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Update 2022-03-02 15:47

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Jean-Sébastien 2022-03-02 15:47:54 +01:00
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4
build/a.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/a_l.html
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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c.html
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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m.html
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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m_cs.html
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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

16
build/c_m_cs_cyl.html
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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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="org5cf5791">
<div class="eqlabel" id="org3f7fe1a">
<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="org8187762">
<div class="alteqlabels" id="orgcf66eee">
<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="org710d9bc">
<div class="eqlabel" id="org77a88dd">
<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="org9273252">
<div class="alteqlabels" id="orga1ce1b4">
<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="orgc433f93">
<div class="eqlabel" id="orgb7e3d3b">
<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="orgd149afb">
<div class="alteqlabels" id="orgd068b29">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</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>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</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>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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="org9ccc564">
<div class="eqlabel" id="org3e4cc27">
<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="org7f8648f">
<div class="alteqlabels" id="org6cfcb56">
</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="org5529443">
<div class="eqlabel" id="org115ed32">
<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="orge91c3eb">
<div class="alteqlabels" id="org965d0cb">
</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="orga22710f">
<div class="eqlabel" id="orgdb194fb">
<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="org31d976f">
<div class="alteqlabels" id="org96ce6ff">
</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="org824a5cb">
<div class="eqlabel" id="org40e244c">
<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="org9e40179">
<div class="alteqlabels" id="org6796383">
</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="orgd062ca2">
<div class="eqlabel" id="org7c9e5d8">
<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="org776b74f">
<div class="alteqlabels" id="org94fd9f5">
</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="orgbff4022">
<div class="eqlabel" id="org25fdd4e">
<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="orgd13f0a2">
<div class="alteqlabels" id="org3d0214d">
</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="org4bf1642">
<div class="eqlabel" id="org082892a">
<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="orge688a5a">
<div class="alteqlabels" id="orge37528b">
</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="org66b8a68">
<div class="eqlabel" id="orgf48e07a">
<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="org25a1307">
<div class="alteqlabels" id="orge21efb8">
</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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</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>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</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>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</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>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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-org27b21d4" class="outline-6">
<h6 id="org27b21d4"><a href="#org27b21d4">Divergence of gradient</a></h6>
<div class="outline-text-6" id="text-org27b21d4">
<div id="outline-container-org3814a3d" class="outline-6">
<h6 id="org3814a3d"><a href="#org3814a3d">Divergence of gradient</a></h6>
<div class="outline-text-6" id="text-org3814a3d">
<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-org8ab79ed" class="outline-6">
<h6 id="org8ab79ed"><a href="#org8ab79ed">Curl of a gradient</a></h6>
<div class="outline-text-6" id="text-org8ab79ed">
<div id="outline-container-orgc8aa2ed" class="outline-6">
<h6 id="orgc8aa2ed"><a href="#orgc8aa2ed">Curl of a gradient</a></h6>
<div class="outline-text-6" id="text-orgc8aa2ed">
<p>
This always vanishes.
</p>
</div>
</div>
<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">
<div id="outline-container-orge3b0dcb" class="outline-6">
<h6 id="orge3b0dcb"><a href="#orge3b0dcb">Gradient of the divergence</a></h6>
<div class="outline-text-6" id="text-orge3b0dcb">
<p>
\({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.
</p>
</div>
</div>
<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">
<div id="outline-container-org1f5474a" class="outline-6">
<h6 id="org1f5474a"><a href="#org1f5474a">Divergence of a curl</a></h6>
<div class="outline-text-6" id="text-org1f5474a">
<p>
This always vanishes.
</p>
</div>
</div>
<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">
<div id="outline-container-org4b8cb79" class="outline-6">
<h6 id="org4b8cb79"><a href="#org4b8cb79">Curl of curl</a></h6>
<div class="outline-text-6" id="text-org4b8cb79">
<div class="eqlabel" id="orgad52ac5">
<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="org8cfc576">
<div class="alteqlabels" id="orgde36146">
</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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m_dc_del.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m_dc_div.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m_dc_g.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

28
build/c_m_dc_pr.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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="org7ccb680">
<div class="eqlabel" id="org4bcb61a">
<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="orga7c6aac">
<div class="alteqlabels" id="org8c15975">
<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="orgbe6a82f">
<div class="eqlabel" id="org7784802">
<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="orgc2565bd">
<div class="alteqlabels" id="orgb3682de">
<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="org2591737">
<div class="eqlabel" id="org96b1b94">
<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="orge3c89ac">
<div class="alteqlabels" id="org29d8167">
<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="org37cc95b">
<div class="eqlabel" id="orgfe2b08f">
<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="org0277164">
<div class="alteqlabels" id="orgdce994d">
<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="org980570b">
<div class="eqlabel" id="orgbb8798b">
<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="org92e3c88">
<div class="alteqlabels" id="org93d7ea5">
<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="orga4932f1">
<div class="eqlabel" id="orgd9fde5c">
<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="org93712f0">
<div class="alteqlabels" id="orgb1d2b5b">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m_dd.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m_dd_1d.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

16
build/c_m_dd_3d.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<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="org21f3d7e">
<div class="eqlabel" id="orgf5d25ef">
<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="orgeaacc49">
<div class="alteqlabels" id="org6fba030">
<ul class="org-ul">
<li>Gr (1.100)</li>
</ul>
@ -1670,14 +1670,14 @@ More generally,
Since
</p>
<div class="eqlabel" id="org2f9beaa">
<div class="eqlabel" id="org9588ee0">
<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="org86371cb">
<div class="alteqlabels" id="orgb0b4434">
<ul class="org-ul">
<li>Gr (1.101)</li>
</ul>
@ -1693,14 +1693,14 @@ Since
<p>
we have that
</p>
<div class="eqlabel" id="org77d9fa8">
<div class="eqlabel" id="orgd223c1e">
<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="org9d70b5c">
<div class="alteqlabels" id="org2ae8bc7">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m_dd_div.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m_ic.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m_ic_ftc.html
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@ -1,7 +1,7 @@
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m_ic_ftg.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1657,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m_ic_gauss.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m_ic_ip.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1658,7 +1658,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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build/c_m_ic_lsv.html
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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
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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-org0446f34" class="outline-6">
<h6 id="org0446f34"><a href="#org0446f34">Line Integrals</a></h6>
<div class="outline-text-6" id="text-org0446f34">
<div id="outline-container-org2486563" class="outline-6">
<h6 id="org2486563"><a href="#org2486563">Line Integrals</a></h6>
<div class="outline-text-6" id="text-org2486563">
<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-org0cc3862" class="outline-6">
<h6 id="org0cc3862"><a href="#org0cc3862">Surface Integrals</a></h6>
<div class="outline-text-6" id="text-org0cc3862">
<div id="outline-container-orgf44a1b8" class="outline-6">
<h6 id="orgf44a1b8"><a href="#orgf44a1b8">Surface Integrals</a></h6>
<div class="outline-text-6" id="text-orgf44a1b8">
<p>
\[
\int_{\cal S} {\bf v} \cdot d{\bf a}
@ -1676,9 +1676,9 @@ Over a closed surface:
</div>
</div>
<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">
<div id="outline-container-org7c26dcf" class="outline-6">
<h6 id="org7c26dcf"><a href="#org7c26dcf">Volume Integrals</a></h6>
<div class="outline-text-6" id="text-org7c26dcf">
<p>
\[
\int_{\cal V} T d\tau
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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build/c_m_ic_stokes.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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>
<div class="outline-text-5" id="text-c_m_ic_stokes">
<div class="eqlabel" id="org2339b1e">
<div class="eqlabel" id="org603ea63">
<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"/>
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</svg></a>
</p>
<div class="alteqlabels" id="orga96f215">
<div class="alteqlabels" id="org95b24de">
<ul class="org-ul">
<li>Gr (1.57)</li>
</ul>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m_uf.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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build/c_m_uf_cyl.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1660,7 +1660,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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build/c_m_uf_sph.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1660,7 +1660,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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build/c_m_uf_vi.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1663,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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build/c_m_va.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1649,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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build/c_m_va_cp.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1678,7 +1678,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m_va_n.html
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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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build/c_m_va_pds.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1707,7 +1707,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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build/c_m_va_sp.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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build/c_m_va_tp.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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build/c_m_vf.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/c_m_vf_helm.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1692,7 +1692,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

