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jscaux/pqed_html:master

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Author SHA1 Message Date
Jean-Sébastien 50704eba07 Update 2022-03-24 08:43 2022-03-24 08:43:21 +01:00
Jean-Sébastien 1d852e7213 Update 2022-03-22 2022-03-22 10:54:46 +01:00
Jean-Sébastien 2d1d8d39a4 Update 2022-03-22 10:53 2022-03-22 10:53:22 +01:00
211 changed files with 1606 additions and 3089 deletions
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build/a.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/a_l.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1720,7 +1720,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c_m.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c_m_cs.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c_m_cs_cyl.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -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"> <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> <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="outline-text-6" id="text-c_m_cs_cyl_grad">
<div class="eqlabel" id="org9ded6d5"> <div class="eqlabel" id="org718e42d">
<p> <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"> <a id="cylgrad"></a><a href="./c_m_cs_cyl.html#cylgrad"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org3066755"> <div class="alteqlabels" id="org9ec66cd">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr4(1.79)</li> <li>Gr4(1.79)</li>
</ul> </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"> <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> <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="outline-text-6" id="text-c_m_cs_cyl_div">
<div class="eqlabel" id="org0eb5cd5"> <div class="eqlabel" id="orgd5ab218">
<p> <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"> <a id="cyl_div"></a><a href="./c_m_cs_cyl.html#cyl_div"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orge14058b"> <div class="alteqlabels" id="orgd93619e">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr4(2.21)</li> <li>Gr4(2.21)</li>
</ul> </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"> <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> <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="outline-text-6" id="text-c_m_cs_cyl_curl">
<div class="eqlabel" id="org34649e5"> <div class="eqlabel" id="orgabcc2d8">
<p> <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"> <a id="cyl_curl"></a><a href="./c_m_cs_cyl.html#cyl_curl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgeec607c"> <div class="alteqlabels" id="orga07a427">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr4(2.21)</li> <li>Gr4(2.21)</li>
</ul> </ul>
@@ -1771,7 +1771,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c_m_cs_hyp.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1632,7 +1632,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c_m_cs_sph.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1623,14 +1623,14 @@ which \(r\) is the distance from the chosen origin,
The usual Cartesian coordinates relate to spherical coordinates The usual Cartesian coordinates relate to spherical coordinates
according to according to
</p> </p>
<div class="eqlabel" id="org309b2b3"> <div class="eqlabel" id="org763df52">
<p> <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"> <a id="sph_xyz"></a><a href="./c_m_cs_sph.html#sph_xyz"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org9ad4ca2"> <div class="alteqlabels" id="orgb59869b">
</div> </div>
@@ -1655,14 +1655,14 @@ A generic vector can be expressed as
where the explicit relation between spherical and where the explicit relation between spherical and
Cartesian unit vectors is Cartesian unit vectors is
</p> </p>
<div class="eqlabel" id="orgbf81903"> <div class="eqlabel" id="org923fbbc">
<p> <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"> <a id="sph_uv"></a><a href="./c_m_cs_sph.html#sph_uv"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org437eb02"> <div class="alteqlabels" id="org2051229">
</div> </div>
@@ -1685,14 +1685,14 @@ and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).
<p> <p>
An infinitesimal displacement \(d{\bf l}\) can be written as An infinitesimal displacement \(d{\bf l}\) can be written as
</p> </p>
<div class="eqlabel" id="orgda909cf"> <div class="eqlabel" id="orgb032a0d">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org02f05bf"> <div class="alteqlabels" id="orgaeea797">
</div> </div>
@@ -1708,14 +1708,14 @@ d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin
<p> <p>
Infinitesimal volume element: Infinitesimal volume element:
</p> </p>
<div class="eqlabel" id="orgd2cfc4a"> <div class="eqlabel" id="orgf42297c">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org0a4c199"> <div class="alteqlabels" id="orgb0160b3">
</div> </div>
@@ -1736,14 +1736,14 @@ Infinitesimal surface element: depends on situation.
<div id="outline-container-c_m_cs_sph_grad" class="outline-6"> <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> <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="outline-text-6" id="text-c_m_cs_sph_grad">
<div class="eqlabel" id="orge44fa24"> <div class="eqlabel" id="orgbf5f354">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org6b20d40"> <div class="alteqlabels" id="orgbda0409">
</div> </div>
@@ -1760,14 +1760,14 @@ Infinitesimal surface element: depends on situation.
<div id="outline-container-c_m_cs_sph_div" class="outline-6"> <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> <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="outline-text-6" id="text-c_m_cs_sph_div">
<div class="eqlabel" id="org3d72303"> <div class="eqlabel" id="orge3c6cf7">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org9b1dfae"> <div class="alteqlabels" id="orged88fee">
</div> </div>
@@ -1784,14 +1784,14 @@ Infinitesimal surface element: depends on situation.
<div id="outline-container-c_m_cs_sph_curl" class="outline-6"> <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> <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="outline-text-6" id="text-c_m_cs_sph_curl">
<div class="eqlabel" id="orge960708"> <div class="eqlabel" id="org34ae3e4">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org0f49079"> <div class="alteqlabels" id="org5f176f6">
</div> </div>
@@ -1810,14 +1810,14 @@ Infinitesimal surface element: depends on situation.
<div id="outline-container-c_m_cs_sph_lap" class="outline-6"> <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> <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="outline-text-6" id="text-c_m_cs_sph_lap">
<div class="eqlabel" id="org1d2bef1"> <div class="eqlabel" id="orgb40c79b">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org6ba97dc"> <div class="alteqlabels" id="org1c581f6">
</div> </div>
@@ -1851,7 +1851,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c_m_dc.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1642,7 +1642,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c_m_dc_curl.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1644,7 +1644,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c_m_dc_d2.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1615,9 +1615,9 @@ Table of contents
<div class="outline-text-5" id="text-c_m_dc_d2"> <div class="outline-text-5" id="text-c_m_dc_d2">
</div> </div>
<div id="outline-container-orgc162c65" class="outline-6"> <div id="outline-container-org873f1bb" class="outline-6">
<h6 id="orgc162c65"><a href="#orgc162c65">Divergence of gradient</a></h6> <h6 id="org873f1bb"><a href="#org873f1bb">Divergence of gradient</a></h6>
<div class="outline-text-6" id="text-orgc162c65"> <div class="outline-text-6" id="text-org873f1bb">
<p> <p>
\({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\). \({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) 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 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> </div>
<div id="outline-container-org2f7ef92" class="outline-6"> <div id="outline-container-orged6374e" class="outline-6">
<h6 id="org2f7ef92"><a href="#org2f7ef92">Curl of a gradient</a></h6> <h6 id="orged6374e"><a href="#orged6374e">Curl of a gradient</a></h6>
<div class="outline-text-6" id="text-org2f7ef92"> <div class="outline-text-6" id="text-orged6374e">
<p> <p>
This always vanishes. This always vanishes.
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-orgceccef1" class="outline-6"> <div id="outline-container-orgcf8bc6c" class="outline-6">
<h6 id="orgceccef1"><a href="#orgceccef1">Gradient of the divergence</a></h6> <h6 id="orgcf8bc6c"><a href="#orgcf8bc6c">Gradient of the divergence</a></h6>
<div class="outline-text-6" id="text-orgceccef1"> <div class="outline-text-6" id="text-orgcf8bc6c">
<p> <p>
\({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name. \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-orgc07e1dc" class="outline-6"> <div id="outline-container-orgc1dc4c7" class="outline-6">
<h6 id="orgc07e1dc"><a href="#orgc07e1dc">Divergence of a curl</a></h6> <h6 id="orgc1dc4c7"><a href="#orgc1dc4c7">Divergence of a curl</a></h6>
<div class="outline-text-6" id="text-orgc07e1dc"> <div class="outline-text-6" id="text-orgc1dc4c7">
<p> <p>
This always vanishes. This always vanishes.
