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Update 2022-03-01 08:15

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Jean-Sébastien 2022-03-01 08:15:26 +01:00
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4
build/a.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-21 Mon 20:41 -->
<!-- 2022-03-01 Tue 08:14 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="validation"></p>
</div>

4
build/a_l.html
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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-21 Mon 20:41 -->
<!-- 2022-03-01 Tue 08:14 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1728,7 +1728,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="validation"></p>
</div>

4
build/c.html
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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-21 Mon 20:41 -->
<!-- 2022-03-01 Tue 08:14 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="validation"></p>
</div>

4
build/c_m.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<!-- 2022-03-01 Tue 08:14 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</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>
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<!-- 2022-03-01 Tue 08:14 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</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>
<!-- 2022-02-21 Mon 20:41 -->
<!-- 2022-03-01 Tue 08:14 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1667,14 +1667,14 @@ 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="org5189029">
<div class="eqlabel" id="org5cf5791">
<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="org04b2fad">
<div class="alteqlabels" id="org8187762">
<ul class="org-ul">
<li>Gr4(1.79)</li>
</ul>
@ -1695,14 +1695,14 @@ 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="orgecdf384">
<div class="eqlabel" id="org710d9bc">
<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="orgb24faa6">
<div class="alteqlabels" id="org9273252">
<ul class="org-ul">
<li>Gr4(2.21)</li>
</ul>
@ -1723,14 +1723,14 @@ 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="orgdad7eec">
<div class="eqlabel" id="orgc433f93">
<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="orgc696d67">
<div class="alteqlabels" id="orgd149afb">
<ul class="org-ul">
<li>Gr4(2.21)</li>
</ul>
@ -1779,7 +1779,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</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>
<!-- 2022-02-21 Mon 20:41 -->
<!-- 2022-03-01 Tue 08:14 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1640,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</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>
<!-- 2022-02-21 Mon 20:41 -->
<!-- 2022-03-01 Tue 08:14 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1631,14 +1631,14 @@ which \(r\) is the distance from the chosen origin,
The usual Cartesian coordinates relate to spherical coordinates
according to
</p>
<div class="eqlabel" id="org082384f">
<div class="eqlabel" id="org9ccc564">
<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="org292e1da">
<div class="alteqlabels" id="org7f8648f">
</div>
@ -1663,14 +1663,14 @@ A generic vector can be expressed as
where the explicit relation between spherical and
Cartesian unit vectors is
</p>
<div class="eqlabel" id="orgfae6fd4">
<div class="eqlabel" id="org5529443">
<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="orga39ef1c">
<div class="alteqlabels" id="orge91c3eb">
</div>
@ -1693,14 +1693,14 @@ and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).
<p>
An infinitesimal displacement \(d{\bf l}\) can be written as
</p>
<div class="eqlabel" id="orgd89e9ad">
<div class="eqlabel" id="orga22710f">
<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="orga6ac01a">
<div class="alteqlabels" id="org31d976f">
</div>
@ -1716,14 +1716,14 @@ d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin
<p>
Infinitesimal volume element:
</p>
<div class="eqlabel" id="orgad9d14d">
<div class="eqlabel" id="org824a5cb">
<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="org57ba9f6">
<div class="alteqlabels" id="org9e40179">
</div>
@ -1744,14 +1744,14 @@ 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="orgce80236">
<div class="eqlabel" id="orgd062ca2">
<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="orgc254698">
<div class="alteqlabels" id="org776b74f">
</div>
@ -1768,14 +1768,14 @@ 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="org4495058">
<div class="eqlabel" id="orgbff4022">
<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="org2548f95">
<div class="alteqlabels" id="orgd13f0a2">
</div>
@ -1792,14 +1792,14 @@ 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="org2fb26ff">
<div class="eqlabel" id="org4bf1642">
<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="org267c962">
<div class="alteqlabels" id="orge688a5a">
</div>
@ -1818,14 +1818,14 @@ 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="org108e271">
<div class="eqlabel" id="org66b8a68">
<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="org65156f6">
<div class="alteqlabels" id="org25a1307">
</div>
@ -1859,7 +1859,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</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>
<!-- 2022-02-21 Mon 20:41 -->
<!-- 2022-03-01 Tue 08:14 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</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>
<!-- 2022-02-21 Mon 20:41 -->
<!-- 2022-03-01 Tue 08:14 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1652,7 +1652,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="validation"></p>
</div>

38
build/c_m_dc_d2.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-21 Mon 20:41 -->
<!-- 2022-03-01 Tue 08:14 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1623,9 +1623,9 @@ Table of contents
<div class="outline-text-5" id="text-c_m_dc_d2">
</div>
<div id="outline-container-org2f2fb63" class="outline-6">
<h6 id="org2f2fb63"><a href="#org2f2fb63">Divergence of gradient</a></h6>
<div class="outline-text-6" id="text-org2f2fb63">
<div id="outline-container-org27b21d4" class="outline-6">
<h6 id="org27b21d4"><a href="#org27b21d4">Divergence of gradient</a></h6>
<div class="outline-text-6" id="text-org27b21d4">
<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
@ -1634,44 +1634,44 @@ given by the Laplacian of the corresponding vector elements.
