diff --git a/build/a.html b/build/a.html
index f6c2424..9466867 100644
--- a/build/a.html
+++ b/build/a.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/a_l.html b/build/a_l.html
index b1f2e59..88cb9b4 100644
--- a/build/a_l.html
+++ b/build/a_l.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c.html b/build/c.html
index 7054128..c78b680 100644
--- a/build/c.html
+++ b/build/c.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m.html b/build/c_m.html
index 7e563a4..f767f07 100644
--- a/build/c_m.html
+++ b/build/c_m.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_cs.html b/build/c_m_cs.html
index dfe3b1d..883f7f2 100644
--- a/build/c_m_cs.html
+++ b/build/c_m_cs.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_cs_cyl.html b/build/c_m_cs_cyl.html
index 78fd440..9e9c0e6 100644
--- a/build/c_m_cs_cyl.html
+++ b/build/c_m_cs_cyl.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1659,14 +1659,14 @@ 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="org9ded6d5">
+<div class="eqlabel" id="org699f087">
<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"/>
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</svg></a>
</p>
-<div class="alteqlabels" id="org3066755">
+<div class="alteqlabels" id="org2197539">
<ul class="org-ul">
<li>Gr4(1.79)</li>
</ul>
@@ -1687,14 +1687,14 @@ 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="org0eb5cd5">
+<div class="eqlabel" id="org02508ba">
<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"/>
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</svg></a>
</p>
-<div class="alteqlabels" id="orge14058b">
+<div class="alteqlabels" id="org7c0369d">
<ul class="org-ul">
<li>Gr4(2.21)</li>
</ul>
@@ -1715,14 +1715,14 @@ 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="org34649e5">
+<div class="eqlabel" id="orgc355dd0">
<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">
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</svg></a>
</p>
-<div class="alteqlabels" id="orgeec607c">
+<div class="alteqlabels" id="org8037114">
<ul class="org-ul">
<li>Gr4(2.21)</li>
</ul>
@@ -1771,7 +1771,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_cs_hyp.html b/build/c_m_cs_hyp.html
index a5ad65e..5e3e1be 100644
--- a/build/c_m_cs_hyp.html
+++ b/build/c_m_cs_hyp.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1632,7 +1632,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_cs_sph.html b/build/c_m_cs_sph.html
index 29dce12..e42448c 100644
--- a/build/c_m_cs_sph.html
+++ b/build/c_m_cs_sph.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1623,14 +1623,14 @@ which \(r\) is the distance from the chosen origin,
The usual Cartesian coordinates relate to spherical coordinates
according to
</p>
-<div class="eqlabel" id="org309b2b3">
+<div class="eqlabel" id="org309b2e9">
<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">
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</svg></a>
</p>
-<div class="alteqlabels" id="org9ad4ca2">
+<div class="alteqlabels" id="org7a6a40a">
</div>
@@ -1655,14 +1655,14 @@ A generic vector can be expressed as
where the explicit relation between spherical and
Cartesian unit vectors is
</p>
-<div class="eqlabel" id="orgbf81903">
+<div class="eqlabel" id="org43b8824">
<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"/>
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</svg></a>
</p>
-<div class="alteqlabels" id="org437eb02">
+<div class="alteqlabels" id="org054cde6">
</div>
@@ -1685,14 +1685,14 @@ and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).
<p>
An infinitesimal displacement \(d{\bf l}\) can be written as
</p>
-<div class="eqlabel" id="orgda909cf">
+<div class="eqlabel" id="org0afd234">
<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="org02f05bf">
+<div class="alteqlabels" id="orgce4cca6">
</div>
@@ -1708,14 +1708,14 @@ d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin
<p>
Infinitesimal volume element:
</p>
-<div class="eqlabel" id="orgd2cfc4a">
+<div class="eqlabel" id="org260d3f6">
<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="org0a4c199">
+<div class="alteqlabels" id="org5ac82f0">
</div>
@@ -1736,14 +1736,14 @@ Infinitesimal surface element: depends on situation.
<div id="outline-container-c_m_cs_sph_grad" class="outline-6">
<h6 id="c_m_cs_sph_grad"><a href="#c_m_cs_sph_grad">Gradient</a></h6>
<div class="outline-text-6" id="text-c_m_cs_sph_grad">
-<div class="eqlabel" id="orge44fa24">
+<div class="eqlabel" id="org0a3cd9b">
<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="org6b20d40">
+<div class="alteqlabels" id="orgdf69aa8">
</div>
@@ -1760,14 +1760,14 @@ Infinitesimal surface element: depends on situation.
<div id="outline-container-c_m_cs_sph_div" class="outline-6">
<h6 id="c_m_cs_sph_div"><a href="#c_m_cs_sph_div">Divergence</a></h6>
<div class="outline-text-6" id="text-c_m_cs_sph_div">
-<div class="eqlabel" id="org3d72303">
+<div class="eqlabel" id="orgc4833f8">
<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="org9b1dfae">
+<div class="alteqlabels" id="org12f3039">
</div>
@@ -1784,14 +1784,14 @@ Infinitesimal surface element: depends on situation.
<div id="outline-container-c_m_cs_sph_curl" class="outline-6">
<h6 id="c_m_cs_sph_curl"><a href="#c_m_cs_sph_curl">Curl</a></h6>
<div class="outline-text-6" id="text-c_m_cs_sph_curl">
-<div class="eqlabel" id="orge960708">
+<div class="eqlabel" id="org6a4b6db">
<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="org0f49079">
+<div class="alteqlabels" id="org47ac74d">
</div>
@@ -1810,14 +1810,14 @@ Infinitesimal surface element: depends on situation.
<div id="outline-container-c_m_cs_sph_lap" class="outline-6">
<h6 id="c_m_cs_sph_lap"><a href="#c_m_cs_sph_lap">Laplacian</a></h6>
<div class="outline-text-6" id="text-c_m_cs_sph_lap">
-<div class="eqlabel" id="org1d2bef1">
+<div class="eqlabel" id="orgb7a43f7">
<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="org6ba97dc">
+<div class="alteqlabels" id="orgcba73ed">
</div>
@@ -1851,7 +1851,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_dc.html b/build/c_m_dc.html
index 83ba3fe..17a790f 100644
--- a/build/c_m_dc.html
+++ b/build/c_m_dc.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1642,7 +1642,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_dc_curl.html b/build/c_m_dc_curl.html
index 394bf5b..afc4ed5 100644
--- a/build/c_m_dc_curl.html
+++ b/build/c_m_dc_curl.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1644,7 +1644,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_dc_d2.html b/build/c_m_dc_d2.html
index 248c1a9..08d4287 100644
--- a/build/c_m_dc_d2.html
+++ b/build/c_m_dc_d2.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1615,9 +1615,9 @@ Table of contents
<div class="outline-text-5" id="text-c_m_dc_d2">
</div>
-<div id="outline-container-orgc162c65" class="outline-6">
-<h6 id="orgc162c65"><a href="#orgc162c65">Divergence of gradient</a></h6>
-<div class="outline-text-6" id="text-orgc162c65">
+<div id="outline-container-org78664e4" class="outline-6">
+<h6 id="org78664e4"><a href="#org78664e4">Divergence of gradient</a></h6>
+<div class="outline-text-6" id="text-org78664e4">
<p>
\({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
@@ -1626,44 +1626,44 @@ given by the Laplacian of the corresponding vector elements.
</div>
</div>
-<div id="outline-container-org2f7ef92" class="outline-6">
-<h6 id="org2f7ef92"><a href="#org2f7ef92">Curl of a gradient</a></h6>
-<div class="outline-text-6" id="text-org2f7ef92">
+<div id="outline-container-orgecb53d1" class="outline-6">
+<h6 id="orgecb53d1"><a href="#orgecb53d1">Curl of a gradient</a></h6>
+<div class="outline-text-6" id="text-orgecb53d1">
<p>
This always vanishes.
</p>
</div>
</div>
-<div id="outline-container-orgceccef1" class="outline-6">
-<h6 id="orgceccef1"><a href="#orgceccef1">Gradient of the divergence</a></h6>
-<div class="outline-text-6" id="text-orgceccef1">
+<div id="outline-container-orgdeab960" class="outline-6">
+<h6 id="orgdeab960"><a href="#orgdeab960">Gradient of the divergence</a></h6>
+<div class="outline-text-6" id="text-orgdeab960">
<p>
\({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.
</p>
</div>
</div>
-<div id="outline-container-orgc07e1dc" class="outline-6">
-<h6 id="orgc07e1dc"><a href="#orgc07e1dc">Divergence of a curl</a></h6>
-<div class="outline-text-6" id="text-orgc07e1dc">
+<div id="outline-container-org11519c8" class="outline-6">
+<h6 id="org11519c8"><a href="#org11519c8">Divergence of a curl</a></h6>
+<div class="outline-text-6" id="text-org11519c8">
<p>
This always vanishes.
</p>
</div>
</div>
-<div id="outline-container-org1a67d52" class="outline-6">
-<h6 id="org1a67d52"><a href="#org1a67d52">Curl of curl</a></h6>
-<div class="outline-text-6" id="text-org1a67d52">
-<div class="eqlabel" id="org5d4ff40">
+<div id="outline-container-org8bb99ae" class="outline-6">
+<h6 id="org8bb99ae"><a href="#org8bb99ae">Curl of curl</a></h6>
+<div class="outline-text-6" id="text-org8bb99ae">
+<div class="eqlabel" id="org38dd889">
<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">
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</svg></a>
</p>
-<div class="alteqlabels" id="orgd2518b8">
+<div class="alteqlabels" id="orgb8e88ad">
</div>
@@ -1694,7 +1694,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_dc_del.html b/build/c_m_dc_del.html
index 3bfe77d..12b1ffa 100644
--- a/build/c_m_dc_del.html
+++ b/build/c_m_dc_del.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_dc_div.html b/build/c_m_dc_div.html
index effcd20..33e2cf7 100644
--- a/build/c_m_dc_div.html
+++ b/build/c_m_dc_div.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_dc_g.html b/build/c_m_dc_g.html
index 069c74a..6816a5c 100644
--- a/build/c_m_dc_g.html
+++ b/build/c_m_dc_g.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_dc_pr.html b/build/c_m_dc_pr.html
index 50a52c1..bdf7e3a 100644
--- a/build/c_m_dc_pr.html
+++ b/build/c_m_dc_pr.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1622,14 +1622,14 @@ explicited as follows:
<p>
<b>Gradient of a product</b>:
</p>
-<div class="eqlabel" id="org824336d">
+<div class="eqlabel" id="org89b1475">
<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">
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</svg></a>
</p>
-<div class="alteqlabels" id="org2bf23af">
+<div class="alteqlabels" id="orgc71740f">
<ul class="org-ul">
<li>Gr (3)</li>
<li>W (1-111)</li>
@@ -1649,14 +1649,14 @@ explicited as follows:
<p>
<b>Gradient of a scalar product</b>:
</p>
-<div class="eqlabel" id="org41d1a68">
+<div class="eqlabel" id="orgdcd103a">
<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">
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org466aa7e">
+<div class="alteqlabels" id="org5dbb766">
<ul class="org-ul">
<li>Gr (4)</li>
<li>W (1-112)</li>
@@ -1676,14 +1676,14 @@ explicited as follows:
<p>
<b>Divergence of a product</b>:
</p>
-<div class="eqlabel" id="org1ad95d1">
+<div class="eqlabel" id="org16aa48b">
<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="org93785a5">
+<div class="alteqlabels" id="org03b99ec">
<ul class="org-ul">
<li>Gr (5)</li>
<li>W (1-115)</li>
@@ -1703,14 +1703,14 @@ explicited as follows:
<p>
<b>Divergence of a cross product</b>:
</p>
-<div class="eqlabel" id="org4333997">
+<div class="eqlabel" id="orga48567e">
<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">
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org2626fe4">
+<div class="alteqlabels" id="org0aece97">
<ul class="org-ul">
<li>Gr (6)</li>
<li>W (1-116)</li>
@@ -1730,14 +1730,14 @@ explicited as follows:
<p>
<b>Curl of a product</b>:
</p>
-<div class="eqlabel" id="orgfa1c2f8">
+<div class="eqlabel" id="org3dec348">
<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">
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="orgb82ff5c">
+<div class="alteqlabels" id="orgd0f6f07">
<ul class="org-ul">
<li>Gr (7)</li>
<li>W (1-118)</li>
@@ -1757,14 +1757,14 @@ explicited as follows:
<p>
<b>Curl of a cross product</b>:
</p>
-<div class="eqlabel" id="org9d0ffb0">
+<div class="eqlabel" id="orga9f7b57">
<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="org6beb5fe">
+<div class="alteqlabels" id="org69dc8a6">
<ul class="org-ul">
<li>Gr (8)</li>
<li>W (1-119)</li>
@@ -1805,7 +1805,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_dd.html b/build/c_m_dd.html
index 4bbeedd..3d54084 100644
--- a/build/c_m_dd.html
+++ b/build/c_m_dd.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_dd_1d.html b/build/c_m_dd_1d.html
index 4d5e0a5..796ce85 100644
--- a/build/c_m_dd_1d.html
+++ b/build/c_m_dd_1d.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_dd_3d.html b/build/c_m_dd_3d.html
index 8c3689d..6a365a5 100644
--- a/build/c_m_dd_3d.html
+++ b/build/c_m_dd_3d.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1636,14 +1636,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
More generally,
</p>
-<div class="eqlabel" id="org501b97b">
+<div class="eqlabel" id="orga7ea793">
<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"/>
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</svg></a>
</p>
-<div class="alteqlabels" id="org2eebb69">
+<div class="alteqlabels" id="org2afb48e">
<ul class="org-ul">
<li>Gr (1.100)</li>
</ul>
@@ -1662,14 +1662,14 @@ More generally,
Since
</p>
-<div class="eqlabel" id="orga23d9bb">
+<div class="eqlabel" id="org0791e6b">
<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">
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org17b7d37">
+<div class="alteqlabels" id="org5b1c286">
<ul class="org-ul">
<li>Gr (1.101)</li>
</ul>
@@ -1685,14 +1685,14 @@ Since
<p>
we have that
</p>
-<div class="eqlabel" id="orgc9c91cd">
+<div class="eqlabel" id="org72632aa">
<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="orgcc41bc0">
+<div class="alteqlabels" id="orgc9783d9">
<ul class="org-ul">
<li>Gr (1.102)</li>
</ul>
@@ -1724,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_dd_div.html b/build/c_m_dd_div.html
index 6c47520..bb36c11 100644
--- a/build/c_m_dd_div.html
+++ b/build/c_m_dd_div.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/c_m_ic.html b/build/c_m_ic.html
index ff7cae8..971b004 100644
--- a/build/c_m_ic.html
+++ b/build/c_m_ic.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
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<p class="author">Author: Jean-Sébastien Caux</p>
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index be85f35..760adf0 100644
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="author">Author: Jean-Sébastien Caux</p>
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index d2e0f0e..0cf78ec 100644
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<title>Pre-Quantum Electrodynamics</title>
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<p class="author">Author: Jean-Sébastien Caux</p>
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index 60d0902..9e4098d 100644
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<p class="author">Author: Jean-Sébastien Caux</p>
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diff --git a/build/c_m_ic_lsv.html b/build/c_m_ic_lsv.html
index 8aa7651..f4d4f3a 100644
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1615,9 +1615,9 @@ Table of contents
<div class="outline-text-5" id="text-c_m_ic_lsv">
</div>
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-<h6 id="org6c52cc6"><a href="#org6c52cc6">Line Integrals</a></h6>
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+<div id="outline-container-orge29f03b" class="outline-6">
+<h6 id="orge29f03b"><a href="#orge29f03b">Line Integrals</a></h6>
+<div class="outline-text-6" id="text-orge29f03b">
<p>
\[
{\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l}
@@ -1646,9 +1646,9 @@ Integral over a closed loop:
</div>
</div>
-<div id="outline-container-org7c42ba4" class="outline-6">
-<h6 id="org7c42ba4"><a href="#org7c42ba4">Surface Integrals</a></h6>
-<div class="outline-text-6" id="text-org7c42ba4">
+<div id="outline-container-org099f03d" class="outline-6">
+<h6 id="org099f03d"><a href="#org099f03d">Surface Integrals</a></h6>
+<div class="outline-text-6" id="text-org099f03d">
<p>
\[
\int_{\cal S} {\bf v} \cdot d{\bf a}
@@ -1668,9 +1668,9 @@ Over a closed surface:
</div>
</div>
-<div id="outline-container-org25bec25" class="outline-6">
-<h6 id="org25bec25"><a href="#org25bec25">Volume Integrals</a></h6>
-<div class="outline-text-6" id="text-org25bec25">
+<div id="outline-container-orgdad74b8" class="outline-6">
+<h6 id="orgdad74b8"><a href="#orgdad74b8">Volume Integrals</a></h6>
+<div class="outline-text-6" id="text-orgdad74b8">
<p>
\[
\int_{\cal V} T d\tau
@@ -1711,7 +1711,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title>
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<p class="validation"></p>
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diff --git a/build/c_m_va.html b/build/c_m_va.html
index beee8d7..ec806b7 100644
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<head>
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+<!-- 2022-03-22 Tue 10:52 -->
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<title>Pre-Quantum Electrodynamics</title>
@@ -1641,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<p class="validation"></p>
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index b1ce0df..c26ce79 100644
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
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diff --git a/build/c_m_va_n.html b/build/c_m_va_n.html
index 3702be0..e89a373 100644
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<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<p class="validation"></p>
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<title>Pre-Quantum Electrodynamics</title>
@@ -1665,7 +1665,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="validation"></p>
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diff --git a/build/c_m_vf.html b/build/c_m_vf.html
index b45e51e..7095019 100644
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<head>
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<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
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diff --git a/build/c_m_vf_helm.html b/build/c_m_vf_helm.html
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<p class="validation"></p>
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diff --git a/build/c_m_vf_pot.html b/build/c_m_vf_pot.html
index fff0045..bc1a15d 100644
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
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diff --git a/build/d.html b/build/d.html
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<title>Pre-Quantum Electrodynamics</title>
@@ -1614,8 +1614,8 @@ Table of contents
</svg></a><span class="headline-id">d</span></h2>
<div class="outline-text-2" id="text-d">
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="validation"></p>
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diff --git a/build/d_emd.html b/build/d_emd.html
index 9f5d5fa..ea4ba30 100644
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<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/d_emd_ce.html b/build/d_emd_ce.html
index a8741ee..6a59509 100644
--- a/build/d_emd_ce.html
+++ b/build/d_emd_ce.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1654,7 +1654,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/d_emd_emw.html b/build/d_emd_emw.html
index d9dd3e6..cf60733 100644
--- a/build/d_emd_emw.html
+++ b/build/d_emd_emw.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/d_emf.html b/build/d_emf.html
index 496bf39..e0b9c5e 100644
--- a/build/d_emf.html
+++ b/build/d_emf.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1654,7 +1654,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/d_ems.html b/build/d_ems.html
index 9f4ce54..9647680 100644
--- a/build/d_ems.html
+++ b/build/d_ems.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1676,7 +1676,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/d_ems_ca.html b/build/d_ems_ca.html
index fffacbe..c2368de 100644
--- a/build/d_ems_ca.html
+++ b/build/d_ems_ca.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/d_ems_ms.html b/build/d_ems_ms.html
index 04db7e4..f048017 100644
--- a/build/d_ems_ms.html
+++ b/build/d_ems_ms.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1656,7 +1656,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/d_emsm.html b/build/d_emsm.html
index f13314a..fbab99f 100644
--- a/build/d_emsm.html
+++ b/build/d_emsm.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/d_emsm_msm.html b/build/d_emsm_msm.html
index ad6aa8e..152ee22 100644
--- a/build/d_emsm_msm.html
+++ b/build/d_emsm_msm.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/d_m.html b/build/d_m.html
index 0ffa14e..5fb3403 100644
--- a/build/d_m.html
+++ b/build/d_m.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/d_red.html b/build/d_red.html
index f640f0e..220efe9 100644
--- a/build/d_red.html
+++ b/build/d_red.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd.html b/build/emd.html
index 59ac15c..6c9fcbf 100644
--- a/build/emd.html
+++ b/build/emd.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1614,8 +1614,8 @@ Table of contents
</svg></a><span class="headline-id">emd</span></h2>
<div class="outline-text-2" id="text-emd">
-<details class="prereq" id="orgeb31271">
-<summary id="org69f893c">
+<details class="prereq" id="orgeb3e60e">
+<summary id="org898f63b">
Prerequisites
</summary>
<ul class="org-ul">
@@ -1624,8 +1624,8 @@ Prerequisites
</ul>
</details>
-<details class="objectives" id="orgd436fef">
-<summary id="orgff05dfe">
+<details class="objectives" id="org11fe244">
+<summary id="org360a5fd">
Objectives
</summary>
<ul class="org-ul">
@@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_Fl.html b/build/emd_Fl.html
index 21a9ed1..86a8ff6 100644
--- a/build/emd_Fl.html
+++ b/build/emd_Fl.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_Fl_Fl.html b/build/emd_Fl_Fl.html
index 2eb510b..4c76482 100644
--- a/build/emd_Fl_Fl.html
+++ b/build/emd_Fl_Fl.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1644,18 +1644,18 @@ Empirically: the changing magnetic field induces an electric current around
the circuit. This current is really driven by an electric field having a component
along the wire. The line integral of this field is called the
</p>
-<div class="core div" id="orgfd2df2d">
+<div class="core div" id="orgc1ab4b8">
<p>
<b>Electromotive force (or electromotance)</b>,
</p>
-<div class="eqlabel" id="org510ac9c">
+<div class="eqlabel" id="org1ce0462">
<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="org85afd05">
+<div class="alteqlabels" id="orge2b1b8b">
<ul class="org-ul">
<li>Gr (7.9)</li>
</ul>
@@ -1680,14 +1680,14 @@ The precise statement associated to Faraday's observations
is that the electromotive force is proportional
to the rate of change of the magnetic flux,
</p>
-<div class="eqlabel" id="org4496ec0">
+<div class="eqlabel" id="orgb4b21bf">
<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="orga00ad3a">
+<div class="alteqlabels" id="org4e36847">
<ul class="org-ul">
<li>Gr (7.14)</li>
</ul>
@@ -1702,18 +1702,18 @@ to the rate of change of the magnetic flux,
\]
so we obtain
</p>
-<div class="core div" id="org35a9412">
+<div class="core div" id="org3b354eb">
<p>
<b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
</p>
-<div class="eqlabel" id="org6646a96">
+<div class="eqlabel" id="org3323690">
<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="org4c06533">
+<div class="alteqlabels" id="orgc991325">
<ul class="org-ul">
<li>Gr (7.15)</li>
</ul>
@@ -1737,15 +1737,15 @@ for any loop (on a wire or not). Using Stokes' theorem,
\]
we obtain
</p>
-<div class="core div" id="orgfef0799">
-<div class="eqlabel" id="org07be4b3">
+<div class="core div" id="org5eb686c">
+<div class="eqlabel" id="org7ca80cd">
<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="orgd458eb7">
+<div class="alteqlabels" id="org859ac4a">
<ul class="org-ul">
<li>Gr (7.16)</li>
</ul>
@@ -1790,7 +1790,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_Fl_e.html b/build/emd_Fl_e.html
index 47d03b3..e02406a 100644
--- a/build/emd_Fl_e.html
+++ b/build/emd_Fl_e.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1639,14 +1639,14 @@ W = \frac{1}{2} I \oint {\bf A} \cdot d{\bf l} = \frac{1}{2} \oint ({\bf A} \cdo
\]
Generalization to volume currents:
</p>
-<div class="eqlabel" id="org9c43b0a">
+<div class="eqlabel" id="org3409c0a">
<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="org1b4fa12">
+<div class="alteqlabels" id="org84f1d51">
<ul class="org-ul">
<li>Gr (7.31)</li>
</ul>
@@ -1681,15 +1681,15 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold
\]
We can integrate over all space: after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with
</p>
-<div class="core div" id="orgc92d34b">
-<div class="eqlabel" id="org7ec4181">
+<div class="core div" id="orgced7757">
+<div class="eqlabel" id="orgd899514">
<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="org073f8ff">
+<div class="alteqlabels" id="orgf7c92a2">
<ul class="org-ul">
<li>Gr (7.34)</li>
</ul>
@@ -1718,7 +1718,7 @@ W_{mag} &= \frac{1}{2} \int d\tau ~({\bf A} \cdot {\bf J}) &= \frac{1}{2
which are equations <a href="./ems_es_e.html#W_vcd">W_vcd</a>, <a href="./ems_es_e.html#W_intEsq">W_intEsq</a>, <a href="./emd_Fl_e.html#W_intAJ">W_intAJ</a> and <a href="./emd_Fl_e.html#W_intBsq">W_intBsq</a>.
</p>
-<div class="example div" id="org60417d4">
+<div class="example div" id="orge558476">
<p>
<b>Example: energy in coaxial cable</b>
</p>
@@ -1768,7 +1768,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_Fl_i.html b/build/emd_Fl_i.html
index 388ff54..3c1850b 100644
--- a/build/emd_Fl_i.html
+++ b/build/emd_Fl_i.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1641,14 +1641,14 @@ so
\]
and we can write the mutual inductance as the <b>Neumann formula</b>,
</p>
-<div class="eqlabel" id="org554c244">
+<div class="eqlabel" id="org3ec4c18">
<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="orgf48cffa">
+<div class="alteqlabels" id="org7ff8d1e">
<ul class="org-ul">
<li>Gr (7.22)</li>
</ul>
@@ -1665,14 +1665,14 @@ M_{21} = \frac{\mu_0}{4\pi} \oint_{{\cal P}_1} \oint_{{\cal P}_2} \frac{d{\bf l}
Two things:
first, \(M_{21}\) is purely geometrical. Second,
</p>
-<div class="eqlabel" id="org12f955a">
+<div class="eqlabel" id="orge7f5edd">
<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="orga59d4b0">
+<div class="alteqlabels" id="org54ddc77">
<ul class="org-ul">
<li>Gr (7.23)</li>
</ul>
@@ -1687,7 +1687,7 @@ M_{12} = M_{21}
\]
</p>
-<div class="example div" id="orgdd5f18b">
+<div class="example div" id="org1bcebe3">
<p>
<b>Example: solenoid in solenoid</b>
</p>
@@ -1735,14 +1735,14 @@ What if we vary current in loop 1? Flux in 2 will vary. Induces EMF in loop 2:
\]
Changing current also induces EMF in the source loop itself:
</p>
-<div class="eqlabel" id="org06772cd">
+<div class="eqlabel" id="orgdaaac4e">
<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="org5ce4f36">
+<div class="alteqlabels" id="orgb8faf94">
<ul class="org-ul">
<li>Gr (7.25)</li>
</ul>
@@ -1765,7 +1765,7 @@ Inductance: measured in <b>henries</b> (\(H\)). \(H = V s/A\).
</p>
-<div class="example div" id="org678dce1">
+<div class="example div" id="org11544ba">
<p>
<b>Example: self-inductance of toroidal coil</b>
</p>
@@ -1804,7 +1804,7 @@ Inductance (like capacitance) is intrinsically positive. Use Lenz law.
Think of <i>back EMF</i>.
</p>
-<div class="example div" id="org42aa8b9">
+<div class="example div" id="org52373ca">
<p>
<b>Example: circuit</b>
</p>
@@ -1853,7 +1853,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_Fl_ief.html b/build/emd_Fl_ief.html
index 3de193e..a17e094 100644
--- a/build/emd_Fl_ief.html
+++ b/build/emd_Fl_ief.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1637,7 +1637,7 @@ law in integral form:
-<div class="example div" id="orgcfd6427">
+<div class="example div" id="orgfcd2e61">
<p>
<b>Example: loop with time-dependent flux</b>
</p>
@@ -1664,7 +1664,7 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
</div>
-<div class="example div" id="orgfc45720">
+<div class="example div" id="orgafdd47c">
<p>
<b>Example: wheel with charged rim traversed by flux</b>
</p>
@@ -1705,7 +1705,7 @@ called the <b>quasistatic</b> approximation, and works provided we deal with
<i>slow enough</i> phenomena.