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-org1a67d52" class="outline-6"> <div id="outline-container-org766740d" class="outline-6">
<h6 id="org1a67d52"><a href="#org1a67d52">Curl of curl</a></h6> <h6 id="org766740d"><a href="#org766740d">Curl of curl</a></h6>
<div class="outline-text-6" id="text-org1a67d52"> <div class="outline-text-6" id="text-org766740d">
<div class="eqlabel" id="org5d4ff40"> <div class="eqlabel" id="orga94a832">
<p> <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"> <a id="curlcurl"></a><a href="./c_m_dc_d2.html#curlcurl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgd2518b8"> <div class="alteqlabels" id="orgc3745e4">
</div> </div>
@@ -1694,7 +1694,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c_m_dc_del.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c_m_dc_div.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1640,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c_m_dc_g.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1664,7 +1664,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c_m_dc_pr.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1622,14 +1622,14 @@ explicited as follows:
<p> <p>
<b>Gradient of a product</b>: <b>Gradient of a product</b>:
</p> </p>
<div class="eqlabel" id="org824336d"> <div class="eqlabel" id="orge6e6a80">
<p> <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"> <a id="grad_prod"></a><a href="./c_m_dc_pr.html#grad_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org2bf23af"> <div class="alteqlabels" id="orgf9842ae">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (3)</li> <li>Gr (3)</li>
<li>W (1-111)</li> <li>W (1-111)</li>
@@ -1649,14 +1649,14 @@ explicited as follows:
<p> <p>
<b>Gradient of a scalar product</b>: <b>Gradient of a scalar product</b>:
</p> </p>
<div class="eqlabel" id="org41d1a68"> <div class="eqlabel" id="orgb8556f5">
<p> <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"> <a id="grad_sprod"></a><a href="./c_m_dc_pr.html#grad_sprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org466aa7e"> <div class="alteqlabels" id="org884e962">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (4)</li> <li>Gr (4)</li>
<li>W (1-112)</li> <li>W (1-112)</li>
@@ -1676,14 +1676,14 @@ explicited as follows:
<p> <p>
<b>Divergence of a product</b>: <b>Divergence of a product</b>:
</p> </p>
<div class="eqlabel" id="org1ad95d1"> <div class="eqlabel" id="org3da56b1">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org93785a5"> <div class="alteqlabels" id="orgeb30387">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (5)</li> <li>Gr (5)</li>
<li>W (1-115)</li> <li>W (1-115)</li>
@@ -1703,14 +1703,14 @@ explicited as follows:
<p> <p>
<b>Divergence of a cross product</b>: <b>Divergence of a cross product</b>:
</p> </p>
<div class="eqlabel" id="org4333997"> <div class="eqlabel" id="orgbfaa39d">
<p> <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"> <a id="div_xprod"></a><a href="./c_m_dc_pr.html#div_xprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org2626fe4"> <div class="alteqlabels" id="orgca8e0c0">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (6)</li> <li>Gr (6)</li>
<li>W (1-116)</li> <li>W (1-116)</li>
@@ -1730,14 +1730,14 @@ explicited as follows:
<p> <p>
<b>Curl of a product</b>: <b>Curl of a product</b>:
</p> </p>
<div class="eqlabel" id="orgfa1c2f8"> <div class="eqlabel" id="org5b32772">
<p> <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"> <a id="curl_prod"></a><a href="./c_m_dc_pr.html#curl_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgb82ff5c"> <div class="alteqlabels" id="org22a6f09">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7)</li> <li>Gr (7)</li>
<li>W (1-118)</li> <li>W (1-118)</li>
@@ -1757,14 +1757,14 @@ explicited as follows:
<p> <p>
<b>Curl of a cross product</b>: <b>Curl of a cross product</b>:
</p> </p>
<div class="eqlabel" id="org9d0ffb0"> <div class="eqlabel" id="org81448bf">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org6beb5fe"> <div class="alteqlabels" id="org5de3f60">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (8)</li> <li>Gr (8)</li>
<li>W (1-119)</li> <li>W (1-119)</li>
@@ -1805,7 +1805,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c_m_dd.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c_m_dd_1d.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1651,7 +1651,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/c_m_dd_3d.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1636,14 +1636,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
More generally, More generally,
</p> </p>
<div class="eqlabel" id="org501b97b"> <div class="eqlabel" id="orgcf3b8a6">
<p> <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"> <a id="divdel"></a><a href="./c_m_dd_3d.html#divdel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org2eebb69"> <div class="alteqlabels" id="orgd295657">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (1.100)</li> <li>Gr (1.100)</li>
</ul> </ul>
@@ -1662,14 +1662,14 @@ More generally,
Since Since
</p> </p>
<div class="eqlabel" id="orga23d9bb"> <div class="eqlabel" id="org8e3f2ff">
<p> <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"> <a id="div1or"></a><a href="./c_m_dd_3d.html#div1or"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org17b7d37"> <div class="alteqlabels" id="orgd9d8d13">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (1.101)</li> <li>Gr (1.101)</li>
</ul> </ul>
@@ -1685,14 +1685,14 @@ Since
<p> <p>
we have that we have that
</p> </p>
<div class="eqlabel" id="orgc9c91cd"> <div class="eqlabel" id="org4584082">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgcc41bc0"> <div class="alteqlabels" id="org4002cb1">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (1.102)</li> <li>Gr (1.102)</li>
</ul> </ul>
@@ -1724,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
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<head> <head>
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<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1657,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
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<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1642,7 +1642,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
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<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1640,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
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<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1649,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
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<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1650,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
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<head> <head>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1615,9 +1615,9 @@ Table of contents
<div class="outline-text-5" id="text-c_m_ic_lsv"> <div class="outline-text-5" id="text-c_m_ic_lsv">
</div> </div>
<div id="outline-container-org6c52cc6" class="outline-6"> <div id="outline-container-org2ee5abe" class="outline-6">
<h6 id="org6c52cc6"><a href="#org6c52cc6">Line Integrals</a></h6> <h6 id="org2ee5abe"><a href="#org2ee5abe">Line Integrals</a></h6>
<div class="outline-text-6" id="text-org6c52cc6"> <div class="outline-text-6" id="text-org2ee5abe">
<p> <p>
\[ \[
{\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l} {\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> </div>
<div id="outline-container-org7c42ba4" class="outline-6"> <div id="outline-container-org5c34665" class="outline-6">
<h6 id="org7c42ba4"><a href="#org7c42ba4">Surface Integrals</a></h6> <h6 id="org5c34665"><a href="#org5c34665">Surface Integrals</a></h6>
<div class="outline-text-6" id="text-org7c42ba4"> <div class="outline-text-6" id="text-org5c34665">
<p> <p>
\[ \[
\int_{\cal S} {\bf v} \cdot d{\bf a} \int_{\cal S} {\bf v} \cdot d{\bf a}
@@ -1668,9 +1668,9 @@ Over a closed surface:
</div> </div>
</div> </div>
<div id="outline-container-org25bec25" class="outline-6"> <div id="outline-container-org3f94a4e" class="outline-6">
<h6 id="org25bec25"><a href="#org25bec25">Volume Integrals</a></h6> <h6 id="org3f94a4e"><a href="#org3f94a4e">Volume Integrals</a></h6>
<div class="outline-text-6" id="text-org25bec25"> <div class="outline-text-6" id="text-org3f94a4e">
<p> <p>
\[ \[
\int_{\cal V} T d\tau \int_{\cal V} T d\tau
@@ -1711,7 +1711,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1613,14 +1613,14 @@ Table of contents
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<div class="outline-text-5" id="text-c_m_ic_stokes"> <div class="outline-text-5" id="text-c_m_ic_stokes">
<div class="eqlabel" id="org936b14a"> <div class="eqlabel" id="org46c1290">
<p> <p>