8
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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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="orgca523dc">
<summary id="orgd4ff14b">
<details class="objectives" id="orge784719">
<summary id="orgfb0d046">
Objectives
</summary>
@ -1687,7 +1687,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<!DOCTYPE html>
<html lang="en">
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1662,7 +1662,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1662,7 +1662,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1664,7 +1664,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1675,7 +1675,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

12
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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>
@ -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="org818864e">
<summary id="org9e6a46d">
<details class="prereq" id="org2aba3cc">
<summary id="org30ca6a6">
Prerequisites
</summary>
<ul class="org-ul">
@ -1632,8 +1632,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="org52c0d63">
<summary id="org4e1bb09">
<details class="objectives" id="org98af4b0">
<summary id="org78e61da">
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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1622,15 +1622,11 @@ Table of contents
</svg></a><span class="headline-id">emd.Fl.Fl</span></h4>
<div class="outline-text-4" id="text-emd_Fl_Fl">
<p>
1831: 3 experiments by Faraday (according to Griffiths! but it's historically incorrect)
\paragraph{1)} Pull a loop of wire through a magnetic field.
\paragraph{2)} Move magnet around a still loop.
\paragraph{3)} Change strength of field, holding magnet and loop still.
Around 1831, Faraday performed a number of experiments pertaining to
the effects of time-dependent fields.
</p>
<p>
Actually, historically, things didn't happen like that.
The first experiment that Faraday performed (1831) involved two metal coils wound
on opposite sides of a metal ring. When a current was turned on through the first
coil, it generated a transient current in the second coil (as measured by a
@ -1647,26 +1643,38 @@ on this idea. Faraday observed transient current in a circuit when:
<p>
Faraday's big insight was to summarize these effects by noticing that
</p>
<p>
\[
\boxed{
\mbox{\bf A changing magnetic field induces an electric field}
\mbox{A changing magnetic field induces an electric field.}
}
\]
</p>
<p>
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="org03c55ba">
<div class="core div" id="orgdfb0aad">
<p>
<b>Electromotive force (or electromotance)</b>,
</p>
<div class="eqlabel" id="org64afdaf">
<p>
<a id="elmofo"></a><a href="./emd_Fl_Fl.html#elmofo"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgb3cda90">
<ul class="org-ul">
<li>Gr (7.9)</li>
</ul>
</div>
</div>
<p>
\[
{\cal E} \equiv \oint_{\cal P} {\bf E} \cdot d{\bf l}.
\tag{elmofo}\label{elmofo}
\]
</p>
@ -1676,20 +1684,55 @@ You can think of the emf in different ways. It's the energy accumulated as a uni
</p>
<p>
The precise statement is that the electromotive force is proportional
The precise statement associated to Faraday's observations
is that the electromotive force is proportional
to the rate of change of the magnetic flux,
</p>
<div class="eqlabel" id="orgec7b520">
<p>
<a id="Fl_flux"></a><a href="./emd_Fl_Fl.html#Fl_flux"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgf70f495">
<ul class="org-ul">
<li>Gr (7.14)</li>
</ul>
</div>
</div>
<p>
\[
{\cal E} = \oint_{\cal P} {\bf E} \cdot d{\bf l} = -\frac{d\Phi}{dt}
\label{Gr(7.14)}
\tag{Fl_flux}\label{Fl_flux}
\]
so we obtain
</p>
<div class="core div" id="org93f9990">
<div class="core div" id="orgdfedc05">
<p>
<b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
</p>
<div class="eqlabel" id="orge87df83">
<p>
<a id="Fl_int"></a><a href="./emd_Fl_Fl.html#Fl_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org7b32fe7">
<ul class="org-ul">
<li>Gr (7.15)</li>
</ul>
</div>
</div>
<p>
\[
\oint_{\cal P} {\bf E} \cdot d{\bf l} = -\int_{\cal S} \frac{\partial {\bf B}}{\partial t} \cdot d{\bf a}
\label{Gr(7.15)}
\tag{Fl_int}\label{Fl_int}
\]
</p>
@ -1702,20 +1745,36 @@ for any loop (on a wire or not). Using Stokes' theorem,
\]
we obtain
</p>
<div class="core div" id="org6046a76">
<div class="core div" id="orgafa0d15">
<div class="eqlabel" id="org13d3c14">
<p>
<a id="Fl"></a><a href="./emd_Fl_Fl.html#Fl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgd662a28">
<ul class="org-ul">
<li>Gr (7.16)</li>
</ul>
</div>
</div>
<p>
<b>Faraday's law</b> (differential form)
\[
{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}
\label{Gr(7.16)}
\tag{Fl}\label{Fl}
\]
</p>
</div>
<p>
Right-hand rule always sorts signs out. Easier rule: {\bf Lenz's law}, which
states that {\bf nature resists a change in flux}. This is in fact just
{\bf Le Ch\^atelier's principle} of any action at an equilibrium point leading
Right-hand rule always sorts signs out. Easier rule: <b>Lenz's law</b>, which
states that physical systems naturally resist a change in flux.
This is in fact just
<b>Le Châtelier's principle</b> of any action at an equilibrium point leading
to an opposing counter-reaction.
</p>
</div>
@ -1739,7 +1798,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1631,7 +1631,7 @@ Start from zero current, integrate in time:
W = \frac{1}{2} L I^2
\label{Gr(7.29)}
\]
Nicer way (generalizable to surface and volume currents): from (\ref{Gr(7.25)}), flux through loop is \(\Phi = L I\). But
Nicer way (generalizable to surface and volume currents): from <a href="./emd_Fl_i.html#PLI">PLI</a>, flux through loop is \(\Phi = L I\). But
\[
\Phi = \int_{\cal S} {\bf B} \cdot d{\bf a} = \int_{\cal S} ({\boldsymbol \nabla} \times {\bf A}) \cdot d{\bf a}
= \oint_{\cal P} {\bf A} \cdot d{\bf l},
@ -1647,16 +1647,16 @@ W = \frac{1}{2} I \oint {\bf A} \cdot d{\bf l} = \frac{1}{2} \oint ({\bf A} \cdo
\]
Generalization to volume currents:
</p>
<div class="eqlabel" id="org9fbfb9d">
<div class="eqlabel" id="org87af1ba">
<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">
<div class="alteqlabels" id="org95d3f8b">
<ul class="org-ul">
<li>gr (7.31)</li>
<li>Gr (7.31)</li>
</ul>
</div>
@ -1664,31 +1664,22 @@ Generalization to volume currents:
</div>
<p>
\[
W = \frac{1}{2} \int_{\cal V} ({\bf A} \cdot {\bf J}) d\tau
W = \frac{1}{2} \int_{\cal V} d\tau ~({\bf A} \cdot {\bf J})
\tag{W_intAJ}\label{W_intAJ}
\]
Even better: use Ampère, \({\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J}\):
\[
W = \frac{1}{2\mu_0} \int_{\cal V} {\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B}) d\tau
W = \frac{1}{2\mu_0} \int_{\cal V} d\tau ~{\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B})
\label{Gr(7.32)}
\]
Integrate by parts using product rule 6:
\[
{\boldsymbol ∇} ⋅ ({\bf A} × {\bf B}) = {\bf B} ⋅ ({\boldsymbol ∇} × {\bf A})
</p>
<ul class="org-ul">
<li>{\bf A} ⋅ ({\boldsymbol ∇} × {\bf B}),</li>
</ul>
<p>
{\boldsymbol \nabla} \cdot ({\bf A} \times {\bf B}) = {\bf B} \cdot ({\boldsymbol \nabla} \times {\bf A}) - {\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B}),
\]
so
\[
{\bf A} ⋅ ({\boldsymbol ∇} × {\bf B}) = {\bf B} ⋅ {\bf B}
</p>
<ul class="org-ul">
<li>{\boldsymbol ∇} ⋅ ({\bf A} × {\bf B}).</li>
</ul>
<p>
{\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B}) =
{\bf B} \cdot {\bf B} - {\boldsymbol \nabla} \cdot ({\bf A} \times {\bf B}).
\]
Then,
\[
@ -1698,11 +1689,26 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold
\]
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="org2f4a453">
<div class="core div" id="org247bd37">
<div class="eqlabel" id="orgdf4c394">
<p>
<a id="W_intBsq"></a><a href="./emd_Fl_e.html#W_intBsq"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orge069b86">
<ul class="org-ul">
<li>Gr (7.34)</li>
</ul>
</div>
</div>
<p>
\[
W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2
\label{Gr(7.34)}
\tag{W_intBsq}\label{W_intBsq}
\]
</p>
@ -1713,17 +1719,29 @@ We can integrate over all space: after neglecting boundary terms (assuming fiel
Summary: energy in electric and magnetic fields:
</p>
\begin{align}
W_{elec} = \frac{1}{2} \int d\tau V\rho = \frac{\varepsilon_0}{2} \int d\tau E^2, \hspace{2cm}
\mbox{(2.43 and 2.45)}, \\
W_{mag} = \frac{1}{2} \int d\tau ({\bf A} \cdot {\bf J}) = \frac{1}{2\mu_0} \int d\tau B^2,
\hspace{2cm} \mbox{(7.31 and 7.34)}
W_{elec} &amp;= \frac{1}{2} \int d\tau ~V\rho &amp;= \frac{\varepsilon_0}{2} \int d\tau ~E^2, \\
W_{mag} &amp;= \frac{1}{2} \int d\tau ~({\bf A} \cdot {\bf J}) &amp;= \frac{1}{2\mu_0} \int d\tau ~B^2,
\end{align}
<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 \(l\).
\paragraph{Solution:} from Ampère,
which are equations <a href="./ems_es_e.html#W_vcd">W_vcd</a>, <a href="./ems_es_e.html#W_intEsq">W_intEsq</a>, <a href="./emd_Fl_e.html#W_intAJ">W_intAJ</a> and <a href="./emd_Fl_e.html#W_intBsq">W_intBsq</a>.
</p>
<div class="example div" id="org03fb4e8">
