3934004">
<p>
{\bf Poynting theorem (differential form)}
\[
@ -1757,7 +1757,7 @@ and has a similar for to the continuity equation
<div class="example div" id="orge461ff2">
<div class="example div" id="orgb4eafb2">
<p>
\paragraph{Example 8.1} Current in a wire: Joule heating. Energy per unit time delivered to wire: from Poynting.
Assuming that the field is uniform, the electric field parallel to the wire is
@ -1802,7 +1802,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">emd.emw</span></h3>
<div class="outline-text-3" id="text-emd_emw">
<details class="prereq" id="org00481ff">
<summary id="org600f412">
<details class="prereq" id="org078565b">
<summary id="org8eb1ce8">
Prerequisites
</summary>
<ul class="org-ul">
@ -1608,8 +1608,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="org060125e">
<summary id="org05ed276">
<details class="objectives" id="orgaefc9c9">
<summary id="org4dd81e4">
Objectives
</summary>
<ul class="org-ul">
@ -1650,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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@ -1628,7 +1628,7 @@ so for a monochromatic EM plan wave,
\]
or more succinctly:
</p>
<div class="main div" id="org2499ef6">
<div class="main div" id="orgbc4e836">
<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="orga85c294">
<div class="main div" id="org1beac86">
<p>
{\bf Momentum density of a monochromatic EM wave}
\[
@ -1695,7 +1695,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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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="org10be9fb">
<div class="core div" id="orgc712ee1">
<p>
{\bf E and B fields for a monochromatic EM plane wave}
\[
@ -1673,7 +1673,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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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="org655a07a">
<div class="core div" id="orgafd96cf">
<p>
{\bf Wave equations for electric and magnetic fields in vacuum}
\[
@ -1682,7 +1682,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1622,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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@ -1622,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1633,7 +1633,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
\]
We therefore have the
</p>
<div class="core div" id="orgd0b6d5d">
<div class="core div" id="orge4132f2">
<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="orgbb9bfed">
<aside id="orgcbc08fb">
<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="orgbd03b99">
<div class="main div" id="org5d3e365">
<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="org3af2590">
<div class="main div" id="org385196e">
<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="org614aca7">
<div class="core div" id="org94769a9">
<p>
\[
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
@ -1737,7 +1737,7 @@ or
\]
where as before
</p>
<div class="core div" id="org1c539e9">
<div class="core div" id="org4d90031">
<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="org251fe8d">
<div class="core div" id="org52f1ad1">
<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="org9471db8">
<div class="main div" id="org7bab0fe">
<p>
\[
{\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
@ -1818,7 +1818,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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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>
Discontinuities between different media, deduced from
</p>
<div class="core div" id="org8a4907f">
<div class="core div" id="org9ce00cc">
<p>
{\bf Maxwell's equations {\it (in matter)}, integral form}
</p>
@ -1691,7 +1691,7 @@ 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: 2022-02-15 Tue 10:14</p>
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@ -1625,7 +1625,7 @@ 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: 2022-02-15 Tue 10:14</p>
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@ -1621,7 +1621,7 @@ 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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@ -1717,7 +1717,7 @@ 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: 2022-02-15 Tue 10:14</p>
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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="org7a1428b">
<div class="main div" id="org3ccae9b">
<p>
{\bf Index of refraction}
\[
@ -1692,7 +1692,7 @@ 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: 2022-02-15 Tue 10:14</p>
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@ -1636,7 +1636,7 @@ 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: 2022-02-15 Tue 10:14</p>
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@ -1616,7 +1616,7 @@ 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: 2022-02-15 Tue 10:14</p>
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@ -1617,7 +1617,7 @@ 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: 2022-02-15 Tue 10:14</p>
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@ -1708,7 +1708,7 @@ 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: 2022-02-15 Tue 10:14</p>
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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="org33d6498">
<div class="core div" id="org77887dc">
<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="org78be6c2">
<div class="core div" id="orgff35843">
<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="org1c50a1f">
<div class="main div" id="orgc0d3690">
<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="org82f70fb">
<div class="main div" id="org40da3bc">
<p>
{\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}}
\[
@ -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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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@ -1617,7 +1617,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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@ -1623,7 +1623,7 @@ 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: 2022-02-15 Tue 10:14</p>
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@ -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: 2022-02-15 Tue 10:14</p>
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@ -1667,7 +1667,7 @@ 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: 2022-02-15 Tue 10:14</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1676,7 +1676,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</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">emf</span></h2>
<div class="outline-text-2" id="text-emf">
<details class="prereq" id="org009413e">
<summary id="org7b71bfd">
<details class="prereq" id="org0117d21">
<summary id="orgd0bf86c">
Prerequisites
</summary>
<ul class="org-ul">
@ -1607,8 +1607,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="org539b7e7">
<summary id="orgd79df87">
<details class="objectives" id="orgd14f948">
<summary id="orgc1f9363">
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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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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>
<p class="date">Created: 2022-02-15 Tue 10:14</p>
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@ -1640,7 +1640,7 @@ 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: 2022-02-15 Tue 10:14</p>
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@ -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="org3d474a8">
<div class="core div" id="org1bdc3b7">
<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="orgf9dae9e">
<div class="core div" id="org0e31a43">
<p>
{\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
\[
@ -1676,7 +1676,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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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="org0b07814">
<div class="core div" id="org04beb23">
<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="orgaa4bf24">
<div class="core div" id="orgd1c129d">
<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="org02abfdf">
<div class="main div" id="orgf163188">
<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="org4727603">
<div class="main div" id="org2b39ef9">
<p>
\[
\left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
@ -1695,7 +1695,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
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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>
@ -1602,8 +1602,8 @@ Table of contents
<li>Gr 3</li>
</ul>
<details class="prereq" id="orgc765c45">
<summary id="orgd67b2d8">
<details class="prereq" id="org97f4b33">
<summary id="org8628991">
Prerequisites
</summary>
<ul class="org-ul">
@ -1611,8 +1611,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="orga647534">
<summary id="org3b3715b">
<details class="objectives" id="orgdf0f685">
<summary id="org9c3f957">
Objectives
</summary>
<ul class="org-ul">
@ -1650,7 +1650,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>