</div>
</div>
<div id="outline-container-org7940ce1" class="outline-6">
<h6 id="org7940ce1"><a href="#org7940ce1">Curl of a gradient</a></h6>
<div class="outline-text-6" id="text-org7940ce1">
<div id="outline-container-org8ab79ed" class="outline-6">
<h6 id="org8ab79ed"><a href="#org8ab79ed">Curl of a gradient</a></h6>
<div class="outline-text-6" id="text-org8ab79ed">
<p>
This always vanishes.
</p>
</div>
</div>
<div id="outline-container-org8fc30e5" class="outline-6">
<h6 id="org8fc30e5"><a href="#org8fc30e5">Gradient of the divergence</a></h6>
<div class="outline-text-6" id="text-org8fc30e5">
<div id="outline-container-org3af2976" class="outline-6">
<h6 id="org3af2976"><a href="#org3af2976">Gradient of the divergence</a></h6>
<div class="outline-text-6" id="text-org3af2976">
<p>
\({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.
</p>
</div>
</div>
<div id="outline-container-org5323eef" class="outline-6">
<h6 id="org5323eef"><a href="#org5323eef">Divergence of a curl</a></h6>
<div class="outline-text-6" id="text-org5323eef">
<div id="outline-container-orgb2afb29" class="outline-6">
<h6 id="orgb2afb29"><a href="#orgb2afb29">Divergence of a curl</a></h6>
<div class="outline-text-6" id="text-orgb2afb29">
<p>
This always vanishes.
</p>
</div>
</div>
<div id="outline-container-org0c116f8" class="outline-6">
<h6 id="org0c116f8"><a href="#org0c116f8">Curl of curl</a></h6>
<div class="outline-text-6" id="text-org0c116f8">
<div class="eqlabel" id="org51a12dc">
<div id="outline-container-orgd5bfcaa" class="outline-6">
<h6 id="orgd5bfcaa"><a href="#orgd5bfcaa">Curl of curl</a></h6>
<div class="outline-text-6" id="text-orgd5bfcaa">
<div class="eqlabel" id="org4a8e6af">
<p>
<a id="curlcurl"></a><a href="./c_m_dc_d2.html#curlcurl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orge1da3d2">
<div class="alteqlabels" id="org8cfc576">
</div>
@ -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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="validation"></p>
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<html lang="en">
<head>
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<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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@ -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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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@ -1672,7 +1672,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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28
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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>
@ -1630,14 +1630,14 @@ explicited as follows:
<p>
<b>Gradient of a product</b>:
</p>
<div class="eqlabel" id="orgfc34aae">
<div class="eqlabel" id="org7ccb680">
<p>
<a id="grad_prod"></a><a href="./c_m_dc_pr.html#grad_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org0981207">
<div class="alteqlabels" id="orga7c6aac">
<ul class="org-ul">
<li>Gr (3)</li>
<li>W (1-111)</li>
@ -1657,14 +1657,14 @@ explicited as follows:
<p>
<b>Gradient of a scalar product</b>:
</p>
<div class="eqlabel" id="org1897a2b">
<div class="eqlabel" id="orgbe6a82f">
<p>
<a id="grad_sprod"></a><a href="./c_m_dc_pr.html#grad_sprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgd1aefe7">
<div class="alteqlabels" id="orgc2565bd">
<ul class="org-ul">
<li>Gr (4)</li>
<li>W (1-112)</li>
@ -1684,14 +1684,14 @@ explicited as follows:
<p>
<b>Divergence of a product</b>:
</p>
<div class="eqlabel" id="orgd81e567">
<div class="eqlabel" id="org2591737">
<p>
<a id="div_prod"></a><a href="./c_m_dc_pr.html#div_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org11cd161">
<div class="alteqlabels" id="orge3c89ac">
<ul class="org-ul">
<li>Gr (5)</li>
<li>W (1-115)</li>
@ -1711,14 +1711,14 @@ explicited as follows:
<p>
<b>Divergence of a cross product</b>:
</p>
<div class="eqlabel" id="orgac85f09">
<div class="eqlabel" id="org37cc95b">
<p>
<a id="div_xprod"></a><a href="./c_m_dc_pr.html#div_xprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org66c049b">
<div class="alteqlabels" id="org0277164">
<ul class="org-ul">
<li>Gr (6)</li>
<li>W (1-116)</li>
@ -1738,14 +1738,14 @@ explicited as follows:
<p>
<b>Curl of a product</b>:
</p>
<div class="eqlabel" id="org18804f3">