<p>
<b>Example: energy in coaxial cable</b>
</p>
<p>
Consider a coaxial cable with inner cylinder radius \(a\), outer \(b\),
carrying current \(I\).
</p>
<p>
<b>Task</b>: find the energy stored in a section of length \(l\).
</p>
<p>
<b>Solution</b>: from Ampère,
\[
{\bf B} = \frac{\mu_0 I}{2\pi s} \hat{\boldsymbol \varphi}, \hspace{1cm} a &lt; s &lt; b, \hspace{1cm}
{\bf B} = 0, \hspace{1cm} s &lt; a ~\mbox{or}~ s &gt; b.
@ -1758,7 +1776,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1629,7 +1629,7 @@ is (using fact that \({\bf B}_1\) is proportional to \(I_1\))
\Phi_2 = \int {\bf B}_1 \cdot d{\bf a}_2 \Longrightarrow
\Phi_2 = M_{21} I_1
\]
where \(M_{21}\) is the {\bf mutual inductance} of the two loops.
where \(M_{21}\) is the <b>mutual inductance</b> of the two loops.
</p>
<p>
@ -1638,7 +1638,7 @@ Useful formula:
\Phi_2 = \int {\bf B}_1 \cdot d{\bf a}_2 = \int ({\boldsymbol \nabla} \times {\bf A}_1) \cdot d{\bf a}_2
= \oint {\bf A}_1 \cdot d{\bf l}_2
\]
But from (\ref{Gr(5.63)}),
But from <a href="./ems_ms_vp_A.html#A_CoulG">A_CoulG</a>,
\[
{\bf A}_1 ({\bf r}) = \frac{\mu_0 I_1}{4\pi} \oint_{{\cal P}_1} \frac{d{\bf l}_1}{|{\bf r} - {\bf r}_1|}
\]
@ -1647,29 +1647,74 @@ so
\Phi_2 = \frac{\mu_0 I_1}{4\pi} \oint_{{\cal P}_2} d{\bf l}_2 \cdot
\left(\oint_{{\cal P}_1} \frac{d{\bf l}_1 }{|{\bf r}_2 - {\bf r}_1|}\right)
\]
and we can write the mutual inductance as the {\bf Neumann formula},
and we can write the mutual inductance as the <b>Neumann formula</b>,
</p>
<div class="eqlabel" id="org197212f">
<p>
<a id="Newmann_M"></a><a href="./emd_Fl_i.html#Newmann_M"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org4709e6a">
<ul class="org-ul">
<li>Gr (7.22)</li>
</ul>
</div>
</div>
<p>
\[
M_{21} = \frac{\mu_0}{4\pi} \oint_{{\cal P}_1} \oint_{{\cal P}_2} \frac{d{\bf l}_1 \cdot d{\bf l}_2}
{|{\bf r}_1 - {\bf r}_2|}
\label{Gr(7.22)}
\tag{Neumann_M}\label{Neumann_M}
\]
Two things:
first, \(M_{21}\) is purely geometrical. Second,
</p>
<div class="eqlabel" id="orgb952a44">
<p>
<a id="Msym"></a><a href="./emd_Fl_i.html#Msym"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org53f3037">
<ul class="org-ul">
<li>Gr (7.23)</li>
</ul>
</div>
</div>
<p>
\[
M_{12} = M_{21}
\label{Gr(7.23)}
\tag{Msym}\label{Msym}
\]
</p>
<div class="example div" id="org34dd058">
<div class="example div" id="org6347791">
<p>
\paragraph{Example 7.10:}
short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) lies concentrically inside
<b>Example: solenoid in solenoid</b>
</p>
<p>
Consider a short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length)
which lies concentrically inside
a very long solenoid (radius \(b\), \(n_2\) turns per unit length). Current \(I\) in short solenoid.
What is flux through long solenoid ?
\paragraph{Solution:} complicated to calculate \({\bf B}_1\). Use mutual inductance, starting from
</p>
<p>
<b>Task</b>: compute the flux through the long solenoid.
</p>
<p>
<b>Solution</b>: it's complicated to calculate \({\bf B}_1\).
Use mutual inductance, starting from
the reverse situation: current \(I\) on outer solenoid, calculate flux through inner one.
Field of outer solenoid: from (\ref{Gr(5.57)}),
Field of outer solenoid: from <a href="./ems_ms_dcB_c.html#Amp_int">Amp_int</a>,
\[
B = \mu_0 n_2 I
\]
@ -1697,26 +1742,54 @@ What if we vary current in loop 1? Flux in 2 will vary. Induces EMF in loop 2:
\label{Gr(7.24)}
\]
Changing current also induces EMF in the source loop itself:
</p>
<div class="eqlabel" id="org667fec5">
<p>
<a id="PLI"></a><a href="./emd_Fl_i.html#PLI"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org28a1a02">
<ul class="org-ul">
<li>Gr (7.25)</li>
</ul>
</div>
</div>
<p>
\[
\Phi = L I
\label{Gr(7.25)}
\tag{PLI}\label{PLI}
\]
where \(L\) is the {\bf self-inductance} (or inductance) of the loop. Depends only on
where \(L\) is the <b>self-inductance</b> (or inductance) of the loop. Depends only on
geometry. Changing current induces EMF of
\[
{\cal E} = -L \frac{dI}{dt}
\label{Gr(7.26)}
\]
Inductance: measured in {\bf henries} (\(H\)). \(H = V s/A\).
Inductance: measured in <b>henries</b> (\(H\)). \(H = V s/A\).
</p>
<div class="example div" id="org7acd05e">
<div class="example div" id="org00e72a3">
<p>
\paragraph{Example 7.11:} find self-inductance of toroidal coil with
<b>Example: self-inductance of toroidal coil</b>
</p>
<p>
Consider a toroidal coil with
rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\))
which carries total of \(N\) turns.
\paragraph{Solution:} magnetic field inside toroid is (\ref{Gr(5.58)})
</p>
<p>
<b>Task</b>: find its self-inductance
</p>
<p>
<b>Solution</b>: magnetic field inside toroid is <a href="./ems_ms_dcB_c.html#Btor">Btor</a>
\[
B = \frac{\mu_0 NI}{2\pi s}
\]
@ -1735,14 +1808,25 @@ Total flux: \(N\) times this, so self-inductance is
</div>
<p>
Inductance (like capacitance) is intrinsically positive. Use Lenz law. Think of {\bf back EMF}.
Inductance (like capacitance) is intrinsically positive. Use Lenz law.
Think of <i>back EMF</i>.
</p>
<div class="example div" id="org9f29434">
<div class="example div" id="org958ea6c">
<p>
\paragraph{Example 7.12:} circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\).
What is the current ?
\paragraph{Solution:}
<b>Example: circuit</b>
</p>
<p>
Consider a circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\).
</p>
<p>
<b>Task</b>: find the current
</p>
<p>
<b>Solution</b>:
Ohm's law:
\[
{\cal E}_0 - L \frac{dI}{dt} = IR \Longrightarrow I(t) = \frac{{\cal E}_0}{R} + k e^{-(R/L)t}.
@ -1752,7 +1836,7 @@ If initial condition: \(I(0) = 0\), then
I(t) = \frac{{\cal E}_0}{R} \left[ 1 - e^{-(R/L)t} \right]
\label{Gr(7.28)}
\]
where \(\tau \equiv L/R\) is the {\bf time constant} of the circuit.
where \(\tau \equiv L/R\) is the <b>time constant</b> of the circuit.
</p>
</div>
@ -1777,7 +1861,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1628,7 +1628,7 @@ Two sources of electric fields: electric charges, and changing magnetic fields.
<p>
Electric fields induced by a changing magnetic field are determined in an exactly
parallel way as magnetostatic fields from the current: exploit parallel
between Ampère and Faraday!
between Ampère and Faraday
\[
{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J}
\hspace{3cm}
@ -1645,11 +1645,22 @@ law in integral form:
<div class="example div" id="org3b5285b">
<div class="example div" id="org045463b">
<p>
{\bf Example 7.7:}
\({\bf B}(t)\) points up in circular region of radius \(R\). What is the induced \({\bf E}(t)\) ?
\paragraph{Solution:}
<b>Example: loop with time-dependent flux</b>
</p>
<p>
Consider a time-dependent magnetic field \({\bf B}(t)\) directed vertically
through a horizontal circular region of radius \(R\).
</p>
<p>
<b>Task</b>: find the induced \({\bf E}(t)\).
</p>
<p>
<b>Solution</b>:
amperian loop of radius \(s\), apply Faraday:
\[
\oint {\bf E} \cdot d{\bf l} = E (2\pi s) = -\frac{d\Phi}{dt} = -\pi s^2 \frac{dB}{dt}
@ -1661,12 +1672,19 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
</div>
<div class="example div" id="org68f8620">
<div class="example div" id="org047ea10">
<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 \(a &lt; b\),
pointing up. Field turned off. What happens ?
\paragraph{Solution:} the wheel starts spinning to compensate the reduction of field.
<b>Example: wheel with charged rim traversed by flux</b>
</p>
<p>
Consider a wheel of radius \(b\) with line charge \(\lambda\) on the rim.
A uniform magnetic field \({\bf B}_0\) pointing up is traversing the central region
up to radius \(a &lt; b\). The field is then turned off. What happens?
</p>
<p>
<b>Solution</b>: the wheel starts spinning to compensate the reduction of field.
Faraday:
\[
\oint {\bf E} \cdot d{\bf l} = -\frac{d\Phi}{dt} = - \pi a^2 \frac{dB}{dt}
@ -1677,7 +1695,7 @@ Total torque:
\[
N = b\lambda \oint E dl = -b \lambda \pi a^2 \frac{dB}{dt}
\]
so total angular momentum imparted is
so total angular momentum imparted to the wheel is
\[
\int N dt = -\lambda \pi a^2 b \int_{B_0}^0 dB = \lambda \pi a^2 b B_0.
\]
@ -1690,16 +1708,27 @@ The precise way the field is turned off doesn't matter. Only electric field doe
</p>
<p>
{\bf N.B.:} we use magnetostatic formulas for changing fields. This is
called the {\bf quasistatic} approximation, and works provided we deal with
'slow enough' phenomena.
<b>N.B.</b>: we use magnetostatic formulas for changing fields. This is
called the <b>quasistatic</b> approximation, and works provided we deal with
<i>slow enough</i> phenomena.
</p>
<div class="example div" id="org5b14490">
<div class="example div" id="org733cbdd">
<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.
\paragraph{Solution:} quasistatic: magnetic field is \(B = \frac{\mu_0 I}{2\pi s}\)
<b>Example: field from wire with time-dependent current</b>
</p>
<p>
Consider an infinitely long straight wire which carries current \(I(t)\).
</p>
<p>
<b>Task</b>: find the induced \({\bf E}\) field as a function of distance \(s\) from wire.