</p>
-<div class="example div" id="org1d9bce6">
+<div class="example div" id="org6631fcc">
<p>
<b>Example: field from wire with time-dependent current</b>
</p>
@@ -1764,7 +1764,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_Me.html b/build/emd_Me.html
index 14162d3..6580b87 100644
--- a/build/emd_Me.html
+++ b/build/emd_Me.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_Me_Me.html b/build/emd_Me_Me.html
index 447b9ef..1944a1a 100644
--- a/build/emd_Me_Me.html
+++ b/build/emd_Me_Me.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1616,18 +1616,18 @@ Table of contents
<p>
Full set of equations for the electromagnetic field:
</p>
-<div class="core div" id="orgee05fb0">
+<div class="core div" id="org71429ce">
<p>
<b>Maxwell's equations</b> <i>(in vacuum)</i>
</p>
-<div class="eqlabel" id="orgf090588">
+<div class="eqlabel" id="org92aa36d">
<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">
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</svg></a>
</p>
-<div class="alteqlabels" id="org538c235">
+<div class="alteqlabels" id="orgd8ea0a7">
</div>
@@ -1644,7 +1644,7 @@ Full set of equations for the electromagnetic field:
<p>
Complement:
</p>
-<div class="core div" id="org14efa58">
+<div class="core div" id="org93d1ad6">
<p>
Force law <a href="./ems_ms_lf_pc.html#LorFo">LorFo</a>
\[
@@ -1667,15 +1667,15 @@ take divergence of \((iv)\) and use \((i)\).
<p>
Better way of writing: all fields on left, all sources on right,
</p>
-<div class="core div" id="orge7a8009">
-<div class="eqlabel" id="orgaf03f6a">
+<div class="core div" id="org82c9083">
+<div class="eqlabel" id="org1583626">
<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="orgc1dd627">
+<div class="alteqlabels" id="orgfc0a16c">
<ul class="org-ul">
<li>Gr (7.42)</li>
</ul>
@@ -1713,7 +1713,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_Me_dc.html b/build/emd_Me_dc.html
index 4d54e09..3b7c053 100644
--- a/build/emd_Me_dc.html
+++ b/build/emd_Me_dc.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1622,15 +1622,15 @@ The term which should be zero (but isn't) in <a href="./emd_Me_ebM.html#divcurlB
\]
The extra term would thus be eliminated if we were to put
</p>
-<div class="core div" id="org9029eb1">
-<div class="eqlabel" id="org7937f05">
+<div class="core div" id="orgfb244bd">
+<div class="eqlabel" id="org02e358a">
<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="orgc4ec53c">
+<div class="alteqlabels" id="orgf266507">
<ul class="org-ul">
<li>Gr (7.36)</li>
</ul>
@@ -1659,18 +1659,18 @@ Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation
<p>
Maxwell baptized this term the
</p>
-<div class="core div" id="orgf5b87fa">
+<div class="core div" id="orgb3e1688">
<p>
<b>Displacement current</b>
</p>
-<div class="eqlabel" id="orgff1f2c8">
+<div class="eqlabel" id="org5bc6ac9">
<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="org87a5cc7">
+<div class="alteqlabels" id="org2273d93">
<ul class="org-ul">
<li>Gr (7.37)</li>
</ul>
@@ -1724,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_Me_ebM.html b/build/emd_Me_ebM.html
index 1c8d345..b3cbec5 100644
--- a/build/emd_Me_ebM.html
+++ b/build/emd_Me_ebM.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1630,14 +1630,14 @@ Fatal inconsistency: div of curl must always vanish. Check on \((iii)\):
\]
But: try same with \((iv)\):
</p>
-<div class="eqlabel" id="org76390cc">
+<div class="eqlabel" id="org1332a90">
<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="org64ca711">
+<div class="alteqlabels" id="orgc919b58">
<ul class="org-ul">
<li>Gr (7.35)</li>
</ul>
@@ -1682,7 +1682,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_Me_mc.html b/build/emd_Me_mc.html
index be16b63..08849f6 100644
--- a/build/emd_Me_mc.html
+++ b/build/emd_Me_mc.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_ce.html b/build/emd_ce.html
index b9a793e..3c77cc0 100644
--- a/build/emd_ce.html
+++ b/build/emd_ce.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1614,8 +1614,8 @@ Table of contents
</svg></a><span class="headline-id">emd.ce</span></h3>
<div class="outline-text-3" id="text-emd_ce">
-<details class="prereq" id="orgd007ccd">
-<summary id="org4b65898">
+<details class="prereq" id="org4aea2a7">
+<summary id="orgbb42f82">
Prerequisites
</summary>
<ul class="org-ul">
@@ -1623,8 +1623,8 @@ Prerequisites
</ul>
</details>
-<details class="objectives" id="org1ca55b0">
-<summary id="org22a98ae">
+<details class="objectives" id="orgacf1d30">
+<summary id="org560d61d">
Objectives
</summary>
<ul class="org-ul">
@@ -1662,7 +1662,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_ce_amom.html b/build/emd_ce_amom.html
index 950723c..16b550a 100644
--- a/build/emd_ce_amom.html
+++ b/build/emd_ce_amom.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1616,18 +1616,18 @@ Table of contents
<p>
The angular momentum of EM fields is directly given by
</p>
-<div class="main div" id="org9f96756">
+<div class="main div" id="org8982815">
<p>
<b>Angular momentum of EM fields</b>
</p>
-<div class="eqlabel" id="org2a11d69">
+<div class="eqlabel" id="org8297ce4">
<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="orgfb3f8f7">
+<div class="alteqlabels" id="orgc1dccb2">
</div>
@@ -1661,7 +1661,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_ce_ce.html b/build/emd_ce_ce.html
index f559746..d5ec10d 100644
--- a/build/emd_ce_ce.html
+++ b/build/emd_ce_ce.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1635,7 +1635,7 @@ This means that
\]
Since this is true for any volume, we have (re)derived the
</p>
-<div class="core div" id="org3f0dec4">
+<div class="core div" id="org1d21244">
<p>
<b>Continuity equation</b> <a href="./ems_ms_ce.html#conteq">conteq</a>
\[
@@ -1675,7 +1675,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_ce_mom.html b/build/emd_ce_mom.html
index aa90916..1510ed0 100644
--- a/build/emd_ce_mom.html
+++ b/build/emd_ce_mom.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1628,18 +1628,18 @@ in which the first integral can be interpreted as the momentum stored in the EM
<p>
This is thus simply a conservation law for momentum, with
</p>
-<div class="main div" id="orgc119c5d">
+<div class="main div" id="orgbd3d67d">
<p>
<b>Momentum density in the EM fields</b>
</p>
-<div class="eqlabel" id="org4c00e17">
+<div class="eqlabel" id="org2b66251">
<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="org62e68b8">
+<div class="alteqlabels" id="org006aaa0">
</div>
@@ -1655,18 +1655,18 @@ This is thus simply a conservation law for momentum, with
<p>
In a region in which the mechanical momentum is not changing due to external influences, we then have the
</p>
-<div class="main div" id="org55398a2">
+<div class="main div" id="orgd647d42">
<p>
<b>Continuity equation for EM momentum</b>
</p>
-<div class="eqlabel" id="org02bb382">
+<div class="eqlabel" id="org84364f8">
<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="orgf040d5c">
+<div class="alteqlabels" id="org03c5d23">
</div>
@@ -1699,7 +1699,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_ce_mst.html b/build/emd_ce_mst.html
index 90f337a..67ffc16 100644
--- a/build/emd_ce_mst.html
+++ b/build/emd_ce_mst.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1667,18 +1667,18 @@ and similarly for \({\boldsymbol B}\). We thus get
<p>
This expression can be greatly simplified by introducing the
</p>
-<div class="main div" id="org6a32dc6">
+<div class="main div" id="org63be843">
<p>
<b>Maxwell stress tensor</b>
</p>
-<div class="eqlabel" id="org46015d8">
+<div class="eqlabel" id="org7c75f9d">
<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="org9675080">
+<div class="alteqlabels" id="orgd84a618">
</div>
@@ -1699,18 +1699,18 @@ The element \(T_{ij}\) represents the force per unit area in the \(i\) direction
<p>
We then obtain the
</p>
-<div class="main div" id="org0171e33">
+<div class="main div" id="orgfeaab67">
<p>
<b>EM force per unit volume</b>
</p>
-<div class="eqlabel" id="org5306def">
+<div class="eqlabel" id="org9fc5057">
<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="orgc0f4f98">
+<div class="alteqlabels" id="orgecbfdae">
</div>
@@ -1726,18 +1726,18 @@ We then obtain the
<p>
where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
</p>
-<div class="main div" id="org28ab853">
+<div class="main div" id="org86829bd">
<p>
<b>Total force on charges in volume</b>
</p>
-<div class="eqlabel" id="orgb821a87">
+<div class="eqlabel" id="org2370cf3">
<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="orgdcbb5c3">
+<div class="alteqlabels" id="orgc55376c">
</div>
@@ -1770,7 +1770,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_ce_poy.html b/build/emd_ce_poy.html
index cada692..75edee8 100644
--- a/build/emd_ce_poy.html
+++ b/build/emd_ce_poy.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1636,14 +1636,14 @@ done by EM forces? From Lorentz force law:
Really, we're looking at a small volume element \(d\tau\) carrying charge \(\rho d\tau\), moving
at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\). Thus,
</p>
-<div class="eqlabel" id="org935dce9">
+<div class="eqlabel" id="orge2567ac">
<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="orgad7d22c">
+<div class="alteqlabels" id="org4d05a80">
<ul class="org-ul">
<li>Gr (8.6)</li>
</ul>
@@ -1683,18 +1683,18 @@ so we get
Substituting this in <a href="./emd_ce_poy.html#dWdt_intEJ">dWdt_intEJ</a> and using the divergence theorem,
we obtain
</p>
-<div class="main div" id="orgdbc3e5c">
+<div class="main div" id="org36db96e">
<p>
<b>Poynting's theorem</b>
</p>
-<div class="eqlabel" id="orgc64c555">
+<div class="eqlabel" id="org21ce6b8">
<p>
<a id="👉Thm"></a><a href="./emd_ce_poy.html#👉Thm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="orgf1ac980">
+<div class="alteqlabels" id="org09a112b">
<ul class="org-ul">
<li>Gr (8.9)</li>
</ul>
@@ -1720,18 +1720,18 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
<p>
Energy per unit time, per unit area carried by EM fields: given by the
</p>
-<div class="core div" id="orga8e7c87">
+<div class="core div" id="org0bf3f6c">
<p>
<b>Poynting vector</b>
</p>
-<div class="eqlabel" id="org7fb23da">
+<div class="eqlabel" id="org327095e">
<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="org8150e29">
+<div class="alteqlabels" id="org7684a8e">
<ul class="org-ul">
<li>Gr (8.10)</li>
</ul>
@@ -1750,18 +1750,18 @@ Energy per unit time, per unit area carried by EM fields: given by the
<p>
We can thus express Poynting's theorem more compactly:
</p>
-<div class="core div" id="org41e1182">
+<div class="core div" id="org4255191">
<p>
<b>Poynting's theorem</b> (integral form)
</p>
-<div class="eqlabel" id="orgfa25d4a">
+<div class="eqlabel" id="orgd0524d3">
<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="org264bb7a">
+<div class="alteqlabels" id="org8acaa16">
<ul class="org-ul">
<li>Gr (8.11)</li>
</ul>
@@ -1780,18 +1780,18 @@ We can thus express Poynting's theorem more compactly:
<p>
where we have defined the total
</p>
-<div class="core div" id="orga6dad30">
+<div class="core div" id="org85e77e2">
<p>
<b>Energy in electromagnetic fields</b>
</p>
-<div class="eqlabel" id="org16b17ef">
+<div class="eqlabel" id="org7d4ae3d">
<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="orgdf4c40f">
+<div class="alteqlabels" id="org7a55647">
<ul class="org-ul">
<li>Gr (8.5)</li>
</ul>
@@ -1820,18 +1820,18 @@ Then,
\]
so we get the
</p>
-<div class="core div" id="orgfd64b94">
+<div class="core div" id="orgbdb096d">
<p>
<b>Poynting theorem</b> (differential form)
</p>
-<div class="eqlabel" id="org47e48c8">
+<div class="eqlabel" id="org9c56948">
<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="orge4a2af6">
+<div class="alteqlabels" id="orgdfe3994">
<ul class="org-ul">
<li>Gr (8.14)</li>
</ul>
@@ -1854,7 +1854,7 @@ and has a similar for to the continuity equation
-<div class="example div" id="org5f795de">
+<div class="example div" id="org15b725b">
<p>
<b>Example: Joule heating</b>
</p>
@@ -1911,7 +1911,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_emw.html b/build/emd_emw.html
index e88cc8b..494963b 100644
--- a/build/emd_emw.html
+++ b/build/emd_emw.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1614,8 +1614,8 @@ Table of contents
</svg></a><span class="headline-id">emd.emw</span></h3>
<div class="outline-text-3" id="text-emd_emw">
-<details class="prereq" id="org81c4de7">
-<summary id="org94e2b41">
+<details class="prereq" id="org17a8cef">
+<summary id="org45e360a">
Prerequisites
</summary>
<ul class="org-ul">
@@ -1624,8 +1624,8 @@ Prerequisites
</ul>
</details>
-<details class="objectives" id="org04214f1">
-<summary id="org840c379">
+<details class="objectives" id="org91615eb">
+<summary id="orgec366bf">
Objectives
</summary>
<ul class="org-ul">
@@ -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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_emw_ep.html b/build/emd_emw_ep.html
index 7f90b97..441d2bb 100644
--- a/build/emd_emw_ep.html
+++ b/build/emd_emw_ep.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1644,18 +1644,18 @@ so for a monochromatic EM plan wave,
\]
or more succinctly:
</p>
-<div class="main div" id="orge6fd2ac">
+<div class="main div" id="orgdf4b8bc">
<p>
<b>Poynting vector of a monochromatic EM wave</b>
</p>
-<div class="eqlabel" id="orgc1c96f7">
+<div class="eqlabel" id="org3913519">
<p>
<a id="Poynting_mpw"></a><a href="./emd_emw_ep.html#Poynting_mpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org907a483">
+<div class="alteqlabels" id="org8b59092">
<ul class="org-ul">
<li>Gr (9.57)</li>
</ul>
@@ -1678,7 +1678,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
<p>
Similary, we get the
</p>
-<div class="main div" id="orga2199bc">
+<div class="main div" id="org21912bb">
<p>
<b>Momentum density of a monochromatic EM wave</b>
\[
@@ -1729,7 +1729,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_emw_mpw.html b/build/emd_emw_mpw.html
index 6b5f01b..665b048 100644
--- a/build/emd_emw_mpw.html
+++ b/build/emd_emw_mpw.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1636,14 +1636,14 @@ From Faraday: \({\boldsymbol \nabla} \times {\bf E} = -\partial {\bf B}/\partia
\]
so \({\bf E}\) and \({\bf B}\) are mutually perpendicular, and
</p>
-<div class="eqlabel" id="org4864134">
+<div class="eqlabel" id="org619c73f">
<p>
<a id="EBmpw"></a><a href="./emd_emw_mpw.html#EBmpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org6d00653">
+<div class="alteqlabels" id="orgaad096d">
<ul class="org-ul">
<li>Gr (9.47)</li>
</ul>
@@ -1662,18 +1662,18 @@ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0.
Generalizing to propagation in the direction of an arbitrary wavevector
\({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
</p>
-<div class="core div" id="org9581019">
+<div class="core div" id="org9c77e17">
<p>
<b>E and B fields for a monochromatic EM plane wave</b>
</p>
-<div class="eqlabel" id="org98b7671">
+<div class="eqlabel" id="org54506e2">
<p>
<a id="mpw"></a><a href="./emd_emw_mpw.html#mpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="orge381054">
+<div class="alteqlabels" id="orgb384c79">
<ul class="org-ul">
<li>Gr (9.49)</li>
</ul>
@@ -1724,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emd_emw_we.html b/build/emd_emw_we.html
index 34cd8e8..f8b6db0 100644
--- a/build/emd_emw_we.html
+++ b/build/emd_emw_we.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1641,18 +1641,18 @@ These take the form of coupled first-order partial differential equations for \(
Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
we get the
</p>
-<div class="core div" id="org672468f">
+<div class="core div" id="org0b3ebdc">
<p>
<b>Wave equations for electric and magnetic fields in vacuum</b>
</p>
-<div class="eqlabel" id="orgf656d76">
+<div class="eqlabel" id="orge465a34">
<p>
<a id="WaveEq"></a><a href="./emd_emw_we.html#WaveEq"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org1ac2ec3">
+<div class="alteqlabels" id="org3ff5ea0">
<ul class="org-ul">
<li>Gr (9.41)</li>
</ul>
@@ -1714,7 +1714,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm.html b/build/emdm.html
index d8eab27..770533d 100644
--- a/build/emdm.html
+++ b/build/emdm.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm_Me.html b/build/emdm_Me.html
index 60938aa..95191e9 100644
--- a/build/emdm_Me.html
+++ b/build/emdm_Me.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm_Me_Mem.html b/build/emdm_Me_Mem.html
index 42906a3..f1e2b53 100644
--- a/build/emdm_Me_Mem.html
+++ b/build/emdm_Me_Mem.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1648,18 +1648,18 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
\]
We therefore have the
</p>
-<div class="core div" id="orge0518b1">
+<div class="core div" id="org64396a0">
<p>
<b>Polarization current density</b>
</p>
-<div class="eqlabel" id="org62db796">
+<div class="eqlabel" id="orgf37fc4f">
<p>
<a id="Jp"></a><a href="./emdm_Me_Mem.html#Jp"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="orgebf1aab">
+<div class="alteqlabels" id="org12af1ad">
<ul class="org-ul">
<li>Gr (7.48)</li>
</ul>
@@ -1683,7 +1683,7 @@ 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="orgfd82638">
+<aside id="org0322793">
<p>
Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current \({\boldsymbol J}_b\) but this is not the common convention.
</p>
@@ -1705,15 +1705,15 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
In view of this: total charge density can be separated into 2 parts,
<b>free</b> and <b>bound</b>:
</p>
-<div class="main div" id="org5bd748a">
-<div class="eqlabel" id="org94dfd02">
+<div class="main div" id="org56027a6">
+<div class="eqlabel" id="org6fd2aed">
<p>
<a id="rhofb"></a><a href="./emdm_Me_Mem.html#rhofb"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="orgd2d4d1d">
+<div class="alteqlabels" id="org9e03f13">
<ul class="org-ul">
<li>Gr (7.49)</li>
</ul>
@@ -1733,15 +1733,15 @@ In view of this: total charge density can be separated into 2 parts,
and current can be separated into three parts, <b>free</b>, <b>bound</b> and
<b>polarization</b>:
</p>
-<div class="main div" id="org7ad8d3b">
-<div class="eqlabel" id="org19a6c97">
+<div class="main div" id="org96443a3">
+<div class="eqlabel" id="orga6b8e4a">
<p>
<a id="Jfbp"></a><a href="./emdm_Me_Mem.html#Jfbp"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="orgac73457">
+<div class="alteqlabels" id="orgddcfaf9">
<ul class="org-ul">
<li>Gr (7.50)</li>
</ul>
@@ -1767,7 +1767,7 @@ Gauss's law: can be rewritten
\]
where (as in static case)
</p>
-<div class="core div" id="org65ac5cc">
+<div class="core div" id="orgdbd92f1">
<p>
\[
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
@@ -1788,7 +1788,7 @@ or
\]
where as before
</p>
-<div class="core div" id="org7e8e1b5">
+<div class="core div" id="org7839f4c">
<p>
\[
{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
@@ -1806,18 +1806,18 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
<p>
In terms of free charges and currents, we thus get
</p>
-<div class="core div" id="org089bb13">
+<div class="core div" id="org6597738">
<p>
<b>Maxwell's equations</b> <i>(in matter)</i>
</p>
-<div class="eqlabel" id="orgfabaa36">
+<div class="eqlabel" id="orgffcbbe9">
<p>
<a id="Max_mat"></a><a href="./emdm_Me_Mem.html#Max_mat"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="orgaba63f2">
+<div class="alteqlabels" id="org5c6c62d">
<ul class="org-ul">
<li>Gr (7.55)</li>
</ul>
@@ -1847,15 +1847,15 @@ This must all be complemented by the <b>constitutive relations</b> giving \({\bf
in terms of \({\bf E}\) and \({\bf B}\).
For the restricted case of linear media:
</p>
-<div class="main div" id="org435d93d">
-<div class="eqlabel" id="orgb5db623">
+<div class="main div" id="org2d4f8dd">
+<div class="eqlabel" id="org03c06a4">
<p>
<a id="consrel"></a><a href="./emdm_Me_Mem.html#consrel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org6a8a0ba">
+<div class="alteqlabels" id="org45b4fff">
<ul class="org-ul">
<li>Gr (7.56,7.57)</li>
</ul>
@@ -1898,7 +1898,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm_Me_bc.html b/build/emdm_Me_bc.html
index 1d23b5f..048eac2 100644
--- a/build/emdm_Me_bc.html
+++ b/build/emdm_Me_bc.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1616,18 +1616,18 @@ Table of contents
<p>
Discontinuities between different media, deduced from
</p>
-<div class="core div" id="org31953cb">
+<div class="core div" id="orgb7aedd2">
<p>
<b>Maxwell's equations</b> <i>(in matter)</i>, <i>integral form</i>
</p>
-<div class="eqlabel" id="org853b142">
+<div class="eqlabel" id="org6c267cc">
<p>
<a id="Max_mat_int"></a><a href="./emdm_Me_bc.html#Max_mat_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org734ccfd">
+<div class="alteqlabels" id="org8952960">
</div>
@@ -1645,15 +1645,15 @@ Discontinuities between different media, deduced from
Applying \((i)\) to wafer-thin Gaussian pillbox straddling boundary between 2 materials:
\({\bf D}_1 \cdot {\bf a} - {\bf D}_2 \cdot {\bf a} = \sigma_f a\) so
</p>
-<div class="main div" id="org74cc983">
-<div class="eqlabel" id="org49821b6">
+<div class="main div" id="orgf92353e">
+<div class="eqlabel" id="orgb9a9a3d">
<p>
<a id="Ddisc"></a><a href="./emdm_Me_bc.html#Ddisc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="orgb4f898e">
+<div class="alteqlabels" id="orga6c409d">
<ul class="org-ul">
<li>Gr (7.60)</li>
</ul>
@@ -1672,9 +1672,9 @@ D^{\perp}_1 - D^{\perp}_2 = \sigma_f
<p>
Same reasoning applied to \((ii)\) gives <a href="./ems_ms_vp_mbc.html#Bdisc">Bdisc</a>
</p>
-<div class="main div" id="org167c3b0">
-<div class="eqlabel" id="orgd15716f">
-<div class="alteqlabels" id="orgc56e074">
+<div class="main div" id="orga22be6d">
+<div class="eqlabel" id="orge0d8321">
+<div class="alteqlabels" id="orgb94a831">
</div>
@@ -1690,7 +1690,7 @@ B^{\perp}_1 - B^{\perp}_2 = 0
For \((iii)\): Amperian loop straddling surface: \({\bf E}_1 \cdot {\bf l} - {\bf E}_2 \cdot {\bf l} =
-\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}\). Limit of small loop: flux vanishes, therefore
</p>
-<div class="main div" id="org944c540">
+<div class="main div" id="org55a1d42">
<p>
\[
{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0
@@ -1704,7 +1704,7 @@ No volume current can contribute, but a surface current can. Can write
\(I_{f_{enc}} = {\bf K}_f \cdot (\hat{\bf n} \times {\bf l}) = ({\bf K}_f \times \hat{\bf n}) \cdot {\bf l}\)
and thus (as we got before in <a href="./emsm_msm_H_A.html#Hdisc">Hdisc</a>)
</p>
-<div class="main div" id="org44aa4fb">
+<div class="main div" id="org4a2ee70">
<p>
\[
{\bf H}_1^{\parallel} - {\bf H}_2^{\parallel} = {\bf K}_f \times \hat{\bf n}
@@ -1720,15 +1720,15 @@ These are the general boundary conditions for electrodynamics.
<p>
In case of linear media: can be expressed in terms of \({\bf E}\) and \({\bf B}\) alone:
</p>
-<div class="main div" id="org74a1c76">
-<div class="eqlabel" id="orgbb2af38">
+<div class="main div" id="org9e90e47">
+<div class="eqlabel" id="org8bbc727">
<p>
<a id="disc_lm"></a><a href="./emdm_Me_bc.html#disc_lm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org98cdce0">
+<div class="alteqlabels" id="orga51b616">
<ul class="org-ul">
<li>Gr (7.64)</li>
</ul>
@@ -1748,14 +1748,14 @@ In case of linear media: can be expressed in terms of \({\bf E}\) and \({\bf B}
<p>
If there is no free charge and no free current at boundary:
</p>
-<div class="eqlabel" id="orga2465cb">
+<div class="eqlabel" id="orgf07dc3e">
<p>
<a id="disc_nfc"></a><a href="./emdm_Me_bc.html#disc_nfc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org06ac001">
+<div class="alteqlabels" id="orgdd74046">
<ul class="org-ul">
<li>Gr (7.64)</li>
</ul>
@@ -1792,7 +1792,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm_emwm.html b/build/emdm_emwm.html
index eb87693..a22cf60 100644
--- a/build/emdm_emwm.html
+++ b/build/emdm_emwm.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1614,8 +1614,8 @@ Table of contents
</svg></a><span class="headline-id">emdm.emwm</span></h3>
<div class="outline-text-3" id="text-emdm_emwm">
-<details class="prereq" id="orgf2f0eda">
-<summary id="orged680b6">
+<details class="prereq" id="org5c9f997">
+<summary id="orgcac0ec5">
Prerequisites
</summary>
<ul class="org-ul">
@@ -1625,8 +1625,8 @@ Prerequisites
</ul>
</details>
-<details class="objectives" id="orge99509d">
-<summary id="orge43e5c1">
+<details class="objectives" id="orgde042c7">
+<summary id="org6ca3c40">
Objectives
</summary>
<ul class="org-ul">
@@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm_emwm_ad.html b/build/emdm_emwm_ad.html
index 0caec91..64f1afc 100644
--- a/build/emdm_emwm_ad.html
+++ b/build/emdm_emwm_ad.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm_emwm_ad_c.html b/build/emdm_emwm_ad_c.html
index 2d7b870..85dd41a 100644
--- a/build/emdm_emwm_ad_c.html
+++ b/build/emdm_emwm_ad_c.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1659,7 +1659,7 @@ Applying \({\boldsymbol \nabla} \times\) to the curl equations gives the modifie
\]
with plane-wave solutions
\[
- {\boldsymbol E} ({\boldsymbol r}, t) = {\boldsymbol E}_0 e^{i \tilde{\boldsymbol k} \cdot {\boldsymbol r} - \omega t}
+ {\boldsymbol E} ({\boldsymbol r}, t) = {\boldsymbol E}_0 e^{i (\tilde{\boldsymbol k} \cdot {\boldsymbol r} - \omega t)}
\]
with complex-valued wavevector
\[
@@ -1733,7 +1733,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm_emwm_plm.html b/build/emdm_emwm_plm.html
index 9c7a8e9..56e9cb1 100644
--- a/build/emdm_emwm_plm.html
+++ b/build/emdm_emwm_plm.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1632,15 +1632,15 @@ For linear medium:
\]
If the medium is homogeneous (no spatial dependence of \(\varepsilon\) or \(\mu\)),
</p>
-<div class="main div" id="org7ad5a88">
-<div class="eqlabel" id="orgde6c1b4">
+<div class="main div" id="org3abedbf">
+<div class="eqlabel" id="org6c3c992">
<p>
<a id="Max_lh"></a><a href="./emdm_emwm_plm.html#Max_lh"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org8b72259">
+<div class="alteqlabels" id="org4747e62">
<ul class="org-ul">
<li>Gr (9.65)</li>
</ul>
@@ -1669,18 +1669,18 @@ v = \frac{1}{\sqrt{\mu \varepsilon}} = \frac{c}{n}
\]
where the index of refraction of the material is defined as
</p>
-<div class="main div" id="org56849ee">
+<div class="main div" id="org92a77a7">
<p>
<b>Index of refraction</b>
</p>
-<div class="eqlabel" id="orgff8c108">
+<div class="eqlabel" id="orgb3ac891">
<p>
<a id="n"></a><a href="./emdm_emwm_plm.html#n"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="orgf8eaa9d">
+<div class="alteqlabels" id="orge10b1d5">
<ul class="org-ul">
<li>Gr (9.69)</li>
</ul>
@@ -1743,7 +1743,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm_emwm_refl.html b/build/emdm_emwm_refl.html
index 076504e..14efd24 100644
--- a/build/emdm_emwm_refl.html
+++ b/build/emdm_emwm_refl.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm_emwm_refl_ni.html b/build/emdm_emwm_refl_ni.html
index dc63179..2675754 100644
--- a/build/emdm_emwm_refl_ni.html
+++ b/build/emdm_emwm_refl_ni.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1671,14 +1671,14 @@ E_{0_I} - E_{0_R} = \beta E_{0_T}, \hspace{1cm}
Solving these coupled equations, we can write
outgoing amplitudes in terms of incident ones:
</p>
-<div class="eqlabel" id="org1b7d37e">
+<div class="eqlabel" id="org0451818">
<p>
<a id="ERT"></a><a href="./emdm_emwm_refl_ni.html#ERT"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org2907578">
+<div class="alteqlabels" id="orgca80cb9">
<ul class="org-ul">
<li>Gr (9.82)</li>
</ul>
@@ -1744,7 +1744,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm_emwm_refl_oi.html b/build/emdm_emwm_refl_oi.html
index 39d1cd4..97ed148 100644
--- a/build/emdm_emwm_refl_oi.html
+++ b/build/emdm_emwm_refl_oi.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1635,14 +1635,14 @@ Transmitted wave:
\]
All waves have the same frequency \(\omega\). Since \(\omega = k v\), the three wavevectors are related by
</p>
-<div class="eqlabel" id="org792ed2b">
+<div class="eqlabel" id="orgd898837">
<p>
<a id="RTobliquek"></a><a href="./emdm_emwm_refl_oi.html#RTobliquek"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org1997374">
+<div class="alteqlabels" id="orgabe9175">
</div>
@@ -1665,7 +1665,7 @@ These forms for incident, reflected and transmitted wave can be substituted in t
<p>
From now on we will orient the axes so that \({\boldsymbol k}_I\) lies in the \(xz\) plane. This means that \({\boldsymbol k}_R\) and \({\boldsymbol k}_T\) also lie in that plane. This is the
</p>
-<div class="core div" id="orgbf226b1">
+<div class="core div" id="org4e7e42a">
<p>
<b>First law of reflection:</b>
the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
@@ -1680,7 +1680,7 @@ Specializing <a href="./emdm_emwm_refl_oi.html#RTobliquek">RTobliquek</a> to our
with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
and the angle of refraction (\(\theta_T\)) obey the following laws:
</p>
-<div class="core div" id="org000f37c">
+<div class="core div" id="org99d6843">
<p>
<b>Law of reflection</b>
\[
@@ -1697,14 +1697,14 @@ and the angle of refraction (\(\theta_T\)) obey the following laws:
<p>
This takes care of the spatially-dependent exponential factors in the boundary conditions. The coefficients must further obey
</p>
-<div class="eqlabel" id="orgdc1db4c">
+<div class="eqlabel" id="orgdc0e7b2">
<p>
<a id="EBRT"></a><a href="./emdm_emwm_refl_oi.html#EBRT"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="orgf69a0d1">
+<div class="alteqlabels" id="org6d39cd4">
</div>
@@ -1750,18 +1750,18 @@ while the third equation becomes
\]
Writing everything in terms of the incident amplitude, we get
</p>
-<div class="main div" id="org7e347a7">
+<div class="main div" id="org1929cd0">
<p>
<b>Fresnel's equations for reflection and transmission amplitudes (parallel case)</b>
</p>
-<div class="eqlabel" id="org5c049e2">
+<div class="eqlabel" id="org0b8be69">
<p>
<a id="Fresnel"></a><a href="./emdm_emwm_refl_oi.html#Fresnel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org50e3086">
+<div class="alteqlabels" id="org617a8e4">
</div>
@@ -1785,15 +1785,15 @@ Amplitudes for transmitted and reflected wave: depend on angle of incidence:
Behaviour: for \(\theta_I = 0\) we recover <a href="./emdm_emwm_refl_ni.html#ERT">ERT</a>.