<a id="Stokes"></a><a href="./c_m_ic_stokes.html#Stokes"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16"> <a id="Stokes"></a><a href="./c_m_ic_stokes.html#Stokes"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgc066803"> <div class="alteqlabels" id="org1c7e7a7">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (1.57)</li> <li>Gr (1.57)</li>
</ul> </ul>
@@ -1665,7 +1665,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
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<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1652,7 +1652,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
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<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
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<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
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Objectives Objectives
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<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1659,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1656,7 +1656,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1659,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1657,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1662,7 +1662,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1663,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1614,8 +1614,8 @@ Table of contents
</svg></a><span class="headline-id">emd</span></h2> </svg></a><span class="headline-id">emd</span></h2>
<div class="outline-text-2" id="text-emd"> <div class="outline-text-2" id="text-emd">
<details class="prereq" id="orgeb31271"> <details class="prereq" id="orgabe80a6">
<summary id="org69f893c"> <summary id="orgccf4253">
Prerequisites Prerequisites
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@@ -1624,8 +1624,8 @@ Prerequisites
</ul> </ul>
</details> </details>
<details class="objectives" id="orgd436fef"> <details class="objectives" id="orgf2d0735">
<summary id="orgff05dfe"> <summary id="orge4261f8">
Objectives Objectives
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1640,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -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 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 along the wire. The line integral of this field is called the
</p> </p>
<div class="core div" id="orgfd2df2d"> <div class="core div" id="orgb0fd5b4">
<p> <p>
<b>Electromotive force (or electromotance)</b>, <b>Electromotive force (or electromotance)</b>,
</p> </p>
<div class="eqlabel" id="org510ac9c"> <div class="eqlabel" id="orgb0e7cde">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org85afd05"> <div class="alteqlabels" id="org516a196">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.9)</li> <li>Gr (7.9)</li>
</ul> </ul>
@@ -1680,14 +1680,14 @@ The precise statement associated to Faraday's observations
is that the electromotive force is proportional is that the electromotive force is proportional
to the rate of change of the magnetic flux, to the rate of change of the magnetic flux,
</p> </p>
<div class="eqlabel" id="org4496ec0"> <div class="eqlabel" id="orgab619b1">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orga00ad3a"> <div class="alteqlabels" id="org969857c">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.14)</li> <li>Gr (7.14)</li>
</ul> </ul>
@@ -1702,18 +1702,18 @@ to the rate of change of the magnetic flux,
\] \]
so we obtain so we obtain
</p> </p>
<div class="core div" id="org35a9412"> <div class="core div" id="org5419feb">
<p> <p>
<b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>) <b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
</p> </p>
<div class="eqlabel" id="org6646a96"> <div class="eqlabel" id="org4696e9f">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org4c06533"> <div class="alteqlabels" id="org439dfef">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.15)</li> <li>Gr (7.15)</li>
</ul> </ul>
@@ -1737,15 +1737,15 @@ for any loop (on a wire or not). Using Stokes' theorem,
\] \]
we obtain we obtain
</p> </p>
<div class="core div" id="orgfef0799"> <div class="core div" id="org49ffd42">
<div class="eqlabel" id="org07be4b3"> <div class="eqlabel" id="org5ecebad">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgd458eb7"> <div class="alteqlabels" id="org6a61522">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.16)</li> <li>Gr (7.16)</li>
</ul> </ul>
@@ -1790,7 +1790,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
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<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -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: Generalization to volume currents:
</p> </p>
<div class="eqlabel" id="org9c43b0a"> <div class="eqlabel" id="orge9580af">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org1b4fa12"> <div class="alteqlabels" id="orgbe0c60a">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.31)</li> <li>Gr (7.31)</li>
</ul> </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 We can integrate over all space: after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with
</p> </p>
<div class="core div" id="orgc92d34b"> <div class="core div" id="org9173f2d">
<div class="eqlabel" id="org7ec4181"> <div class="eqlabel" id="org717e156">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org073f8ff"> <div class="alteqlabels" id="org0d5a0b3">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.34)</li> <li>Gr (7.34)</li>
</ul> </ul>
@@ -1718,7 +1718,7 @@ W_{mag} &amp;= \frac{1}{2} \int d\tau ~({\bf A} \cdot {\bf J}) &amp;= \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>. 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> </p>
<div class="example div" id="org60417d4"> <div class="example div" id="org34d5a7d">
<p> <p>
<b>Example: energy in coaxial cable</b> <b>Example: energy in coaxial cable</b>
</p> </p>
@@ -1768,7 +1768,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_Fl_i.html
+12 -12
View File
@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1641,14 +1641,14 @@ so
\] \]
and we can write the mutual inductance as the <b>Neumann formula</b>, and we can write the mutual inductance as the <b>Neumann formula</b>,
</p> </p>
<div class="eqlabel" id="org554c244"> <div class="eqlabel" id="org1f70ff2">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgf48cffa"> <div class="alteqlabels" id="org57de753">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.22)</li> <li>Gr (7.22)</li>
</ul> </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: Two things:
first, \(M_{21}\) is purely geometrical. Second, first, \(M_{21}\) is purely geometrical. Second,
</p> </p>
<div class="eqlabel" id="org12f955a"> <div class="eqlabel" id="org43668cc">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orga59d4b0"> <div class="alteqlabels" id="org7b1e4fa">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.23)</li> <li>Gr (7.23)</li>
</ul> </ul>
@@ -1687,7 +1687,7 @@ M_{12} = M_{21}
\] \]
</p> </p>
<div class="example div" id="orgdd5f18b"> <div class="example div" id="orgecf7176">
<p> <p>
<b>Example: solenoid in solenoid</b> <b>Example: solenoid in solenoid</b>
</p> </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: Changing current also induces EMF in the source loop itself:
</p> </p>
<div class="eqlabel" id="org06772cd"> <div class="eqlabel" id="org2512f69">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org5ce4f36"> <div class="alteqlabels" id="org080d683">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.25)</li> <li>Gr (7.25)</li>
</ul> </ul>
@@ -1765,7 +1765,7 @@ Inductance: measured in <b>henries</b> (\(H\)). \(H = V s/A\).
</p> </p>
<div class="example div" id="org678dce1"> <div class="example div" id="orgce99b7c">
<p> <p>
<b>Example: self-inductance of toroidal coil</b> <b>Example: self-inductance of toroidal coil</b>
</p> </p>
@@ -1804,7 +1804,7 @@ Inductance (like capacitance) is intrinsically positive. Use Lenz law.
Think of <i>back EMF</i>. Think of <i>back EMF</i>.
</p> </p>
<div class="example div" id="org42aa8b9"> <div class="example div" id="orgb4d543b">
<p> <p>
<b>Example: circuit</b> <b>Example: circuit</b>
</p> </p>
@@ -1853,7 +1853,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_Fl_ief.html
+6 -6
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1637,7 +1637,7 @@ law in integral form:
<div class="example div" id="orgcfd6427"> <div class="example div" id="org2badd21">
<p> <p>
<b>Example: loop with time-dependent flux</b> <b>Example: loop with time-dependent flux</b>
</p> </p>
@@ -1664,7 +1664,7 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
</div> </div>
<div class="example div" id="orgfc45720"> <div class="example div" id="org6994570">
<p> <p>
<b>Example: wheel with charged rim traversed by flux</b> <b>Example: wheel with charged rim traversed by flux</b>
</p> </p>
@@ -1705,7 +1705,7 @@ called the <b>quasistatic</b> approximation, and works provided we deal with
<i>slow enough</i> phenomena. <i>slow enough</i> phenomena.