</p>
<p>
<b>Solution</b>: assuming we can use the quasistatic approximation, the
magnetic field is \(B = \frac{\mu_0 I}{2\pi s}\)
and circles the wire. Like \({\bf B}\) field of solenoid, \({\bf E}\) runs parallel
to wire. Amperian loop with sides at distances \(s_0\) and \(s\):
\[
@ -1716,7 +1745,7 @@ where \(K\) is a constant (depends on the history of \(I(t)\)).
</p>
<p>
{\bf N.B.:} this can't be true always, since it blows up as \(s \rightarrow \infty\).
<b>N.B.</b>: this can't be true always, since it blows up as \(s \rightarrow \infty\).
Reason: in this case, we've overstepped the quasistatic limit. We need
\(s \ll c\tau\) where \(\tau\) is a typical time scale for change of \(I(t)\).
</p>
@ -1743,7 +1772,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1624,25 +1624,37 @@ Table of contents
<p>
Full set of equations for the electromagnetic field:
</p>
<div class="core div" id="org883ebf1">
<div class="core div" id="org40e4789">
<p>
{\bf Maxwell's equations} {\it (in vacuum)}
<b>Maxwell's equations</b> <i>(in vacuum)</i>
</p>
<div class="eqlabel" id="org3e8ad9b">
<p>
<a id="Max_vac"></a><a href="./emd_Me_Me.html#Max_vac"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgc606bec">
</div>
</div>
\begin{align}
(i) {\boldsymbol \nabla} \cdot {\bf E} &amp;= \frac{\rho}{\varepsilon_0}, \hspace{1cm} &amp;\mbox{Gauss}, \nonumber \\
(ii) {\boldsymbol \nabla} \cdot {\bf B} &amp;= 0, \hspace{1cm} &amp;\mbox{anonymous} \nonumber \\
(iii) {\boldsymbol \nabla} \times {\bf E} &amp;= -\frac{\partial {\bf B}}{\partial t}, \hspace{1cm} &amp;\mbox{Faraday}, \nonumber \\
(iv) {\boldsymbol \nabla} \times {\bf B} &amp;= \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, \hspace{1cm} &amp;\mbox{Ampère + Maxwell}.
\label{Gr(7.39)}
(i)~ {\boldsymbol \nabla} \cdot {\bf E} &amp;= \frac{\rho}{\varepsilon_0}, \hspace{1cm} &amp;\mbox{Gauss}, \nonumber \\
(ii)~ {\boldsymbol \nabla} \cdot {\bf B} &amp;= 0, &amp;\mbox{anonymous} \nonumber \\
(iii)~ {\boldsymbol \nabla} \times {\bf E} &amp;= -\frac{\partial {\bf B}}{\partial t}, &amp;\mbox{Faraday}, \nonumber \\
(iv)~ {\boldsymbol \nabla} \times {\bf B} &amp;= \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, &amp;\mbox{Ampère + Maxwell}.
\tag{Max_vac}\label{Max_vac}
\end{align}
</div>
<p>
Complement:
</p>
<div class="core div" id="org2481694">
<div class="core div" id="org28daec0">
<p>
{\bf Force law}
Force law <a href="./ems_ms_lf_pc.html#LorFo">LorFo</a>
\[
{\bf F} = q ({\bf E} + {\bf v} \times {\bf B}).
\label{Gr(7.40)}
@ -1651,26 +1663,40 @@ Complement:
</div>
<p>
These equations summarize the {\bf entire content of classical electrodynamics}.
These equations contain the <b>entirety of pre-quantum electrodynamics</b>.
</p>
<p>
\paragraph{Note:} even the continuity equation can be derived from Maxwell's equations:
take divergence of \((iv)\).
<b>Note</b>: even the continuity equation can be derived from Maxwell's equations:
take divergence of \((iv)\) and use \((i)\).
</p>
<p>
Better way of writing: all fields on left, all sources on right,
</p>
<div class="core div" id="org47f05c2">
<div class="core div" id="org4bb3e78">
<div class="eqlabel" id="org05437e5">
<p>
<a id="Max_vac_s"></a><a href="./emd_Me_Me.html#Max_vac_s"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org61742b2">
<ul class="org-ul">
<li>Gr (7.42)</li>
</ul>
</div>
</div>
\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, \\
(ii) &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0,
&amp;(iv) {\boldsymbol \nabla} \times {\bf B} - \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} = \mu_0 {\bf J},
\label{Gr(7.42)}
(i)~ {\boldsymbol \nabla} \cdot {\bf E} &amp;= \frac{\rho}{\varepsilon_0}, \nonumber \\
(ii)~{\boldsymbol \nabla} \cdot {\bf B} &amp;= 0, \nonumber \\
(iii)~ {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} &amp;= 0, \nonumber \\
(iv)~ {\boldsymbol \nabla} \times {\bf B} - \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} &amp;= \mu_0 {\bf J},
\tag{Max_vac_s}\label{Max_vac_s}
\end{align}
</div>
@ -1679,6 +1705,7 @@ Better way of writing: all fields on left, all sources on right,
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Me_dc.html">Maxwell's Correction to Ampère's Law; the Displacement Current&emsp;<small>[emd.Me.dc]</small></a></li><li>Next:&nbsp;<a href="emd_Me_mc.html">Magnetic Charge&emsp;<small>[emd.Me.mc]</small></a></li><li>Up:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li></ul>
<br>
<hr>
@ -1694,7 +1721,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1622,8 +1622,7 @@ Table of contents
</svg></a><span class="headline-id">emd.Me.dc</span></h4>
<div class="outline-text-4" id="text-emd_Me_dc">
<p>
The term which should be zero (but isn't) in (\ref{Gr(7.35)}) can be rewritten using
the continuity equation as
The term which should be zero (but isn't) in <a href="./emd_Me_ebM.html#divcurlB">divcurlB</a> can be rewritten using the continuity equation as
\[
{\boldsymbol \nabla} \cdot {\bf J} = -\frac{\partial \rho}{\partial t} = - \frac{\partial}{\partial t}
(\varepsilon_0 {\boldsymbol \nabla} \cdot {\bf E}) = -{\boldsymbol \nabla} \cdot \left(
@ -1631,17 +1630,32 @@ the continuity equation as
\]
The extra term would thus be eliminated if we were to put
</p>
<div class="core div" id="orgb95d862">
<div class="core div" id="orgdf7b3f9">
<div class="eqlabel" id="org5844c7b">
<p>
<a id="AmpMax"></a><a href="./emd_Me_dc.html#AmpMax"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org1e192cc">
<ul class="org-ul">
<li>Gr (7.36)</li>
</ul>
</div>
</div>
<p>
\[
{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}
\label{Gr(7.36)}
\tag{AmpMax}\label{AmpMax}
\]
</p>
</div>
<p>
\paragraph{Note:} this changes nothing in magnetostatics. Aesthetic appeal:
<b>Note</b>: this changes nothing in magnetostatics. Aesthetic appeal:
\[
\boxed{
\mbox{A changing electric field induces a magnetic field.}
@ -1650,17 +1664,32 @@ The extra term would thus be eliminated if we were to put
Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation of electromagnetic waves.
</p>
<p>
Maxwell baptized this term the
</p>
<div class="core div" id="orgb8b014a">
<div class="core div" id="orgb28580f">
<p>
<b>Displacement current</b>
</p>
<div class="eqlabel" id="org4a72dcc">
<p>
<a id="Jd"></a><a href="./emd_Me_dc.html#Jd"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org57849e1">
<ul class="org-ul">
<li>Gr (7.37)</li>
</ul>
</div>
</div>
<p>
{\bf Displacement current}
\[
{\bf J}_d \equiv \varepsilon_0 \frac{\partial {\bf E}}{\partial t}.
\label{Gr(7.37)}
\tag{Jd}\label{Jd}
\]
</p>
@ -1675,7 +1704,7 @@ where \(A\) is the area. Between the plates,
\[
\frac{\partial E}{\partial t} = \frac{1}{\varepsilon_0 A} \frac{dQ}{dt} = \frac{1}{\varepsilon_0 A} I.
\]
Checking (\ref{Gr(7.36)}),
Checking <a href="./emd_Me_dc.html#AmpMax">AmpMax</a>,
\[
\oint {\bf B} \cdot d{\bf l} = \mu_0 I_{\mbox{enc}} + \mu_0 \varepsilon_0 \int d{\bf a} \cdot \frac{\partial {\bf E}}{\partial t}
\label{Gr(7.38)}
@ -1703,7 +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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1625,10 +1625,10 @@ Table of contents
We've encountered:
</p>
\begin{align}
(i) &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0}, \hspace{1cm} &amp;\mbox{Gauss}, \nonumber \\
(ii) &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \hspace{1cm} &amp;\mbox{anonymous} \nonumber \\
(iii) &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \hspace{1cm} &amp;\mbox{Faraday}, \nonumber \\
(iv) &amp;{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J}, \hspace{1cm} &amp;\mbox{Ampère}.
(i)~ {\boldsymbol \nabla} \cdot {\bf E} &amp;= \frac{\rho}{\varepsilon_0}, \hspace{1cm} &amp;\mbox{Gauss}, \nonumber \\
(ii)~ {\boldsymbol \nabla} \cdot {\bf B} &amp;= 0, &amp;\mbox{anonymous} \nonumber \\
(iii)~ {\boldsymbol \nabla} \times {\bf E} &amp;= -\frac{\partial {\bf B}}{\partial t}, &amp;\mbox{Faraday}, \nonumber \\
(iv)~ {\boldsymbol \nabla} \times {\bf B} &amp;= \mu_0 {\bf J}, &amp;\mbox{Ampère}.
\end{align}
<p>
Fatal inconsistency: div of curl must always vanish. Check on \((iii)\):
@ -1637,9 +1637,26 @@ Fatal inconsistency: div of curl must always vanish. Check on \((iii)\):
= {\boldsymbol \nabla} \cdot \left( -\frac{\partial {\bf B}}{\partial t} \right) = -\frac{\partial}{\partial t} ({\boldsymbol \nabla} \cdot {\bf B}) = 0.
\]
But: try same with \((iv)\):
</p>
<div class="eqlabel" id="org764ac3e">
<p>
<a id="divcurlB"></a><a href="./emd_Me_ebM.html#divcurlB"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org0f4f12d">
<ul class="org-ul">
<li>Gr (7.35)</li>
</ul>
</div>
</div>
<p>
\[
{\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} \times {\bf B}) = \mu_0 {\boldsymbol \nabla} \cdot {\bf J}
\label{Gr(7.35)}
\tag{divcurlB}\label{divcurlB}
\]
LHS must be zero, but RHS is not zero for non-steady currents. Cannot be right!
</p>
@ -1673,7 +1690,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