For grazing waves \(\theta_I \rightarrow \pi/2\) we have that \(\alpha \rightarrow \infty\) and the wave is totally reflected. The most interesting angle is the one at which \(\alpha = \beta\) and the reflected wave has zero amplitude. This is known as
</p>
-<div class="main div" id="org73df67c">
-<div class="eqlabel" id="org8295eee">
+<div class="main div" id="org087c316">
+<div class="eqlabel" id="orgb61a423">
<p>
<a id="Brewster"></a><a href="./emdm_emwm_refl_oi.html#Brewster"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<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="org2f65b27">
+<div class="alteqlabels" id="org4e4ad5e">
</div>
@@ -1846,7 +1846,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm_emwm_refr.html b/build/emdm_emwm_refr.html
index 71e62ab..cc40ee4 100644
--- a/build/emdm_emwm_refr.html
+++ b/build/emdm_emwm_refr.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1633,7 +1633,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm_emwm_wg.html b/build/emdm_emwm_wg.html
index 385a43e..53ea6a8 100644
--- a/build/emdm_emwm_wg.html
+++ b/build/emdm_emwm_wg.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm_emwm_wg_c.html b/build/emdm_emwm_wg_c.html
index 5368752..4248574 100644
--- a/build/emdm_emwm_wg_c.html
+++ b/build/emdm_emwm_wg_c.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm_emwm_wg_gw.html b/build/emdm_emwm_wg_gw.html
index 8cb5150..24bf043 100644
--- a/build/emdm_emwm_wg_gw.html
+++ b/build/emdm_emwm_wg_gw.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1683,7 +1683,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emdm_emwm_wg_r.html b/build/emdm_emwm_wg_r.html
index 74c9297..056ce29 100644
--- a/build/emdm_emwm_wg_r.html
+++ b/build/emdm_emwm_wg_r.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1693,7 +1693,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emf.html b/build/emf.html
index 7dc6be0..de417fd 100644
--- a/build/emf.html
+++ b/build/emf.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1614,8 +1614,8 @@ Table of contents
</svg></a><span class="headline-id">emf</span></h2>
<div class="outline-text-2" id="text-emf">
-<details class="prereq" id="org101ede2">
-<summary id="org9fe7d9c">
+<details class="prereq" id="org85fb173">
+<summary id="orgdb11421">
Prerequisites
</summary>
<ul class="org-ul">
@@ -1623,8 +1623,8 @@ Prerequisites
</ul>
</details>
-<details class="objectives" id="org01ce394">
-<summary id="org119d5b0">
+<details class="objectives" id="orga59b9df">
+<summary id="orgf26745f">
Objectives
</summary>
<ul class="org-ul">
@@ -1660,7 +1660,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emf_g.html b/build/emf_g.html
index c773a78..db433bb 100644
--- a/build/emf_g.html
+++ b/build/emf_g.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emf_g_Cg.html b/build/emf_g_Cg.html
index 01d5f79..fb98451 100644
--- a/build/emf_g_Cg.html
+++ b/build/emf_g_Cg.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1656,7 +1656,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emf_g_Lg.html b/build/emf_g_Lg.html
index 7a2ef13..2f14faf 100644
--- a/build/emf_g_Lg.html
+++ b/build/emf_g_Lg.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1616,14 +1616,14 @@ Table of contents
<p>
A more aesthetic choice is the <b>Lorenz gauge</b>:
</p>
-<div class="eqlabel" id="orgb11b2c1">
+<div class="eqlabel" id="orgb29b121">
<p>
<a id="LorenzG"></a><a href="./emf_g_Lg.html#LorenzG"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org9cfa6f0">
+<div class="alteqlabels" id="org2006eb5">
</div>
@@ -1635,14 +1635,14 @@ A more aesthetic choice is the <b>Lorenz gauge</b>:
\]
which is chosen to put the second term in the left-hand side of <a href="./emf_svp.html#LapA">LapA</a> to zero. What remains is then
</p>
-<div class="eqlabel" id="org8bdd983">
+<div class="eqlabel" id="orgcba4a5d">
<p>
<a id="ALorenzG"></a><a href="./emf_g_Lg.html#ALorenzG"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="orgda10464">
+<div class="alteqlabels" id="org2d4cecb">
</div>
@@ -1654,14 +1654,14 @@ which is chosen to put the second term in the left-hand side of <a href="./emf_s
\]
while the equation for \(\phi\) becomes
</p>
-<div class="eqlabel" id="org92ed269">
+<div class="eqlabel" id="org7c6aebc">
<p>
<a id="phiLorenzG"></a><a href="./emf_g_Lg.html#phiLorenzG"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="orgf853d29">
+<div class="alteqlabels" id="org8b3cda4">
</div>
@@ -1673,18 +1673,18 @@ while the equation for \(\phi\) becomes
\]
These can be written compactly upon introducing a new operator: the
</p>
-<div class="core div" id="orgd77f9a6">
+<div class="core div" id="orgd056ba8">
<p>
<b>d'Alembertian operator</b>
</p>
-<div class="eqlabel" id="orgc902ec7">
+<div class="eqlabel" id="orgfd937d0">
<p>
<a id="dAl"></a><a href="./emf_g_Lg.html#dAl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="orgac35877">
+<div class="alteqlabels" id="org57101d6">
</div>
@@ -1700,18 +1700,18 @@ These can be written compactly upon introducing a new operator: the
<p>
so we get the
</p>
-<div class="core div" id="org12b2ff9">
+<div class="core div" id="org037b830">
<p>
<b>Inhomogeneous Maxwell equations</b> <i>(Lorenz gauge)</i>
</p>
-<div class="eqlabel" id="orge48b8c0">
+<div class="eqlabel" id="org9dce32f">
<p>
<a id="MaxLor"></a><a href="./emf_g_Lg.html#MaxLor"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org140821b">
+<div class="alteqlabels" id="org22f6c85">
</div>
@@ -1764,7 +1764,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emf_svp.html b/build/emf_svp.html
index 3f58c0b..30770aa 100644
--- a/build/emf_svp.html
+++ b/build/emf_svp.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1631,7 +1631,7 @@ Useful strategy: represent fields in terms of potentials.
Easiest: as we already saw (<a href="./ems_ms_vp_A.html#BcurlA">BcurlA</a>), we can write the magnetic
field as a pure curl (since its divergence always vanishes):
</p>
-<div class="core div" id="orgb63eb69">
+<div class="core div" id="org0868f0a">
<p>
\[
{\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
@@ -1648,15 +1648,15 @@ Putting this into Faraday's law <a href="./emd_Fl_Fl.html#Fl">Fl</a> gives
so this can be written as the gradient of a scalar.
Making the choice \(-{\boldsymbol \nabla} \phi\) for this, we get
</p>
-<div class="core div" id="org23c61e0">
-<div class="eqlabel" id="org51cbd1b">
+<div class="core div" id="org5ea4990">
+<div class="eqlabel" id="orgccc2c0e">
<p>
<a id="E_phiA"></a><a href="./emf_svp.html#E_phiA"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org25412c6">
+<div class="alteqlabels" id="orgc2e0dbe">
</div>
@@ -1673,15 +1673,15 @@ Making the choice \(-{\boldsymbol \nabla} \phi\) for this, we get
<p>
Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting <a href="./emf_svp.html#E_phiA">E_phiA</a> into Gauss's law gives
</p>
-<div class="main div" id="orgf26abb4">
-<div class="eqlabel" id="org986b9b3">
+<div class="main div" id="org7ca77db">
+<div class="eqlabel" id="org49309f0">
<p>
<a id="Lapphi"></a><a href="./emf_svp.html#Lapphi"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="orgcd04ad9">
+<div class="alteqlabels" id="org1b03eff">
</div>
@@ -1701,15 +1701,15 @@ whereas Ampère-Maxwell becomes
\]
which becomes after simple rearrangement and use of the <a href="./c_m_dc_d2.html#curlcurl">curlcurl</a> identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
</p>
-<div class="main div" id="org135124e">
-<div class="eqlabel" id="org73667ac">
+<div class="main div" id="org930bf11">
+<div class="eqlabel" id="org7818374">
<p>
<a id="LapA"></a><a href="./emf_svp.html#LapA"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org3f09a58">
+<div class="alteqlabels" id="org9e9918d">
</div>
@@ -1744,7 +1744,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems.html b/build/ems.html
index c19a647..24ab814 100644
--- a/build/ems.html
+++ b/build/ems.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca.html b/build/ems_ca.html
index 561fd07..aad9162 100644
--- a/build/ems_ca.html
+++ b/build/ems_ca.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1618,8 +1618,8 @@ Table of contents
<li>Gr 3</li>
</ul>
-<details class="prereq" id="org1512362">
-<summary id="orgf95a20b">
+<details class="prereq" id="orgd0e4e8d">
+<summary id="org949c0df">
Prerequisites
</summary>
<ul class="org-ul">
@@ -1627,8 +1627,8 @@ Prerequisites
</ul>
</details>
-<details class="objectives" id="org60b3244">
-<summary id="org14c49b9">
+<details class="objectives" id="orgfb8d074">
+<summary id="org39be3a5">
Objectives
</summary>
<ul class="org-ul">
@@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_fe.html b/build/ems_ca_fe.html
index d38b3c6..b6c14d5 100644
--- a/build/ems_ca_fe.html
+++ b/build/ems_ca_fe.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1624,7 +1624,7 @@ A generic configuration of static charges coupled via the Coulomb interaction
defines an electrostatic problem, whose solution is in principle obtained
from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
</p>
-<div class="main div" id="org91ccc03">
+<div class="main div" id="org8236a16">
<p>
</p>
@@ -1638,7 +1638,7 @@ from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_v
or (often simpler) by calculating the electrostatic potential, using either the
explicit construction <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
</p>
-<div class="main div" id="orge494253">
+<div class="main div" id="orgd333b49">
<p>
</p>
@@ -1658,7 +1658,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
<a href="./ems_es_ep_PL.html#Poi">🐟</a>
</p>
-<div class="core div" id="org4db04a9">
+<div class="core div" id="orge399a4f">
<p>
</p>
@@ -1674,7 +1674,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
<p>
In the specific case where the charge density vanishes, we fall back onto the simpler Laplace equation <a href="./ems_es_ep_PL.html#Lap">Lap</a>
</p>
-<div class="core div" id="orgd28c357">
+<div class="core div" id="orgea54c28">
<p>
</p>
@@ -1712,7 +1712,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_fe_L.html b/build/ems_ca_fe_L.html
index 7c40eaf..1ad2b11 100644
--- a/build/ems_ca_fe_L.html
+++ b/build/ems_ca_fe_L.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1630,14 +1630,14 @@ In one dimension, the potential is a single-variable
function \(\phi (x)\) and the Laplace equation reads
</p>
-<div class="eqlabel" id="orgc211f0a">
+<div class="eqlabel" id="orgd14f5e0">
<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="org36bff82">
+<div class="alteqlabels" id="orgdd168eb">
</div>
@@ -1652,14 +1652,14 @@ function \(\phi (x)\) and the Laplace equation reads
<p>
The solution to this is
</p>
-<div class="eqlabel" id="orgf5f202e">
+<div class="eqlabel" id="org9e3195b">
<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="org25a65ef">
+<div class="alteqlabels" id="org7a39d73">
<ul class="org-ul">
<li>Gr (3.6)</li>
</ul>
@@ -1718,14 +1718,14 @@ In two dimensions, the potential becomes a function
of two variables (here: \(x\) and \(y\)), so Laplace's
equation now reads
</p>
-<div class="eqlabel" id="org4bb6b92">
+<div class="eqlabel" id="org01996f8">
<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="org482a454">
+<div class="alteqlabels" id="org518eefc">
</div>
@@ -1778,14 +1778,14 @@ a point equals its value averaged over a sphere
\(S_R({\bf r})\) of any radius \(R\) centered on this point
(and of course not containing any charges),
</p>
-<div class="eqlabel" id="org96c7b40">
+<div class="eqlabel" id="org1ec3447">
<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="org550802d">
+<div class="alteqlabels" id="org832c6a2">
</div>
@@ -1797,8 +1797,8 @@ a point equals its value averaged over a sphere
\]
</p>
-<details id="org8644d79">
-<summary id="orgbe634df">
+<details id="org653c3ca">
+<summary id="org9063816">
<strong>Physicist's proof</strong>
</summary>
<p>
@@ -1860,8 +1860,8 @@ proving the theorem.
</p>
</details>
-<details id="orgb4be215">
-<summary id="orgb09f919">
+<details id="org086b4a6">
+<summary id="orgd2d3c6e">
<strong>Formal proof</strong>
</summary>
@@ -1911,14 +1911,14 @@ we get the following general
<p>
<b>Theorem</b>:
</p>
-<div class="eqlabel" id="org51f491a">
+<div class="eqlabel" id="org6d0244a">
<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="org9bebf97">
+<div class="alteqlabels" id="org227c883">
</div>
@@ -1971,19 +1971,19 @@ are necessarily positive, we thus require \(f_x > 0\), \(f_y > 0\) and \(f
of the \(f_x + f_y + f_z = 0\) condition above.
</p>
-<div class="eqlabel" id="org02999c2">
+<div class="eqlabel" id="orga00aa93">
<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="org3cd554f">
+<div class="alteqlabels" id="orge0f050e">
</div>
</div>
-<div class="info div" id="org8708715">
+<div class="info div" id="org81fcf04">
<p>
<b>Earnshaw's theorem (physical version)</b> <br>
</p>
@@ -2102,7 +2102,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_fe_g.html b/build/ems_ca_fe_g.html
index 6b9c3a4..d8e622c 100644
--- a/build/ems_ca_fe_g.html
+++ b/build/ems_ca_fe_g.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1613,11 +1613,11 @@ Table of contents
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a><span class="headline-id">ems.ca.fe.g</span></h5>
<div class="outline-text-5" id="text-ems_ca_fe_g">
-<div class="info div" id="orgce63324">
+<div class="info div" id="orgef35f51">
<p>
<b>George Green</b>
</p>
-<aside id="org3d8c3b4">
+<aside id="org432d494">
<p>
See a <a href="https://en.wikipedia.org/wiki/George%5C_Green%5C_(mathematician)">short bio on wikipedia</a>
</p>
@@ -1650,14 +1650,14 @@ and
\]
Substituting this in the divergence theorem gives <b>Green's first identity</b>
</p>
-<div class="eqlabel" id="org86176bc">
+<div class="eqlabel" id="orgfd30f5d">
<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="org8e9389d">
+<div class="alteqlabels" id="org3125160">
<ul class="org-ul">
<li>J (1.34)</li>
</ul>
@@ -1675,14 +1675,14 @@ As an aside for now, for completeness, if we do the same thing again but with \(
interchanged, and subtract the result, we obtain another useful result known as
<b>Green's second identity</b> or <b>Green's theorem</b>
</p>
-<div class="eqlabel" id="orgee2e293">
+<div class="eqlabel" id="org028fffd">
<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="org20f4498">
+<div class="alteqlabels" id="org9ea869b">
<ul class="org-ul">
<li>J (1.35)</li>
</ul>
@@ -1718,7 +1718,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_fe_uP.html b/build/ems_ca_fe_uP.html
index 5ae2da6..65304e2 100644
--- a/build/ems_ca_fe_uP.html
+++ b/build/ems_ca_fe_uP.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1635,14 +1635,14 @@ This yields
The first term on the left-hand side vanishes since \(\Phi\) satisfies Laplace.
The right-hand side can be made to vanish if \(\Phi\) obeys either
</p>
-<div class="eqlabel" id="org13e35fc">
+<div class="eqlabel" id="orgfa460b1">
<p>
<a id="Dirichlet"></a><a href="./ems_ca_fe_uP.html#Dirichlet"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="orgccd338e">
+<div class="alteqlabels" id="org14f7efc">
</div>
@@ -1654,14 +1654,14 @@ The right-hand side can be made to vanish if \(\Phi\) obeys either
<p>
or
</p>
-<div class="eqlabel" id="org68968dd">
+<div class="eqlabel" id="orgcd08c73">
<p>
<a id="Neumann"></a><a href="./ems_ca_fe_uP.html#Neumann"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org062fea2">
+<div class="alteqlabels" id="org6976f14">
</div>
@@ -1689,19 +1689,19 @@ additive constant.
<p>
We can thus finally state the
</p>
-<div class="eqlabel" id="org154b9ea">
+<div class="eqlabel" id="orgd45117f">
<p>
<a id="uniq_thm"></a><a href="./ems_ca_fe_uP.html#uniq_thm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org43b26fb">
+<div class="alteqlabels" id="org2d283cb">
</div>
</div>
-<div class="core div" id="orgc742981">
+<div class="core div" id="orgf2a5ea9">
<p>
<b>Uniqueness Theorem</b>
</p>
@@ -1740,7 +1740,7 @@ Reading other books, you might be misled into thinking that there are numerous c
and corollaries and that things are
</p>
-<div class="info div" id="org2adbb79">
+<div class="info div" id="orgd298dad">
<p>
<b>Comment/warning</b>: <b>uniqueness theorem on uniqueness theorems</b> <br>
Do not be misled: there is a <i>unique</i> uniqueness theorem for the
@@ -1767,7 +1767,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_me.html b/build/ems_ca_me.html
index d64bdc8..afe4669 100644
--- a/build/ems_ca_me.html
+++ b/build/ems_ca_me.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_me_Ed.html b/build/ems_ca_me_Ed.html
index a43dbdf..afef1aa 100644
--- a/build/ems_ca_me_Ed.html
+++ b/build/ems_ca_me_Ed.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1634,14 +1634,14 @@ E_\phi &= -\frac{1}{r \sin \theta} \frac{\partial \phi}{\partial \varphi} =
<p>
we get
</p>
-<div class="eqlabel" id="orgde20f17">
+<div class="eqlabel" id="orgb63ca95">
<p>
<a id="E_di_1"></a><a href="./ems_ca_me_Ed.html#E_di_1"><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="org6716c9d">
+<div class="alteqlabels" id="orgcc89131">
<ul class="org-ul">
<li>Gr (3.103)</li>
</ul>
@@ -1663,14 +1663,14 @@ E_d ({\bf r}) = - \frac{1}{4\pi \varepsilon_0} \nabla \frac{{\bf p} \cdot {\bf r
\]
immediately yielding the coordinate-free expression
</p>
-<div class="eqlabel" id="org9343029">
+<div class="eqlabel" id="org186e651">
<p>
<a id="E_di"></a><a href="./ems_ca_me_Ed.html#E_di"><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="orgc63b0fa">
+<div class="alteqlabels" id="org06f6a5c">
<ul class="org-ul">
<li>Gr (3.104)</li>
</ul>
@@ -1710,7 +1710,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_me_Eq.html b/build/ems_ca_me_Eq.html
index a095f02..0195130 100644
--- a/build/ems_ca_me_Eq.html
+++ b/build/ems_ca_me_Eq.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_me_a.html b/build/ems_ca_me_a.html
index 812de72..94e5a86 100644
--- a/build/ems_ca_me_a.html
+++ b/build/ems_ca_me_a.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1628,14 +1628,14 @@ By Taylor expanding, we get
Formally, we could do this for any vector \({\bf r}_s\) such that \(|{\bf r}_s| < |{\bf r}|\) by
Taylor expanding with the \({\boldsymbol \nabla}\) operator,
</p>
-<div class="eqlabel" id="org2bce040">
+<div class="eqlabel" id="org7b106bc">
<p>
<a id="1or_grad"></a><a href="./ems_ca_me_a.html#1or_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="org262a4dc">
+<div class="alteqlabels" id="orgf9dea72">
</div>
@@ -1653,14 +1653,14 @@ the potential takes the form of the general solution of Laplace's equation <a hr
Reading the parameters, we get \(A_l = 0\), \(B_l = r_s^l\). Putting back a generic angle
(the coefficients remain the same), we thus get
</p>
-<div class="eqlabel" id="org6b30a55">
+<div class="eqlabel" id="org68023ff">
<p>
<a id="1or_Leg"></a><a href="./ems_ca_me_a.html#1or_Leg"><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="org97b5686">
+<div class="alteqlabels" id="org593b10c">
<ul class="org-ul">
<li>Gr(3.94)</li>
</ul>
@@ -1682,14 +1682,14 @@ we can expand the potential at a point \({\bf r}\) outside \({\cal V}\) accordin
(here, we put the origin of our coordinate system closer to all points in \({\cal V}\) than to \({\bf r}\)
to ensure convergence)
</p>
-<div class="eqlabel" id="orgb92deaa">
+<div class="eqlabel" id="orgaf6dad4">
<p>
<a id="p_Leg"></a><a href="./ems_ca_me_a.html#p_Leg"><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="orgc46288e">
+<div class="alteqlabels" id="org0904818">
<ul class="org-ul">
<li>Gr (3.95)</li>
</ul>
@@ -1752,14 +1752,14 @@ For \(r \gg d\), we can expand (immediately dropping terms of order \(d^2/r^2\))
Putting things together, the leading term in the expansion <a href="./ems_ca_me_a.html#p_Leg">p_Leg</a> for the
potential of the physical dipole is given by
</p>
-<div class="eqlabel" id="orga3ff152">
+<div class="eqlabel" id="org0b3de33">
<p>
<a id="p_physdi"></a><a href="./ems_ca_me_a.html#p_physdi"><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="org099e31e">
+<div class="alteqlabels" id="org1803e09">
<ul class="org-ul">
<li>Gr (3.90)</li>
</ul>
@@ -1806,7 +1806,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_me_h.html b/build/ems_ca_me_h.html
index 603c6cd..12bfdb6 100644
--- a/build/ems_ca_me_h.html
+++ b/build/ems_ca_me_h.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1621,19 +1621,19 @@ The next terms in the expansion are obtained similarly: the <b>quadrupole</b> te
\]
and can be rewritten as
</p>
-<div class="eqlabel" id="orgc0ad248">
+<div class="eqlabel" id="org9893011">
<p>
<a id="p_quad"></a><a href="./ems_ca_me_h.html#p_quad"><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="orgfb585e2">
+<div class="alteqlabels" id="org8d2b01e">
</div>
</div>
-<div class="main div" id="org1a7deb8">
+<div class="main div" id="org3dd96b8">
<p>
\[
\phi_q ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab}
@@ -1645,19 +1645,19 @@ and can be rewritten as
<p>
in terms of the <b>quadrupole moment</b>
</p>
-<div class="eqlabel" id="orgbc8de26">
+<div class="eqlabel" id="org476da28">
<p>
<a id="quadmom"></a><a href="./ems_ca_me_h.html#quadmom"><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="orge3da8dd">
+<div class="alteqlabels" id="org3a377ca">
</div>
</div>
-<div class="main div" id="orgbc87048">
+<div class="main div" id="orgf571492">
<p>
\[
Q_{ab} = \int_{\cal V} d\tau_s (3 r_{s,a} r_{s,b} - r_s^2 \delta_{a,b}) \rho ({\bf r}_s).
@@ -1676,19 +1676,19 @@ This is a symmetric, traceless rank \(2\) tensor: \(Q_{ab} = Q_{ba}\) and
<p>
Our expansion for the potential, with terms up to quadrupole ones, thus looks like
</p>
-<div class="eqlabel" id="org2c48c0e">
+<div class="eqlabel" id="org2f1183a">
<p>
<a id="p_mdq"></a><a href="./ems_ca_me_h.html#p_mdq"><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="orgfce9466">
+<div class="alteqlabels" id="orgae17c12">
</div>
</div>
-<div class="main div" id="org930eead">
+<div class="main div" id="org3db4f46">
<p>
\[
\phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \left( \frac{Q}{r} + \sum_a \frac{r_a}{r^3} p_a + \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab} + ... \right)
@@ -1709,7 +1709,7 @@ the leading nonvanishing multipole moment is independent of the chosen location
the origin of the coordinate system (see Jackson Prob. 4.4).
</p>
-<div class="info div" id="orgf830029">
+<div class="info div" id="org6c4c92f">
<p>
<b>A consistent nomenclature for the multipole expansion?</b>
</p>
@@ -1858,7 +1858,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_me_md.html b/build/ems_ca_me_md.html
index 838f20b..9884374 100644
--- a/build/ems_ca_me_md.html
+++ b/build/ems_ca_me_md.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1617,14 +1617,14 @@ Table of contents
The series <a href="./ems_ca_me_a.html#p_Leg">p_Leg</a> is organized in increasing powers of inverse distance.