<div class="eqlabel" id="org980570b">
<p>
<a id="curl_prod"></a><a href="./c_m_dc_pr.html#curl_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org7cfdedf">
<div class="alteqlabels" id="org92e3c88">
<ul class="org-ul">
<li>Gr (7)</li>
<li>W (1-118)</li>
@ -1765,14 +1765,14 @@ explicited as follows:
<p>
<b>Curl of a cross product</b>:
</p>
<div class="eqlabel" id="orgb314b2b">
<div class="eqlabel" id="orga4932f1">
<p>
<a id="curl_xprod"></a><a href="./c_m_dc_pr.html#curl_xprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org27d2d68">
<div class="alteqlabels" id="org93712f0">
<ul class="org-ul">
<li>Gr (8)</li>
<li>W (1-119)</li>
@ -1813,7 +1813,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="validation"></p>
</div>

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@ -1647,7 +1647,7 @@ 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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@ -1659,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1644,14 +1644,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
More generally,
</p>
<div class="eqlabel" id="org0433d75">
<div class="eqlabel" id="org21f3d7e">
<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="org68675a6">
<div class="alteqlabels" id="orgeaacc49">
<ul class="org-ul">
<li>Gr (1.100)</li>
</ul>
@ -1670,14 +1670,14 @@ More generally,
Since
</p>
<div class="eqlabel" id="orgd02de7a">
<div class="eqlabel" id="org2f9beaa">
<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="orgc48b943">
<div class="alteqlabels" id="org86371cb">
<ul class="org-ul">
<li>Gr (1.101)</li>
</ul>
@ -1693,14 +1693,14 @@ Since
<p>
we have that
</p>
<div class="eqlabel" id="orgb494e7d">
<div class="eqlabel" id="org77d9fa8">
<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="orgc599cf6">
<div class="alteqlabels" id="org9d70b5c">
<ul class="org-ul">
<li>Gr (1.102)</li>
</ul>
@ -1732,7 +1732,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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@ -1650,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-21 Mon 20:41</p>
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<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-21 Mon 20:41</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: 2022-02-21 Mon 20:41</p>
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@ -1658,7 +1658,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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build/c_m_ic_lsv.html
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1623,9 +1623,9 @@ Table of contents
<div class="outline-text-5" id="text-c_m_ic_lsv">
</div>
<div id="outline-container-orgde47b1c" class="outline-6">
<h6 id="orgde47b1c"><a href="#orgde47b1c">Line Integrals</a></h6>
<div class="outline-text-6" id="text-orgde47b1c">
<div id="outline-container-org0446f34" class="outline-6">
<h6 id="org0446f34"><a href="#org0446f34">Line Integrals</a></h6>
<div class="outline-text-6" id="text-org0446f34">
<p>
\[
{\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l}
@ -1654,9 +1654,9 @@ Integral over a closed loop:
</div>
</div>
<div id="outline-container-org5884ec7" class="outline-6">
<h6 id="org5884ec7"><a href="#org5884ec7">Surface Integrals</a></h6>
<div class="outline-text-6" id="text-org5884ec7">
<div id="outline-container-org0cc3862" class="outline-6">
<h6 id="org0cc3862"><a href="#org0cc3862">Surface Integrals</a></h6>
<div class="outline-text-6" id="text-org0cc3862">
<p>
\[
\int_{\cal S} {\bf v} \cdot d{\bf a}
@ -1676,9 +1676,9 @@ Over a closed surface:
</div>
</div>
<div id="outline-container-org8682a2f" class="outline-6">
<h6 id="org8682a2f"><a href="#org8682a2f">Volume Integrals</a></h6>
<div class="outline-text-6" id="text-org8682a2f">
<div id="outline-container-orga0f7b3b" class="outline-6">
<h6 id="orga0f7b3b"><a href="#orga0f7b3b">Volume Integrals</a></h6>
<div class="outline-text-6" id="text-orga0f7b3b">
<p>
\[
\int_{\cal V} T d\tau
@ -1719,7 +1719,7 @@ 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: 2022-02-21 Mon 20:41</p>
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@ -1621,14 +1621,14 @@ 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>
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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="org2ffa281">
<div class="alteqlabels" id="orga96f215">