8
build/emd_Me_mc.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1633,7 +1633,7 @@ In free space, where \(\rho\) and \({\bf J}\) vanish:
<p>
Symmetry: replace \({\bf E}\) by \({\bf B}\) and \({\bf B}\) by \(-\mu_0 \varepsilon_0{\bf E}\) in the first pair.
They turn into the second pair. This symmetry is spoiled by \(\rho\) and \({\bf J}\). What if we had
a truly symmetric situation, {\it i.e.}
a truly symmetric situation, <i>i.e.</i>
</p>
\begin{align}
(i) &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho_e}{\varepsilon_0},
@ -1650,7 +1650,7 @@ of magnetic charge. Both charges would be conserved:
{\boldsymbol \nabla} \cdot {\bf J}_e = -\frac{\partial \rho_e}{\partial t}.
\label{Gr(7.44)}
\]
Maxwell's equations {\bf beg} for magnetic charges. But we've never found any!
Maxwell's equations <i>beg</i> for magnetic charges. But we've never found any!
</p>
</div>
</div>
@ -1671,7 +1671,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

12
build/emd_ce.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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="org70c43f6">
<summary id="org3522c33">
<details class="prereq" id="orgca631bc">
<summary id="org2ad112f">
Prerequisites
</summary>
<ul class="org-ul">
@ -1631,8 +1631,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="orgead6b14">
<summary id="orgf683e76">
<details class="objectives" id="orgbdb48d8">
<summary id="org495ffbc">
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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

25
build/emd_ce_amom.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1624,13 +1624,28 @@ Table of contents
<p>
The angular momentum of EM fields is directly given by
</p>
<div class="main div" id="orgaa57ede">
<div class="main div" id="org6e52344">
<p>
<b>Angular momentum of EM fields</b>
</p>
<div class="eqlabel" id="org9b249c7">
<p>
<a id="lrxg"></a><a href="./emd_ce_amom.html#lrxg"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org9913b19">
</div>
</div>
<p>
{\bf Angular momentum of EM fields}
\[
{\boldsymbol l} = {\boldsymbol r} \times {\boldsymbol g}
= \varepsilon_0 {\boldsymbol r} \times
= \varepsilon_0 ~{\boldsymbol r} \times
\left({\boldsymbol E} \times {\boldsymbol B}\right)
\tag{lrxg}\label{lrxg}
\]
</p>
@ -1654,7 +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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