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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-15 Tue 10:14 -->
<!-- 2022-02-17 Thu 08:42 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1608,7 +1608,7 @@ 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="org58a5d64">
<div class="main div" id="org0a614c7">
<p>
</p>
@ -1622,7 +1622,7 @@ 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="orgf8b994c">
<div class="main div" id="org6e195dd">
<p>
</p>
@ -1642,7 +1642,7 @@ 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="org5224cd4">
<div class="core div" id="orgef7da18">
<p>
</p>
@ -1658,7 +1658,7 @@ 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="org2a317e6">
<div class="core div" id="orgf5b1b63">
<p>
</p>
@ -1696,7 +1696,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>

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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>
@ -1614,14 +1614,14 @@ In one dimension, the potential is a single-variable
function \(\phi (x)\) and the Laplace equation reads
</p>
<div class="eqlabel" id="orgdfca409">
<div class="eqlabel" id="org20933f1">
<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="org36d5863">
<div class="alteqlabels" id="org1705801">
</div>
@ -1636,14 +1636,14 @@ function \(\phi (x)\) and the Laplace equation reads
<p>
The solution to this is
</p>
<div class="eqlabel" id="orga37bc11">
<div class="eqlabel" id="orgc9124d1">
<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="orgc5114c3">
<div class="alteqlabels" id="org566337b">
<ul class="org-ul">
<li>Gr (3.6)</li>
</ul>
@ -1702,14 +1702,14 @@ 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="orgc587278">
<div class="eqlabel" id="org0af82b1">
<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="orgd96eb38">
<div class="alteqlabels" id="orgf05a1be">
</div>
@ -1762,14 +1762,14 @@ 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="orgf7426f4">
<div class="eqlabel" id="org45554bc">
<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="org1c3162d">
<div class="alteqlabels" id="org242a219">
</div>
@ -1781,8 +1781,8 @@ a point equals its value averaged over a sphere
\]
</p>
<details id="orgbc3026b">
<summary id="orge4e657e">
<details id="orgeb8b5b8">
<summary id="org92de027">
<strong>Physicist's proof</strong>
</summary>
<p>
@ -1844,8 +1844,8 @@ proving the theorem.
</p>
</details>
<details id="orgf289197">
<summary id="orge3fd0c7">
<details id="org2eee056">
<summary id="orgf338007">
<strong>Formal proof</strong>
</summary>
@ -1895,14 +1895,14 @@ we get the following general
<p>
<b>Theorem</b>:
</p>
<div class="eqlabel" id="orgd8bf4d9">
<div class="eqlabel" id="org723ed6d">
<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="org49a36be">
<div class="alteqlabels" id="org447b5bf">
</div>
@ -1955,19 +1955,19 @@ 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="eqlabel" id="orgfeb6aae">
<div class="eqlabel" id="org7f931db">
<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="org7f4d8f6">
<div class="alteqlabels" id="orgfc31737">
</div>
</div>
<div class="info div" id="orgf970a43">
<div class="info div" id="org181f62a">
<p>
<b>Earnshaw's theorem (physical version)</b> <br>
</p>
@ -2086,7 +2086,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>

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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-15 Tue 10:14 -->
<!-- 2022-02-17 Thu 08:42 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<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="org99000c7">
<div class="info div" id="org28a8993">
<p>
<b>George Green</b>
</p>
<aside id="orga8c84e7">
<aside id="org0ab9bd7">
<p>
See a <a href="https://en.wikipedia.org/wiki/George%5C_Green%5C_(mathematician)">short bio on wikipedia</a>
</p>
@ -1634,14 +1634,14 @@ and
\]
Substituting this in the divergence theorem gives <b>Green's first identity</b>
</p>
<div class="eqlabel" id="org6de2e63">
<div class="eqlabel" id="orgbbf64af">
<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="org6c786de">
<div class="alteqlabels" id="orga3318b9">
<ul class="org-ul">
<li>J (1.34)</li>
</ul>
@ -1659,14 +1659,14 @@ 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="org0307c8b">
<div class="eqlabel" id="org785b1da">
<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="org3cf622d">
<div class="alteqlabels" id="org08207ba">
<ul class="org-ul">
<li>J (1.35)</li>
</ul>
@ -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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>

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