<ul class="org-ul">
<li>Gr (1.57)</li>
</ul>
@ -1673,7 +1673,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="author">Author: Jean-Sébastien Caux</p>
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@ -1622,8 +1622,8 @@ Table of contents
</svg></a><span class="headline-id">d</span></h2>
<div class="outline-text-2" id="text-d">
<details class="objectives" id="org6aaebc8">
<summary id="org80e2c68">
<details class="objectives" id="orgca523dc">
<summary id="orgd4ff14b">
Objectives
</summary>
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@ -1669,7 +1669,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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@ -1622,8 +1622,8 @@ Table of contents
</svg></a><span class="headline-id">emd</span></h2>
<div class="outline-text-2" id="text-emd">
<details class="prereq" id="org3aa6e54">
<summary id="orgf9c9f1a">
<details class="prereq" id="org818864e">
<summary id="org9e6a46d">
Prerequisites
</summary>
<ul class="org-ul">
@ -1632,8 +1632,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="org933e70a">
<summary id="orgfee67cd">
<details class="objectives" id="org52c0d63">
<summary id="org4e1bb09">
Objectives
</summary>
<ul class="org-ul">
@ -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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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@ -1648,7 +1648,7 @@ 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: 2022-02-21 Mon 20:41</p>
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@ -1662,7 +1662,7 @@ 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="orgec9bb01">
<div class="core div" id="org03c55ba">
<p>
<b>Electromotive force (or electromotance)</b>,
\[
@ -1684,7 +1684,7 @@ to the rate of change of the magnetic flux,
\]
so we obtain
</p>
<div class="core div" id="org9b4af23">
<div class="core div" id="org93f9990">
<p>
<b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
\[
@ -1702,7 +1702,7 @@ for any loop (on a wire or not). Using Stokes' theorem,
\]
we obtain
</p>
<div class="core div" id="org91ad35a">
<div class="core div" id="org6046a76">
<p>
<b>Faraday's law</b> (differential form)
\[
@ -1739,7 +1739,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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@ -1646,9 +1646,26 @@ W = \frac{1}{2} I \oint {\bf A} \cdot d{\bf l} = \frac{1}{2} \oint ({\bf A} \cdo
\label{Gr(7.30)}
\]
Generalization to volume currents:
</p>
<div class="eqlabel" id="org9fbfb9d">
<p>
<a id="W_intAJ"></a><a href="./emd_Fl_e.html#W_intAJ"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org6b3d2a5">
<ul class="org-ul">
<li>gr (7.31)</li>
</ul>
</div>
</div>
<p>
\[
W = \frac{1}{2} \int_{\cal V} ({\bf A} \cdot {\bf J}) d\tau
\label{Gr(7.31)}
\tag{W_intAJ}\label{W_intAJ}
\]
Even better: use Ampère, \({\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J}\):
\[
@ -1681,7 +1698,7 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold
\]
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="orgac0c4b7">
<div class="core div" id="org2f4a453">
<p>
\[
W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2
@ -1702,7 +1719,7 @@ 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="org763563a">
<div class="example div" id="orgeb4514a">
<p>
\paragraph{Example 7.13:} coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\).
Find energy stored in section of length \(l\).
@ -1741,7 +1758,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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@ -1661,7 +1661,7 @@ M_{12} = M_{21}
\]
</p>
<div class="example div" id="orgf0e7804">
<div class="example div" id="org34dd058">
<p>
\paragraph{Example 7.10:}
short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) lies concentrically inside
@ -1711,7 +1711,7 @@ Inductance: measured in {\bf henries} (\(H\)). \(H = V s/A\).
</p>
<div class="example div" id="orgf51dfda">
<div class="example div" id="org7acd05e">
<p>
\paragraph{Example 7.11:} find self-inductance of toroidal coil with
rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\))
@ -1738,7 +1738,7 @@ 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="orgef41138">
<div class="example div" id="org9f29434">
<p>
\paragraph{Example 7.12:} circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\).
What is the current ?