</p> </p>
<div class="example div" id="org1d9bce6"> <div class="example div" id="orgddce4f1">
<p> <p>
<b>Example: field from wire with time-dependent current</b> <b>Example: field from wire with time-dependent current</b>
</p> </p>
@@ -1764,7 +1764,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_Me.html
+3 -3
View File
@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1640,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_Me_Me.html
+10 -10
View File
@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1616,18 +1616,18 @@ Table of contents
<p> <p>
Full set of equations for the electromagnetic field: Full set of equations for the electromagnetic field:
</p> </p>
<div class="core div" id="orgee05fb0"> <div class="core div" id="org96eda60">
<p> <p>
<b>Maxwell's equations</b> <i>(in vacuum)</i> <b>Maxwell's equations</b> <i>(in vacuum)</i>
</p> </p>
<div class="eqlabel" id="orgf090588"> <div class="eqlabel" id="org99bc512">
<p> <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"> <a id="Max_vac"></a><a href="./emd_Me_Me.html#Max_vac"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org538c235"> <div class="alteqlabels" id="org0385504">
</div> </div>
@@ -1644,7 +1644,7 @@ Full set of equations for the electromagnetic field:
<p> <p>
Complement: Complement:
</p> </p>
<div class="core div" id="org14efa58"> <div class="core div" id="org8bd62bd">
<p> <p>
Force law <a href="./ems_ms_lf_pc.html#LorFo">LorFo</a> Force law <a href="./ems_ms_lf_pc.html#LorFo">LorFo</a>
\[ \[
@@ -1667,15 +1667,15 @@ take divergence of \((iv)\) and use \((i)\).
<p> <p>
Better way of writing: all fields on left, all sources on right, Better way of writing: all fields on left, all sources on right,
</p> </p>
<div class="core div" id="orge7a8009"> <div class="core div" id="org3b3efc0">
<div class="eqlabel" id="orgaf03f6a"> <div class="eqlabel" id="orgaa87be2">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgc1dd627"> <div class="alteqlabels" id="org3e69b1d">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.42)</li> <li>Gr (7.42)</li>
</ul> </ul>
@@ -1713,7 +1713,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_Me_dc.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -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 The extra term would thus be eliminated if we were to put
</p> </p>
<div class="core div" id="org9029eb1"> <div class="core div" id="org65d0a3c">
<div class="eqlabel" id="org7937f05"> <div class="eqlabel" id="org2343c92">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgc4ec53c"> <div class="alteqlabels" id="org09fe122">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.36)</li> <li>Gr (7.36)</li>
</ul> </ul>
@@ -1659,18 +1659,18 @@ Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation
<p> <p>
Maxwell baptized this term the Maxwell baptized this term the
</p> </p>
<div class="core div" id="orgf5b87fa"> <div class="core div" id="org9a1c650">
<p> <p>
<b>Displacement current</b> <b>Displacement current</b>
</p> </p>
<div class="eqlabel" id="orgff1f2c8"> <div class="eqlabel" id="orgea96ca5">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org87a5cc7"> <div class="alteqlabels" id="org147727a">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.37)</li> <li>Gr (7.37)</li>
</ul> </ul>
@@ -1724,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_Me_ebM.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1630,14 +1630,14 @@ Fatal inconsistency: div of curl must always vanish. Check on \((iii)\):
\] \]
But: try same with \((iv)\): But: try same with \((iv)\):
</p> </p>
<div class="eqlabel" id="org76390cc"> <div class="eqlabel" id="org5adad0c">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org64ca711"> <div class="alteqlabels" id="orga04de46">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.35)</li> <li>Gr (7.35)</li>
</ul> </ul>
@@ -1682,7 +1682,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_Me_mc.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1663,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_ce.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
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<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1614,8 +1614,8 @@ Table of contents
</svg></a><span class="headline-id">emd.ce</span></h3> </svg></a><span class="headline-id">emd.ce</span></h3>
<div class="outline-text-3" id="text-emd_ce"> <div class="outline-text-3" id="text-emd_ce">
<details class="prereq" id="orgd007ccd"> <details class="prereq" id="orgbcfd9aa">
<summary id="org4b65898"> <summary id="orgfd4a1b6">
Prerequisites Prerequisites
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@@ -1623,8 +1623,8 @@ Prerequisites
</ul> </ul>
</details> </details>
<details class="objectives" id="org1ca55b0"> <details class="objectives" id="orgb58d097">
<summary id="org22a98ae"> <summary id="org5cd5f85">
Objectives Objectives
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@@ -1662,7 +1662,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_ce_amom.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
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<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1616,18 +1616,18 @@ Table of contents
<p> <p>
The angular momentum of EM fields is directly given by The angular momentum of EM fields is directly given by
</p> </p>
<div class="main div" id="org9f96756"> <div class="main div" id="orgc2d5343">
<p> <p>
<b>Angular momentum of EM fields</b> <b>Angular momentum of EM fields</b>
</p> </p>
<div class="eqlabel" id="org2a11d69"> <div class="eqlabel" id="orgd4d0682">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgfb3f8f7"> <div class="alteqlabels" id="orgf246c58">
</div> </div>
@@ -1661,7 +1661,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_ce_ce.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1635,7 +1635,7 @@ This means that
\] \]
Since this is true for any volume, we have (re)derived the Since this is true for any volume, we have (re)derived the
</p> </p>
<div class="core div" id="org3f0dec4"> <div class="core div" id="orgc746662">
<p> <p>
<b>Continuity equation</b> <a href="./ems_ms_ce.html#conteq">conteq</a> <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>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_ce_mom.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1628,18 +1628,18 @@ in which the first integral can be interpreted as the momentum stored in the EM
<p> <p>
This is thus simply a conservation law for momentum, with This is thus simply a conservation law for momentum, with
</p> </p>
<div class="main div" id="orgc119c5d"> <div class="main div" id="org9032358">
<p> <p>
<b>Momentum density in the EM fields</b> <b>Momentum density in the EM fields</b>
</p> </p>
<div class="eqlabel" id="org4c00e17"> <div class="eqlabel" id="org0649d73">
<p> <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"> <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">
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</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org62e68b8"> <div class="alteqlabels" id="org7dc7b00">
</div> </div>
@@ -1655,18 +1655,18 @@ This is thus simply a conservation law for momentum, with
<p> <p>
In a region in which the mechanical momentum is not changing due to external influences, we then have the In a region in which the mechanical momentum is not changing due to external influences, we then have the
</p> </p>
<div class="main div" id="org55398a2"> <div class="main div" id="org72dfdb8">
<p> <p>
<b>Continuity equation for EM momentum</b> <b>Continuity equation for EM momentum</b>
</p> </p>
<div class="eqlabel" id="org02bb382"> <div class="eqlabel" id="org89a6b5f">
<p> <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"> <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">
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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"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgf040d5c"> <div class="alteqlabels" id="orgd3a1580">
</div> </div>
@@ -1699,7 +1699,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_ce_mst.html
+12 -12
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1667,18 +1667,18 @@ and similarly for \({\boldsymbol B}\). We thus get
<p> <p>
This expression can be greatly simplified by introducing the This expression can be greatly simplified by introducing the
</p> </p>
<div class="main div" id="org6a32dc6"> <div class="main div" id="orge483101">
<p> <p>
<b>Maxwell stress tensor</b> <b>Maxwell stress tensor</b>
</p> </p>
<div class="eqlabel" id="org46015d8"> <div class="eqlabel" id="org42e9f16">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org9675080"> <div class="alteqlabels" id="orgf3e9098">
</div> </div>
@@ -1699,18 +1699,18 @@ The element \(T_{ij}\) represents the force per unit area in the \(i\) direction
<p> <p>
We then obtain the We then obtain the
</p> </p>
<div class="main div" id="org0171e33"> <div class="main div" id="org3664d29">
<p> <p>
<b>EM force per unit volume</b> <b>EM force per unit volume</b>
</p> </p>
<div class="eqlabel" id="org5306def"> <div class="eqlabel" id="org6136353">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgc0f4f98"> <div class="alteqlabels" id="org7273a9f">
</div> </div>
@@ -1726,18 +1726,18 @@ We then obtain the