The leading term is called the *monopole term, and is
</p>
-<div class="eqlabel" id="orgb777718">
+<div class="eqlabel" id="org0adcdb6">
<p>
<a id="p_mono"></a><a href="./ems_ca_me_md.html#p_mono"><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="orgce09c64">
+<div class="alteqlabels" id="orgc659d6a">
<ul class="org-ul">
<li>Gr (3.97)</li>
</ul>
@@ -1632,7 +1632,7 @@ The leading term is called the *monopole term, and is
</div>
</div>
-<div class="core div" id="org9169015">
+<div class="core div" id="org46f41ca">
<p>
\[
\phi_m ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{Q}{r}, \hspace{1cm}
@@ -1653,19 +1653,19 @@ The next term is the <b>dipole</b> term: by using \(P_1 (x) = x\), we have
\]
This can be written
</p>
-<div class="eqlabel" id="org63e0b61">
+<div class="eqlabel" id="orgfa57fa4">
<p>
<a id="p_di"></a><a href="./ems_ca_me_md.html#p_di"><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="org07072c1">
+<div class="alteqlabels" id="org486bae0">
</div>
</div>
-<div class="core div" id="orgf15593a">
+<div class="core div" id="org1702f06">
<p>
\[
\phi_d ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{\hat{\bf r} \cdot {\bf p}}{r^2},
@@ -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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_mi.html b/build/ems_ca_mi.html
index 8d96c7f..460047c 100644
--- a/build/ems_ca_mi.html
+++ b/build/ems_ca_mi.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1646,14 +1646,14 @@ by distance \(d\). For definiteness, we put a charge \(q\) at coordinate
<p>
By superposition, we have that
</p>
-<div class="eqlabel" id="org1e6390c">
+<div class="eqlabel" id="org86b5563">
<p>
<a id="p_di_z"></a><a href="./ems_ca_mi.html#p_di_z"><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="org36c8ccf">
+<div class="alteqlabels" id="org66c454d">
<ul class="org-ul">
<li>FLS II (6.8)</li>
<li>Gr (3.9)</li>
@@ -1711,7 +1711,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_mi_fe.html b/build/ems_ca_mi_fe.html
index 3ac3967..33f75b1 100644
--- a/build/ems_ca_mi_fe.html
+++ b/build/ems_ca_mi_fe.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1617,14 +1617,14 @@ Table of contents
The force between the point charge and the plane
is the same as in the case of two point charges:
</p>
-<div class="eqlabel" id="org2dabd48">
+<div class="eqlabel" id="orga85414d">
<p>
<a id="F_dip_z"></a><a href="./ems_ca_mi_fe.html#F_dip_z"><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="org96001c6">
+<div class="alteqlabels" id="org46b95a1">
<ul class="org-ul">
<li>Gr (3.12)</li>
</ul>
@@ -1677,7 +1677,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_mi_isc.html b/build/ems_ca_mi_isc.html
index 227a040..34ceef7 100644
--- a/build/ems_ca_mi_isc.html
+++ b/build/ems_ca_mi_isc.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1617,14 +1617,14 @@ Table of contents
Use <a href="./ems_es_c_sc.html#scd_cond">scd_cond</a>, with the normal direction now
being \(\hat{\bf z}\):
</p>
-<div class="eqlabel" id="orgbd23e13">
+<div class="eqlabel" id="org3efc1b8">
<p>
<a id="scd_dip_z"></a><a href="./ems_ca_mi_isc.html#scd_dip_z"><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="orgfce0891">
+<div class="alteqlabels" id="orgf89ac22">
<ul class="org-ul">
<li>Gr (3.10)</li>
</ul>
@@ -1645,14 +1645,14 @@ The total induced charge can be obtained by simple integration as
Using planar coordinates,
\(\sigma(r) = \frac{-qd}{4\pi (r^2 + (d/2)^2)^{3/2}}\), so
</p>
-<div class="eqlabel" id="org94fe5ca">
+<div class="eqlabel" id="orgbab70df">
<p>
<a id="sc_dip_z"></a><a href="./ems_ca_mi_isc.html#sc_dip_z"><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="orgca80b29">
+<div class="alteqlabels" id="orgf3efaa2">
<ul class="org-ul">
<li>Gr (3.11)</li>
</ul>
@@ -1693,7 +1693,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_mi_o.html b/build/ems_ca_mi_o.html
index 33ada8d..e8897a5 100644
--- a/build/ems_ca_mi_o.html
+++ b/build/ems_ca_mi_o.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1676,7 +1676,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_sv.html b/build/ems_ca_sv.html
index 99586c0..3776a6a 100644
--- a/build/ems_ca_sv.html
+++ b/build/ems_ca_sv.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1656,7 +1656,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_sv_car.html b/build/ems_ca_sv_car.html
index a2869a2..eaf5037 100644
--- a/build/ems_ca_sv_car.html
+++ b/build/ems_ca_sv_car.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1613,7 +1613,7 @@ 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.sv.car</span></h5>
<div class="outline-text-5" id="text-ems_ca_sv_car">
-<div class="example div" id="org6fa8ae7">
+<div class="example div" id="org98b3764">
<p>
</p>
@@ -1659,14 +1659,14 @@ This thus falls back onto a 2d problem. We need to solve the 2d Laplace equatio
<p>
Let us look for solutions in the form
</p>
-<div class="eqlabel" id="orga36e788">
+<div class="eqlabel" id="orgc28ff62">
<p>
<a id="Lap_sv_car"></a><a href="./ems_ca_sv_car.html#Lap_sv_car"><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="orge29141d">
+<div class="alteqlabels" id="org1acbd09">
<ul class="org-ul">
<li>Gr (3.23)</li>
</ul>
@@ -1685,14 +1685,14 @@ individual term in the Laplace equation equals a constant,
and that these constants add up to zero. We can thus put (the sign choice anticipates
the solution somewhat)
</p>
-<div class="eqlabel" id="orgc324eae">
+<div class="eqlabel" id="org8e7ab92">
<p>
<a id="Lap_sv_car_sep"></a><a href="./ems_ca_sv_car.html#Lap_sv_car_sep"><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="org653d57a">
+<div class="alteqlabels" id="org86925b3">
<ul class="org-ul">
<li>Gr (3.26)</li>
</ul>
@@ -1716,14 +1716,14 @@ Let's look first of all at the solutions of <a href="./ems_ca_sv_car.html#Lap_sv
Since this is a second-order linear differential equation, there are two linearly
independent solutions. The most general solution for \(X\) and \(Y\) can be written
</p>
-<div class="eqlabel" id="orgb25d4e1">
+<div class="eqlabel" id="orgd404b3a">
<p>
<a id="Lap_sv_car_solXY"></a><a href="./ems_ca_sv_car.html#Lap_sv_car_solXY"><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="org2316d74">
+<div class="alteqlabels" id="org30886f9">
<ul class="org-ul">
<li>Gr (3.27)</li>
</ul>
@@ -1780,14 +1780,14 @@ C_n = \frac{2}{a} \int_0^a dy ~\phi_0(y) \sin(n\pi y/a)
<p>
<b>Specific example</b>: say that \(\phi_0(y) = \phi_0\), <i>i.e.</i> just a constant. Then,
</p>
-<div class="eqlabel" id="org6d2dab3">
+<div class="eqlabel" id="orgf5c8f3f">
<p>
<a id="Cn"></a><a href="./ems_ca_sv_car.html#Cn"><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="org58a2ab8">
+<div class="alteqlabels" id="org5ad8bf2">
<ul class="org-ul">
<li>Gr (3.35)</li>
</ul>
@@ -1841,7 +1841,7 @@ The solution for the specific case \(\phi_0 (y) = \phi_0\) is thus
</p>
-<div class="example div" id="org1ba9ed2">
+<div class="example div" id="org8b40fc1">
<p>
<b>Example: rectangular pipe</b>
</p>
@@ -1898,14 +1898,14 @@ The full solution is then a linear combination of complete set of functions,
The coefficients must be chosen such that \((iii)\) is fulfilled, \(\phi(b,y) = \phi_0\).
This simple case of a constant value \(\phi_0\) gives us the same relation as <a href="./ems_ca_sv_car.html#Cn">Cn</a>, so
</p>
-<div class="eqlabel" id="org3d51969">
+<div class="eqlabel" id="org4c670ac">
<p>
<a id="p_recpipe"></a><a href="./ems_ca_sv_car.html#p_recpipe"><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="orgc8e3689">
+<div class="alteqlabels" id="orgdedd5fd">
<ul class="org-ul">
<li>Gr (3.42)</li>
</ul>
@@ -1941,7 +1941,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_sv_cyl.html b/build/ems_ca_sv_cyl.html
index 9fc7110..c13bd7b 100644
--- a/build/ems_ca_sv_cyl.html
+++ b/build/ems_ca_sv_cyl.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1633,7 +1633,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ca_sv_sph.html b/build/ems_ca_sv_sph.html
index da95e36..2f8be20 100644
--- a/build/ems_ca_sv_sph.html
+++ b/build/ems_ca_sv_sph.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1617,14 +1617,14 @@ Table of contents
In spherical coordinates, the Laplace equation takes the following form
(using <a href="./c_m_cs_sph.html#sph_Lap">sph_Lap</a>):
</p>
-<div class="eqlabel" id="org8944b83">
+<div class="eqlabel" id="org960f8d0">
<p>
<a id="Lap_sph"></a><a href="./ems_ca_sv_sph.html#Lap_sph"><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="org1d52bd8">
+<div class="alteqlabels" id="org1c959d0">
<ul class="org-ul">
<li>Gr (3.53)</li>
<li>W (11-86)</li>
@@ -1633,7 +1633,7 @@ In spherical coordinates, the Laplace equation takes the following form
</div>
</div>
-<div class="main div" id="org9f248f2">
+<div class="main div" id="org973c5fc">
<p>
</p>
@@ -1650,14 +1650,14 @@ In spherical coordinates, the Laplace equation takes the following form
If you are dealing with a problem having <b>azimuthal symmetry</b>,
\(\phi\) is independent of \(\varphi\) and the equation simplifies to:
</p>
-<div class="eqlabel" id="orgb5d38b5">
+<div class="eqlabel" id="orgb049674">
<p>
<a id="Lap_sph_az"></a><a href="./ems_ca_sv_sph.html#Lap_sph_az"><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="orgbe80ac3">
+<div class="alteqlabels" id="org954a313">
<ul class="org-ul">
<li>Gr (3.54)</li>
<li>W (11-87)</li>
@@ -1755,7 +1755,7 @@ This equation is solved by <b>Legendre polynomials</b> of the variable \(\cos \t
</p>
-<div class="info div" id="org168cefa">
+<div class="info div" id="org2ecd100">
<p>
<b>Legendre polynomials</b>
</p>
@@ -1776,14 +1776,14 @@ and conveniently defined (for trigonometric arguments) to obey the orthogonality
relationship (the reason for the normalization on the right-hand side will become clear later)
</p>
-<div class="eqlabel" id="orgc073888">
+<div class="eqlabel" id="org4d778e7">
<p>
<a id="Leg_orth_trig"></a><a href="./ems_ca_sv_sph.html#Leg_orth_trig"><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="org7bd920b">
+<div class="alteqlabels" id="org27ae25a">
</div>
@@ -1798,14 +1798,14 @@ relationship (the reason for the normalization on the right-hand side will becom
<p>
This same relation can be more simply written by using the variable \(x = \cos \theta\),
</p>
-<div class="eqlabel" id="org782f2a5">
+<div class="eqlabel" id="orgaf08cf2">
<p>
<a id="Leg_orth"></a><a href="./ems_ca_sv_sph.html#Leg_orth"><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="orga8724bc">
+<div class="alteqlabels" id="org7afced6">
</div>
@@ -1823,14 +1823,14 @@ To get started, we need to define the "seed" polynomial (carrying label \(l=0\))
To make life easy, we set \(P_0 (x) = 1\). Higher polynomials are then sought in the
form of power series in \(x\). This leads to the first few Legendre polynomials being:
</p>
-<div class="eqlabel" id="orga793c4e">
+<div class="eqlabel" id="org5045860">
<p>
<a id="Leg_pols"></a><a href="./ems_ca_sv_sph.html#Leg_pols"><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="org4f6a95e">
+<div class="alteqlabels" id="org0da2636">
</div>
@@ -1850,14 +1850,14 @@ P_5 (x) &= \frac{1}{8} (63x^5 - 70x^3 + 15x).
The prefactor (and thus the factor in the orthogonality relations <a href="./ems_ca_sv_sph.html#Leg_orth_trig">Leg_orth_trig</a> and the equivalent <a href="./ems_ca_sv_sph.html#Leg_orth">Leg_orth</a> is chosen for convenience such that each polynomial
takes the value \(1\) when evaluated at argument \(x = 1\),
</p>
-<div class="eqlabel" id="orga351757">
+<div class="eqlabel" id="org234cd5e">
<p>
<a id="Pl_1_1"></a><a href="./ems_ca_sv_sph.html#Pl_1_1"><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="org6763018">
+<div class="alteqlabels" id="orgd9a5971">
</div>
@@ -1873,14 +1873,14 @@ P_l(1) = 1
<p>
The Legendre polynomial \(P_l\) obeys the differential equation
</p>
-<div class="eqlabel" id="orgc481719">
+<div class="eqlabel" id="org20460a3">
<p>
<a id="Leg_de_trig"></a><a href="./ems_ca_sv_sph.html#Leg_de_trig"><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="org2141b06">
+<div class="alteqlabels" id="orgf97a64a">
</div>
@@ -1892,14 +1892,14 @@ The Legendre polynomial \(P_l\) obeys the differential equation
\]
or equivalently
</p>
-<div class="eqlabel" id="orga7bc01b">
+<div class="eqlabel" id="org25bc290">
<p>
<a id="Leg_de"></a><a href="./ems_ca_sv_sph.html#Leg_de"><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="org423560c">
+<div class="alteqlabels" id="org3caf6ac">
</div>
@@ -1915,14 +1915,14 @@ or equivalently
A particularly convenient formula for deriving \(P_l(x)\)
is the <b>Rodrigues formula</b>:
</p>
-<div class="eqlabel" id="orgdd028ca">
+<div class="eqlabel" id="orgb07c11e">
<p>
<a id="Rodrigues"></a><a href="./ems_ca_sv_sph.html#Rodrigues"><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="orgcd24176">
+<div class="alteqlabels" id="orgff8e527">
</div>
@@ -1938,14 +1938,14 @@ P_l(x) = \frac{1}{2^l l!} \left( \frac{d}{dx} \right)^l (x^2 - 1)^l
<p>
Actually, a more practical formula is <b>Bonnet's recursion relation</b>
</p>
-<div class="eqlabel" id="org28cb378">
+<div class="eqlabel" id="org91981a9">
<p>
<a id="Bonnet"></a><a href="./ems_ca_sv_sph.html#Bonnet"><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="org199bde7">
+<div class="alteqlabels" id="org1046201">
</div>
@@ -1981,14 +1981,14 @@ We therefore come to the culmination of our efforts here, and write
the general solution to <i>any</i> problem with azimuthal symmetry
(for which the potential takes a finite value for \(\theta = 0, \pi\)) as
</p>
-<div class="eqlabel" id="orgdb33d5d">
+<div class="eqlabel" id="org3bd5c2b">
<p>
<a id="Lap_sph_az_sol"></a><a href="./ems_ca_sv_sph.html#Lap_sph_az_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="orgcd5cf30">
+<div class="alteqlabels" id="orgd15cd63">
<ul class="org-ul">
<li>Gr (3.65)</li>
</ul>
@@ -1996,7 +1996,7 @@ the general solution to <i>any</i> problem with azimuthal symmetry
</div>
</div>
-<div class="main div" id="org55e5667">
+<div class="main div" id="orgfae9cbf">
<p>
\[
\phi(r,\theta) = \sum_{l=0}^{\infty} \left( A_l r^l + \frac{B_l}{r^{l+1}} \right) P_l (\cos \theta)
@@ -2008,7 +2008,7 @@ the general solution to <i>any</i> problem with azimuthal symmetry
-<div class="example div" id="orgbfafc9e">
+<div class="example div" id="orgdf8fb7a">
<p>
<b>Example: potential inside a hollow sphere</b>
</p>
@@ -2091,7 +2091,7 @@ Thus, \(A_0 = k/2\), \(A_1 = -k/2\), and all others are zero, so
</div>
-<div class="example div" id="org1180317">
+<div class="example div" id="org7d1e98d">
<p>
<b>Example: surface charge density on sphere</b>
</p>
@@ -2221,14 +2221,14 @@ A_1^i = \frac{k}{2\varepsilon_0} \int_0^\pi d\theta \sin \theta [P_l(\cos \theta
<p>
The potential inside/outside the sphere is then
</p>
-<div class="eqlabel" id="org206a11d">
+<div class="eqlabel" id="orgf34445a">
<p>
<a id="p_uni_ch_sph"></a><a href="./ems_ca_sv_sph.html#p_uni_ch_sph"><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="org28b3018">
+<div class="alteqlabels" id="org5858176">
</div>
@@ -2261,7 +2261,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es.html b/build/ems_es.html
index d1b5824..b88131a 100644
--- a/build/ems_es.html
+++ b/build/ems_es.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1620,8 +1620,8 @@ Table of contents
<li>PM 1</li>
</ul>
-<details class="prereq" id="org96a1136">
-<summary id="org4eaf2d9">
+<details class="prereq" id="org28f38e1">
+<summary id="org4ff5949">
Prerequisites
</summary>
<ul class="org-ul">
@@ -1629,8 +1629,8 @@ Prerequisites
</ul>
</details>
-<details class="objectives" id="org52afd24">
-<summary id="orgc0f7348">
+<details class="objectives" id="org562fcf2">
+<summary id="org99ee0cc">
Objectives
</summary>
<ul class="org-ul">
@@ -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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_c.html b/build/ems_es_c.html
index 32c552a..14958c4 100644
--- a/build/ems_es_c.html
+++ b/build/ems_es_c.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_c_cap.html b/build/ems_es_c_cap.html
index 4240173..23c5e85 100644
--- a/build/ems_es_c_cap.html
+++ b/build/ems_es_c_cap.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1635,7 +1635,7 @@ C \equiv \frac{Q}{V}
\]
</p>
-<div class="example div" id="orgaa30121">
+<div class="example div" id="org1297454">
<p>
<b>Concentric shells</b>
</p>
@@ -1686,7 +1686,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_c_ic.html b/build/ems_es_c_ic.html
index 93bc562..d89f17a 100644
--- a/build/ems_es_c_ic.html
+++ b/build/ems_es_c_ic.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1618,7 +1618,7 @@ Charge in hollow conductor.
Induced surface charges on inside and outside surfaces.
</p>
-<div class="example div" id="org0dc11f2">
+<div class="example div" id="org0ce1dfe">
<p>
<b>Field outside spherical conductor with irregular cavity</b>
</p>
@@ -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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_c_p.html b/build/ems_es_c_p.html
index a502192..57d169f 100644
--- a/build/ems_es_c_p.html
+++ b/build/ems_es_c_p.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1613,7 +1613,7 @@ 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.es.c.p</span></h5>
<div class="outline-text-5" id="text-ems_es_c_p">
-<div class="main div" id="orgb87699d">
+<div class="main div" id="org44b23ff">
<p>
<b>Basic Properties of a Conductor</b>
</p>
@@ -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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_c_sc.html b/build/ems_es_c_sc.html
index be3419d..169bccc 100644
--- a/build/ems_es_c_sc.html
+++ b/build/ems_es_c_sc.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1618,14 +1618,14 @@ For a conductor, we can exploit the fact that electrical fields vanish on the in
get a proper boundary condition for the potential. Namely, here, boundary condition
<a href="./ems_es_ep_bc.html#Edisc">Edisc</a> yields
</p>
-<div class="eqlabel" id="orgcb4ae52">
+<div class="eqlabel" id="orge4aa51f">
<p>
<a id="E_sur_cond"></a><a href="./ems_es_c_sc.html#E_sur_cond"><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="orgc0d8168">
+<div class="alteqlabels" id="orgbc9f5e5">
<ul class="org-ul">
<li>Gr (2.48)</li>
</ul>
@@ -1643,14 +1643,14 @@ get a proper boundary condition for the potential. Namely, here, boundary condi
<p>
which in terms of potential reads
</p>
-<div class="eqlabel" id="orgb2ad74d">
+<div class="eqlabel" id="org4bb3570">
<p>
<a id="scd_cond"></a><a href="./ems_es_c_sc.html#scd_cond"><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="org85ed0a5">
+<div class="alteqlabels" id="org82a270e">
<ul class="org-ul">
<li>Gr (2.49)</li>
</ul>
@@ -1712,7 +1712,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_e.html b/build/ems_es_e.html
index 44ef56e..f492c66 100644
--- a/build/ems_es_e.html
+++ b/build/ems_es_e.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1634,14 +1634,14 @@ W_u = -\int_{{\bf a}}^{{\bf b}} {\bf E} \cdot d{\bf l}
For a point charge distribution, the total work required to assemble it
can also be rewritten in terms of the potential by using <a href="./ems_es_ep_d.html#p">p</a>,
</p>
-<div class="eqlabel" id="org8fc89a8">
+<div class="eqlabel" id="org8f28b0d">
<p>
<a id="W_pcd"></a><a href="./ems_es_e.html#W_pcd"><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="org5f75aee">
+<div class="alteqlabels" id="org7f0bee7">
<ul class="org-ul">
<li>Gr (2.42)</li>
</ul>
@@ -1660,14 +1660,14 @@ W = \frac{1}{2} \sum_{i=1}^m q_i \phi({\bf r}_i)
For a volume charge distribution, this becomes
</p>
-<div class="eqlabel" id="orgb540630">
+<div class="eqlabel" id="org8496249">
<p>
<a id="W_vcd"></a><a href="./ems_es_e.html#W_vcd"><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="orgcdfbd12">
+<div class="alteqlabels" id="orgd44ae8e">
<ul class="org-ul">
<li>Gr (2.43)</li>
</ul>
@@ -1709,14 +1709,14 @@ W = \frac{\varepsilon_0}{2} \left( \int_{\cal V} E^2 d\tau + \oint_{\cal S} \phi
Integrating over all space (and thus dropping the surface integrale whose integrand is assumed
to vanish at infinity), we get
</p>
-<div class="eqlabel" id="org3d801e6">
+<div class="eqlabel" id="org4f4c7f0">
<p>
<a id="W_intEsq"></a><a href="./ems_es_e.html#W_intEsq"><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="org3c999e9">
+<div class="alteqlabels" id="orgeb19c6f">
<ul class="org-ul">
<li>Gr (2.45)</li>
</ul>
@@ -1724,7 +1724,7 @@ to vanish at infinity), we get
</div>
</div>
-<div class="core div" id="org9197121">
+<div class="core div" id="org61da001">
<p>
</p>
@@ -1738,7 +1738,7 @@ W = \frac{\varepsilon_0}{2} \int E^2 d\tau
</div>
-<div class="example div" id="orgbfb4a03">
+<div class="example div" id="org84561d4">
<p>
<b>Energy of spherical shell</b>
</p>
@@ -1811,7 +1811,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ec.html b/build/ems_es_ec.html
index afa7ad6..cf9d228 100644
--- a/build/ems_es_ec.html
+++ b/build/ems_es_ec.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ec_b.html b/build/ems_es_ec_b.html
index c170c42..458100c 100644
--- a/build/ems_es_ec_b.html
+++ b/build/ems_es_ec_b.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1636,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ec_c.html b/build/ems_es_ec_c.html
index 4c97c76..fa2d931 100644
--- a/build/ems_es_ec_c.html
+++ b/build/ems_es_ec_c.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1636,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ec_q.html b/build/ems_es_ec_q.html
index a8ea181..3ea49c9 100644
--- a/build/ems_es_ec_q.html
+++ b/build/ems_es_ec_q.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1636,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ec_s.html b/build/ems_es_ec_s.html
index 767423b..f0de517 100644
--- a/build/ems_es_ec_s.html
+++ b/build/ems_es_ec_s.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1635,7 +1635,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ef.html b/build/ems_es_ef.html
index 4c98ff2..db306f5 100644
--- a/build/ems_es_ef.html
+++ b/build/ems_es_ef.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ef_Gl.html b/build/ems_es_ef_Gl.html
index 1798291..6b90e43 100644
--- a/build/ems_es_ef_Gl.html
+++ b/build/ems_es_ef_Gl.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1658,14 +1658,14 @@ sphere of radius \(r\) around the charge,
<p>
so by superposition, we obtain
</p>
-<div class="eqlabel" id="orgdb1b683">
+<div class="eqlabel" id="org91cf7d7">
<p>
<a id="Gl_i"></a><a href="./ems_es_ef_Gl.html#Gl_i"><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="orgce2000b">
+<div class="alteqlabels" id="org6ea6288">
<ul class="org-ul">
<li>Gr (2.13)</li>
</ul>
@@ -1673,7 +1673,7 @@ so by superposition, we obtain
</div>
</div>
-<div class="core div" id="org2a79f58">
+<div class="core div" id="org0decae8">
<p>
<b>Gauss' law (in integral form)</b>
</p>
@@ -1705,14 +1705,14 @@ By applying the divergence theorem,
and using \(Q_{\mbox{enc}} = \int_{\cal V} \rho d\tau\), and using the fact the the choice of volume
is arbitrary, we get
</p>
-<div class="eqlabel" id="org8cb0bb4">
+<div class="eqlabel" id="org99cb3fa">
<p>
<a id="Gl_d"></a><a href="./ems_es_ef_Gl.html#Gl_d"><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="org0bf7bc4">
+<div class="alteqlabels" id="org26da670">
<ul class="org-ul">
<li>Gr (2.14)</li>
</ul>
@@ -1720,7 +1720,7 @@ is arbitrary, we get
</div>
</div>
-<div class="core div" id="org03bfea1">
+<div class="core div" id="orga601c85">
<p>
<b>Gauss' law in differential form</b>
</p>
@@ -1779,7 +1779,7 @@ cylindrical or plane symmetry.
Gaussian surfaces: respectively, concentric sphere, coaxial cylinder, pillbox.
</p>
-<div class="example div" id="org86862f2">
+<div class="example div" id="org5c01f30">
<p>
<a id="E_uni_sph"></a><a href="./ems_es_ef_Gl.html#E_uni_sph"><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"/>
@@ -1814,7 +1814,7 @@ Same as point charge at origin!
</div>
-<div class="example div" id="orge7c3996">
+<div class="example div" id="org15c018b">
<p>
<b>Example 2.3</b>: infinitely long cylinder carrying charge density \(\rho = k s\) for some constant \(k\). Find \({\bf E}\) within the cylinder.
</p>
@@ -1853,7 +1853,7 @@ Therefore,
</div>
-<div class="example div" id="org2465dd4">
+<div class="example div" id="org83647ca">
<p>
<b>Example 2.4</b>: infinite plane (defined by \(z = 0\)) with uniform surface charge density \(\sigma\). Find \({\bf E}\).
</p>
@@ -1879,7 +1879,7 @@ where \(\hat{\bf n}\) is a unit vector extending away from the plane. Independe
</div>
-<div class="example div" id="org2090e4b">
+<div class="example div" id="org6d1758e">
<p>
<b>Example 2.5</b>: two infinite planes (put them vertical) carrying equal but opposite uniform surface charge densities \(\pm \sigma\).