@ -1777,7 +1777,7 @@ 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: 2022-02-21 Mon 20:41</p>
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@ -1645,7 +1645,7 @@ law in integral form:
<div class="example div" id="orge69c6d6">
<div class="example div" id="org3b5285b">
<p>
{\bf Example 7.7:}
\({\bf B}(t)\) points up in circular region of radius \(R\). What is the induced \({\bf E}(t)\) ?
@ -1661,7 +1661,7 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
</div>
<div class="example div" id="org3276236">
<div class="example div" id="org68f8620">
<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 \(a &lt; b\),
@ -1695,7 +1695,7 @@ called the {\bf quasistatic} approximation, and works provided we deal with
'slow enough' phenomena.
</p>
<div class="example div" id="org91337cc">
<div class="example div" id="org5b14490">
<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.
@ -1743,7 +1743,7 @@ 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: 2022-02-21 Mon 20:41</p>
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<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-21 Mon 20:41</p>
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@ -1624,7 +1624,7 @@ Table of contents
<p>
Full set of equations for the electromagnetic field:
</p>
<div class="core div" id="org7b46da1">
<div class="core div" id="org883ebf1">
<p>
{\bf Maxwell's equations} {\it (in vacuum)}
</p>
@ -1640,7 +1640,7 @@ Full set of equations for the electromagnetic field:
<p>
Complement:
</p>
<div class="core div" id="org6297b6f">
<div class="core div" id="org2481694">
<p>
{\bf Force law}
\[
@ -1664,7 +1664,7 @@ take divergence of \((iv)\).
<p>
Better way of writing: all fields on left, all sources on right,
</p>
<div class="core div" id="org6d46b26">
<div class="core div" id="org47f05c2">
\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, \\
@ -1694,7 +1694,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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@ -1631,7 +1631,7 @@ the continuity equation as
\]
The extra term would thus be eliminated if we were to put
</p>
<div class="core div" id="orgc070790">
<div class="core div" id="orgb95d862">
<p>
\[
{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}
@ -1655,7 +1655,7 @@ Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation
<p>
Maxwell baptized this term the
</p>
<div class="core div" id="orgb352387">
<div class="core div" id="orgb8b014a">
<p>
{\bf Displacement current}
\[
@ -1703,7 +1703,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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@ -1673,7 +1673,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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@ -1671,7 +1671,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="date">Created: 2022-02-21 Mon 20:41</p>
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@ -1622,8 +1622,8 @@ 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="org805c0a9">
<summary id="org5cff90a">
<details class="prereq" id="org70c43f6">
<summary id="org3522c33">
Prerequisites
</summary>
<ul class="org-ul">
@ -1631,8 +1631,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="org33a8e47">
<summary id="orga264cda">
<details class="objectives" id="orgead6b14">
<summary id="orgf683e76">
Objectives
</summary>
<ul class="org-ul">
@ -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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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@ -1624,7 +1624,7 @@ Table of contents
<p>
The angular momentum of EM fields is directly given by
</p>
<div class="main div" id="orgd257fc0">
<div class="main div" id="orgaa57ede">
<p>
{\bf Angular momentum of EM fields}
\[
@ -1654,7 +1654,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1643,7 +1643,7 @@ This means that
\]
Since this is true for any volume, we have (re)derived the
</p>
<div class="core div" id="org822d00a">
<div class="core div" id="orgfb5850b">
<p>
{\bf Continuity equation}
\[
@ -1684,7 +1684,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1636,7 +1636,7 @@ 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="org2961159">
<div class="main div" id="orgc75d196">
<p>
{\bf Momentum density in the EM fields}
\[
@ -1648,7 +1648,7 @@ 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="org1aacec6">
<div class="main div" id="orgc680b5b">
<p>
{\bf Continuity equation for EM momentum}
\[
@ -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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="validation"></p>
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@ -1678,7 +1678,7 @@ and similarly for \({\boldsymbol B}\). We thus get
<p>
This expression can be greatly simplified by introducing the
</p>