<p> <p>
where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
</p> </p>
<div class="main div" id="org28ab853"> <div class="main div" id="org2a4f340">
<p> <p>
<b>Total force on charges in volume</b> <b>Total force on charges in volume</b>
</p> </p>
<div class="eqlabel" id="orgb821a87"> <div class="eqlabel" id="org0298272">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgdcbb5c3"> <div class="alteqlabels" id="org1024eff">
</div> </div>
@@ -1770,7 +1770,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_ce_poy.html
+21 -21
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -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 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, at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\). Thus,
</p> </p>
<div class="eqlabel" id="org935dce9"> <div class="eqlabel" id="org0c693e0">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgad7d22c"> <div class="alteqlabels" id="orge34a4b7">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (8.6)</li> <li>Gr (8.6)</li>
</ul> </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, Substituting this in <a href="./emd_ce_poy.html#dWdt_intEJ">dWdt_intEJ</a> and using the divergence theorem,
we obtain we obtain
</p> </p>
<div class="main div" id="orgdbc3e5c"> <div class="main div" id="orgb7d7869">
<p> <p>
<b>Poynting's theorem</b> <b>Poynting's theorem</b>
</p> </p>
<div class="eqlabel" id="orgc64c555"> <div class="eqlabel" id="org1a4b16b">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgf1ac980"> <div class="alteqlabels" id="org4feadff">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (8.9)</li> <li>Gr (8.9)</li>
</ul> </ul>
@@ -1720,18 +1720,18 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
<p> <p>
Energy per unit time, per unit area carried by EM fields: given by the Energy per unit time, per unit area carried by EM fields: given by the
</p> </p>
<div class="core div" id="orga8e7c87"> <div class="core div" id="orgdc56ce9">
<p> <p>
<b>Poynting vector</b> <b>Poynting vector</b>
</p> </p>
<div class="eqlabel" id="org7fb23da"> <div class="eqlabel" id="orga61d997">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org8150e29"> <div class="alteqlabels" id="orgb7a4cb9">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (8.10)</li> <li>Gr (8.10)</li>
</ul> </ul>
@@ -1750,18 +1750,18 @@ Energy per unit time, per unit area carried by EM fields: given by the
<p> <p>
We can thus express Poynting's theorem more compactly: We can thus express Poynting's theorem more compactly:
</p> </p>
<div class="core div" id="org41e1182"> <div class="core div" id="orgd48d57f">
<p> <p>
<b>Poynting's theorem</b> (integral form) <b>Poynting's theorem</b> (integral form)
</p> </p>
<div class="eqlabel" id="orgfa25d4a"> <div class="eqlabel" id="org32c9daf">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org264bb7a"> <div class="alteqlabels" id="org59d2761">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (8.11)</li> <li>Gr (8.11)</li>
</ul> </ul>
@@ -1780,18 +1780,18 @@ We can thus express Poynting's theorem more compactly:
<p> <p>
where we have defined the total where we have defined the total
</p> </p>
<div class="core div" id="orga6dad30"> <div class="core div" id="org5e10531">
<p> <p>
<b>Energy in electromagnetic fields</b> <b>Energy in electromagnetic fields</b>
</p> </p>
<div class="eqlabel" id="org16b17ef"> <div class="eqlabel" id="orgad6f50f">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgdf4c40f"> <div class="alteqlabels" id="orga3df7bc">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (8.5)</li> <li>Gr (8.5)</li>
</ul> </ul>
@@ -1820,18 +1820,18 @@ Then,
\] \]
so we get the so we get the
</p> </p>
<div class="core div" id="orgfd64b94"> <div class="core div" id="orgb7a11e9">
<p> <p>
<b>Poynting theorem</b> (differential form) <b>Poynting theorem</b> (differential form)
</p> </p>
<div class="eqlabel" id="org47e48c8"> <div class="eqlabel" id="orga2e5c79">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orge4a2af6"> <div class="alteqlabels" id="org0cf780d">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (8.14)</li> <li>Gr (8.14)</li>
</ul> </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="org1083447">
<p> <p>
<b>Example: Joule heating</b> <b>Example: Joule heating</b>
</p> </p>
@@ -1911,7 +1911,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1614,8 +1614,8 @@ Table of contents
</svg></a><span class="headline-id">emd.emw</span></h3> </svg></a><span class="headline-id">emd.emw</span></h3>
<div class="outline-text-3" id="text-emd_emw"> <div class="outline-text-3" id="text-emd_emw">
<details class="prereq" id="org81c4de7"> <details class="prereq" id="org92d8d0c">
<summary id="org94e2b41"> <summary id="orgb97228e">
Prerequisites Prerequisites
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@@ -1624,8 +1624,8 @@ Prerequisites
</ul> </ul>
</details> </details>
<details class="objectives" id="org04214f1"> <details class="objectives" id="org5c24096">
<summary id="org840c379"> <summary id="orge645495">
Objectives Objectives
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@@ -1659,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_emw_ep.html
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<html lang="en"> <html lang="en">
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1644,18 +1644,18 @@ so for a monochromatic EM plan wave,
\] \]
or more succinctly: or more succinctly:
</p> </p>
<div class="main div" id="orge6fd2ac"> <div class="main div" id="orga5c9575">
<p> <p>
<b>Poynting vector of a monochromatic EM wave</b> <b>Poynting vector of a monochromatic EM wave</b>
</p> </p>
<div class="eqlabel" id="orgc1c96f7"> <div class="eqlabel" id="org72a5a32">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org907a483"> <div class="alteqlabels" id="org7e5d7a7">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (9.57)</li> <li>Gr (9.57)</li>
</ul> </ul>
@@ -1678,7 +1678,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
<p> <p>
Similary, we get the Similary, we get the
</p> </p>
<div class="main div" id="orga2199bc"> <div class="main div" id="orgd6eada9">
<p> <p>
<b>Momentum density of a monochromatic EM wave</b> <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>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_emw_mpw.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
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<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -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 so \({\bf E}\) and \({\bf B}\) are mutually perpendicular, and
</p> </p>
<div class="eqlabel" id="org4864134"> <div class="eqlabel" id="org44bb1e2">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org6d00653"> <div class="alteqlabels" id="org0064233">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (9.47)</li> <li>Gr (9.47)</li>
</ul> </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 Generalizing to propagation in the direction of an arbitrary wavevector
\({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
</p> </p>
<div class="core div" id="org9581019"> <div class="core div" id="org83aeece">
<p> <p>
<b>E and B fields for a monochromatic EM plane wave</b> <b>E and B fields for a monochromatic EM plane wave</b>
</p> </p>
<div class="eqlabel" id="org98b7671"> <div class="eqlabel" id="org82e71ea">
<p> <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"> <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">
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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"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orge381054"> <div class="alteqlabels" id="orgb1b901e">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (9.49)</li> <li>Gr (9.49)</li>
</ul> </ul>
@@ -1724,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emd_emw_we.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1641,18 +1641,18 @@ These take the form of coupled first-order partial differential equations for \(
Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\), Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
we get the we get the
</p> </p>
<div class="core div" id="org672468f"> <div class="core div" id="orge19d256">
<p> <p>
<b>Wave equations for electric and magnetic fields in vacuum</b> <b>Wave equations for electric and magnetic fields in vacuum</b>
</p> </p>
<div class="eqlabel" id="orgf656d76"> <div class="eqlabel" id="orgff19c15">
<p> <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"> <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">
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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"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org1ac2ec3"> <div class="alteqlabels" id="orga8c7165">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (9.41)</li> <li>Gr (9.41)</li>
</ul> </ul>
@@ -1714,7 +1714,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emdm.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
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<!DOCTYPE html> <!DOCTYPE html>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1648,18 +1648,18 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
\] \]
We therefore have the We therefore have the
</p> </p>
<div class="core div" id="orge0518b1"> <div class="core div" id="orgc00904a">