</p>
@@ -1909,7 +1909,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ef_cE.html b/build/ems_es_ef_cE.html
index cca0ffe..bfc48b6 100644
--- a/build/ems_es_ef_cE.html
+++ b/build/ems_es_ef_cE.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1617,14 +1617,14 @@ Table of contents
Since the work done when moving on a closed path vanishes, we have
</p>
-<div class="eqlabel" id="orgae91a8d">
+<div class="eqlabel" id="org2b0c542">
<p>
<a id="ointE0"></a><a href="./ems_es_ef_cE.html#ointE0"><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="orgafa4040">
+<div class="alteqlabels" id="org02ac6b2">
<ul class="org-ul">
<li>Gr (2.19)</li>
</ul>
@@ -1632,7 +1632,7 @@ Since the work done when moving on a closed path vanishes, we have
</div>
</div>
-<div class="core div" id="org4168973">
+<div class="core div" id="orga54176a">
<p>
</p>
@@ -1650,14 +1650,14 @@ Since the work done when moving on a closed path vanishes, we have
which by <a href="./c_m_ic_stokes.html#Stokes">Stokes' theorem</a> implies that the electrostatic field is curlless,
</p>
-<div class="eqlabel" id="org0ef32fd">
+<div class="eqlabel" id="orgd47b068">
<p>
<a id="curlE0"></a><a href="./ems_es_ef_cE.html#curlE0"><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="org696251f">
+<div class="alteqlabels" id="org6235c0c">
<ul class="org-ul">
<li>Gr (2.20)</li>
</ul>
@@ -1665,7 +1665,7 @@ which by <a href="./c_m_ic_stokes.html#Stokes">Stokes' theorem</a> implies that
</div>
</div>
-<div class="core div" id="orgb57e45b">
+<div class="core div" id="orgf1c6e91">
<p>
</p>
@@ -1711,7 +1711,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ef_ccd.html b/build/ems_es_ef_ccd.html
index 779961e..3365b1c 100644
--- a/build/ems_es_ef_ccd.html
+++ b/build/ems_es_ef_ccd.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1619,14 +1619,14 @@ calculated from Coulomb's law using the superposition principle. Since each inf
volume element \(d\tau' = dx' dy' dz'\) contains a charge \(dq' = \rho({\bf r}') d\tau'\), we have
</p>
-<div class="eqlabel" id="org3f167fe">
+<div class="eqlabel" id="org0685ec2">
<p>
<a id="E_vcd"></a><a href="./ems_es_ef_ccd.html#E_vcd"><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="orgea0c302">
+<div class="alteqlabels" id="org2a6f970">
<ul class="org-ul">
<li>Gr4 (2.8)</li>
</ul>
@@ -1634,7 +1634,7 @@ volume element \(d\tau' = dx' dy' dz'\) contains a charge \(dq' = \rho({\bf r}')
</div>
</div>
-<div class="main div" id="orgb8d314c">
+<div class="main div" id="org6569d43">
<p>
</p>
@@ -1654,14 +1654,14 @@ Similarly, if the charge is spread out over a two-dimensional surface \({\cal S}
\(\sigma({\bf r})\), we have over an infinitesimal area \(da'\) a charge \(dq' = \sigma({\bf r}') da'\), so
</p>
-<div class="eqlabel" id="org88e8806">
+<div class="eqlabel" id="org463d4a8">
<p>
<a id="E_scd"></a><a href="./ems_es_ef_ccd.html#E_scd"><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="orgb53c96f">
+<div class="alteqlabels" id="org1997b74">
<ul class="org-ul">
<li>Gr4(2.7)</li>
</ul>
@@ -1669,7 +1669,7 @@ Similarly, if the charge is spread out over a two-dimensional surface \({\cal S}
</div>
</div>
-<div class="main div" id="org9052c26">
+<div class="main div" id="org0795fa2">
<p>
</p>
@@ -1686,14 +1686,14 @@ Similarly, if the charge is spread out over a two-dimensional surface \({\cal S}
Finally, for a line path \({\cal P}\) with linear charge density \(\lambda({\bf r}')\),
</p>
-<div class="eqlabel" id="org047ff2c">
+<div class="eqlabel" id="org5ea882d">
<p>
<a id="E_lcd"></a><a href="./ems_es_ef_ccd.html#E_lcd"><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="org285c4e1">
+<div class="alteqlabels" id="org18ca9b1">
<ul class="org-ul">
<li>Gr (2.6)</li>
</ul>
@@ -1701,7 +1701,7 @@ Finally, for a line path \({\cal P}\) with linear charge density \(\lambda({\bf
</div>
</div>
-<div class="main div" id="orgc32983b">
+<div class="main div" id="org8e461ef">
<p>
</p>
@@ -1715,7 +1715,7 @@ Finally, for a line path \({\cal P}\) with linear charge density \(\lambda({\bf
</div>
-<div class="example div" id="org8bade7b">
+<div class="example div" id="orge5b772b">
<p>
<b>Example</b>
</p>
@@ -1749,7 +1749,7 @@ most easily by observing that \(\frac{d}{dx} \left( \frac{x}{\sqrt{z^2 + x^2}} \
= \frac{1}{\sqrt{z^2 + x^2}} - \frac{x^2}{(z^2 + x^2)^{3/2}} = \frac{z^2}{(z^2 + x^2)^{3/2}}\),
leading to
</p>
-<aside id="org726fb53">
+<aside id="org371c5b7">
<p>
You could alternately proceed by using changes of variables \(y = zx\) followed by \(y = \tanh \alpha\):
\(\int_{-L}^L \frac{dx}{(z^2 + x^2)^{3/2}} = \frac{1}{z^2}
@@ -1806,7 +1806,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ef_pc.html b/build/ems_es_ef_pc.html
index 0d6fb41..9869cce 100644
--- a/build/ems_es_ef_pc.html
+++ b/build/ems_es_ef_pc.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1627,14 +1627,14 @@ an electric field originating from the source charge.
Invoking the superposition principle, we can thus write
</p>
-<div class="eqlabel" id="org9bda26a">
+<div class="eqlabel" id="orge6e38f0">
<p>
<a id="FqE"></a><a href="./ems_es_ef_pc.html#FqE"><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="org7d8c617">
+<div class="alteqlabels" id="org066e788">
<ul class="org-ul">
<li>Gr (2.3)</li>
</ul>
@@ -1642,7 +1642,7 @@ Invoking the superposition principle, we can thus write
</div>
</div>
-<div class="core div" id="org0b469c8">
+<div class="core div" id="org171997a">
<p>
</p>
@@ -1658,14 +1658,14 @@ Invoking the superposition principle, we can thus write
<p>
with the electric field of a point charge distribution being
</p>
-<div class="eqlabel" id="org2649550">
+<div class="eqlabel" id="org87ef17e">
<p>
<a id="E_pcd"></a><a href="./ems_es_ef_pc.html#E_pcd"><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="org5bdbd95">
+<div class="alteqlabels" id="org2002a6f">
<ul class="org-ul">
<li>Gr (2.4)</li>
</ul>
@@ -1673,7 +1673,7 @@ with the electric field of a point charge distribution being
</div>
</div>
-<div class="core div" id="org1890d9d">
+<div class="core div" id="org38b7915">
<p>
</p>
@@ -1691,7 +1691,7 @@ The electric field \({\bf E} ({\bf r})\) is thus the force per unit charge that
be exerted if you put a test charge at position \({\bf r}\).
</p>
-<div class="example div" id="orgcd62ca3">
+<div class="example div" id="org93d7f68">
<p>
<b>Example</b>
</p>
@@ -1735,7 +1735,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_efo.html b/build/ems_es_efo.html
index e8ad618..686d4c7 100644
--- a/build/ems_es_efo.html
+++ b/build/ems_es_efo.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_efo_cl.html b/build/ems_es_efo_cl.html
index 2377102..2cbf3a9 100644
--- a/build/ems_es_efo_cl.html
+++ b/build/ems_es_efo_cl.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1619,14 +1619,14 @@ Table of contents
<li>PM 1.4</li>
</ul>
-<div class="eqlabel" id="org7d49383">
+<div class="eqlabel" id="orgcad778a">
<p>
<a id="Cl"></a><a href="./ems_es_efo_cl.html#Cl"><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="orgc7a0b3a">
+<div class="alteqlabels" id="orge60882d">
<ul class="org-ul">
<li>FLS II (4.9)</li>
<li>Gr (2.1)</li>
@@ -1638,7 +1638,7 @@ Table of contents
</div>
</div>
-<div class="core div" id="orgd0b0f26">
+<div class="core div" id="org5f65836">
<p>
The force \(F_{t\leftarrow s}\) exerted by a point source charge \(q_s\) sitting at \({\bf r}_s\)
on a point test charge \(q_t\) sitting at \({\bf r}_t\) is given by Coulomb's law,
@@ -1655,7 +1655,7 @@ on a point test charge \(q_t\) sitting at \({\bf r}_t\) is given by Coulomb's la
</div>
-<aside id="org91e3080">
+<aside id="org82fc785">
<p>
In case you're wondering: the exact number in the denominator
of the first equality is just \(4 \pi c^2\), so this equation really
@@ -1696,7 +1696,7 @@ in which \(\mu_0\) is the <b>vacuum permeability</b> (or alternately <i>permeabi
with the <i>henry</i> \(H = kg~m^2 / s^2 A^2\) being the unit for inductance.
</p>
-<div class="info div" id="org6f4ee6e">
+<div class="info div" id="org739c878">
<p>
<b>Info: the International System of Units</b>
</p>
@@ -1753,7 +1753,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_efo_e.html b/build/ems_es_efo_e.html
index 89f663b..e5e9b9a 100644
--- a/build/ems_es_efo_e.html
+++ b/build/ems_es_efo_e.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1656,14 +1656,14 @@ For the path, the 'angular' part is not contributing (see drawings in
FLS II 4-3). The integral is thus purely radial,
</p>
-<div class="eqlabel" id="orgdc45157">
+<div class="eqlabel" id="org1d38281">
<p>
<a id="Wab"></a><a href="./ems_es_efo_e.html#Wab"><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="org1dc3b03">
+<div class="alteqlabels" id="orga342766">
<ul class="org-ul">
<li>Gr (2.18)</li>
</ul>
@@ -1750,7 +1750,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_efo_exp.html b/build/ems_es_efo_exp.html
index 031f258..e8730ae 100644
--- a/build/ems_es_efo_exp.html
+++ b/build/ems_es_efo_exp.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1615,17 +1615,17 @@ Table of contents
<div class="outline-text-5" id="text-ems_es_efo_exp">
</div>
-<div id="outline-container-org0e4492e" class="outline-6">
-<h6 id="org0e4492e"><a href="#org0e4492e">Before Coulomb</a></h6>
+<div id="outline-container-org49d790a" class="outline-6">
+<h6 id="org49d790a"><a href="#org49d790a">Before Coulomb</a></h6>
</div>
-<div id="outline-container-orgfa0f166" class="outline-6">
-<h6 id="orgfa0f166"><a href="#orgfa0f166">Cavendish's experiment</a></h6>
+<div id="outline-container-orgd713e5c" class="outline-6">
+<h6 id="orgd713e5c"><a href="#orgd713e5c">Cavendish's experiment</a></h6>
</div>
-<div id="outline-container-orgad3be86" class="outline-6">
-<h6 id="orgad3be86"><a href="#orgad3be86">Coulomb</a></h6>
+<div id="outline-container-orga0bbfa0" class="outline-6">
+<h6 id="orga0bbfa0"><a href="#orga0bbfa0">Coulomb</a></h6>
</div>
-<div id="outline-container-orgf9342f9" class="outline-6">
-<h6 id="orgf9342f9"><a href="#orgf9342f9">Current status</a></h6>
+<div id="outline-container-org9f40a7a" class="outline-6">
+<h6 id="org9f40a7a"><a href="#org9f40a7a">Current status</a></h6>
</div>
</div>
@@ -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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_efo_ps.html b/build/ems_es_efo_ps.html
index e09319c..93b5b23 100644
--- a/build/ems_es_efo_ps.html
+++ b/build/ems_es_efo_ps.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1619,7 +1619,7 @@ Table of contents
<li>PM 1.4</li>
</ul>
-<div class="core div" id="orgabf6f0f">
+<div class="core div" id="org4c6f42c">
<p>
Electrical forces obey the <b>principle of superposition</b>:
the electrostatic interaction between two charged particles is left entirely
@@ -1655,7 +1655,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ep.html b/build/ems_es_ep.html
index 1dae905..1fd28de 100644
--- a/build/ems_es_ep.html
+++ b/build/ems_es_ep.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ep_PL.html b/build/ems_es_ep_PL.html
index 52d227d..e3d1c13 100644
--- a/build/ems_es_ep_PL.html
+++ b/build/ems_es_ep_PL.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1622,14 +1622,14 @@ Our two fundamental equations for the electrostatic field are
\]
For the electrostatic potential, by <a href="./ems_es_ep_fp.html#Emgp">Emgp</a>, Gauss's law becomes
</p>
-<div class="eqlabel" id="org5858f81">
+<div class="eqlabel" id="orga248783">
<p>
<a id="Poi"></a><a href="./ems_es_ep_PL.html#Poi"><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="org179b8e7">
+<div class="alteqlabels" id="org2e27297">
<ul class="org-ul">
<li>Gr (2.24)</li>
<li>PM (2.72)</li>
@@ -1638,7 +1638,7 @@ For the electrostatic potential, by <a href="./ems_es_ep_fp.html#Emgp">Emgp</a>,
</div>
</div>
-<div class="core div" id="org52f9c95">
+<div class="core div" id="org2b5be1d">
<p>
<b>Poisson's equation</b>
\[
@@ -1651,14 +1651,14 @@ For the electrostatic potential, by <a href="./ems_es_ep_fp.html#Emgp">Emgp</a>,
<p>
When the charge density vanishes, it becomes more simply
</p>
-<div class="eqlabel" id="org66aff4c">
+<div class="eqlabel" id="org98e969b">
<p>
<a id="Lap"></a><a href="./ems_es_ep_PL.html#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="orgd0696f8">
+<div class="alteqlabels" id="org1b9ea13">
<ul class="org-ul">
<li>Gr (2.25)</li>
<li>PM (2.75)</li>
@@ -1667,7 +1667,7 @@ When the charge density vanishes, it becomes more simply
</div>
</div>
-<div class="core div" id="orgd89910d">
+<div class="core div" id="orgdc0fc61">
<p>
<b>Laplace's Equation</b>
\[
@@ -1703,7 +1703,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ep_bc.html b/build/ems_es_ep_bc.html
index 174a986..2e0a3b3 100644
--- a/build/ems_es_ep_bc.html
+++ b/build/ems_es_ep_bc.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1650,14 +1650,14 @@ applied to a small loop straddling the surface:
<p>
We can unify both boundary conditions into a single equation:
</p>
-<div class="eqlabel" id="orga24969d">
+<div class="eqlabel" id="org017eb44">
<p>
<a id="Edisc"></a><a href="./ems_es_ep_bc.html#Edisc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org8aab62c">
+<div class="alteqlabels" id="org5a5e4ce">
<ul class="org-ul">
<li>Gr (2.33)</li>
</ul>
@@ -1692,14 +1692,14 @@ The gradient however inherits the discontinuity of the electrostatic field, sinc
\]
or
</p>
-<div class="eqlabel" id="org590ed85">
+<div class="eqlabel" id="orge3f6c85">
<p>
<a id="dpdisc"></a><a href="./ems_es_ep_bc.html#dpdisc"><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="org6ac014a">
+<div class="alteqlabels" id="org2c83462">
<ul class="org-ul">
<li>Gr (2.36)</li>
</ul>
@@ -1747,7 +1747,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ep_d.html b/build/ems_es_ep_d.html
index 89e5c90..f8ad046 100644
--- a/build/ems_es_ep_d.html
+++ b/build/ems_es_ep_d.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1620,14 +1620,14 @@ field is independent of the path, and that it is proportional to the test charge
We can thus define a function representing the work per unit charge and
called the <b>electrostatic potential</b>
</p>
-<div class="eqlabel" id="orgfcac90f">
+<div class="eqlabel" id="org39ed3cc">
<p>
<a id="p"></a><a href="./ems_es_ep_d.html#p"><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="orgab3c157">
+<div class="alteqlabels" id="orgfc6116b">
<ul class="org-ul">
<li>Gr (2.21)</li>
</ul>
@@ -1643,7 +1643,7 @@ called the <b>electrostatic potential</b>
\]
</p>
-<aside id="orgc523465">
+<aside id="org5587716">
<p>
\(\phi\) carries as unit: Newton-meters per coulomb or joules per coulomb, which is called a <b>volt</b>.
In the SI system where the ampere is a base unit, a volt is a \(kg m^2/s^3 A\).
@@ -1660,14 +1660,14 @@ want; it's energy <i>differences</i> which matter).
The potential difference between two arbitrary points \({\bf a}\) and \({\bf b}\) is
well-defined without the need to specify the reference point,
</p>
-<div class="eqlabel" id="orge35b00f">
+<div class="eqlabel" id="org63f500f">
<p>
<a id="p_diff"></a><a href="./ems_es_ep_d.html#p_diff"><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="org21f750c">
+<div class="alteqlabels" id="org026dcd6">
<ul class="org-ul">
<li>Gr (2.21)</li>
</ul>
@@ -1675,7 +1675,7 @@ well-defined without the need to specify the reference point,
</div>
</div>
-<div class="main div" id="orgfcd367f">
+<div class="main div" id="orgf050485">
<p>
\[
\phi({\bf b}) - \phi({\bf a}) = - \int_{\bf a}^{\bf b} {\bf E} \cdot d{\bf l}
@@ -1692,14 +1692,14 @@ The reference point is typically put at infinity. The electrostatic potential c
point charge \(q\) at the origin can then be calculated (taking for convenience \(d{\bf l} = \hat{\bf r} dr\),
<i>i.e.</i> moving in purely radially) as
</p>
-<div class="eqlabel" id="org7062925">
+<div class="eqlabel" id="orga100469">
<p>
<a id="p_pc"></a><a href="./ems_es_ep_d.html#p_pc"><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="org35b2cdc">
+<div class="alteqlabels" id="org83e9a3a">
<ul class="org-ul">
<li>FLS II (4.20,4.23)</li>
<li>Gr (2.26)</li>
@@ -1722,14 +1722,14 @@ point charge \(q\) at the origin can then be calculated (taking for convenience
The electrostatic potential moreover inherits the superposition principle from the electric field,
so for a distribution of point charges \(q_i\) at positions \({\bf r}_i\), we have
</p>
-<div class="eqlabel" id="orgf111ed3">
+<div class="eqlabel" id="org8fb4f47">
<p>
<a id="p_pcd"></a><a href="./ems_es_ep_d.html#p_pcd"><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="org89071e0">
+<div class="alteqlabels" id="orgd4b75d4">
<ul class="org-ul">
<li>FLS II (4.24)</li>
<li>Gr (2.27)</li>
@@ -1739,7 +1739,7 @@ so for a distribution of point charges \(q_i\) at positions \({\bf r}_i\), we ha
</div>
</div>
-<div class="core div" id="org4ac5fc6">
+<div class="core div" id="org0feaada">
<p>
</p>
@@ -1755,14 +1755,14 @@ so for a distribution of point charges \(q_i\) at positions \({\bf r}_i\), we ha
<p>
For a continuous charge density in a volume \({\cal V}\), we have
</p>
-<div class="eqlabel" id="orgd4c214c">
+<div class="eqlabel" id="org4f7acf1">
<p>
<a id="p_vcd"></a><a href="./ems_es_ep_d.html#p_vcd"><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="org8fb75e5">
+<div class="alteqlabels" id="org89a4a17">
<ul class="org-ul">
<li>Gr (2.29)</li>
<li>PM (2.18)</li>
@@ -1771,7 +1771,7 @@ For a continuous charge density in a volume \({\cal V}\), we have
</div>
</div>
-<div class="main div" id="org0561934">
+<div class="main div" id="orgce6daed">
<p>
</p>
@@ -1788,14 +1788,14 @@ For a continuous charge density in a volume \({\cal V}\), we have
<p>
whereas for a surface or line charge distribution, respectively,
</p>
-<div class="eqlabel" id="org886392d">
+<div class="eqlabel" id="orgcac3c57">
<p>
<a id="p_scd"></a><a href="./ems_es_ep_d.html#p_scd"><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="org488ec6b">
+<div class="alteqlabels" id="org5fbbe53">
<ul class="org-ul">
<li>Gr (2.30)b</li>
</ul>
@@ -1803,7 +1803,7 @@ whereas for a surface or line charge distribution, respectively,
</div>
</div>
-<div class="main div" id="org21be33b">
+<div class="main div" id="org7d0d5f8">
<p>
\[
\phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\cal S} da' \frac{\sigma({\bf r}')}{|{\bf r} - {\bf r}'|}
@@ -1813,14 +1813,14 @@ whereas for a surface or line charge distribution, respectively,
</div>
-<div class="eqlabel" id="orgf80de2f">
+<div class="eqlabel" id="org49407d6">
<p>
<a id="p_lcd"></a><a href="./ems_es_ep_d.html#p_lcd"><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="orgc67dd5f">
+<div class="alteqlabels" id="orgfd9b5c6">
<ul class="org-ul">
<li>Gr (2.30)a</li>
</ul>
@@ -1828,7 +1828,7 @@ whereas for a surface or line charge distribution, respectively,
</div>
</div>
-<div class="main div" id="orgaeed524">
+<div class="main div" id="orgfc25e94">
<p>
</p>
@@ -1861,7 +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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ep_ex.html b/build/ems_es_ep_ex.html
index 63f68ae..d4609df 100644
--- a/build/ems_es_ep_ex.html
+++ b/build/ems_es_ep_ex.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1613,7 +1613,7 @@ 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.es.ep.ex</span></h5>
<div class="outline-text-5" id="text-ems_es_ep_ex">
-<div class="example div" id="org1f04fcd">
+<div class="example div" id="orgcaa3290">
<p>
<b>Spherical shell: via \({\bf E}\)</b>
</p>
@@ -1661,7 +1661,7 @@ and using spherical symmetry, we get:
</div>
-<div class="example div" id="orgc15b358">
+<div class="example div" id="orgb08b399">
<p>
<b>Spherical shell: direct calculation</b>
</p>
@@ -1738,7 +1738,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_es_ep_fp.html b/build/ems_es_ep_fp.html
index 1d4b3fa..7ad47af 100644
--- a/build/ems_es_ep_fp.html
+++ b/build/ems_es_ep_fp.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1639,14 +1639,14 @@ so
<p>
Since this is true for any choice of \({\bf a}\) and \({\bf b}\), we have
</p>
-<div class="eqlabel" id="org284663d">
+<div class="eqlabel" id="orgd7cc06c">
<p>
<a id="Emgp"></a><a href="./ems_es_ep_fp.html#Emgp"><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="org1ee291c">
+<div class="alteqlabels" id="org4f178ad">
<ul class="org-ul">
<li>Gr (2.23)</li>
</ul>
@@ -1654,7 +1654,7 @@ Since this is true for any choice of \({\bf a}\) and \({\bf b}\), we have
</div>
</div>
-<div class="core div" id="org8378290">
+<div class="core div" id="org07c0b3a">
<p>
\[
{\bf E} = -{\boldsymbol \nabla} \phi
@@ -1725,7 +1725,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms.html b/build/ems_ms.html
index 6c8b2b6..fef5307 100644
--- a/build/ems_ms.html
+++ b/build/ems_ms.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1620,8 +1620,8 @@ Table of contents
<li>PM 6</li>
</ul>
-<details class="prereq" id="orge84c2ba">
-<summary id="orgf8fa4bb">
+<details class="prereq" id="orge2c497a">
+<summary id="org76315f8">
Prerequisites
</summary>
<ul class="org-ul">
@@ -1629,8 +1629,8 @@ Prerequisites
</ul>
</details>
-<details class="objectives" id="org9c57226">
-<summary id="org07e81b9">
+<details class="objectives" id="org1724943">
+<summary id="org51e4b43">
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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms_BS.html b/build/ems_ms_BS.html
index a93f9b2..7a13160 100644
--- a/build/ems_ms_BS.html
+++ b/build/ems_ms_BS.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1617,18 +1617,18 @@ Table of contents
The magnetic field issuing from a steady surrent
is given experimentally (around 1820) by the
</p>
-<div class="core div" id="org51f208a">
+<div class="core div" id="orga827512">
<p>
<b>Biot-Savart law</b>
</p>
-<div class="eqlabel" id="org7066903">
+<div class="eqlabel" id="org52886c6">
<p>
<a id="BiotSavart"></a><a href="./ems_ms_BS.html#BiotSavart"><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="orged15a3a">
+<div class="alteqlabels" id="orgdc1e7ba">
<ul class="org-ul">
<li>FLS II (14.43)</li>
<li>Gr (5.34)</li>
@@ -1658,15 +1658,15 @@ with the <i>henry</i> \(H = kg ~m^2 / s^2 A^2\) being the unit for inductance.
<p>
For surface and volume density currents:
</p>
-<div class="main div" id="org93fdedb">
-<div class="eqlabel" id="org7ca46e7">
+<div class="main div" id="org7a35caa">
+<div class="eqlabel" id="org23a2575">
<p>
<a id="BiotSavart_s"></a><a href="./ems_ms_BS.html#BiotSavart_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="org96cf80f">
+<div class="alteqlabels" id="org85a286c">
</div>
@@ -1677,14 +1677,14 @@ For surface and volume density currents:
\tag{BiotSavart_s}\label{BiotSavart_s}
\]
</p>
-<div class="eqlabel" id="orgcc6c1f1">
+<div class="eqlabel" id="org938c8cc">
<p>
<a id="BiotSavart_v"></a><a href="./ems_ms_BS.html#BiotSavart_v"><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="org15cd877">
+<div class="alteqlabels" id="orgd7e4616">
</div>
@@ -1709,7 +1709,7 @@ The <b>superposition principle</b> applies here as well: a collection of curren
the vector sum of the fields generated by the individual currents.
</p>
-<div class="example div" id="orgcb01a42">
+<div class="example div" id="org01ad376">
<p>
<b>Example: \({\bf B}\) from long straight wire</b>
</p>
@@ -1732,14 +1732,14 @@ B = \frac{\mu_0}{4\pi} I \int_{\theta_1}^{\theta_2} d\theta \cos \theta \frac{\c
\]
For infinite wire: \(\theta_1 = -\pi/2\), \(\theta_2 = \pi/2\), so
</p>
-<div class="eqlabel" id="org4013eb6">
+<div class="eqlabel" id="org76e595d">
<p>
<a id="Bwire1"></a><a href="./ems_ms_BS.html#Bwire1"><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="org18cf243">
+<div class="alteqlabels" id="org1b67385">
<ul class="org-ul">
<li>Gr (5.38)</li>
</ul>
@@ -1768,7 +1768,7 @@ f = \frac{\mu_0}{2\pi} \frac{I_1 I_2}{d}
(like currents attract).
</p>
-<div class="example div" id="org3d9edf3">
+<div class="example div" id="org961db5f">
<p>
<b>Example: \({\bf B}\) above a circular loop</b>
</p>
@@ -1812,7 +1812,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms_ce.html b/build/ems_ms_ce.html
index 22831b4..769e916 100644
--- a/build/ems_ms_ce.html
+++ b/build/ems_ms_ce.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1635,18 +1635,18 @@ through the surface is (minus) the change of charge contained inside the surface
\]
Since this is valid for any volume, we get the
</p>
-<div class="core div" id="org88ef2b3">
+<div class="core div" id="orga21a4ef">
<p>
<b>Continuity Equation</b>
</p>
-<div class="eqlabel" id="org34b3d80">
+<div class="eqlabel" id="org32cd8f0">
<p>
<a id="conteq"></a><a href="./ems_ms_ce.html#conteq"><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="org6c0a202">
+<div class="alteqlabels" id="orged6767a">
<ul class="org-ul">
<li>FLS II (13.8)</li>
<li>Gr (5.29)</li>
@@ -1693,7 +1693,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms_dcB.html b/build/ems_ms_dcB.html
index 2bf8523..059ae43 100644
--- a/build/ems_ms_dcB.html
+++ b/build/ems_ms_dcB.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms_dcB_c.html b/build/ems_ms_dcB_c.html
index 1471d71..d431c14 100644
--- a/build/ems_ms_dcB_c.html
+++ b/build/ems_ms_dcB_c.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1663,18 +1663,18 @@ at infinity), and in the third step we have used the assumption of steady-state
<p>
We thus obtain
</p>
-<div class="core div" id="org6a024aa">
+<div class="core div" id="org783256e">
<p>
<b>Ampère's law</b>
</p>
-<div class="eqlabel" id="orgb881792">
+<div class="eqlabel" id="orge745f25">
<p>
<a id="Amp"></a><a href="./ems_ms_dcB_c.html#Amp"><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="org54bd1f6">
+<div class="alteqlabels" id="orgda8539d">
<ul class="org-ul">
<li>FLS II (13.13)</li>
<li>Gr (5.56)</li>
@@ -1699,18 +1699,18 @@ We thus obtain
\]
yielding
</p>
-<div class="core div" id="orge0c90b7">
+<div class="core div" id="org354b44c">
<p>
<b>Ampère's law (integral form)</b>
</p>
-<div class="eqlabel" id="org07d6c1e">
+<div class="eqlabel" id="orgd83eec1">
<p>
<a id="Amp_int"></a><a href="./ems_ms_dcB_c.html#Amp_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="orgb68c754">
+<div class="alteqlabels" id="orgfb58e7c">
<ul class="org-ul">
<li>FLS II (13.16)</li>
<li>Gr (5.57)</li>
@@ -1741,7 +1741,7 @@ Sign ambiguity: resolved by right-hand rule as usual.
Ampère's law in magnetostatics takes a parallel role to Gauss's law in electrostatics.
</p>
-<div class="example div" id="org3d7cd57">
+<div class="example div" id="org228e2e4">
<p>
<b>Example: infinite wire</b>
</p>
@@ -1760,7 +1760,7 @@ choosing an amperian loop at a fixed radius \(r\), we get
</div>
-<div class="example div" id="org8d8b261">
+<div class="example div" id="org6ded1ae">
<p>
<b>Example: uniform sheet current</b>
</p>
@@ -1789,7 +1789,7 @@ and along \(\hat{\bf y}\) for \(z < 0\). Amperian loop of width \(l\) punchi
</div>
-<div class="example div" id="org2c3c8bc">
+<div class="example div" id="org79e3791">
<p>
<b>Example: solenoid</b>
</p>
@@ -1809,14 +1809,14 @@ Outside: infinitely far away, \({\bf B}\) must vanish.
But an amperian loop outside gives zero always, so \({\bf B}\) vanishes everywhere outside the solenoid.