<div class="main div" id="orgcd94992">
<div class="main div" id="org41d984c">
<p>
{\bf Maxwell stress tensor}
\[
@ -1701,7 +1701,7 @@ 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="org0f23aa8">
<div class="main div" id="orgefc4ae2">
<p>
{\bf EM force per unit volume}
\[
@ -1713,7 +1713,7 @@ We then obtain
<p>
where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
</p>
<div class="main div" id="org2d0e69d">
<div class="main div" id="orgc9bf6dd">
<p>
{\bf Total force on charges in volume}
\[
@ -1742,7 +1742,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="validation"></p>
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@ -1690,7 +1690,7 @@ so we get
Substituting this in \ref{Gr(8.6)} and using the divergence theorem,
we obtain
</p>
<div class="main div" id="org22fe564">
<div class="main div" id="orgda8af3a">
<p>
{\bf Poynting's theorem}
\[
@ -1715,7 +1715,7 @@ 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="orgdc206f6">
<div class="core div" id="org8a1c10e">
<p>
{\bf Poynting vector}
\[
@ -1728,7 +1728,7 @@ 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="org9460249">
<div class="core div" id="org57576be">
<p>
{\bf Poynting's theorem}
\[
@ -1741,7 +1741,7 @@ We can thus express Poynting's theorem more compactly:
<p>
where we have defined the total
</p>
<div class="core div" id="orgeb66148">
<div class="core div" id="org43eb64b">
<p>
{\bf Energy in electromagnetic fields}
\[
@ -1764,7 +1764,7 @@ Then,
\]
so we get the
</p>
<div class="core div" id="org8be7377">
<div class="core div" id="org29b53c2">
<p>
{\bf Poynting theorem (differential form)}
\[
@ -1781,7 +1781,7 @@ and has a similar for to the continuity equation
<div class="example div" id="org69805a3">
<div class="example div" id="orgbb7ac4b">
<p>
\paragraph{Example 8.1} Current in a wire: Joule heating. Energy per unit time delivered to wire: from Poynting.
Assuming that the field is uniform, the electric field parallel to the wire is
@ -1826,7 +1826,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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@ -1622,8 +1622,8 @@ Table of contents
</svg></a><span class="headline-id">emd.emw</span></h3>
<div class="outline-text-3" id="text-emd_emw">
<details class="prereq" id="orgccb0742">
<summary id="org65b5674">
<details class="prereq" id="org9d74d32">
<summary id="org175f28d">
Prerequisites
</summary>
<ul class="org-ul">
@ -1632,8 +1632,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="org0b2c32c">
<summary id="orgabe83b8">
<details class="objectives" id="org73ffbc8">
<summary id="org638d300">
Objectives
</summary>
<ul class="org-ul">
@ -1674,7 +1674,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
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<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-21 Mon 20:41</p>
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@ -1652,7 +1652,7 @@ so for a monochromatic EM plan wave,
\]
or more succinctly:
</p>
<div class="main div" id="org3c34193">
<div class="main div" id="org677cdf7">
<p>
{\bf Poynting vector of a monochromatic EM wave}
\[
@ -1668,7 +1668,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
<p>
Similary, we get the
</p>
<div class="main div" id="org361020e">
<div class="main div" id="orgc6fc8e6">
<p>
{\bf Momentum density of a monochromatic EM wave}
\[
@ -1719,7 +1719,7 @@ 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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@ -1653,7 +1653,7 @@ 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="orgaf2015e">
<div class="core div" id="orgd2950da">
<p>
{\bf E and B fields for a monochromatic EM plane wave}
\[
@ -1697,7 +1697,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
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@ -1650,7 +1650,7 @@ 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="orgc1dcb4b">
<div class="core div" id="org88fc1e4">
<p>
{\bf Wave equations for electric and magnetic fields in vacuum}
\[
@ -1706,7 +1706,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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@ -1646,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
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</div>
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<p class="author">Author: Jean-Sébastien Caux</p>
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@ -1657,7 +1657,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
\]
We therefore have the
</p>
<div class="core div" id="org51f1815">
<div class="core div" id="org9606db6">
<p>
{\bf Polarization current density}
\[
@ -1675,7 +1675,7 @@ 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="org467ee7a">
<aside id="orge1ae7f1">
<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.