<p> <p>
<b>Polarization current density</b> <b>Polarization current density</b>
</p> </p>
<div class="eqlabel" id="org62db796"> <div class="eqlabel" id="org3742df8">
<p> <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"> <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">
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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"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgebf1aab"> <div class="alteqlabels" id="org6073635">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.48)</li> <li>Gr (7.48)</li>
</ul> </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 polarization changes). We can check consistency with the continuity equation
associated to the conservation of bound charges: associated to the conservation of bound charges:
</p> </p>
<aside id="orgfd82638"> <aside id="orgff2c75f">
<p> <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. 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> </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, In view of this: total charge density can be separated into 2 parts,
<b>free</b> and <b>bound</b>: <b>free</b> and <b>bound</b>:
</p> </p>
<div class="main div" id="org5bd748a"> <div class="main div" id="org7866a36">
<div class="eqlabel" id="org94dfd02"> <div class="eqlabel" id="orge94a27e">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgd2d4d1d"> <div class="alteqlabels" id="orgbb91da6">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.49)</li> <li>Gr (7.49)</li>
</ul> </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 and current can be separated into three parts, <b>free</b>, <b>bound</b> and
<b>polarization</b>: <b>polarization</b>:
</p> </p>
<div class="main div" id="org7ad8d3b"> <div class="main div" id="org765eeee">
<div class="eqlabel" id="org19a6c97"> <div class="eqlabel" id="org49803b2">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgac73457"> <div class="alteqlabels" id="orgc87ffc7">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.50)</li> <li>Gr (7.50)</li>
</ul> </ul>
@@ -1767,7 +1767,7 @@ Gauss's law: can be rewritten
\] \]
where (as in static case) where (as in static case)
</p> </p>
<div class="core div" id="org65ac5cc"> <div class="core div" id="org91a1e38">
<p> <p>
\[ \[
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P} {\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
@@ -1788,7 +1788,7 @@ or
\] \]
where as before where as before
</p> </p>
<div class="core div" id="org7e8e1b5"> <div class="core div" id="org0c2e84f">
<p> <p>
\[ \[
{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M} {\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> <p>
In terms of free charges and currents, we thus get In terms of free charges and currents, we thus get
</p> </p>
<div class="core div" id="org089bb13"> <div class="core div" id="orgeeb0b35">
<p> <p>
<b>Maxwell's equations</b> <i>(in matter)</i> <b>Maxwell's equations</b> <i>(in matter)</i>
</p> </p>
<div class="eqlabel" id="orgfabaa36"> <div class="eqlabel" id="orgb9e6675">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgaba63f2"> <div class="alteqlabels" id="org6367cef">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.55)</li> <li>Gr (7.55)</li>
</ul> </ul>
@@ -1847,15 +1847,15 @@ This must all be complemented by the <b>constitutive relations</b> giving \({\bf
in terms of \({\bf E}\) and \({\bf B}\). in terms of \({\bf E}\) and \({\bf B}\).
For the restricted case of linear media: For the restricted case of linear media:
</p> </p>
<div class="main div" id="org435d93d"> <div class="main div" id="orgd78e0b5">
<div class="eqlabel" id="orgb5db623"> <div class="eqlabel" id="org86e1a29">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org6a8a0ba"> <div class="alteqlabels" id="org7ea2c8c">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.56,7.57)</li> <li>Gr (7.56,7.57)</li>
</ul> </ul>
@@ -1898,7 +1898,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emdm_Me_bc.html
+19 -19
View File
@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1616,18 +1616,18 @@ Table of contents
<p> <p>
Discontinuities between different media, deduced from Discontinuities between different media, deduced from
</p> </p>
<div class="core div" id="org31953cb"> <div class="core div" id="org876a60b">
<p> <p>
<b>Maxwell's equations</b> <i>(in matter)</i>, <i>integral form</i> <b>Maxwell's equations</b> <i>(in matter)</i>, <i>integral form</i>
</p> </p>
<div class="eqlabel" id="org853b142"> <div class="eqlabel" id="orga776629">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org734ccfd"> <div class="alteqlabels" id="org6c05186">
</div> </div>
@@ -1645,15 +1645,15 @@ Discontinuities between different media, deduced from
Applying \((i)\) to wafer-thin Gaussian pillbox straddling boundary between 2 materials: 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 \({\bf D}_1 \cdot {\bf a} - {\bf D}_2 \cdot {\bf a} = \sigma_f a\) so
</p> </p>
<div class="main div" id="org74cc983"> <div class="main div" id="org8d20540">
<div class="eqlabel" id="org49821b6"> <div class="eqlabel" id="org7926d65">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgb4f898e"> <div class="alteqlabels" id="orge1fcf9c">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.60)</li> <li>Gr (7.60)</li>
</ul> </ul>
@@ -1672,9 +1672,9 @@ D^{\perp}_1 - D^{\perp}_2 = \sigma_f
<p> <p>
Same reasoning applied to \((ii)\) gives <a href="./ems_ms_vp_mbc.html#Bdisc">Bdisc</a> Same reasoning applied to \((ii)\) gives <a href="./ems_ms_vp_mbc.html#Bdisc">Bdisc</a>
</p> </p>
<div class="main div" id="org167c3b0"> <div class="main div" id="orgfa3145b">
<div class="eqlabel" id="orgd15716f"> <div class="eqlabel" id="orgddc00f3">
<div class="alteqlabels" id="orgc56e074"> <div class="alteqlabels" id="org80b4003">
</div> </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} = 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 -\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}\). Limit of small loop: flux vanishes, therefore
</p> </p>
<div class="main div" id="org944c540"> <div class="main div" id="org2dd2bbb">
<p> <p>
\[ \[
{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0 {\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
\(I_{f_{enc}} = {\bf K}_f \cdot (\hat{\bf n} \times {\bf l}) = ({\bf K}_f \times \hat{\bf n}) \cdot {\bf l}\) \(I_{f_{enc}} = {\bf K}_f \cdot (\hat{\bf n} \times {\bf l}) = ({\bf K}_f \times \hat{\bf n}) \cdot {\bf l}\)
and thus (as we got before in <a href="./emsm_msm_H_A.html#Hdisc">Hdisc</a>) and thus (as we got before in <a href="./emsm_msm_H_A.html#Hdisc">Hdisc</a>)
</p> </p>
<div class="main div" id="org44aa4fb"> <div class="main div" id="orgd223e2f">
<p> <p>
\[ \[
{\bf H}_1^{\parallel} - {\bf H}_2^{\parallel} = {\bf K}_f \times \hat{\bf n} {\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> <p>
In case of linear media: can be expressed in terms of \({\bf E}\) and \({\bf B}\) alone: In case of linear media: can be expressed in terms of \({\bf E}\) and \({\bf B}\) alone:
</p> </p>
<div class="main div" id="org74a1c76"> <div class="main div" id="orgaafde6e">
<div class="eqlabel" id="orgbb2af38"> <div class="eqlabel" id="orge514e83">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org98cdce0"> <div class="alteqlabels" id="org786e6ec">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.64)</li> <li>Gr (7.64)</li>
</ul> </ul>
@@ -1748,14 +1748,14 @@ In case of linear media: can be expressed in terms of \({\bf E}\) and \({\bf B}
<p> <p>
If there is no free charge and no free current at boundary: If there is no free charge and no free current at boundary:
</p> </p>
<div class="eqlabel" id="orga2465cb"> <div class="eqlabel" id="orga74521d">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org06ac001"> <div class="alteqlabels" id="orgac68624">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.64)</li> <li>Gr (7.64)</li>
</ul> </ul>
@@ -1792,7 +1792,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emdm_emwm.html
+7 -7
View File
@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1614,8 +1614,8 @@ Table of contents
</svg></a><span class="headline-id">emdm.emwm</span></h3> </svg></a><span class="headline-id">emdm.emwm</span></h3>
<div class="outline-text-3" id="text-emdm_emwm"> <div class="outline-text-3" id="text-emdm_emwm">
<details class="prereq" id="orgf2f0eda"> <details class="prereq" id="org499ec58">
<summary id="orged680b6"> <summary id="org504cdbb">
Prerequisites Prerequisites
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@@ -1625,8 +1625,8 @@ Prerequisites
</ul> </ul>
</details> </details>
<details class="objectives" id="orge99509d"> <details class="objectives" id="org76e4576">
<summary id="orge43e5c1"> <summary id="org77d80f7">
Objectives Objectives
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emdm_emwm_ad.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emdm_emwm_ad_c.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1659,7 +1659,7 @@ Applying \({\boldsymbol \nabla} \times\) to the curl equations gives the modifie
\] \]
with plane-wave solutions 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 with complex-valued wavevector
\[ \[
@@ -1733,7 +1733,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emdm_emwm_plm.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1632,15 +1632,15 @@ For linear medium:
\] \]
If the medium is homogeneous (no spatial dependence of \(\varepsilon\) or \(\mu\)), If the medium is homogeneous (no spatial dependence of \(\varepsilon\) or \(\mu\)),
</p> </p>
<div class="main div" id="org7ad5a88"> <div class="main div" id="orgead2b1c">
<div class="eqlabel" id="orgde6c1b4"> <div class="eqlabel" id="org9916855">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org8b72259"> <div class="alteqlabels" id="org5111273">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (9.65)</li> <li>Gr (9.65)</li>
</ul> </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 where the index of refraction of the material is defined as
</p> </p>
<div class="main div" id="org56849ee"> <div class="main div" id="org5237a1a">
<p> <p>
<b>Index of refraction</b> <b>Index of refraction</b>
</p> </p>
<div class="eqlabel" id="orgff8c108"> <div class="eqlabel" id="org6cb7362">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgf8eaa9d"> <div class="alteqlabels" id="orgc12f464">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (9.69)</li> <li>Gr (9.69)</li>
</ul> </ul>
@@ -1743,7 +1743,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emdm_emwm_refl.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1649,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emdm_emwm_refl_ni.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1671,14 +1671,14 @@ E_{0_I} - E_{0_R} = \beta E_{0_T}, \hspace{1cm}
Solving these coupled equations, we can write Solving these coupled equations, we can write
outgoing amplitudes in terms of incident ones: outgoing amplitudes in terms of incident ones:
</p> </p>
<div class="eqlabel" id="org1b7d37e"> <div class="eqlabel" id="orgf6ae25d">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org2907578"> <div class="alteqlabels" id="org17eb76a">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (9.82)</li> <li>Gr (9.82)</li>
</ul> </ul>
@@ -1744,7 +1744,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emdm_emwm_refl_oi.html
+15 -15
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1635,14 +1635,14 @@ Transmitted wave:
\] \]
All waves have the same frequency \(\omega\). Since \(\omega = k v\), the three wavevectors are related by All waves have the same frequency \(\omega\). Since \(\omega = k v\), the three wavevectors are related by
</p> </p>
<div class="eqlabel" id="org792ed2b"> <div class="eqlabel" id="org512368e">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org1997374"> <div class="alteqlabels" id="org1361b67">
</div> </div>
@@ -1665,7 +1665,7 @@ These forms for incident, reflected and transmitted wave can be substituted in t
<p> <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 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> </p>
<div class="core div" id="orgbf226b1"> <div class="core div" id="org86dbe79">
<p> <p>
<b>First law of reflection:</b> <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. 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 with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
and the angle of refraction (\(\theta_T\)) obey the following laws: and the angle of refraction (\(\theta_T\)) obey the following laws:
</p> </p>
<div class="core div" id="org000f37c"> <div class="core div" id="orge55fc76">
<p> <p>
<b>Law of reflection</b> <b>Law of reflection</b>
\[ \[
@@ -1697,14 +1697,14 @@ and the angle of refraction (\(\theta_T\)) obey the following laws:
<p> <p>
This takes care of the spatially-dependent exponential factors in the boundary conditions. The coefficients must further obey This takes care of the spatially-dependent exponential factors in the boundary conditions. The coefficients must further obey
</p> </p>
<div class="eqlabel" id="orgdc1db4c"> <div class="eqlabel" id="orga2e8f02">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgf69a0d1"> <div class="alteqlabels" id="org12decff">
</div> </div>
@@ -1750,18 +1750,18 @@ while the third equation becomes
\] \]
Writing everything in terms of the incident amplitude, we get Writing everything in terms of the incident amplitude, we get
</p> </p>
<div class="main div" id="org7e347a7"> <div class="main div" id="org29006a2">
<p> <p>
<b>Fresnel's equations for reflection and transmission amplitudes (parallel case)</b> <b>Fresnel's equations for reflection and transmission amplitudes (parallel case)</b>
</p> </p>
<div class="eqlabel" id="org5c049e2"> <div class="eqlabel" id="orgd431e6b">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org50e3086"> <div class="alteqlabels" id="orgd3db507">
</div> </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>. 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 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
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<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
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</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1633,7 +1633,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
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<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
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</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
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<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
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</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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</summary> </summary>
<ul> <ul>
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<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
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</svg></a><span class="headline-id">emf</span></h2> </svg></a><span class="headline-id">emf</span></h2>
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Prerequisites Prerequisites
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Objectives Objectives
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<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
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<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
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</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
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</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
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</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
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<p> <p>
A more aesthetic choice is the <b>Lorenz gauge</b>: A more aesthetic choice is the <b>Lorenz gauge</b>:
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<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"> <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">
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@@ -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 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
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<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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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\] \]
while the equation for \(\phi\) becomes while the equation for \(\phi\) becomes
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<div class="eqlabel" id="org92ed269"> <div class="eqlabel" id="org29d69e0">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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\] \]
These can be written compactly upon introducing a new operator: the These can be written compactly upon introducing a new operator: the
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<div class="core div" id="orgd77f9a6"> <div class="core div" id="orga269826">
<p> <p>
<b>d'Alembertian operator</b> <b>d'Alembertian operator</b>
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<div class="eqlabel" id="orgc902ec7"> <div class="eqlabel" id="org39fa795">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgac35877"> <div class="alteqlabels" id="orgf4aa848">
</div> </div>
@@ -1700,18 +1700,18 @@ These can be written compactly upon introducing a new operator: the
<p> <p>
so we get the so we get the
</p> </p>
<div class="core div" id="org12b2ff9"> <div class="core div" id="org65e01f8">
<p> <p>
<b>Inhomogeneous Maxwell equations</b> <i>(Lorenz gauge)</i> <b>Inhomogeneous Maxwell equations</b> <i>(Lorenz gauge)</i>
</p> </p>
<div class="eqlabel" id="orge48b8c0"> <div class="eqlabel" id="orgc262f70">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org140821b"> <div class="alteqlabels" id="orgc540066">
</div> </div>
@@ -1764,7 +1764,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/emf_svp.html
+13 -13
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -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 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): field as a pure curl (since its divergence always vanishes):
</p> </p>
<div class="core div" id="orgb63eb69"> <div class="core div" id="org913ae4e">
<p> <p>
\[ \[
{\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A} {\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. so this can be written as the gradient of a scalar.