Amperian loop of length \(l\), half-inside and half-outside:
</p>
-<div class="eqlabel" id="orgdc9d8f5">
+<div class="eqlabel" id="orgca26b03">
<p>
<a id="Bsol"></a><a href="./ems_ms_dcB_c.html#Bsol"><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="orgaf8209e">
+<div class="alteqlabels" id="orgc2fa23f">
<ul class="org-ul">
<li>Gr (5.59)</li>
</ul>
@@ -1841,7 +1841,7 @@ form of symmetry, for example
i) infinite straight lines, ii) infinite planes, iii) infinite solenoids, iv) toroids.
</p>
-<div class="example div" id="orgc74ca44">
+<div class="example div" id="org0bb7ec0">
<p>
<b>Example: toroidal coil</b>
</p>
@@ -1863,14 +1863,14 @@ Outside the coil, the field must again be zero.
<p>
Consider now an Amperian loop which is just a circle at radius \(r\):
</p>
-<div class="eqlabel" id="org79fc555">
+<div class="eqlabel" id="orgbea10d8">
<p>
<a id="Btor"></a><a href="./ems_ms_dcB_c.html#Btor"><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="org44e5c13">
+<div class="alteqlabels" id="org8bc0f73">
<ul class="org-ul">
<li>Gr (5.60)</li>
</ul>
@@ -1907,7 +1907,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms_dcB_d.html b/build/ems_ms_dcB_d.html
index 61508d4..3e00bab 100644
--- a/build/ems_ms_dcB_d.html
+++ b/build/ems_ms_dcB_d.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1642,15 +1642,15 @@ Note that we can write (using <a href="./c_m_dd_3d.html#div1or">div1or</a>) \(\f
But \({\bf J}\) depends only on \({\bf r}'\) so \({\boldsymbol \nabla} \times {\bf J} ({\bf r}') = 0\), and since
the curl of a gradient always vanishes, we obtain
</p>
-<div class="core div" id="orgcc7d0d1">
-<div class="eqlabel" id="org5a5f4dd">
+<div class="core div" id="orgdba520c">
+<div class="eqlabel" id="orgd9bb35e">
<p>
<a id="divB0"></a><a href="./ems_ms_dcB_d.html#divB0"><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="org269c203">
+<div class="alteqlabels" id="orge93ce7b">
<ul class="org-ul">
<li>FLS II (13.12)</li>
<li>Gr (5.50)</li>
@@ -1693,7 +1693,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms_dcB_iw.html b/build/ems_ms_dcB_iw.html
index 5a93f15..634061c 100644
--- a/build/ems_ms_dcB_iw.html
+++ b/build/ems_ms_dcB_iw.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1617,14 +1617,14 @@ Table of contents
Let us start with the simple case of the field \({\bf B}\) coming from an infinite wire, as
given by <a href="./ems_ms_BS.html#Bwire1">Bwire1</a>, rewritten here in cylindrical coordinates along the axis of the wire:
</p>
-<div class="eqlabel" id="org71698f7">
+<div class="eqlabel" id="orgcbf3a0a">
<p>
<a id="Bwire"></a><a href="./ems_ms_dcB_iw.html#Bwire"><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="org28e6010">
+<div class="alteqlabels" id="org7898995">
<ul class="org-ul">
<li>Gr (5.39)</li>
</ul>
@@ -1705,7 +1705,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms_lf.html b/build/ems_ms_lf.html
index 67f2c9f..809599f 100644
--- a/build/ems_ms_lf.html
+++ b/build/ems_ms_lf.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms_lf_pc.html b/build/ems_ms_lf_pc.html
index ae9950f..e88354d 100644
--- a/build/ems_ms_lf_pc.html
+++ b/build/ems_ms_lf_pc.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1617,7 +1617,7 @@ Table of contents
The magnetic force on a point charge \(q\) moving at velocity \({\bf v}\) in magnetic field \({\bf B}\)
is given by
</p>
-<div class="core div" id="org30f0e3f">
+<div class="core div" id="org868db15">
<p>
\[
{\bf F}_{mag} = q {\bf v} \times {\bf B}
@@ -1631,14 +1631,14 @@ Units of \({\bf B}\): \(N/(A~m)\) is called a <b>tesla</b> (symbol: \(T\)).
The total electromagnetic force on a point charge \(q\) moving at velocity
\({\bf v}\) is given by the <b>Lorentz force</b>:
</p>
-<div class="eqlabel" id="orga80999e">
+<div class="eqlabel" id="orga1b18db">
<p>
<a id="LorFo"></a><a href="./ems_ms_lf_pc.html#LorFo"><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="orgaa242da">
+<div class="alteqlabels" id="orgb36876d">
<ul class="org-ul">
<li>FLS II (13.1)</li>
<li>Gr (5.2)</li>
@@ -1648,7 +1648,7 @@ The total electromagnetic force on a point charge \(q\) moving at velocity
</div>
</div>
-<div class="core div" id="org5d8be1f">
+<div class="core div" id="org1cce15c">
<p>
\[
{\bf F} = q ({\bf E} + {\bf v} \times {\bf B})
@@ -1658,7 +1658,7 @@ The total electromagnetic force on a point charge \(q\) moving at velocity
</div>
-<div class="example div" id="org0806d35">
+<div class="example div" id="org45d1c2b">
<p>
<b>Cyclotron motion</b>
</p>
@@ -1717,7 +1717,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms_lf_sc.html b/build/ems_ms_lf_sc.html
index 53583f4..561b5ad 100644
--- a/build/ems_ms_lf_sc.html
+++ b/build/ems_ms_lf_sc.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1632,14 +1632,14 @@ This leads us to define a <b>current density</b> \({\bf J}\) as the vector repre
charge flowing through a unit area per unit time, with direction along the motion of the charges.
Then, \(\Delta q = \rho {\bf v} \cdot {\bf n} ~\Delta {\cal S} ~\Delta t\) so we can identify
</p>
-<div class="eqlabel" id="orgaccd76e">
+<div class="eqlabel" id="org634eed3">
<p>
<a id="vcurd"></a><a href="./ems_ms_lf_sc.html#vcurd"><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="org5e4014c">
+<div class="alteqlabels" id="orgec85e6b">
<ul class="org-ul">
<li>Gr (5.26)</li>
<li>PM (4.4)</li>
@@ -1659,14 +1659,14 @@ Then, \(\Delta q = \rho {\bf v} \cdot {\bf n} ~\Delta {\cal S} ~\Delta t\) so we
Similarly, a current running down a wire can be described by a linear charge density \(\lambda\)
moving at velocity \({\bf v}\), giving the current
</p>
-<div class="eqlabel" id="org613b2d4">
+<div class="eqlabel" id="orgb7eeef6">
<p>
<a id="lcurd"></a><a href="./ems_ms_lf_sc.html#lcurd"><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="org5012b81">
+<div class="alteqlabels" id="orgf8d5d1a">
<ul class="org-ul">
<li>Gr (5.14)</li>
</ul>
@@ -1685,14 +1685,14 @@ moving at velocity \({\bf v}\), giving the current
For charge flowing on a surface, we can describe the <b>surface current density</b> as a surface charge
density \(\sigma\) moving at velocity \({\bf v}\),
</p>
-<div class="eqlabel" id="orga0d42bc">
+<div class="eqlabel" id="org84167b3">
<p>
<a id="scurd"></a><a href="./ems_ms_lf_sc.html#scurd"><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="orgcd09447">
+<div class="alteqlabels" id="orgb6d5c96">
<ul class="org-ul">
<li>Gr (5.23)</li>
</ul>
@@ -1707,18 +1707,18 @@ density \(\sigma\) moving at velocity \({\bf v}\),
\]
</p>
-<div class="main div" id="orgf7b1e52">
+<div class="main div" id="org9ca056c">
<p>
Magnetic forces on wire, surface and volume carrying current densities:
</p>
-<div class="eqlabel" id="org9d77d7d">
+<div class="eqlabel" id="org92b3fea">
<p>
<a id="Fmagl"></a><a href="./ems_ms_lf_sc.html#Fmagl"><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="org3d85d43">
+<div class="alteqlabels" id="orgad8b186">
<ul class="org-ul">
<li>Gr (5.16)</li>
</ul>
@@ -1732,14 +1732,14 @@ Magnetic forces on wire, surface and volume carrying current densities:
\tag{Fmagl}\label{Fmagl}
\]
</p>
-<div class="eqlabel" id="orgada65a5">
+<div class="eqlabel" id="org5ab1162">
<p>
<a id="Fmags"></a><a href="./ems_ms_lf_sc.html#Fmags"><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="org581bcbc">
+<div class="alteqlabels" id="org5599c69">
<ul class="org-ul">
<li>Gr (5.24)</li>
</ul>
@@ -1753,14 +1753,14 @@ Magnetic forces on wire, surface and volume carrying current densities:
\tag{Fmags}\label{Fmags}
\]
</p>
-<div class="eqlabel" id="orge1c412b">
+<div class="eqlabel" id="orgde4d414">
<p>
<a id="Fmagv"></a><a href="./ems_ms_lf_sc.html#Fmagv"><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="orgf7d34a2">
+<div class="alteqlabels" id="org7f23ff0">
<ul class="org-ul">
<li>Gr (5.27)</li>
</ul>
@@ -1795,7 +1795,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms_vp.html b/build/ems_ms_vp.html
index bc06662..e335a38 100644
--- a/build/ems_ms_vp.html
+++ b/build/ems_ms_vp.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms_vp_A.html b/build/ems_ms_vp_A.html
index 11e700d..1876789 100644
--- a/build/ems_ms_vp_A.html
+++ b/build/ems_ms_vp_A.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1617,15 +1617,15 @@ Table of contents
Since \({\boldsymbol \nabla} \cdot {\bf B} = 0\) in magnetostatics, following Helmholtz's theorem
(see <a href="./c_m_vf_helm.html#c_m_vf_helm">c_m_vf_helm</a>) we can write
</p>
-<div class="core div" id="org0e39e79">
-<div class="eqlabel" id="org373dcc3">
+<div class="core div" id="org43892ce">
+<div class="eqlabel" id="org3a3bc3a">
<p>
<a id="BcurlA"></a><a href="./ems_ms_vp_A.html#BcurlA"><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="org19a8207">
+<div class="alteqlabels" id="orgb09d3b2">
<ul class="org-ul">
<li>Gr (5.61)</li>
</ul>
@@ -1661,18 +1661,18 @@ add any curlless function (said otherwise: the gradient of any scalar field) to
without changing the magnetic field. Making a specific choice is called making a <b>gauge choice</b>.
For example, we can always choose to eliminate the divergence of \({\bf A}\), giving us the example of the
</p>
-<div class="main div" id="orgc170b1d">
+<div class="main div" id="org6e595ac">
<p>
<b>Coulomb gauge</b>
</p>
-<div class="eqlabel" id="org31d1543">
+<div class="eqlabel" id="orgff30682">
<p>
<a id="CoulG"></a><a href="./ems_ms_vp_A.html#CoulG"><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="org3a1cdeb">
+<div class="alteqlabels" id="org9e044e6">
<ul class="org-ul">
<li>Gr (5.63)</li>
</ul>
@@ -1711,15 +1711,15 @@ zero at infinity,
<p>
Under this gauge choice, Ampère's law becomes
</p>
-<div class="main div" id="orgf96dc74">
-<div class="eqlabel" id="orgb96c94b">
+<div class="main div" id="org764e2b0">
+<div class="eqlabel" id="org3f693e5">
<p>
<a id="Amp_CoulG"></a><a href="./ems_ms_vp_A.html#Amp_CoulG"><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="org7571aa0">
+<div class="alteqlabels" id="org74208de">
<ul class="org-ul">
<li>Gr (5.64)</li>
</ul>
@@ -1739,15 +1739,15 @@ Under this gauge choice, Ampère's law becomes
Again this is precisely like the Poisson equation, with one separate equation for each component.
For currents falling off sufficiently rapidly at infinity, we thus have
</p>
-<div class="core div" id="org5ec98ad">
-<div class="eqlabel" id="org0962fcb">
+<div class="core div" id="orga51c6d0">
+<div class="eqlabel" id="orgf1fa3d4">
<p>
<a id="A_CoulG"></a><a href="./ems_ms_vp_A.html#A_CoulG"><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="org8ff7eb6">
+<div class="alteqlabels" id="org35d9537">
<ul class="org-ul">
<li>Gr (5.65)</li>
</ul>
@@ -1766,15 +1766,15 @@ For currents falling off sufficiently rapidly at infinity, we thus have
<p>
For line and surface currents,
</p>
-<div class="main div" id="org602b0f8">
-<div class="eqlabel" id="orgf1afc13">
+<div class="main div" id="org6da036c">
+<div class="eqlabel" id="orge6299eb">
<p>
<a id="A_CoulG_l"></a><a href="./ems_ms_vp_A.html#A_CoulG_l"><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="org7f3875b">
+<div class="alteqlabels" id="orga10cfa7">
</div>
@@ -1785,14 +1785,14 @@ For line and surface currents,
\tag{A_CoulG_l}\label{A_CoulG_l}
\]
</p>
-<div class="eqlabel" id="org805a994">
+<div class="eqlabel" id="orgae969c5">
<p>
<a id="A_CoulG_s"></a><a href="./ems_ms_vp_A.html#A_CoulG_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="orgfe08271">
+<div class="alteqlabels" id="org3392659">
</div>
@@ -1807,7 +1807,7 @@ For line and surface currents,
</div>
-<div class="example div" id="org5085a2c">
+<div class="example div" id="org277d07b">
<p>
<b>Example: infinite solenoid</b>
</p>
@@ -1888,7 +1888,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms_vp_LC.html b/build/ems_ms_vp_LC.html
index c887431..5a37ebd 100644
--- a/build/ems_ms_vp_LC.html
+++ b/build/ems_ms_vp_LC.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1618,18 +1618,18 @@ Similarly to the electrostatic dipole,
the magnetic field of a dipole can be written in coordinate-free form.
For this: we define a useful object, the <br>
</p>
-<div class="info div" id="orgc596f77">
+<div class="info div" id="org05a2609">
<p>
<b>Levi-Civita symbol</b>
</p>
-<div class="eqlabel" id="orgfd892d9">
+<div class="eqlabel" id="orgce508dc">
<p>
<a id="LeviCivita"></a><a href="./ems_ms_vp_LC.html#LeviCivita"><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="orgf39dca9">
+<div class="alteqlabels" id="org0208d2c">
</div>
@@ -1652,14 +1652,14 @@ Using this: we can for example rewrite the cross product as
\]
Important identity:
</p>
-<div class="eqlabel" id="org02cff7d">
+<div class="eqlabel" id="org7163578">
<p>
<a id="sumLeviCivita"></a><a href="./ems_ms_vp_LC.html#sumLeviCivita"><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="orge6e67bb">
+<div class="alteqlabels" id="orgad94abf">
</div>
@@ -1698,14 +1698,14 @@ and \(\epsilon_{ijk} \epsilon_{klm} = \epsilon_{ijk} \epsilon_{lmk} = \delta_{il
<p>
Putting back the vector notation, and the \(m\) notation for the magnetic dipole (instead of \(M\)), we obtain
</p>
-<div class="eqlabel" id="org1c483db">
+<div class="eqlabel" id="org8d308be">
<p>
<a id="B_di"></a><a href="./ems_ms_vp_LC.html#B_di"><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="orgd1ea2aa">
+<div class="alteqlabels" id="org51c2391">
<ul class="org-ul">
<li>Gr (5.89)</li>
</ul>
@@ -1743,7 +1743,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms_vp_comp.html b/build/ems_ms_vp_comp.html
index a5cf32b..e0633df 100644
--- a/build/ems_ms_vp_comp.html
+++ b/build/ems_ms_vp_comp.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1683,7 +1683,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms_vp_mbc.html b/build/ems_ms_vp_mbc.html
index c0a33a4..fc1a741 100644
--- a/build/ems_ms_vp_mbc.html
+++ b/build/ems_ms_vp_mbc.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1649,15 +1649,15 @@ B^{\parallel}_{above} - B^{\parallel}_{below} = \mu_0 K
Thus, the component of \({\bf B}\) that is parallel to surface but perpendicular to the current flow
is discontinuous, whereas the one parallel to the flow is continuous. In vector notation, this becomes
</p>
-<div class="main div" id="org12096f9">
-<div class="eqlabel" id="org7bfa150">
+<div class="main div" id="org12d7a29">
+<div class="eqlabel" id="org788cf9f">
<p>
<a id="Bdisc"></a><a href="./ems_ms_vp_mbc.html#Bdisc"><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="orge2b7b81">
+<div class="alteqlabels" id="orgce8ac7d">
<ul class="org-ul">
<li>Gr (5.76)</li>
</ul>
@@ -1676,15 +1676,15 @@ is discontinuous, whereas the one parallel to the flow is continuous. In vector
<p>
where \(\hat{\bf n}\) points upwards. For the vector potential, the relations are
</p>
-<div class="main div" id="org705dee9">
-<div class="eqlabel" id="orgf58e297">
+<div class="main div" id="org98afe60">
+<div class="eqlabel" id="org1152f89">
<p>
<a id="Adisc"></a><a href="./ems_ms_vp_mbc.html#Adisc"><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="org5f2b272">
+<div class="alteqlabels" id="orgacd4a93">
<ul class="org-ul">
<li>Gr (5.77)</li>
</ul>
@@ -1737,15 +1737,15 @@ What is left is
\]
Therefore, reidentifying the normal component, we get
</p>
-<div class="main div" id="org42afa62">
-<div class="eqlabel" id="org076eaf5">
+<div class="main div" id="orgcfe3f01">
+<div class="eqlabel" id="org11b9194">
<p>
<a id="dAdisc"></a><a href="./ems_ms_vp_mbc.html#dAdisc"><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="orga61fbea">
+<div class="alteqlabels" id="org41f86e1">
<ul class="org-ul">
<li>Gr (5.78)</li>
</ul>
@@ -1782,7 +1782,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/ems_ms_vp_me.html b/build/ems_ms_vp_me.html
index af5ea4c..4c68607 100644
--- a/build/ems_ms_vp_me.html
+++ b/build/ems_ms_vp_me.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1629,14 +1629,14 @@ can thus be written
\]
Explicitly, the first few terms are (letting \(\hat{\bf r} \cdot \hat{\bf r}_s = \cos \theta\))
</p>
-<div class="eqlabel" id="org7ea630d">
+<div class="eqlabel" id="org5130ba2">
<p>
<a id="A_Leg"></a><a href="./ems_ms_vp_me.html#A_Leg"><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="orgf544c92">
+<div class="alteqlabels" id="org4bcf6bb">
<ul class="org-ul">
<li>Gr (5.80)</li>
</ul>
@@ -1663,14 +1663,14 @@ monopoles (also from Maxwell's equation <a href="./ems_ms_dcB_d.html#divB0">divB
<p>
The dominant term is thus the dipole,
</p>
-<div class="eqlabel" id="org1d49e4f">
+<div class="eqlabel" id="orgba02679">
<p>
<a id="A_di1"></a><a href="./ems_ms_vp_me.html#A_di1"><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="org0f2e926">
+<div class="alteqlabels" id="orga27ade1">
<ul class="org-ul">
<li>Gr (5.85)</li>
</ul>
@@ -1685,14 +1685,14 @@ The dominant term is thus the dipole,
\]
Using the identity (see Gr Problem 1.62),
</p>
-<div class="eqlabel" id="org62a7346">
+<div class="eqlabel" id="org3f6dca3">
<p>
<a id="intcr"></a><a href="./ems_ms_vp_me.html#intcr"><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="orga73669a">
+<div class="alteqlabels" id="orge0834a1">
<ul class="org-ul">
<li>Gr (1.108)</li>
</ul>
@@ -1709,7 +1709,7 @@ Using the identity (see Gr Problem 1.62),
with \({\bf c} = \hat {\bf r}\),
</p>
-<div class="info div" id="orgd9261d2">
+<div class="info div" id="org04c6a44">
<p>
<b>Parenthesis: derivation of</b> <a href="./ems_ms_vp_me.html#intcr">intcr</a>
</p>
@@ -1752,18 +1752,18 @@ Back to our problem:
\]
and defining the
</p>
-<div class="core div" id="org252bb34">
+<div class="core div" id="orgbe06c08">
<p>
<b>magnetic dipole moment</b>
</p>
-<div class="eqlabel" id="org52e910f">
+<div class="eqlabel" id="orgdd422db">
<p>
<a id="magdim"></a><a href="./ems_ms_vp_me.html#magdim"><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="org521ff8a">
+<div class="alteqlabels" id="org6cd583f">
<ul class="org-ul">
<li>Gr (5.86)</li>
</ul>
@@ -1782,18 +1782,18 @@ and defining the
<p>
we obtain the convenient expression for the
</p>
-<div class="main div" id="org919de64">
+<div class="main div" id="org066e08b">
<p>
<b>dipole term of the vector potential</b>
</p>
-<div class="eqlabel" id="orga93a621">
+<div class="eqlabel" id="org7c02396">
<p>
<a id="A_di"></a><a href="./ems_ms_vp_me.html#A_di"><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="orgc5bc8d7">
+<div class="alteqlabels" id="orgf4eddbd">
<ul class="org-ul">
<li>Gr (5.85)</li>
</ul>
@@ -1810,7 +1810,7 @@ we obtain the convenient expression for the
</div>
-<div class="example div" id="org6a86d5a">
+<div class="example div" id="org70776ed">
<p>
<b>Example: bookend shape</b>
</p>
@@ -1864,7 +1864,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm.html b/build/emsm.html
index d128eca..cfc8d2f 100644
--- a/build/emsm.html
+++ b/build/emsm.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1614,8 +1614,8 @@ Table of contents
</svg></a><span class="headline-id">emsm</span></h2>
<div class="outline-text-2" id="text-emsm">
-<details class="prereq" id="orgdf4df90">
-<summary id="org09e4816">
+<details class="prereq" id="org013ef1b">
+<summary id="orgdf86daf">
Prerequisites
</summary>
<ul class="org-ul">
@@ -1624,8 +1624,8 @@ Prerequisites
</ul>
</details>
-<details class="objectives" id="org82a8bfb">
-<summary id="orge50ecf8">
+<details class="objectives" id="org6f330fd">
+<summary id="org5ce87ab">
Objectives
</summary>
<ul class="org-ul">
@@ -1663,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_esm.html b/build/emsm_esm.html
index 8fa1ba8..f1f7963 100644
--- a/build/emsm_esm.html
+++ b/build/emsm_esm.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_esm_D.html b/build/emsm_esm_D.html
index f03b9fd..73c1bce 100644
--- a/build/emsm_esm_D.html
+++ b/build/emsm_esm_D.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1617,7 +1617,7 @@ Table of contents
<p>
Field due to polarization: effectively comes from bound surface and volume charges,
</p>
-<div class="main div" id="orgce7147d">
+<div class="main div" id="orgadd4b69">
<p>
\[
\rho_b = -{\boldsymbol \nabla} \cdot {\bf P}, \hspace{1cm} \sigma_b = {\bf P} \cdot \hat{\bf n}.
@@ -1646,18 +1646,18 @@ Convenient way of writing:
\]
Defining the
</p>
-<div class="core div" id="org72b7779">
+<div class="core div" id="org1bfbe0c">
<p>
<b>Electric displacement</b> \({\bf D}\)
</p>
-<div class="eqlabel" id="orgdd5a6d2">
+<div class="eqlabel" id="orgadf914a">
<p>
<a id="D"></a><a href="./emsm_esm_D.html#D"><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="orgc4d9540">
+<div class="alteqlabels" id="org289b58b">
<ul class="org-ul">
<li>Gr (4.21)</li>
</ul>
@@ -1676,15 +1676,15 @@ Defining the
<p>
Gauss's law becomes
</p>
-<div class="core div" id="org3a2f5a3">
-<div class="eqlabel" id="org7059860">
+<div class="core div" id="org3c3599d">
+<div class="eqlabel" id="org5565708">
<p>
<a id="GlD"></a><a href="./emsm_esm_D.html#GlD"><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="org3ac64ba">
+<div class="alteqlabels" id="org06803ca">
<ul class="org-ul">
<li>Gr (4.22)</li>
</ul>
@@ -1703,15 +1703,15 @@ Gauss's law becomes
<p>
or in integral form
</p>
-<div class="core div" id="orga6a5d6d">
-<div class="eqlabel" id="orgf14e923">
+<div class="core div" id="org81fd8e3">
+<div class="eqlabel" id="org6289c7f">
<p>
<a id="GlD_int"></a><a href="./emsm_esm_D.html#GlD_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="orgc23cc6a">
+<div class="alteqlabels" id="org0d6b00c">
<ul class="org-ul">
<li>Gr (4.23)</li>
</ul>
@@ -1728,7 +1728,7 @@ or in integral form
</div>
-<div class="example div" id="org367061a">
+<div class="example div" id="orge0e4a91">
<p>
<b>Example: long insulated wire</b>
</p>
@@ -1773,14 +1773,14 @@ Inside: can't know the electric field, since \({\bf P}\) isn't known.
but this is <i>not</i> correct, because the curl of the displacement isn't necessarily zero,
</p>
-<div class="eqlabel" id="orgc3d1f52">
+<div class="eqlabel" id="orgb86dac9">
<p>
<a id="curlDcurlP"></a><a href="./emsm_esm_D.html#curlDcurlP"><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="org6a08571">
+<div class="alteqlabels" id="org6510384">
<ul class="org-ul">
<li>Gr (4.25)</li>
</ul>
@@ -1820,7 +1820,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_esm_D_bc.html b/build/emsm_esm_D_bc.html
index cc1f401..ace3b86 100644
--- a/build/emsm_esm_D_bc.html
+++ b/build/emsm_esm_D_bc.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1616,14 +1616,14 @@ Table of contents
<p>
From <a href="./emsm_esm_D.html#GlD_int">GlD_int</a>, using the usual Gaussian pillbox logic:
</p>
-<div class="eqlabel" id="orgc43edb3">
+<div class="eqlabel" id="org8ff3eb0">
<p>
<a id="Dperp_disc"></a><a href="./emsm_esm_D_bc.html#Dperp_disc"><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="org9daf211">
+<div class="alteqlabels" id="org338781a">
<ul class="org-ul">
<li>Gr (4.26)</li>
</ul>
@@ -1638,15 +1638,15 @@ D^{\perp}_{above} - D^{\perp}_{below} = \sigma_f
\]
Then, <a href="./emsm_esm_D.html#curlDcurlP">curlDcurlP</a> gives
</p>
-<div class="main div" id="org2e69bea">
-<div class="eqlabel" id="org3f04d96">
+<div class="main div" id="org5bce686">
+<div class="eqlabel" id="org278337e">
<p>
<a id="Dpar_disc"></a><a href="./emsm_esm_D_bc.html#Dpar_disc"><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="orgacc1079">
+<div class="alteqlabels" id="orgeca80db">
<ul class="org-ul">
<li>Gr (4.27)</li>
</ul>
@@ -1681,7 +1681,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_esm_di.html b/build/emsm_esm_di.html
index ed8e86c..fc0b8ef 100644
--- a/build/emsm_esm_di.html
+++ b/build/emsm_esm_di.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1648,7 +1648,7 @@ This is common in crystals, leading to phenomena such as double refraction.
the dielectric are independent of the direction of \({\bf E}\) (isotropy). Liquids fall into
this category. We here have
</p>
-<div class="main div" id="org066a3be">
+<div class="main div" id="org31ece11">
<p>
\[
{\bf P} = \varepsilon_0 \chi_e {\bf E}
@@ -1677,7 +1677,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_esm_ld.html b/build/emsm_esm_ld.html
index d95cd15..d927b2f 100644
--- a/build/emsm_esm_ld.html
+++ b/build/emsm_esm_ld.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_esm_ld_bvp.html b/build/emsm_esm_ld_bvp.html
index a7321a5..4e43212 100644
--- a/build/emsm_esm_ld_bvp.html
+++ b/build/emsm_esm_ld_bvp.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1617,14 +1617,14 @@ Table of contents
In homogeneous linear dielectric, the bound charge density is directly
proportional to the free charge:
</p>
-<div class="eqlabel" id="orgb8ac21d">
+<div class="eqlabel" id="org5a60e41">
<p>
<a id="rhob_ld"></a><a href="./emsm_esm_ld_bvp.html#rhob_ld"><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="orgd134564">
+<div class="alteqlabels" id="orgf2e150e">
<ul class="org-ul">
<li>Gr (4.39)</li>
</ul>
@@ -1646,15 +1646,15 @@ and the potential then obeys the Laplace equation.