@ -1698,7 +1698,7 @@ 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="org29c3c4f">
<div class="main div" id="orgb0d00db">
<p>
\[
\rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
@ -1711,7 +1711,7 @@ 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="orgee8dfce">
<div class="main div" id="org89c562c">
<p>
\[
{\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
@ -1735,7 +1735,7 @@ Gauss's law: can be rewritten
\]
where (as in static case)
</p>
<div class="core div" id="orgaa4fe44">
<div class="core div" id="org0196779">
<p>
\[
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
@ -1761,7 +1761,7 @@ or
\]
where as before
</p>
<div class="core div" id="org952d9d3">
<div class="core div" id="org2b5f6d5">
<p>
\[
{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
@ -1779,7 +1779,7 @@ 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="orgdc9fcaf">
<div class="core div" id="org2c2cd2a">
<p>
{\bf Maxwell's equations {\it (in matter)}}
</p>
@ -1805,7 +1805,7 @@ 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="org4946062">
<div class="main div" id="orgcaf59d5">
<p>
\[
{\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
@ -1842,7 +1842,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
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<p class="author">Author: Jean-Sébastien Caux</p>
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@ -1624,7 +1624,7 @@ Table of contents
<p>
Discontinuities between different media, deduced from
</p>
<div class="core div" id="orgdf7437e">
<div class="core div" id="orgc82cb6f">
<p>
{\bf Maxwell's equations {\it (in matter)}, integral form}
</p>
@ -1715,7 +1715,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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@ -1659,7 +1659,7 @@ v = \frac{1}{\sqrt{\mu \varepsilon}} = \frac{c}{n}
\]
where the index of refraction of the material is defined as
</p>
<div class="main div" id="org74f7e8d">
<div class="main div" id="orga0b7fe5">
<p>
{\bf Index of refraction}
\[
@ -1716,7 +1716,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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@ -1659,7 +1659,7 @@ 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="org5257e23">
<div class="core div" id="orgab5561c">
<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.
@ -1674,7 +1674,7 @@ 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="org0362172">
<div class="core div" id="orgfe980d3">
<p>
{\bf Law of reflection}
\[
@ -1732,7 +1732,7 @@ while the third equation becomes
\]
Writing everything in terms of the incident amplitude, we get
</p>
<div class="main div" id="org11aebdb">
<div class="main div" id="org3d560e5">
<p>
{\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)}
\[
@ -1752,7 +1752,7 @@ 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="org432eb4f">
<div class="main div" id="org88a1e03">
<p>
{\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}}
\[
@ -1802,7 +1802,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<title>Pre-Quantum Electrodynamics</title>
@ -1667,7 +1667,7 @@ 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: 2022-02-21 Mon 20:41</p>
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@ -1691,7 +1691,7 @@ 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: 2022-02-21 Mon 20:41</p>
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@ -1700,7 +1700,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1622,8 +1622,8 @@ Table of contents
</svg></a><span class="headline-id">emf</span></h2>
<div class="outline-text-2" id="text-emf">
<details class="prereq" id="orga060b72">
<summary id="org694c2b7">
<details class="prereq" id="orgb740c0e">
<summary id="org919d213">
Prerequisites
</summary>
<ul class="org-ul">
@ -1631,8 +1631,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="orgdf6ad7f">
<summary id="org404ebae">
<details class="objectives" id="orgd6caf8f">
<summary id="org998f38c">
Objectives
</summary>
<ul class="org-ul">
@ -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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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@ -1664,7 +1664,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1637,7 +1637,7 @@ while the equation for \(V\) becomes
\]
These can be written compactly upon introducing a new operator: the
</p>
<div class="core div" id="org60d2ea9">
<div class="core div" id="org892bf92">
<p>
{\bf d'Alembertian operator}
\[
@ -1650,7 +1650,7 @@ These can be written compactly upon introducing a new operator: the
<p>
so we get the
</p>
<div class="core div" id="org8c509da">
<div class="core div" id="org2f9d40b">
<p>
{\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
\[
@ -1700,7 +1700,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1638,7 +1638,7 @@ Useful strategy: represent fields in terms of potentials.