17
build/emd_ce_ce.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1622,13 +1622,13 @@ Table of contents
</svg></a><span class="headline-id">emd.ce.ce</span></h4>
<div class="outline-text-4" id="text-emd_ce_ce">
<p>
Very important distinction: {\bf global} versus {\bf local} conservation of charge.
Very important distinction: <b>global</b> versus <b>local</b> conservation of charge.
</p>
<p>
Charge in a volume {\cal V}:
Charge in a volume \({\cal V}\):
\[
Q_{\cal V} (t) = \int_{\cal V} d\tau \rho ({\bf r}, t)
Q_{\cal V} (t) = \int_{\cal V} d\tau~ \rho ({\bf r}, t)
\label{Gr(8.1)}
\]
Current \({\bf J}\) flowing out through boundary \({\cal S}\) of \({\cal V}\): conservation of charge means
@ -1643,12 +1643,11 @@ This means that
\]
Since this is true for any volume, we have (re)derived the
</p>
<div class="core div" id="orgfb5850b">
<div class="core div" id="orgf1c2b2a">
<p>
{\bf Continuity equation}
<b>Continuity equation</b> <a href="./ems_ms_ce.html#conteq">conteq</a>
\[
\frac{\partial \rho}{\partial t} + {\boldsymbol \nabla} \cdot {\bf J} = 0
\label{Gr(8.4)}
\]
</p>
@ -1660,7 +1659,7 @@ Therefore, conservation of charge is a direct consequence of Maxwell's equations
<p>
One thing to note: we have viewed \(\rho\) and \({\boldsymbol J}\) as sources
(the ''right-hand side'') of Maxwell's equations. The continuity equation thus
imposes a functional constraint on these sources: not {\it any} \(\rho\) and
imposes a functional constraint on these sources: not <i>any</i> \(\rho\) and
\({\boldsymbol J}\) will do the trick, only the ones what obey it.
</p>
</div>
@ -1684,7 +1683,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

42
build/emd_ce_mom.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1636,11 +1636,26 @@ 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="orgc75d196">
<div class="main div" id="orgecd6647">
<p>
<b>Momentum density in the EM fields</b>
</p>
<div class="eqlabel" id="orga1d9e2c">
<p>
<a id="gExB"></a><a href="./emd_ce_mom.html#gExB"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgf73a06d">
</div>
</div>
<p>
{\bf Momentum density in the EM fields}
\[
{\boldsymbol g} = \varepsilon_0 \mu_0 {\boldsymbol S} = \varepsilon_0 {\boldsymbol E} \times {\boldsymbol B}
\tag{gExB}\label{gExB}
\]
</p>
@ -1648,11 +1663,26 @@ 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="orgc680b5b">
<div class="main div" id="org8cc6bb8">
<p>
<b>Continuity equation for EM momentum</b>
</p>
<div class="eqlabel" id="orgcad090e">
<p>
<a id="contg"></a><a href="./emd_ce_mom.html#contg"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org69cf28d">
</div>
</div>
<p>
{\bf Continuity equation for EM momentum}
\[
\frac{\partial}{\partial t} {\boldsymbol g} - {\boldsymbol \nabla} \cdot {\boldsymbol T} = 0
\tag{contg}\label{contg}
\]
</p>
@ -1677,7 +1707,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

96
build/emd_ce_mst.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1638,30 +1638,27 @@ Substitute for \(\rho\) and \({\boldsymbol J}\) using Maxwell (Gauss and Ampère
<p>
On the other hand we have
\[
\frac{\partial }{\partial t} \left( {\boldsymbol E} × {\boldsymbol B} \right)
= \frac{∂ {\boldsymbol E}}{∂ t} × {\boldsymbol B}
</p>
<ul class="org-ul">
<li>{\boldsymbol E} × \frac{∂ {\boldsymbol B}}{∂ t}.</li>
</ul>
\begin{equation}
\frac{\partial }{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B} \right)
= \frac{\partial {\boldsymbol E}}{\partial t} \times {\boldsymbol B}
+ {\boldsymbol E} \times \frac{\partial {\boldsymbol B}}{\partial t}.
\end{equation}
<p>
\]
Using Faraday to substitute for \(\frac{\partial {\boldsymbol B}}{\partial t}\),
\[
\frac{ ∂ {\boldsymbol E}}{∂ t} × {\boldsymbol B}
= \frac{\partial }{\partial t} \left( {\boldsymbol E} × {\boldsymbol B}\right)
</p>
<ul class="org-ul">
<li>{\boldsymbol E} × \left({\boldsymbol ∇} × {\boldsymbol E} \right)</li>
</ul>
\begin{equation}
\frac{ \partial {\boldsymbol E}}{\partial t} \times {\boldsymbol B}
= \frac{\partial }{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B}\right)
+ {\boldsymbol E} \times \left({\boldsymbol \nabla} \times {\boldsymbol E} \right)
\end{equation}
<p>
\]
so
\[
{\boldsymbol f} = \varepsilon_0 \left( \left( {\boldsymbol \nabla} \cdot {\boldsymbol E} \right) {\boldsymbol E} - {\boldsymbol E} \times \left( {\boldsymbol \nabla} \times {\boldsymbol E} \right) \right) - \frac{1}{\mu_0} \left( {\boldsymbol B} \times \left( {\boldsymbol \nabla} \times {\boldsymbol B} \right) \right) - \varepsilon_0 \frac{\partial}{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B} \right).
\]
Since \({\boldsymbol \nabla} \cdot {\boldsymbol B} = 0\), we can symmetrize the expression in \({\boldsymbol E}\) and \({\boldsymbol B}\). Moreover, by product rule 4,
Since \({\boldsymbol \nabla} \cdot {\boldsymbol B} = 0\), we can symmetrize the expression in \({\boldsymbol E}\) and \({\boldsymbol B}\).
Moreover, by <a href="./c_m_dc_pr.html#grad_sprod">grad_sprod</a>,
\[
\frac{1}{2}{\boldsymbol \nabla} \left( E^2 \right) = \left( {\boldsymbol E} \cdot {\boldsymbol \nabla} \right) {\boldsymbol E} + {\boldsymbol E} \times \left( {\boldsymbol \nabla} \times {\boldsymbol E} \right)
\]
@ -1678,18 +1675,27 @@ and similarly for \({\boldsymbol B}\). We thus get
<p>
This expression can be greatly simplified by introducing the
</p>
<div class="main div" id="org41d984c">
<div class="main div" id="orga7d370d">
<p>
{\bf Maxwell stress tensor}
\[
T_{ij} ≡ ε_0 \left( E_i E_j - \frac{1}{2} δ_{ij} E^2\right)
<b>Maxwell stress tensor</b>
</p>
<ul class="org-ul">
<li>\frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} δ_{ij} B^2 \right)</li>
</ul>
<div class="eqlabel" id="orge5429c3">
<p>
\]
<a id="MaxST"></a><a href="./emd_ce_mst.html#MaxST"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org0306495">
</div>
</div>
\begin{equation}
T_{ij} \equiv \varepsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2\right)
+ \frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} \delta_{ij} B^2 \right)
\tag{MaxST}\label{MaxST}
\end{equation}
</div>
<p>
@ -1699,13 +1705,28 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
<p>
We then obtain
We then obtain the
</p>
<div class="main div" id="orgefc4ae2">
<div class="main div" id="org4781d10">
<p>
<b>EM force per unit volume</b>
</p>
<div class="eqlabel" id="orgd9f91b7">
<p>
<a id="fT"></a><a href="./emd_ce_mst.html#fT"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org7d4a06b">
</div>
</div>
<p>
{\bf EM force per unit volume}
\[
{\boldsymbol f} = {\boldsymbol \nabla} \cdot {\boldsymbol T} - \varepsilon_0 \mu_0 \frac{\partial {\boldsymbol S}}{\partial t}
\tag{fT}\label{fT}
\]
</p>
@ -1713,11 +1734,26 @@ We then obtain
<p>
where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
</p>
<div class="main div" id="orgc9bf6dd">
<div class="main div" id="orgea18677">
<p>
<b>Total force on charges in volume</b>
</p>
<div class="eqlabel" id="orgef98657">
<p>
<a id="totFo"></a><a href="./emd_ce_mst.html#totFo"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org4c9c200">
</div>
</div>
<p>
{\bf Total force on charges in volume}
\[
{\boldsymbol F} = \oint_S {\boldsymbol T} \cdot d{\boldsymbol a} - \varepsilon_0 \mu_0 \frac{d}{dt} \int_{\cal V} {\boldsymbol S} d\tau.
\tag{totFo}\label{totFo}
\]
</p>
@ -1742,7 +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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