It is convenient to rewrite our boundary conditions in terms of free charge densities:
from <a href="./emsm_esm_D_bc.html#Dperp_disc">Dperp_disc</a>,
</p>
-<div class="main div" id="org92be963">
-<div class="eqlabel" id="org351f3b1">
+<div class="main div" id="orgf7ac2ac">
+<div class="eqlabel" id="org15ad919">
<p>
<a id="Edisc_f"></a><a href="./emsm_esm_ld_bvp.html#Edisc_f"><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="org327e7e2">
+<div class="alteqlabels" id="org2eadaf5">
<ul class="org-ul">
<li>Gr (4.40)</li>
</ul>
@@ -1673,15 +1673,15 @@ from <a href="./emsm_esm_D_bc.html#Dperp_disc">Dperp_disc</a>,
<p>
or in terms of the potential,
</p>
-<div class="main div" id="org50c6309">
-<div class="eqlabel" id="org003fd14">
+<div class="main div" id="org6b3545e">
+<div class="eqlabel" id="org2ef4740">
<p>
<a id="dpdisc_f"></a><a href="./emsm_esm_ld_bvp.html#dpdisc_f"><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="org64ea719">
+<div class="alteqlabels" id="orgd5ad929">
<ul class="org-ul">
<li>Gr (4.41)</li>
</ul>
@@ -1701,15 +1701,15 @@ or in terms of the potential,
<p>
The potential itself is continuous,
</p>
-<div class="main div" id="org2468d64">
-<div class="eqlabel" id="org8ac41e2">
+<div class="main div" id="org3ca662e">
+<div class="eqlabel" id="org833e744">
<p>
<a id="pcont"></a><a href="./emsm_esm_ld_bvp.html#pcont"><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="org07943aa">
+<div class="alteqlabels" id="org27d0122">
<ul class="org-ul">
<li>Gr (4.42)</li>
</ul>
@@ -1727,7 +1727,7 @@ The potential itself is continuous,
</div>
-<div class="example div" id="org7a347dd">
+<div class="example div" id="org34078f8">
<p>
<b>Example: sphere of homogeneous dielectric material in uniform electric field</b>
</p>
@@ -1794,7 +1794,7 @@ Thus,
</div>
-<div class="example div" id="org6f154b9">
+<div class="example div" id="orgbbf1e7e">
<p>
<b>Example: point charge above dielectric plane</b>
</p>
@@ -1876,7 +1876,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_esm_ld_e.html b/build/emsm_esm_ld_e.html
index 66403a7..0440168 100644
--- a/build/emsm_esm_ld_e.html
+++ b/build/emsm_esm_ld_e.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1659,15 +1659,15 @@ Special case of linear isotropic dielectric: \({\bf D} = \varepsilon {\bf E}\),
\]
Total work done:
</p>
-<div class="main div" id="orgecef8d0">
-<div class="eqlabel" id="org4ae32b1">
+<div class="main div" id="org80ef55b">
+<div class="eqlabel" id="org713a49e">
<p>
<a id="WintDE"></a><a href="./emsm_esm_ld_e.html#WintDE"><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="orgbb9bbc8">
+<div class="alteqlabels" id="org23ade8c">
<ul class="org-ul">
<li>Gr (4.58)</li>
</ul>
@@ -1704,7 +1704,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_esm_ld_f.html b/build/emsm_esm_ld_f.html
index d453856..ae60515 100644
--- a/build/emsm_esm_ld_f.html
+++ b/build/emsm_esm_ld_f.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1637,14 +1637,14 @@ F = -\frac{dW}{dx}
\]
Energy stored in capacitor:
</p>
-<div class="eqlabel" id="orgd9c66de">
+<div class="eqlabel" id="org8800210">
<p>
<a id="WcapV"></a><a href="./emsm_esm_ld_f.html#WcapV"><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="org9e432a9">
+<div class="alteqlabels" id="org812fe97">
<ul class="org-ul">
<li>Gr (4.61)</li>
</ul>
@@ -1667,14 +1667,14 @@ C = \frac{\varepsilon_0 w}{d} (\varepsilon_r l - \chi_e x)
where \(l\) is the length of the plates, and \(w\) is their width. Assume total charge \(Q\) on each
plate is held constant as \(x\) changes. In terms of \(Q\),
</p>
-<div class="eqlabel" id="org44a3c0e">
+<div class="eqlabel" id="orgb61bde4">
<p>
<a id="WcapQ"></a><a href="./emsm_esm_ld_f.html#WcapQ"><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="org3c8f1f3">
+<div class="alteqlabels" id="org312899c">
<ul class="org-ul">
<li>Gr (4.63)</li>
</ul>
@@ -1748,7 +1748,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_esm_ld_sp.html b/build/emsm_esm_ld_sp.html
index a4aecdc..1401dcd 100644
--- a/build/emsm_esm_ld_sp.html
+++ b/build/emsm_esm_ld_sp.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1616,15 +1616,15 @@ Table of contents
<p>
For many substances: polarization is proportional to field, at least if the latter isn't too strong:
</p>
-<div class="main div" id="orgee624c4">
-<div class="eqlabel" id="org4464ce0">
+<div class="main div" id="orgca50808">
+<div class="eqlabel" id="org6b1486e">
<p>
<a id="PchiE"></a><a href="./emsm_esm_ld_sp.html#PchiE"><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="orgb90581c">
+<div class="alteqlabels" id="org0fb5cad">
<ul class="org-ul">
<li>Gr (4.30)</li>
</ul>
@@ -1661,15 +1661,15 @@ In linear dielectrics:
\]
so
</p>
-<div class="main div" id="orge16cf8e">
-<div class="eqlabel" id="org38822cc">
+<div class="main div" id="orgac075e9">
+<div class="eqlabel" id="orge6ef08e">
<p>
<a id="DepsE"></a><a href="./emsm_esm_ld_sp.html#DepsE"><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="org37fa3a7">
+<div class="alteqlabels" id="orgd3687f6">
<ul class="org-ul">
<li>Gr (4.32)</li>
</ul>
@@ -1690,14 +1690,14 @@ where \(\varepsilon\) is called the <b>permittivity</b> of the material.
In vacuum, susceptibility is zero,
permittivity is \(\varepsilon_0\). Also,
</p>
-<div class="eqlabel" id="org08b41c7">
+<div class="eqlabel" id="org7b81d68">
<p>
<a id="epsr"></a><a href="./emsm_esm_ld_sp.html#epsr"><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="org843db03">
+<div class="alteqlabels" id="orgd11960a">
<ul class="org-ul">
<li>Gr (4.34)</li>
</ul>
@@ -1716,7 +1716,7 @@ This is all just nomenclature, everything is already in <a href="./emsm_esm_ld_s
-<div class="example div" id="orgdf2a597">
+<div class="example div" id="org94c117c">
<p>
<b>Example: metal sphere with insulator shell</b>
</p>
@@ -1790,7 +1790,7 @@ entirely filled with a homogeneous linear dielectric, but then this is just like
a vacuum with a different permittivity.
</p>
-<div class="example div" id="orgec6cfdf">
+<div class="example div" id="org9b0a9fc">
<p>
<b>Example: parallel-plate capacitor filled with dielectric material</b>
</p>
@@ -1839,7 +1839,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_esm_mE.html b/build/emsm_esm_mE.html
index 4572fc2..6e34083 100644
--- a/build/emsm_esm_mE.html
+++ b/build/emsm_esm_mE.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_esm_mE_P.html b/build/emsm_esm_mE_P.html
index b3bfa93..de0c9d1 100644
--- a/build/emsm_esm_mE_P.html
+++ b/build/emsm_esm_mE_P.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_esm_mE_o.html b/build/emsm_esm_mE_o.html
index a158c41..144c193 100644
--- a/build/emsm_esm_mE_o.html
+++ b/build/emsm_esm_mE_o.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1695,18 +1695,18 @@ dipole moment pointing from \(O^-\) along bisector.
<p>
Torque on dipole: if field is uniform, overall force on dipole cancels, but torque remains:
</p>
-<div class="main div" id="org747795f">
+<div class="main div" id="orgfb793ed">
<p>
<b>Torque on a dipole</b>
</p>
-<div class="eqlabel" id="org84c3434">
+<div class="eqlabel" id="org697cf52">
<p>
<a id="di_N"></a><a href="./emsm_esm_mE_o.html#di_N"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org58588b8">
+<div class="alteqlabels" id="org4d4be74">
<ul class="org-ul">
<li>Gr (4.4)</li>
</ul>
@@ -1729,15 +1729,15 @@ If field is non-uniform,
\]
so
</p>
-<div class="main div" id="org389568d">
-<div class="eqlabel" id="orge247238">
+<div class="main div" id="org035b61e">
+<div class="eqlabel" id="org4d19e6c">
<p>
<a id="di_F"></a><a href="./emsm_esm_mE_o.html#di_F"><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="org4048910">
+<div class="alteqlabels" id="org2c810a1">
<ul class="org-ul">
<li>Gr (4.5)</li>
</ul>
@@ -1756,14 +1756,14 @@ so
<p>
The energy of a dipole in an electric field can be shown to be
</p>
-<div class="eqlabel" id="org772b7d0">
+<div class="eqlabel" id="org17f8e9c">
<p>
<a id="di_E"></a><a href="./emsm_esm_mE_o.html#di_E"><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="org5280ad4">
+<div class="alteqlabels" id="org79b39b0">
<ul class="org-ul">
<li>Gr (4.6)</li>
</ul>
@@ -1816,7 +1816,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_esm_po.html b/build/emsm_esm_po.html
index 30c2300..2a6c0c3 100644
--- a/build/emsm_esm_po.html
+++ b/build/emsm_esm_po.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1624,14 +1624,14 @@ For a single dipole, we have <a href="./ems_ca_me_md.html#p_di">p_di</a>
\]
Considering a dipole moment per unit volume \({\bf P}\) (our definition of the polarization), we get
</p>
-<div class="eqlabel" id="org948739c">
+<div class="eqlabel" id="org0258040">
<p>
<a id="p_P"></a><a href="./emsm_esm_po.html#p_P"><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="org8984a1c">
+<div class="alteqlabels" id="orgf8823e1">
<ul class="org-ul">
<li>Gr (4.9)</li>
</ul>
@@ -1666,15 +1666,15 @@ Using the divergence theorem, this becomes
<p>
Interpretation: first terms is like contribution of a surface charge,
</p>
-<div class="main div" id="org3d4f6b0">
-<div class="eqlabel" id="org16ac6ae">
+<div class="main div" id="org0d67ad6">
+<div class="eqlabel" id="org6822ee4">
<p>
<a id="sigmab"></a><a href="./emsm_esm_po.html#sigmab"><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="orgd583273">
+<div class="alteqlabels" id="orgf05f86c">
<ul class="org-ul">
<li>Gr (4.11)</li>
</ul>
@@ -1693,15 +1693,15 @@ Interpretation: first terms is like contribution of a surface charge,
<p>
and second term looks like contribution of a volume charge,
</p>
-<div class="main div" id="orgd219e12">
-<div class="eqlabel" id="org8236389">
+<div class="main div" id="org6d35d8a">
+<div class="eqlabel" id="org869cfc2">
<p>
<a id="rhob"></a><a href="./emsm_esm_po.html#rhob"><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="org3ad58e2">
+<div class="alteqlabels" id="org0b847eb">
<ul class="org-ul">
<li>Gr (4.12)</li>
</ul>
@@ -1720,15 +1720,15 @@ and second term looks like contribution of a volume charge,
<p>
Using these definitions,
</p>
-<div class="main div" id="org496287f">
-<div class="eqlabel" id="orga8a4e9a">
+<div class="main div" id="org8829df8">
+<div class="eqlabel" id="orgd8388a5">
<p>
<a id="p_bound"></a><a href="./emsm_esm_po.html#p_bound"><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="orgfa407ba">
+<div class="alteqlabels" id="org2103784">
<ul class="org-ul">
<li>Gr (4.13)</li>
</ul>
@@ -1749,7 +1749,7 @@ These <b>bound charges</b> faithfully represent the object's sources for electri
</p>
-<div class="example div" id="org5b2586b">
+<div class="example div" id="orga521891">
<p>
<b>Example: uniformly polarized sphere</b>
</p>
@@ -1827,7 +1827,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_esm_po_fid.html b/build/emsm_esm_po_fid.html
index c63d964..52bc5bd 100644
--- a/build/emsm_esm_po_fid.html
+++ b/build/emsm_esm_po_fid.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1638,14 +1638,14 @@ Therefore, \({\bf E}_{out}\) is the field at \({\bf r}\)
from dipoles exterior to the sphere, for which we can use <a href="./emsm_esm_po.html#p_P">p_P</a>:
</p>
-<div class="eqlabel" id="org6e655ec">
+<div class="eqlabel" id="orgd0b898b">
<p>
<a id="p_out"></a><a href="./emsm_esm_po_fid.html#p_out"><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="orgdbffbd1">
+<div class="alteqlabels" id="orgb11243d">
<ul class="org-ul">
<li>Gr (4.17)</li>
</ul>
@@ -1663,14 +1663,14 @@ The dipoles inside the sphere however cannot be treated in this fashion.
However, their average field is all we need, and
this is given by (see Example below for the derivation)
</p>
-<div class="eqlabel" id="orge81d9c5">
+<div class="eqlabel" id="orgfbeede1">
<p>
<a id="Eavg_p"></a><a href="./emsm_esm_po_fid.html#Eavg_p"><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="org8c41e6e">
+<div class="alteqlabels" id="orge636edb">
<ul class="org-ul">
<li>Gr (3.105)</li>
</ul>
@@ -1686,7 +1686,7 @@ this is given by (see Example below for the derivation)
irrespective of charge distribution within sphere.
</p>
-<div class="example div" id="org5bbc6be">
+<div class="example div" id="org662d9cb">
<p>
<b>Average field inside a sphere</b>
</p>
@@ -1823,7 +1823,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_esm_po_pibc.html b/build/emsm_esm_po_pibc.html
index 40dde5c..d791222 100644
--- a/build/emsm_esm_po_pibc.html
+++ b/build/emsm_esm_po_pibc.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1656,7 +1656,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_msm.html b/build/emsm_msm.html
index 79b74d3..c5b8e0b 100644
--- a/build/emsm_msm.html
+++ b/build/emsm_msm.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1614,8 +1614,8 @@ Table of contents
</svg></a><span class="headline-id">emsm.msm</span></h3>
<div class="outline-text-3" id="text-emsm_msm">
-<details class="prereq" id="org3af6e97">
-<summary id="org10dea3b">
+<details class="prereq" id="org0cc6ee8">
+<summary id="org869f211">
Prerequisites
</summary>
<ul class="org-ul">
@@ -1624,8 +1624,8 @@ Prerequisites
</ul>
</details>
-<details class="objectives" id="orgb6f2be0">
-<summary id="org3077e11">
+<details class="objectives" id="org2f9a388">
+<summary id="org66eb2cc">
Objectives
</summary>
<ul class="org-ul">
@@ -1664,7 +1664,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_msm_H.html b/build/emsm_msm_H.html
index d31326b..31c4785 100644
--- a/build/emsm_msm_H.html
+++ b/build/emsm_msm_H.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_msm_H_A.html b/build/emsm_msm_H_A.html
index ffa87c3..9697695 100644
--- a/build/emsm_msm_H_A.html
+++ b/build/emsm_msm_H_A.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1617,14 +1617,14 @@ Table of contents
Nomenclature: as in electric case, we have bound currents, and everything else, which we call the <b>free current</b>.
Total current:
</p>
-<div class="eqlabel" id="orge7be78f">
+<div class="eqlabel" id="org2f67ed9">
<p>
<a id="JbJf"></a><a href="./emsm_msm_H_A.html#JbJf"><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="org9ebfec7">
+<div class="alteqlabels" id="org8bc9595">
<ul class="org-ul">
<li>Gr (6.17)</li>
</ul>
@@ -1644,15 +1644,15 @@ Ampère's law:
\]
so we can define
</p>
-<div class="core div" id="org41742e5">
-<div class="eqlabel" id="org225e925">
+<div class="core div" id="org1215d01">
+<div class="eqlabel" id="orgd5d9468">
<p>
<a id="HBM"></a><a href="./emsm_msm_H_A.html#HBM"><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="org3480310">
+<div class="alteqlabels" id="org9ba5a4d">
<ul class="org-ul">
<li>Gr (6.18)</li>
</ul>
@@ -1671,15 +1671,15 @@ so we can define
<p>
and rewrite Ampère's law as
</p>
-<div class="core div" id="orgee0a44d">
-<div class="eqlabel" id="org54efeb1">
+<div class="core div" id="org18518e4">
+<div class="eqlabel" id="orga0d403e">
<p>
<a id="curlHJf"></a><a href="./emsm_msm_H_A.html#curlHJf"><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="org5731b5f">
+<div class="alteqlabels" id="orgec3bba4">
<ul class="org-ul">
<li>Gr (6.19)</li>
</ul>
@@ -1698,15 +1698,15 @@ and rewrite Ampère's law as
<p>
or in integral form,
</p>
-<div class="core div" id="orgf8bb637">
-<div class="eqlabel" id="orgc93a7f0">
+<div class="core div" id="org288c4e9">
+<div class="eqlabel" id="orgccd63b9">
<p>
<a id="intHdl"></a><a href="./emsm_msm_H_A.html#intHdl"><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="org32a85da">
+<div class="alteqlabels" id="org9025220">
<ul class="org-ul">
<li>Gr (6.20)</li>
</ul>
@@ -1727,7 +1727,7 @@ or in integral form,
rewrite Ampère's law in terms of free currents alone. Bound currents come along for the ride.
</p>
-<div class="example div" id="orgf5dc4f3">
+<div class="example div" id="org2556e80">
<p>
<b>Example: copper rod</b>
</p>
@@ -1746,14 +1746,14 @@ Bound currents antiparallel to \(I\) in bulk and parallel at surface. All curre
so \({\bf B}, {\bf M}, {\bf H}\) are circumferential. Apply integral form of Ampère's law with
radius \(s < R\): \(H (2\pi s) = I_{f_{enc}} = I \frac{\pi s^2}{\pi R^2}\) so
</p>
-<div class="eqlabel" id="orgecb73e9">
+<div class="eqlabel" id="orga2d5f0f">
<p>
<a id="Hrod_in"></a><a href="./emsm_msm_H_A.html#Hrod_in"><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="org55b6cb9">
+<div class="alteqlabels" id="org6c30162">
<ul class="org-ul">
<li>Gr (6.21)</li>
</ul>
@@ -1768,14 +1768,14 @@ radius \(s < R\): \(H (2\pi s) = I_{f_{enc}} = I \frac{\pi s^2}{\pi R^2}\) s
\]
Outside,
</p>
-<div class="eqlabel" id="org341a0f3">
+<div class="eqlabel" id="org3469141">
<p>
<a id="Hrod_out"></a><a href="./emsm_msm_H_A.html#Hrod_out"><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="org5c52141">
+<div class="alteqlabels" id="org617ce83">
<ul class="org-ul">
<li>Gr (6.22)</li>
</ul>
@@ -1801,14 +1801,14 @@ There, \({\bf M} = 0\) so \({\bf B} = \mu_0 {\bf H}\).
Similarly to electric case: cannot assume that \({\bf H}\) is like \({\bf B}\).
\({\bf H}\) might have a divergence,
</p>
-<div class="eqlabel" id="org0cc843b">
+<div class="eqlabel" id="org9d34023">
<p>
<a id="divH"></a><a href="./emsm_msm_H_A.html#divH"><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="org8f63b28">
+<div class="alteqlabels" id="orgd156a25">
<ul class="org-ul">
<li>Gr (6.23)</li>
</ul>
@@ -1869,14 +1869,14 @@ W_{mag} = \int_{all~space} d\tau \frac{1}{2} {\bf H} \cdot {\bf B}
<p>
Can rewrite BCs in terms of \({\bf H}\): from <a href="./emsm_msm_H_A.html#divH">divH</a>,
</p>
-<div class="eqlabel" id="org569f440">
+<div class="eqlabel" id="orge31639d">
<p>
<a id="HdiscM"></a><a href="./emsm_msm_H_A.html#HdiscM"><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="orgd4b0dc7">
+<div class="alteqlabels" id="orga7ad559">
<ul class="org-ul">
<li>Gr (6.24)</li>
</ul>
@@ -1891,14 +1891,14 @@ H^{\perp}_{above} - H^{\perp}_{below} = -(M^{\perp}_{above} - M^{\perp}_{below})
<p>
while <a href="./emsm_msm_H_A.html#curlHJf">curlHJf</a> gives
</p>
-<div class="eqlabel" id="org60d2a08">
+<div class="eqlabel" id="org8258597">
<p>
<a id="Hdisc"></a><a href="./emsm_msm_H_A.html#Hdisc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org3f93727">
+<div class="alteqlabels" id="org086a72c">
<ul class="org-ul">
<li>Gr (6.25)</li>
</ul>
@@ -1943,7 +1943,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_msm_a.html b/build/emsm_msm_a.html
index cfbcc5a..4b3c4b0 100644
--- a/build/emsm_msm_a.html
+++ b/build/emsm_msm_a.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1680,7 +1680,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_msm_fmo.html b/build/emsm_msm_fmo.html
index a8e6d30..ce931a3 100644
--- a/build/emsm_msm_fmo.html
+++ b/build/emsm_msm_fmo.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_msm_fmo_bc.html b/build/emsm_msm_fmo_bc.html
index b1324ef..604cb85 100644
--- a/build/emsm_msm_fmo_bc.html
+++ b/build/emsm_msm_fmo_bc.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1624,15 +1624,15 @@ For a single dipole: refer to <a href="./ems_ms_vp_me.html#A_di">A_di</a> (vect
For a chunk of material with local magnetization \({\bf M} ({\bf r})\),
by the principle of superposition we thus have:
</p>
-<div class="main div" id="orga0403e6">
-<div class="eqlabel" id="org682bc5c">
+<div class="main div" id="org2949db0">
+<div class="eqlabel" id="org8ef6e18">
<p>
<a id="A_M"></a><a href="./emsm_msm_fmo_bc.html#A_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="orgdb5a685">
+<div class="alteqlabels" id="orgc1360ca">
<ul class="org-ul">
<li>Gr (6.11)</li>
</ul>
@@ -1688,15 +1688,15 @@ leads to
<p>
Reinterpretation: first term: potential from volume current,
</p>
-<div class="main div" id="org5339db7">
-<div class="eqlabel" id="org0ef43f0">
+<div class="main div" id="org4073ff1">
+<div class="eqlabel" id="orgcc588cc">
<p>
<a id="JbcurlM"></a><a href="./emsm_msm_fmo_bc.html#JbcurlM"><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="org15b1a59">
+<div class="alteqlabels" id="org428f7f2">
<ul class="org-ul">
<li>Gr (6.13)</li>
</ul>
@@ -1715,15 +1715,15 @@ Reinterpretation: first term: potential from volume current,
<p>
second term: potential from surface current,
</p>
-<div class="main div" id="org93bb63f">
-<div class="eqlabel" id="org218fb15">
+<div class="main div" id="org23bdcca">
+<div class="eqlabel" id="orgfa9f958">
<p>
<a id="KbM"></a><a href="./emsm_msm_fmo_bc.html#KbM"><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="org753350b">
+<div class="alteqlabels" id="org5e854f1">
<ul class="org-ul">
<li>Gr (6.14)</li>
</ul>
@@ -1742,7 +1742,7 @@ second term: potential from surface current,
<p>
With these definitions,
</p>
-<div class="main div" id="org9816118">
+<div class="main div" id="org28ad325">
\begin{equation}
{\bf A} ({\bf r}) = \frac{\mu_0}{4\pi} \int_{\cal V} d\tau' \frac{{\bf J}_b ({\bf r}')}{|{\bf r} - {\bf r}'|}
+ \frac{\mu_0}{4\pi} \oint_{\cal S} da' \frac{{\bf K}_b ({\bf r}')}{|{\bf r} - {\bf r}'|}
@@ -1775,7 +1775,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_msm_fmo_fim.html b/build/emsm_msm_fmo_fim.html
index 08d0442..9005b95 100644
--- a/build/emsm_msm_fmo_fim.html
+++ b/build/emsm_msm_fmo_fim.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1635,7 +1635,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_msm_fmo_pibc.html b/build/emsm_msm_fmo_pibc.html
index 4b48c9b..778363c 100644
--- a/build/emsm_msm_fmo_pibc.html
+++ b/build/emsm_msm_fmo_pibc.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_msm_lnlm.html b/build/emsm_msm_lnlm.html
index f421281..6e84863 100644
--- a/build/emsm_msm_lnlm.html
+++ b/build/emsm_msm_lnlm.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_msm_lnlm_fm.html b/build/emsm_msm_lnlm_fm.html
index 7665c54..3a09717 100644
--- a/build/emsm_msm_lnlm_fm.html
+++ b/build/emsm_msm_lnlm_fm.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1636,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_msm_lnlm_sp.html b/build/emsm_msm_lnlm_sp.html
index 29c3ec5..6b42dfd 100644
--- a/build/emsm_msm_lnlm_sp.html
+++ b/build/emsm_msm_lnlm_sp.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1618,15 +1618,15 @@ In para/diamagnets: when \({\bf B}\) is removed, \({\bf M}\) disappears. For n
fields, proportionality. Custom (slightly different than for dielectrics):
<b>magnetic susceptibility</b> \(\chi_m\) defined as
</p>
-<div class="core div" id="org690f8cb">
-<div class="eqlabel" id="org530a5bc">
+<div class="core div" id="org93a7b9d">
+<div class="eqlabel" id="orge26d08e">
<p>
<a id="chim"></a><a href="./emsm_msm_lnlm_sp.html#chim"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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</svg></a>
</p>
-<div class="alteqlabels" id="org9587edf">
+<div class="alteqlabels" id="org1848ac6">
<ul class="org-ul">
<li>Gr (6.29)</li>
</ul>
@@ -1646,14 +1646,14 @@ fields, proportionality. Custom (slightly different than for dielectrics):
(and <b>not</b> \({\bf M} = \frac{1}{\mu_0} \chi_m {\bf B}\) had the electrostatics parallel
been followed historically). Materials that obey this are called <i>linear media</i>. Then,
</p>
-<div class="eqlabel" id="org92427e8">
+<div class="eqlabel" id="org7b5932a">
<p>
<a id="BHM"></a><a href="./emsm_msm_lnlm_sp.html#BHM"><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="org93c272c">
+<div class="alteqlabels" id="org8cddbc7">
<ul class="org-ul">
<li>Gr (6.30)</li>
</ul>
@@ -1668,14 +1668,14 @@ been followed historically). Materials that obey this are called <i>linear medi
\]
and thus
</p>
-<div class="eqlabel" id="org75fad04">
+<div class="eqlabel" id="org5774258">
<p>
<a id="BH"></a><a href="./emsm_msm_lnlm_sp.html#BH"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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</svg></a>
</p>
-<div class="alteqlabels" id="orgff9b371">
+<div class="alteqlabels" id="org95fbc74">
<ul class="org-ul">
<li>Gr (6.31)</li>
</ul>
@@ -1700,14 +1700,14 @@ have nonzero divergence.