<p>
Easiest:
</p>
<div class="core div" id="org213e22d">
<div class="core div" id="org94d8659">
<p>
\[
{\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
@ -1654,7 +1654,7 @@ Putting this into Faraday's law gives
\]
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="org37d42c9">
<div class="core div" id="org61bab66">
<p>
\[
{\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}
@ -1667,7 +1667,7 @@ 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="org2bb1f3a">
<div class="main div" id="orga59d0cc">
<p>
\[
{\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
@ -1683,7 +1683,7 @@ whereas Amp{\`ere}-Maxwell becomes
\]
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="org5c628f6">
<div class="main div" id="org514d3f1">
<p>
\[
\left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
@ -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>
<p class="date">Created: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-02-21 Mon 20:41</p>
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<title>Pre-Quantum Electrodynamics</title>
@ -1626,8 +1626,8 @@ Table of contents
<li>Gr 3</li>
</ul>
<details class="prereq" id="org6e27ef2">
<summary id="org5cb0144">
<details class="prereq" id="org3402c83">
<summary id="org4cf0fe6">
Prerequisites
</summary>
<ul class="org-ul">
@ -1635,8 +1635,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="org66a1990">
<summary id="org6dae755">
<details class="objectives" id="org9645228">
<summary id="orgc4935b9">
Objectives
</summary>
<ul class="org-ul">
@ -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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="validation"></p>
</div>

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@ -1,7 +1,7 @@
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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>
@ -1632,7 +1632,7 @@ 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="org140ccb6">
<div class="main div" id="orgca99451">
<p>
</p>
@ -1646,7 +1646,7 @@ 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="orgc35ae72">
<div class="main div" id="orge64dc41">
<p>
</p>
@ -1666,7 +1666,7 @@ 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="org7a45415">
<div class="core div" id="orgd88f849">
<p>
</p>
@ -1682,7 +1682,7 @@ 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="orgb0ca256">
<div class="core div" id="orgfefb816">
<p>
</p>
@ -1720,7 +1720,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="validation"></p>
</div>

38
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<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1638,14 +1638,14 @@ In one dimension, the potential is a single-variable
function \(\phi (x)\) and the Laplace equation reads
</p>
<div class="eqlabel" id="orgc3104bc">
<div class="eqlabel" id="org46aafa7">
<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="org7cf3c4b">
<div class="alteqlabels" id="org459093f">
</div>
@ -1660,14 +1660,14 @@ function \(\phi (x)\) and the Laplace equation reads
<p>
The solution to this is
</p>
<div class="eqlabel" id="org9c7d537">
<div class="eqlabel" id="orged9e79a">
<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="org9c28632">
<div class="alteqlabels" id="org599dad5">
<ul class="org-ul">
<li>Gr (3.6)</li>
</ul>
@ -1726,14 +1726,14 @@ 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="orgcb6bc04">
<div class="eqlabel" id="orgdc4453f">
<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="orgfa108be">
<div class="alteqlabels" id="org259268b">
</div>
@ -1786,14 +1786,14 @@ 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="orgfa4a823">
<div class="eqlabel" id="org822a974">
<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="org576e24c">
<div class="alteqlabels" id="org57d10c8">
</div>
@ -1805,8 +1805,8 @@ a point equals its value averaged over a sphere
\]
</p>
<details id="org6ed4b45">
<summary id="org088d2fd">
<details id="org0d796b5">
<summary id="org6d53cda">
<strong>Physicist's proof</strong>
</summary>
<p>
@ -1868,8 +1868,8 @@ proving the theorem.
</p>
</details>
<details id="orgb01e597">
<summary id="orgf937e13">
<details id="org68c57fd">
<summary id="org5db3c01">
<strong>Formal proof</strong>
</summary>
@ -1919,14 +1919,14 @@ we get the following general
<p>
<b>Theorem</b>:
</p>
<div class="eqlabel" id="org5dbce0a">
<div class="eqlabel" id="org59c453b">
<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="org73f4d24">
<div class="alteqlabels" id="org9481971">
</div>
@ -1979,19 +1979,19 @@ 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="org3cd4c43">
<div class="eqlabel" id="org81bf520">
<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="org61c0e88">
<div class="alteqlabels" id="orgf2a161c">
</div>
</div>
<div class="info div" id="org9b11c3d">
<div class="info div" id="orgd9c5641">
<p>
<b>Earnshaw's theorem (physical version)</b> <br>
</p>
@ -2110,7 +2110,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="validation"></p>
</div>

16
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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>
@ -1621,11 +1621,11 @@ 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="org1f63c73">
<div class="info div" id="org1205a85">
<p>
<b>George Green</b>
</p>
<aside id="orgf860ba9">
<aside id="orgadbe90f">
<p>
See a <a href="https://en.wikipedia.org/wiki/George%5C_Green%5C_(mathematician)">short bio on wikipedia</a>
</p>
@ -1658,14 +1658,14 @@ and
\]
Substituting this in the divergence theorem gives <b>Green's first identity</b>
</p>
<div class="eqlabel" id="org72aea90">
<div class="eqlabel" id="org8bf3d02">
<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="org28d41a7">
<div class="alteqlabels" id="org6723606">
<ul class="org-ul">
<li>J (1.34)</li>
</ul>
@ -1683,14 +1683,14 @@ 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="orgaa4afed">
<div class="eqlabel" id="org6f94224">
<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="org7c8256e">
<div class="alteqlabels" id="org75ac930">
<ul class="org-ul">
<li>J (1.35)</li>
</ul>
@ -1726,7 +1726,7 @@ 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: 2022-02-21 Mon 20:41</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="validation"></p>
</div>

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