186
build/emd_ce_poy.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1644,32 +1644,39 @@ done by EM forces ? From Lorentz force law:
\]
Really, we're looking at a small volume element \(d\tau\) carrying charge \(\rho d\tau\), moving
at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\). Thus,
</p>
<div class="eqlabel" id="org0197499">
<p>
<a id="dWdt_intEJ"></a><a href="./emd_ce_poy.html#dWdt_intEJ"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org9aacea3">
<ul class="org-ul">
<li>Gr (8.6)</li>
</ul>
</div>
</div>
<p>
\[
\frac{dW}{dt} = \int_{\cal V} d\tau ~ {\bf E} \cdot {\bf J}
\label{Gr(8.6)}
\tag{dWdt_intEJ}\label{dWdt_intEJ}
\]
The integrand is the work done per unit time, per unit volume, {\it i.e.} the power delivered per unit volume.
In terms of fields alone: use Ampère-Maxwell:
\[
{\bf E} \cdot {\bf J} = \frac{1}{\mu_0} {\bf E} \cdot ({\boldsymbol \nabla} \times {\bf B}) - \varepsilon_0 {\bf E} \cdot \frac{\partial {\bf E}}{\partial t}
\]
Using product rule 6,
Using <a href="./c_m_dc_pr.html#div_xprod">div_xprod</a>,
\[
{\boldsymbol ∇} ⋅ ({\bf E} × {\bf B}) = {\bf B} ⋅ ({\boldsymbol ∇} × {\bf E})
</p>
<ul class="org-ul">
<li>{\bf E} ⋅ ({\boldsymbol ∇} × {\bf B}),</li>
</ul>
<p>
{\boldsymbol \nabla} \cdot ({\bf E} \times {\bf B}) = {\bf B} \cdot ({\boldsymbol \nabla} \times {\bf E}) - {\bf E} \cdot ({\boldsymbol \nabla} \times {\bf B}),
\]
Invoking Faraday \({\boldsymbol \nabla} \times {\bf E} = - \partial {\bf B}/\partial t\),
\[
{\bf E} ⋅ ({\boldsymbol ∇} × {\bf B}) = - {\bf B} ⋅ \frac{∂ {\bf B}}{∂ t}
</p>
<ul class="org-ul">
<li>{\boldsymbol ∇} ⋅ ({\bf E} × {\bf B}).</li>
</ul>
<p>
{\bf E} \cdot ({\boldsymbol \nabla} \times {\bf B}) = - {\bf B} \cdot \frac{\partial {\bf B}}{\partial t} - {\boldsymbol \nabla} \cdot ({\bf E} \times {\bf B}).
\]
But obviously,
\[
@ -1679,28 +1686,35 @@ But obviously,
\]
so we get
\[
{\bf E} ⋅ {\bf J} = -\frac{1}{2} \frac{\partial}{\partial t} \left( ε_0 E^2 + \frac{1}{\mu_0} B^2 \right)
</p>
<ul class="org-ul">
<li>\frac{1}{\mu_0} {\boldsymbol ∇} ⋅ ({\bf E} × {\bf B}).</li>
</ul>
<p>
{\bf E} \cdot {\bf J} = -\frac{1}{2} \frac{\partial}{\partial t} \left( \varepsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) - \frac{1}{\mu_0} {\boldsymbol \nabla} \cdot ({\bf E} \times {\bf B}).
\label{Gr(8.8)}
\]
Substituting this in \ref{Gr(8.6)} and using the divergence theorem,
Substituting this in <a href="./emd_ce_poy.html#dWdt_intEJ">dWdt_intEJ</a> and using the divergence theorem,
we obtain
</p>
<div class="main div" id="orgda8af3a">
<div class="main div" id="orgf118f4f">
<p>
{\bf Poynting's theorem}
\[
\frac{dW}{dt} = -\frac{d}{d t} ∫_{\cal V} dτ \frac{1}{2} \left( ε_0 E^2 + \frac{1}{\mu_0} B^2 \right)
<b>Poynting's theorem</b>
</p>
<ul class="org-ul">
<li>\frac{1}{\mu_0} \oint_{\cal S} d{\bf a} ⋅ ({\bf E} × {\bf B})</li>
</ul>
<div class="eqlabel" id="org1b7cdac">
<p>
\label{Gr(8.9)}
<a id="👉Thm"></a><a href="./emd_ce_poy.html#👉Thm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgd7b6cac">
<ul class="org-ul">
<li>Gr (8.9)</li>
</ul>
</div>
</div>
<p>
\[
\frac{dW}{dt} = -\frac{d}{d t} \int_{\cal V} d\tau \frac{1}{2} \left( \varepsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) - \frac{1}{\mu_0} \oint_{\cal S} d{\bf a} \cdot ({\bf E} \times {\bf B})
\tag{👉Thm}\label{👉Thm}
\]
</p>
@ -1713,14 +1727,31 @@ 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:
Energy per unit time, per unit area carried by EM fields: given by the
</p>
<div class="core div" id="org8a1c10e">
<div class="core div" id="orgf3198a5">
<p>
<b>Poynting vector</b>
</p>
<div class="eqlabel" id="org8725431">
<p>
<a id="PoyntingVec"></a><a href="./emd_ce_poy.html#PoyntingVec"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org6f2879d">
<ul class="org-ul">
<li>Gr (8.10)</li>
</ul>
</div>
</div>
<p>
{\bf Poynting vector}
\[
{\bf S} \equiv \frac{1}{\mu_0} ({\bf E} \times {\bf B})
\label{Gr(8.10)}
\tag{PoyntingVec}\label{PoyntingVec}
\]
</p>
@ -1728,12 +1759,29 @@ 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="org57576be">
<div class="core div" id="org3a4bb91">
<p>
<b>Poynting's theorem</b> (integral form)
</p>
<div class="eqlabel" id="orgbf2cc63">
<p>
<a id="PoyntingThm_int"></a><a href="./emd_ce_poy.html#PoyntingThm_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org5f484e3">
<ul class="org-ul">
<li>Gr (8.11)</li>
</ul>
</div>
</div>
<p>
{\bf Poynting's theorem}
\[
\frac{dW}{dt} = - \frac{dU_{em}}{dt} - \oint_{\cal S} d{\bf a} \cdot {\bf S}.
\label{Gr(8.11)}
\tag{PoyntingThm_int}\label{PoyntingThm_int}
\]
</p>
@ -1741,12 +1789,29 @@ We can thus express Poynting's theorem more compactly:
<p>
where we have defined the total
</p>
<div class="core div" id="org43eb64b">
<div class="core div" id="orgbc7eb16">
<p>
<b>Energy in electromagnetic fields</b>
</p>
<div class="eqlabel" id="org89872c3">
<p>
<a id="Uem"></a><a href="./emd_ce_poy.html#Uem"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org67613ca">
<ul class="org-ul">
<li>Gr (8.5)</li>
</ul>
</div>
</div>
<p>
{\bf Energy in electromagnetic fields}
\[
U_{em} \equiv \frac{1}{2} \int d\tau \left( \varepsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right)
\label{Gr(8.5)}
\tag{Uem}\label{Uem}
\]
</p>
@ -1764,12 +1829,29 @@ Then,
\]
so we get the
</p>
<div class="core div" id="org29b53c2">
<div class="core div" id="org487db23">
<p>
<b>Poynting theorem</b> (differential form)
</p>
<div class="eqlabel" id="org9593699">
<p>
<a id="PoyntingThm"></a><a href="./emd_ce_poy.html#PoyntingThm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org129becb">
<ul class="org-ul">
<li>Gr (8.14)</li>
</ul>
</div>
</div>
<p>
{\bf Poynting theorem (differential form)}
\[
\frac{\partial}{\partial t} u_{em} + {\boldsymbol \nabla} \cdot {\bf S} = 0
\label{Gr(8.14)}
\tag{PoyntingThm}\label{PoyntingThm}
\]
</p>
@ -1781,9 +1863,21 @@ and has a similar for to the continuity equation
<div class="example div" id="orgbb7ac4b">
<div class="example div" id="orgd9e0ab5">
<p>
<b>Example: Joule heating</b>
</p>
<p>
<b>Task</b>: characterize the energy flow for a current-carrying wire.
</p>
<p>
<b>Solution</b>: the energy per unit time delivered to wire the wire can
be obtained from Poynting's theorem.
</p>
<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
\[
{\boldsymbol E} = \frac{V}{L} \hat{\boldsymbol x},
@ -1826,7 +1920,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