<p>
In linear medium: bound volume current proportional to free current,
</p>
-<div class="eqlabel" id="org6b477dd">
+<div class="eqlabel" id="orgff08ab2">
<p>
<a id="JbchimJf"></a><a href="./emsm_msm_lnlm_sp.html#JbchimJf"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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</svg></a>
</p>
-<div class="alteqlabels" id="orge73774d">
+<div class="alteqlabels" id="org560487f">
<ul class="org-ul">
<li>Gr (6.33)</li>
</ul>
@@ -1742,7 +1742,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_msm_m.html b/build/emsm_msm_m.html
index 4d39276..fd8f721 100644
--- a/build/emsm_msm_m.html
+++ b/build/emsm_msm_m.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_msm_m_dpf.html b/build/emsm_msm_m_dpf.html
index 0acd0de..ff3ecd2 100644
--- a/build/emsm_msm_m_dpf.html
+++ b/build/emsm_msm_m_dpf.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1624,9 +1624,9 @@ Table of contents
</p>
</div>
-<div id="outline-container-orgc8b9a5a" class="outline-6">
-<h6 id="orgc8b9a5a"><a href="#orgc8b9a5a">Why is Ferromagnetism such an intriguing phenomenon?</a></h6>
-<div class="outline-text-6" id="text-orgc8b9a5a">
+<div id="outline-container-org169405c" class="outline-6">
+<h6 id="org169405c"><a href="#org169405c">Why is Ferromagnetism such an intriguing phenomenon?</a></h6>
+<div class="outline-text-6" id="text-org169405c">
<p>
[in construction]
</p>
@@ -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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/emsm_msm_m_fdi.html b/build/emsm_msm_m_fdi.html
index 0687f91..44fcb3c 100644
--- a/build/emsm_msm_m_fdi.html
+++ b/build/emsm_msm_m_fdi.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1616,15 +1616,15 @@ Table of contents
<p>
Torque on magnetic dipole:
</p>
-<div class="core div" id="org3b1d61f">
-<div class="eqlabel" id="org81fee68">
+<div class="core div" id="orgbecae08">
+<div class="eqlabel" id="org86842e9">
<p>
<a id="NmB"></a><a href="./emsm_msm_m_fdi.html#NmB"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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</svg></a>
</p>
-<div class="alteqlabels" id="orgdb1db96">
+<div class="alteqlabels" id="org9fae006">
<ul class="org-ul">
<li>Gr (6.1)</li>
</ul>
@@ -1649,15 +1649,15 @@ Similar to electric analogue, <a href="./emsm_esm_mE_o.html#di_N">di_N</a>, \({\
<p>
In a non-uniform field:
</p>
-<div class="main div" id="orga621956">
-<div class="eqlabel" id="org3ffe3ca">
+<div class="main div" id="orgf91bfde">
+<div class="eqlabel" id="orgcba6a69">
<p>
<a id="FgradmB"></a><a href="./emsm_msm_m_fdi.html#FgradmB"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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</svg></a>
</p>
-<div class="alteqlabels" id="orgf09869c">
+<div class="alteqlabels" id="org2c23a3b">
<ul class="org-ul">
<li>Gr (6.3)</li>
</ul>
@@ -1693,7 +1693,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/in.html b/build/in.html
index 3e76237..a6a177d 100644
--- a/build/in.html
+++ b/build/in.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/in_p.html b/build/in_p.html
index 2ddeb1a..22e7798 100644
--- a/build/in_p.html
+++ b/build/in_p.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1661,7 +1661,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/in_t.html b/build/in_t.html
index bbde4f6..14e3e36 100644
--- a/build/in_t.html
+++ b/build/in_t.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/in_t_c.html b/build/in_t_c.html
index fc4a705..41053c5 100644
--- a/build/in_t_c.html
+++ b/build/in_t_c.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1618,8 +1618,8 @@ Throughout these lecture notes, boxes with contextual colors are used,
serving specific purposes. These are:
</p>
-<details class="prereq" id="org9bc1cec">
-<summary id="org204db54">
+<details class="prereq" id="orgcd04899">
+<summary id="orgdb0c3c0">
Prerequisites
</summary>
<p>
@@ -1632,8 +1632,8 @@ Specific prerequisites for understanding the upcoming material
</ul>
</details>
-<details class="objectives" id="org1dca112">
-<summary id="org4785e3c">
+<details class="objectives" id="org1a05669">
+<summary id="org7569856">
Objectives
</summary>
<p>
@@ -1646,8 +1646,8 @@ Objectives for this part: what you should learn by reading this
</ul>
</details>
-<details class="core" id="org53de9ed">
-<summary id="org661ea9f">
+<details class="core" id="orgc21a410">
+<summary id="org28ec08c">
Core
</summary>
<p>
@@ -1655,8 +1655,8 @@ Core material: you have to know this by heart, and how to use it.
</p>
</details>
-<details class="main" id="org63b8804">
-<summary id="orgefaee06">
+<details class="main" id="org365e779">
+<summary id="org7113f4e">
Main
</summary>
<p>
@@ -1664,8 +1664,8 @@ Main matter which you should know how to use.
</p>
</details>
-<details class="example" id="orgc81ee4f">
-<summary id="orgd38f23f">
+<details class="example" id="org8ce185f">
+<summary id="orgd184e8d">
Example
</summary>
<p>
@@ -1673,8 +1673,8 @@ Example of the concepts just covered.
</p>
</details>
-<details class="info" id="orgbc275a7">
-<summary id="orgce74c54">
+<details class="info" id="orgdecff2e">
+<summary id="orgd30b12a">
Info
</summary>
<p>
@@ -1682,8 +1682,8 @@ Additional (contextual) information.
</p>
</details>
-<details class="context" id="org23d498e">
-<summary id="orgbd54818">
+<details class="context" id="org6b4612a">
+<summary id="org42a56ce">
Historical Context
</summary>
<p>
@@ -1709,7 +1709,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/in_t_l.html b/build/in_t_l.html
index ad19a5a..1f66f80 100644
--- a/build/in_t_l.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/index.html b/build/index.html
index 8950ce3..8567ba5 100644
--- a/build/index.html
+++ b/build/index.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -2091,7 +2091,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/qed.html b/build/qed.html
index 3771f87..2643c3d 100644
--- a/build/qed.html
+++ b/build/qed.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/qed_t.html b/build/qed_t.html
index b0d9469..bfd295c 100644
--- a/build/qed_t.html
+++ b/build/qed_t.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1632,7 +1632,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/red.html b/build/red.html
index 90680f4..56d39f8 100644
--- a/build/red.html
+++ b/build/red.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1614,8 +1614,8 @@ Table of contents
</svg></a><span class="headline-id">red</span></h2>
<div class="outline-text-2" id="text-red">
-<details class="prereq" id="org0b1770f">
-<summary id="org33bef36">
+<details class="prereq" id="org4920c42">
+<summary id="org6b1c40c">
Prerequisites
</summary>
<ul class="org-ul">
@@ -1623,8 +1623,8 @@ Prerequisites
</ul>
</details>
-<details class="objectives" id="orga9a7e27">
-<summary id="org4ea017f">
+<details class="objectives" id="orgaf026c5">
+<summary id="org4ca34fc">
Objectives
</summary>
<ul class="org-ul">
@@ -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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/red_rem.html b/build/red_rem.html
index 326f459..28cb851 100644
--- a/build/red_rem.html
+++ b/build/red_rem.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/red_rem_Fmunu.html b/build/red_rem_Fmunu.html
index 319cf0d..46dc070 100644
--- a/build/red_rem_Fmunu.html
+++ b/build/red_rem_Fmunu.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1683,18 +1683,18 @@ how the nonvanishing elements of an antisymmetric tensor transform:
<p>
Comparing with the transformation rules for the electromagnetic field which we obtained in <a href="./red_rem_Ltf.html#EMtr">EMtr</a>, we can define the
</p>
-<div class="core div" id="org5361587">
+<div class="core div" id="org117de8f">
<p>
<b>Electromagnetic Field Tensor</b>
</p>
-<div class="eqlabel" id="orgdddfe8b">
+<div class="eqlabel" id="org358ea94">
<p>
<a id="Fmunu"></a><a href="./red_rem_Fmunu.html#Fmunu"><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="org9e15d6e">
+<div class="alteqlabels" id="org597a4d9">
</div>
@@ -1714,18 +1714,18 @@ F^{\mu \nu} = \left( \begin{array}{cccc}
<p>
together with the handy
</p>
-<div class="main div" id="org53973f7">
+<div class="main div" id="orgc56f47d">
<p>
<b>Dual Field Tensor</b>
</p>
-<div class="eqlabel" id="orgae9a2b2">
+<div class="eqlabel" id="orgda1a353">
<p>
<a id="Gmunu"></a><a href="./red_rem_Fmunu.html#Gmunu"><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="org9c07047">
+<div class="alteqlabels" id="org31f1240">
</div>
@@ -1753,18 +1753,18 @@ obtained from the field tensor by the substitution
<p>
Our electromagnetic field transformation laws then become the simple
</p>
-<div class="core div" id="org840afc2">
+<div class="core div" id="org0e51dab">
<p>
<b>Lorentz Transformation Rules for EM Fields</b>
</p>
-<div class="eqlabel" id="orga8672c3">
+<div class="eqlabel" id="org41f872c">
<p>
<a id="LorF"></a><a href="./red_rem_Fmunu.html#LorF"><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="org49ca23d">
+<div class="alteqlabels" id="orgbfcae60">
</div>
@@ -1798,7 +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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/red_rem_Ltf.html b/build/red_rem_Ltf.html
index 4107cba..9fab40f 100644
--- a/build/red_rem_Ltf.html
+++ b/build/red_rem_Ltf.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1736,18 +1736,18 @@ These factors cancel so \(\bar{B}_x = B_x\).
<p>
We thus obtain the
</p>
-<div class="core div" id="org24ecb59">
+<div class="core div" id="orgf512c1f">
<p>
<b>EM field transformation laws</b> <i>(motion along \(x\) with velocity \(v\))</i>
</p>
-<div class="eqlabel" id="orgd271f80">
+<div class="eqlabel" id="org93bd238">
<p>
<a id="EMtr"></a><a href="./red_rem_Ltf.html#EMtr"><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="orgec94183">
+<div class="alteqlabels" id="orgf5395d1">
</div>
@@ -1805,7 +1805,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/red_rem_Me.html b/build/red_rem_Me.html
index 1f0d8a7..5b0096f 100644
--- a/build/red_rem_Me.html
+++ b/build/red_rem_Me.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1637,18 +1637,18 @@ We thus obtain
\]
Recognizing the proper velocity, we thus get that charge density and current density can together form the
</p>
-<div class="core div" id="org94723ec">
+<div class="core div" id="org4a23d61">
<p>
<b>Current density 4-vector</b>
</p>
-<div class="eqlabel" id="org479bb40">
+<div class="eqlabel" id="org8494a4a">
<p>
<a id="Jmu"></a><a href="./red_rem_Me.html#Jmu"><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="org324820b">
+<div class="alteqlabels" id="org63c21fa">
</div>
@@ -1665,18 +1665,18 @@ J^\mu = \left( c\rho, J_x, J_y, J_z \right)
<p>
The continuity equation <a href="./ems_ms_ce.html#conteq">conteq</a> takes the simple form
</p>
-<div class="core div" id="org5885f61">
+<div class="core div" id="org2a3444b">
<p>
<b>Continuity equation</b>
</p>
-<div class="eqlabel" id="org52b2d08">
+<div class="eqlabel" id="org2db5045">
<p>
<a id="conteq_rel"></a><a href="./red_rem_Me.html#conteq_rel"><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="org8386e2d">
+<div class="alteqlabels" id="org830511b">
</div>
@@ -1692,18 +1692,18 @@ The continuity equation <a href="./ems_ms_ce.html#conteq">conteq</a> takes the s
<p>
while similarly the notation simplifies for
</p>
-<div class="main div" id="orgdb33787">
+<div class="main div" id="org0816f0a">
<p>
<b>Maxwell's equations</b>
</p>
-<div class="eqlabel" id="org26adf05">
+<div class="eqlabel" id="org3cc1fa8">
<p>
<a id="Max_rel"></a><a href="./red_rem_Me.html#Max_rel"><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="orge69d726">
+<div class="alteqlabels" id="org77d9519">
</div>
@@ -1722,18 +1722,18 @@ while similarly the notation simplifies for
<p>
In terms of \(F^{\mu \nu}\) and the proper velocity \(\eta^\mu\), we also have the
</p>
-<div class="main div" id="org1aadbdc">
+<div class="main div" id="orge8ec801">
<p>
<b>Minkowski force on a charge</b> \(q\)
</p>
-<div class="eqlabel" id="org8ed30ad">
+<div class="eqlabel" id="org467f39b">
<p>
<a id="MinkF_rel"></a><a href="./red_rem_Me.html#MinkF_rel"><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="org4770a30">
+<div class="alteqlabels" id="orgf90a88d">
</div>
@@ -1782,14 +1782,14 @@ We can now exploit gauge invariance
which leaves \(F^{\mu \nu}\) invariant. In particular, we can choose
the Lorenz gauge <a href="./emf_g_Lg.html#LorenzG">LorenzG</a> here expressed as
</p>
-<div class="eqlabel" id="org02fe36f">
+<div class="eqlabel" id="org6a52a14">
<p>
<a id="LorenzG_4v"></a><a href="./red_rem_Me.html#LorenzG_4v"><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="org162a638">
+<div class="alteqlabels" id="orgf61a44d">
</div>
@@ -1802,15 +1802,15 @@ the Lorenz gauge <a href="./emf_g_Lg.html#LorenzG">LorenzG</a> here expressed as
Using the d'Alembertian operator introduced in <a href="./emf_g_Lg.html#dAl">dAl</a>
here rewritten as
</p>
-<div class="main div" id="org4189168">
-<div class="eqlabel" id="org347c7ab">
+<div class="main div" id="orgf152aad">
+<div class="eqlabel" id="orgf826c45">
<p>
<a id="dAl_4v"></a><a href="./red_rem_Me.html#dAl_4v"><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="org373726e">
+<div class="alteqlabels" id="orgea79545">
</div>
@@ -1827,18 +1827,18 @@ here rewritten as
we obtain the final form of
</p>
-<div class="core div" id="org1c23781">
+<div class="core div" id="org7a7f039">
<p>
<b>Maxwell's equations</b> <i>(Lorenz gauge, 4-vector notation)</i>
</p>
-<div class="eqlabel" id="org7c865aa">
+<div class="eqlabel" id="org5b1a56e">
<p>
<a id="Max_Lor_4v"></a><a href="./red_rem_Me.html#Max_Lor_4v"><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="org5e9be4d">
+<div class="alteqlabels" id="orga66f820">
</div>
@@ -1874,7 +1874,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/red_rem_mre.html b/build/red_rem_mre.html
index 7aae8b6..23a9de7 100644
--- a/build/red_rem_mre.html
+++ b/build/red_rem_mre.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1622,7 +1622,7 @@ implies the existence of magnetism, and vice-versa.
<p>
To illustrate this, we take a simple example
</p>
-<aside id="orgbe19a87">
+<aside id="org964fb67">
<p>
See Purcell \& Morin, section 5.9.
</p>
@@ -1663,7 +1663,7 @@ The dilation factors in the test particle's frame are thus
\gamma_\pm &= \frac{1}{\sqrt{1 - (v \mp u)^2/(c^2 \mp uv)^2}}
= \frac{c^2 \mp uv}{\sqrt{(c^2 \mp uv)^2 - c^2 (v \mp u)^2}} \nonumber \\
&= \frac{c^2 \mp uv}{\sqrt{(c^2 - v^2)(c^2 - u^2)}}
- = \gamma \frac{1 \mp uv/c^2}{\sqrt{q - u^2/c^2}}
+ = \gamma \frac{1 \mp uv/c^2}{\sqrt{1 - u^2/c^2}}
\end{align}
<p>
so the resultant line charge in the test frame is
@@ -1680,25 +1680,25 @@ neutral in one reference frame, can appear to be charged in another.
In the frame of the test particle, there is an electric field
equal to that of a uniformly charged wire:
\[
- E = \frac{\lambda_{\mbox{test}}}{2\pi \epsilon_0 s}
+ E = \frac{\lambda_{\mbox{test}}}{2\pi \epsilon_0 r}
\]
so the force (in the test frame) is
\[
- F_{\mbox{test}} = q E = -\frac{\lambda v}{\pi \epsilon_0 c^2 s} \frac{q u}{\sqrt{1 - u^2/c^2}}.
+ F_{\mbox{test}} = q E = -\frac{\lambda v}{\pi \epsilon_0 c^2 r} \frac{q u}{\sqrt{1 - u^2/c^2}}.
\]
If there is a force on our test charge in this test frame, there must
also be one in the lab frame. Using equation <a href="./red_rm_Mf.html#Ftr0">Ftr0</a>,
\[
F = \sqrt{1 - u^2/c^2} ~F_{\mbox{test}}
- = -\frac{\lambda v}{\pi \varepsilon_0 c^2} \frac{qu}{s}
+ = -\frac{\lambda v}{\pi \varepsilon_0 c^2} \frac{qu}{r}
\]
which upon recognizing \(c^2 = \frac{1}{\varepsilon_0 \mu_0}\)
and the current \(I = 2 \lambda v\) becomes
\[
- F = - q u ~\frac{\mu_0 I}{2\pi s}
+ F = - q u ~\frac{\mu_0 I}{2\pi r}
\]
which you will recognize as the magnetic part of the Lorentz force
-for a charge \(q\) moving at velocity \(v\) in the presence of the
+for a charge \(q\) moving at velocity \(u\) in the presence of the
magnetic field of a long straight wire carrying current \(I\).
</p>
</div>
@@ -1722,7 +1722,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/red_rm.html b/build/red_rm.html
index 7a25f14..e214256 100644
--- a/build/red_rm.html
+++ b/build/red_rm.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/red_rm_Mf.html b/build/red_rm_Mf.html
index 3dccdf3..a237edb 100644
--- a/build/red_rm_Mf.html
+++ b/build/red_rm_Mf.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1616,7 +1616,7 @@ Table of contents
<p>
Newton's second law remains valid provided we use the relativistic momentum:
</p>
-<div class="core div" id="orgf962475">
+<div class="core div" id="org11c93db">
<p>
<b>Newton's law</b> <i>(relativistic case)</i>
\[
@@ -1626,7 +1626,7 @@ Newton's second law remains valid provided we use the relativistic momentum:
</div>
-<div class="example div" id="orgac47d6f">
+<div class="example div" id="org4050d88">
<p>
<b>Example: motion under a constant force</b>
</p>
@@ -1645,7 +1645,7 @@ p = Ft
\]
and thus
\[
-p(t) = \frac{m u(t)}{1 - u(t)^2/c^2} = Ft
+p(t) = \frac{m u(t)}{\sqrt{1 - u(t)^2/c^2}} = Ft
~~\longrightarrow~~ u(t) = \frac{(F/m)t}{\sqrt{1 + (Ft/mc)^2}}.
\]
Integrating again to get the displacement,
@@ -1656,6 +1656,14 @@ x(t) = \frac{F}{m}\int_0^t dt' \frac{t'}{\sqrt{1 + (Ft'/mc)^2}}
The particle's world line thus shows <b>hyperbolic motion</b>.
</p>
+
+<figure id="orgd090155">
+<object type="image/svg+xml" data="./fig/red/red_motion_under_cst_F.svg" class="org-svg" alt="Relativistic motion under constant force">
+Sorry, your browser does not support SVG.</object>
+
+<figcaption><span class="figure-number">Figure 1: </span>A particle at rest, accelerated from \(t=0\) onwards by a constant force (classical versus relativistic motion).</figcaption>
+</figure>
+
</div>
@@ -1673,8 +1681,8 @@ In the context of relativity, work is still the line integral of the force:
The relationship between work done and increased energy also still holds:
\[
W = \int d{\boldsymbol l} \cdot \frac{d{\boldsymbol p}}{dt}
- = \int dt \frac{d {\boldsymbol l}}{dt} \cdot \frac{d{\boldsymbol p}}{dt}
- = \int dt {\boldsymbol u} \cdot \frac{d{\boldsymbol p}}{dt}
+ = \int dt ~\frac{d {\boldsymbol l}}{dt} \cdot \frac{d{\boldsymbol p}}{dt}
+ = \int dt ~{\boldsymbol u} \cdot \frac{d{\boldsymbol p}}{dt}
\]
but
\[
@@ -1718,14 +1726,15 @@ whereas the longitudinal component transforms in a complicated way:
For the specific case where the particle is instantaneously at rest in
the original frame, then
</p>
-<div class="eqlabel" id="orgb7d57c3">
+<div class="main div" id="org7c7f004">
+<div class="eqlabel" id="org37e2037">
<p>
<a id="Ftr0"></a><a href="./red_rm_Mf.html#Ftr0"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
-<div class="alteqlabels" id="org2309428">
+<div class="alteqlabels" id="orgbf7064d">
</div>
@@ -1739,23 +1748,25 @@ the original frame, then
\]
</p>
+</div>
+
<p>
The way to avoid complicated transformation rules is to define a four-vector
as the derivative of momentum with respect to proper time, which leads to
the definition of the
</p>
-<div class="main div" id="orgb93dc36">
+<div class="main div" id="orgc529917">
<p>
<b>Minkowski force</b>
</p>
-<div class="eqlabel" id="orgca23908">
+<div class="eqlabel" id="org24e429b">
<p>
<a id="MinkF"></a><a href="./red_rm_Mf.html#MinkF"><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="orgd690932">
+<div class="alteqlabels" id="org6498650">
</div>
@@ -1795,7 +1806,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/red_rm_pt.html b/build/red_rm_pt.html
index bc1bbf7..28712cc 100644
--- a/build/red_rm_pt.html
+++ b/build/red_rm_pt.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1629,7 +1629,7 @@ and this is called the <b>ordinary velocity</b>.
The <b>proper velocity</b> is defined as the hybrid-frame quantity
(distance as measured on the ground) divided by (proper time interval):
</p>
-<div class="core div" id="orgc6b970d">
+<div class="core div" id="org6b9040e">
<p>
<b>Proper velocity</b>
\[
@@ -1651,7 +1651,7 @@ By adding the zeroth component
\]
we can define the
</p>
-<div class="core div" id="orgc1354f8">
+<div class="core div" id="org75d23d9">
<p>
<b>four-velocity</b> or <b>proper velocity four-vector</b>
\[
@@ -1659,7 +1659,7 @@ we can define the
\]
which transforms as
\[
-\bar{\eta}^\mu = \Lambda^\mu_\nu \eta^\nu
+\bar{\eta}^\mu = \Lambda^\mu{}_\nu \eta^\nu
\]
</p>
@@ -1673,9 +1673,9 @@ invariant).
By contrast, the ordinary velocities obey cumbersome transformation rules:
for our usual relative frame velocity of \(v\) in the \(x\) direction,
\[
- \bar{u}_x = \frac{d\bar{x}}{dt} = \frac{u_x - v}{1-v u_x/c^2}, \hspace{10mm}
- \bar{u}_y = \frac{d\bar{y}}{dt} = \frac{u_y}{1-v u_x/c^2}, \hspace{10mm}
- \bar{u}_z = \frac{d\bar{z}}{dt} = \frac{u_z}{1-v u_x/c^2}.
+ \bar{u}_x = \frac{d\bar{x}}{d\bar{t}} = \frac{u_x - v}{1-v u_x/c^2}, \hspace{10mm}
+ \bar{u}_y = \frac{d\bar{y}}{d\bar{t}} = \frac{u_y}{\gamma(1-v u_x/c^2)}, \hspace{10mm}
+ \bar{u}_z = \frac{d\bar{z}}{d\bar{t}} = \frac{u_z}{\gamma(1-v u_x/c^2)}.
\]
</p>
</div>
@@ -1699,7 +1699,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/red_rm_rme.html b/build/red_rm_rme.html
index 86659f9..ade68fb 100644
--- a/build/red_rm_rme.html
+++ b/build/red_rm_rme.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1613,20 +1613,20 @@ 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">red.rm.rme</span></h4>
<div class="outline-text-4" id="text-red_rm_rme">
-<div class="core div" id="org37554a5">
+<div class="core div" id="org1fef50a">
<p>
The <b>relativistic momentum</b> \({\boldsymbol p}\) is defined as
\[
{\boldsymbol p} \equiv m {\boldsymbol \eta} =
-\frac{m {\boldsymbol u}}{1 - u^2/c^2}.
+\frac{m {\boldsymbol u}}{\sqrt{1 - u^2/c^2}}.
\]
The <b>relativistic energy</b> is defined as
\[
-E \equiv \frac{m c^2}{\sqrt{1 - u^2/c^2}}.
+E \equiv \frac{m c^2}{\sqrt{1 - u^2/c^2}} \equiv c p^0.
\]
These can be combined into the <b>energy-momentum four-vector</b>
\[
-p^\mu \equiv m \eta^\mu.
+p^\mu \equiv m \eta^\mu
\]
</p>
@@ -1635,7 +1635,7 @@ p^\mu \equiv m \eta^\mu.
<p>
When the object is stationary, its energy is the
</p>
-<div class="core div" id="orgf6b283a">
+<div class="core div" id="org2cb0822">
<p>
<b>Rest energy</b>
\[
@@ -1648,7 +1648,7 @@ E_{\mbox{rest}} \equiv m c^2.
When moving, the difference between relativistic and rest energies
is the
</p>
-<div class="core div" id="org8d28e60">
+<div class="core div" id="org59d0366">
<p>
<b>Kinetic energy</b>
\[
@@ -1661,14 +1661,14 @@ E_{\mbox{kin}} \equiv E - mc^2 = mc^2 \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \rig
For velocities much smaller than the speed of light, we can expand
this to
\[
- E_{\mbox{kin}} = \frac{1}{2} mu^2 + \frac{3}{8} \frac{mu^2}{c^2} + ...
+ E_{\mbox{kin}} = \frac{1}{2} mu^2 + \frac{3}{8} \frac{mu^4}{c^2} + ...
\]
</p>
<p>
In a closed system,
</p>
-<div class="core div" id="org0b3cfe4">
+<div class="core div" id="orge0a6a35">
<p>
<b>Total relativistic energy and momentum is conserved</b>
\[
@@ -1702,7 +1702,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/red_sr.html b/build/red_sr.html
index 37f95b5..2238caf 100644
--- a/build/red_sr.html
+++ b/build/red_sr.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/red_sr_4v.html b/build/red_sr_4v.html
index f0e47c6..a6aa9be 100644
--- a/build/red_sr_4v.html
+++ b/build/red_sr_4v.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1621,7 +1621,7 @@ Table of contents
\]
The Lorentz transformation then reads
</p>
-<div class="core div" id="orgf4d6e6e">
+<div class="core div" id="org2ede351">
<p>
<b>Lorentz transformation</b> (motion along \(x\) at velocity \(v\))
\[
@@ -1657,7 +1657,7 @@ upper index are called <b>contravariant</b>. Their lower-index
counterparts are called <b>covariant</b> vectors and are obtained by
using the Minkowski metric \(g_{\mu \nu}\) according to
</p>
-<div class="main div" id="orge005c2f">
+<div class="main div" id="org1cd4b37">
<p>
\[
a_\mu = \sum_{\nu = 0}^3 g_{\mu \nu} a^\nu, \hspace{10mm}
@@ -1673,7 +1673,7 @@ using the Minkowski metric \(g_{\mu \nu}\) according to
<p>
<b>Scalar products</b> are defined as the in-product of covariant/contravariant four-vectors,
</p>
-<div class="main div" id="org411b956">
+<div class="main div" id="orgbb985f0">
<p>
\[
\sum_{\mu = 0}^3 a^\mu b_\mu \equiv a^\mu b_\mu
@@ -1708,17 +1708,20 @@ can take positive or negative values. The nomenclature goes as follows:
<tbody>
<tr>
<td class="org-left">\(a^\mu a_\mu > 0\)</td>
-<td class="org-left">\(a^\mu\) is {\it spacelike}</td>
+<td class="org-left">\(a^\mu\) is <b>spacelike</b>
+</td>
</tr>
<tr>
<td class="org-left">\(a^\mu a_\mu < 0\)</td>
-<td class="org-left">\(a^\mu\) is {\it timelike}</td>
+<td class="org-left">\(a^\mu\) is <b>timelike</b>
+</td>
</tr>
<tr>
<td class="org-left">\(a^\mu a_\mu = 0\)</td>
-<td class="org-left">\(a^\mu\) is {\it lightlike}</td>
+<td class="org-left">\(a^\mu\) is <b>lightlike</b>
+</td>
</tr>
</tbody>
</table>
@@ -1766,7 +1769,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/red_sr_Lt.html b/build/red_sr_Lt.html
index 26ad2f6..aaf67a5 100644
--- a/build/red_sr_Lt.html
+++ b/build/red_sr_Lt.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1670,7 +1670,7 @@ where \(\bar{d}\) is the distance from \({\cal O}\) to \({\cal A}\) at time
\]
Solving these relations yields the dictionary for
</p>
-<div class="core div" id="org585b32e">
+<div class="core div" id="org65a91be">
<p>
<b>Lorentz transformations</b> (motion along \(x\) at velocity \(v\))
</p>
@@ -1709,7 +1709,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
-<p class="date">Created: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>
diff --git a/build/red_sr_p.html b/build/red_sr_p.html
index cfc9abc..395d4c2 100644
--- a/build/red_sr_p.html
+++ b/build/red_sr_p.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2022-03-15 Tue 08:10 -->
+<!-- 2022-03-22 Tue 10:52 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1648,18 +1648,18 @@ and we get
\]
so the time interval in the train is shorter, namely there is a
</p>
-<div class="core div" id="org2b8c90f">
+<div class="core div" id="org04cf730">
<p>
<b>Time dilation factor</b>
</p>
-<div class="eqlabel" id="orgf672abf">
+<div class="eqlabel" id="org37dbf9e">
<p>
<a id="gamma"></a><a href="./red_sr_p.html#gamma"><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="org0f03bec">
+<div class="alteqlabels" id="org7936dae">
</div>
@@ -1699,7 +1699,7 @@ and thus
\]
Using the time dilation relation then gives
</p>
-<div class="core div" id="orge1bda84">
+<div class="core div" id="orgd363ff1">
<p>
<b>Lorentz contraction</b>
\[
@@ -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: 2022-03-15 Tue 08:10</p>
+<p class="date">Created: 2022-03-22 Tue 10:52</p>
<p class="validation"></p>
</div>