12
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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="org9d74d32">
<summary id="org175f28d">
<details class="prereq" id="orgd522e84">
<summary id="org3c8d630">
Prerequisites
</summary>
<ul class="org-ul">
@ -1632,8 +1632,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="org73ffbc8">
<summary id="org638d300">
<details class="objectives" id="org98124eb">
<summary id="org6546918">
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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

8
build/emd_emw_ep.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1652,7 +1652,7 @@ so for a monochromatic EM plan wave,
\]
or more succinctly:
</p>
<div class="main div" id="org677cdf7">
<div class="main div" id="org0e45759">
<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="orgc6fc8e6">
<div class="main div" id="org4b55ff9">
<p>
{\bf Momentum density of a monochromatic EM wave}
\[
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

6
build/emd_emw_mpw.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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="orgd2950da">
<div class="core div" id="org53e84bf">
<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>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

6
build/emd_emw_we.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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="org88fc1e4">
<div class="core div" id="orgb3cd985">
<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>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/emdm.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/emdm_Me.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

20
build/emdm_Me_Mem.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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="org9606db6">
<div class="core div" id="org0421a72">
<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="orge1ae7f1">
<aside id="org642846e">
<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="orgb0d00db">
<div class="main div" id="orgba471ef">
<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="org89c562c">
<div class="main div" id="org2ffd81b">
<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="org0196779">
<div class="core div" id="org88bb3c5">
<p>
\[
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
@ -1761,7 +1761,7 @@ or
\]
where as before
</p>
<div class="core div" id="org2b5f6d5">
<div class="core div" id="org90cea20">
<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="org2c2cd2a">
<div class="core div" id="orgd6526ab">
<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="orgcaf59d5">
<div class="main div" id="orgd345cd6">
<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>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

6
build/emdm_Me_bc.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1624,7 +1624,7 @@ Table of contents
<p>
Discontinuities between different media, deduced from
</p>
<div class="core div" id="orgc82cb6f">
<div class="core div" id="org48bc400">
<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>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
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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>
@ -1649,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/emdm_emwm_ad.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1645,7 +1645,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/emdm_emwm_ad_c.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<title>Pre-Quantum Electrodynamics</title>
@ -1741,7 +1741,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

6
build/emdm_emwm_plm.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
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<title>Pre-Quantum Electrodynamics</title>
@ -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="orga0b7fe5">
<div class="main div" id="orgd0bce92">
<p>
{\bf Index of refraction}
\[
@ -1716,7 +1716,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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build/emdm_emwm_refl.html
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<!DOCTYPE html>
<html lang="en">
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<meta charset="utf-8">
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<title>Pre-Quantum Electrodynamics</title>
@ -1660,7 +1660,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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build/emdm_emwm_refl_Ba.html
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<!DOCTYPE html>
<html lang="en">
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<meta charset="utf-8">
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<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/emdm_emwm_refl_Fe.html
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<!DOCTYPE html>
<html lang="en">
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1641,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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build/emdm_emwm_refl_ni.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

12
build/emdm_emwm_refl_oi.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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="orgab5561c">
<div class="core div" id="orge681e56">
<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="orgfe980d3">
<div class="core div" id="orgfbf5032">
<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="org3d560e5">
<div class="main div" id="orgcf2803c">
<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="org88a1e03">
<div class="main div" id="org7503b43">
<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>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/emdm_emwm_refr.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1641,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/emdm_emwm_wg.html
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<!DOCTYPE html>
<html lang="en">
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<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/emdm_emwm_wg_c.html
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<!DOCTYPE html>
<html lang="en">
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<meta charset="utf-8">
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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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/emdm_emwm_wg_gw.html
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<!DOCTYPE html>
<html lang="en">
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1691,7 +1691,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/emdm_emwm_wg_r.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

12
build/emf.html
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<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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="orgb740c0e">
<summary id="org919d213">
<details class="prereq" id="orgb93b9c3">
<summary id="org621593e">
Prerequisites
</summary>
<ul class="org-ul">
@ -1631,8 +1631,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="orgd6caf8f">
<summary id="org998f38c">
<details class="objectives" id="org179e7b9">
<summary id="orgb47e5d6">
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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/emf_g.html
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@ -1,7 +1,7 @@
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<html lang="en">
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<meta charset="utf-8">
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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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/emf_g_Cg.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1664,7 +1664,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

8
build/emf_g_Lg.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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="org892bf92">
<div class="core div" id="org242c17e">
<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="org2f9d40b">
<div class="core div" id="orgb00ab8a">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

12
build/emf_svp.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1638,7 +1638,7 @@ Useful strategy: represent fields in terms of potentials.
<p>
Easiest:
</p>
<div class="core div" id="org94d8659">
<div class="core div" id="orgf88fb96">
<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="org61bab66">
<div class="core div" id="org2f23671">
<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="orga59d0cc">
<div class="main div" id="org5127f9c">
<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="org514d3f1">
<div class="main div" id="org3f2c1f0">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

4
build/ems.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

12
build/ems_ca.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1626,8 +1626,8 @@ Table of contents
<li>Gr 3</li>
</ul>
<details class="prereq" id="org3402c83">
<summary id="org4cf0fe6">
<details class="prereq" id="org3b926ab">
<summary id="org2707d3f">
Prerequisites
</summary>
<ul class="org-ul">
@ -1635,8 +1635,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="org9645228">
<summary id="orgc4935b9">
<details class="objectives" id="orgfae12c2">
<summary id="org18515f0">
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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

12
build/ems_ca_fe.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -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="orgca99451">
<div class="main div" id="orgbb90bd9">
<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="orge64dc41">
<div class="main div" id="org9f1e898">
<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="orgd88f849">
<div class="core div" id="org7c70185">
<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="orgfefb816">
<div class="core div" id="org1935b19">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

38
build/ems_ca_fe_L.html
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<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="org46aafa7">
<div class="eqlabel" id="org84cc03f">
<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="org459093f">
<div class="alteqlabels" id="org61fa4ec">
</div>
@ -1660,14 +1660,14 @@ function \(\phi (x)\) and the Laplace equation reads
<p>
The solution to this is
</p>
<div class="eqlabel" id="orged9e79a">
<div class="eqlabel" id="orgb8c4067">
<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="org599dad5">
<div class="alteqlabels" id="org6b2acba">
<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="orgdc4453f">
<div class="eqlabel" id="orgf3e1009">
<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="org259268b">
<div class="alteqlabels" id="org696fc74">
</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="org822a974">
<div class="eqlabel" id="org9f81e63">
<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="org57d10c8">
<div class="alteqlabels" id="org66bdc2f">
</div>
@ -1805,8 +1805,8 @@ a point equals its value averaged over a sphere
\]
</p>
<details id="org0d796b5">
<summary id="org6d53cda">
<details id="org0f27c64">
<summary id="org881e712">
<strong>Physicist's proof</strong>
</summary>
<p>
@ -1868,8 +1868,8 @@ proving the theorem.
</p>
</details>
<details id="org68c57fd">
<summary id="org5db3c01">
<details id="org0dbbb13">
<summary id="orgf2b7c76">
<strong>Formal proof</strong>
</summary>
@ -1919,14 +1919,14 @@ we get the following general
<p>
<b>Theorem</b>:
</p>
<div class="eqlabel" id="org59c453b">
<div class="eqlabel" id="org230e535">
<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="org9481971">
<div class="alteqlabels" id="org37e5ede">
</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="org81bf520">
<div class="eqlabel" id="org8692599">
<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="orgf2a161c">
<div class="alteqlabels" id="orgcae1a27">
</div>
</div>
<div class="info div" id="orgd9c5641">
<div class="info div" id="orgab38e5b">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

16
build/ems_ca_fe_g.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<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="org1205a85">
<div class="info div" id="org5424abd">
<p>
<b>George Green</b>
</p>
<aside id="orgadbe90f">
<aside id="org7013f8e">
<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="org8bf3d02">
<div class="eqlabel" id="org17f1f87">
<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="org6723606">
<div class="alteqlabels" id="org9716c8b">
<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="org6f94224">
<div class="eqlabel" id="orgdecb68b">
<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="org75ac930">
<div class="alteqlabels" id="org1abae59">
<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: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>

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