Making the choice \(-{\boldsymbol \nabla} \phi\) for this, we get Making the choice \(-{\boldsymbol \nabla} \phi\) for this, we get
</p> </p>
<div class="core div" id="org23c61e0"> <div class="core div" id="orgcc3047e">
<div class="eqlabel" id="org51cbd1b"> <div class="eqlabel" id="org50da26e">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org25412c6"> <div class="alteqlabels" id="org3a04a5a">
</div> </div>
@@ -1673,15 +1673,15 @@ Making the choice \(-{\boldsymbol \nabla} \phi\) for this, we get
<p> <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 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> </p>
<div class="main div" id="orgf26abb4"> <div class="main div" id="orge407b00">
<div class="eqlabel" id="org986b9b3"> <div class="eqlabel" id="orgad33676">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgcd04ad9"> <div class="alteqlabels" id="org4f2824a">
</div> </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 \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\), 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> </p>
<div class="main div" id="org135124e"> <div class="main div" id="orga4d93c1">
<div class="eqlabel" id="org73667ac"> <div class="eqlabel" id="orgbbedeec">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org3f09a58"> <div class="alteqlabels" id="orgb9a7aee">
</div> </div>
@@ -1744,7 +1744,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/ems.html
+3 -3
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1640,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/ems_ca.html
+7 -7
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1618,8 +1618,8 @@ Table of contents
<li>Gr 3</li> <li>Gr 3</li>
</ul> </ul>
<details class="prereq" id="org1512362"> <details class="prereq" id="org587aea3">
<summary id="orgf95a20b"> <summary id="orgf03fd4d">
Prerequisites Prerequisites
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@@ -1627,8 +1627,8 @@ Prerequisites
</ul> </ul>
</details> </details>
<details class="objectives" id="org60b3244"> <details class="objectives" id="orgedf1d47">
<summary id="org14c49b9"> <summary id="orgbfc64ab">
Objectives Objectives
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/ems_ca_fe.html
+7 -7
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -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 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> from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
</p> </p>
<div class="main div" id="org91ccc03"> <div class="main div" id="orgce1c4bf">
<p> <p>
</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 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> explicit construction <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
</p> </p>
<div class="main div" id="orge494253"> <div class="main div" id="orge8cb758">
<p> <p>
</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> <a href="./ems_es_ep_PL.html#Poi">🐟</a>
</p> </p>
<div class="core div" id="org4db04a9"> <div class="core div" id="org65cda5b">
<p> <p>
</p> </p>
@@ -1674,7 +1674,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
<p> <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> 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> </p>
<div class="core div" id="orgd28c357"> <div class="core div" id="org9318a15">
<p> <p>
</p> </p>
@@ -1712,7 +1712,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/ems_ca_fe_L.html
+20 -20
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1630,14 +1630,14 @@ In one dimension, the potential is a single-variable
function \(\phi (x)\) and the Laplace equation reads function \(\phi (x)\) and the Laplace equation reads
</p> </p>
<div class="eqlabel" id="orgc211f0a"> <div class="eqlabel" id="orgd38ad90">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org36bff82"> <div class="alteqlabels" id="org23686dc">
</div> </div>
@@ -1652,14 +1652,14 @@ function \(\phi (x)\) and the Laplace equation reads
<p> <p>
The solution to this is The solution to this is
</p> </p>
<div class="eqlabel" id="orgf5f202e"> <div class="eqlabel" id="org78665cb">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org25a65ef"> <div class="alteqlabels" id="orge0dbdb5">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (3.6)</li> <li>Gr (3.6)</li>
</ul> </ul>
@@ -1718,14 +1718,14 @@ In two dimensions, the potential becomes a function
of two variables (here: \(x\) and \(y\)), so Laplace's of two variables (here: \(x\) and \(y\)), so Laplace's
equation now reads equation now reads
</p> </p>
<div class="eqlabel" id="org4bb6b92"> <div class="eqlabel" id="orgd4e7f28">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org482a454"> <div class="alteqlabels" id="orga64f9ad">
</div> </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 \(S_R({\bf r})\) of any radius \(R\) centered on this point
(and of course not containing any charges), (and of course not containing any charges),
</p> </p>
<div class="eqlabel" id="org96c7b40"> <div class="eqlabel" id="org598b2e6">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org550802d"> <div class="alteqlabels" id="orgdb48832">
</div> </div>
@@ -1797,8 +1797,8 @@ a point equals its value averaged over a sphere
\] \]
</p> </p>
<details id="org8644d79"> <details id="orgd997cbe">
<summary id="orgbe634df"> <summary id="org6b22e3f">
<strong>Physicist's proof</strong> <strong>Physicist's proof</strong>
</summary> </summary>
<p> <p>
@@ -1860,8 +1860,8 @@ proving the theorem.
</p> </p>
</details> </details>
<details id="orgb4be215"> <details id="org2ff5eec">
<summary id="orgb09f919"> <summary id="orgc0446a1">
<strong>Formal proof</strong> <strong>Formal proof</strong>
</summary> </summary>
@@ -1911,14 +1911,14 @@ we get the following general
<p> <p>
<b>Theorem</b>: <b>Theorem</b>:
</p> </p>
<div class="eqlabel" id="org51f491a"> <div class="eqlabel" id="orge37257a">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org9bebf97"> <div class="alteqlabels" id="orgb3a7f07">
</div> </div>
@@ -1971,19 +1971,19 @@ are necessarily positive, we thus require \(f_x &gt; 0\), \(f_y &gt; 0\) and \(f
of the \(f_x + f_y + f_z = 0\) condition above. of the \(f_x + f_y + f_z = 0\) condition above.
</p> </p>
<div class="eqlabel" id="org02999c2"> <div class="eqlabel" id="org39eb572">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org3cd554f"> <div class="alteqlabels" id="org2cc4d11">
</div> </div>
</div> </div>
<div class="info div" id="org8708715"> <div class="info div" id="orgcd59178">
<p> <p>
<b>Earnshaw's theorem (physical version)</b> <br> <b>Earnshaw's theorem (physical version)</b> <br>
</p> </p>
@@ -2102,7 +2102,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/ems_ca_fe_g.html
+9 -9
View File
@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -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"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 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> </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="outline-text-5" id="text-ems_ca_fe_g">
<div class="info div" id="orgce63324"> <div class="info div" id="org60c1b11">
<p> <p>
<b>George Green</b> <b>George Green</b>
</p> </p>
<aside id="org3d8c3b4"> <aside id="org0b1da41">
<p> <p>
See a <a href="https://en.wikipedia.org/wiki/George%5C_Green%5C_(mathematician)">short bio on wikipedia</a> See a <a href="https://en.wikipedia.org/wiki/George%5C_Green%5C_(mathematician)">short bio on wikipedia</a>
</p> </p>
@@ -1650,14 +1650,14 @@ and
\] \]
Substituting this in the divergence theorem gives <b>Green's first identity</b> Substituting this in the divergence theorem gives <b>Green's first identity</b>
</p> </p>
<div class="eqlabel" id="org86176bc"> <div class="eqlabel" id="org3d01833">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org8e9389d"> <div class="alteqlabels" id="orga726fcd">
<ul class="org-ul"> <ul class="org-ul">
<li>J (1.34)</li> <li>J (1.34)</li>
</ul> </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 interchanged, and subtract the result, we obtain another useful result known as
<b>Green's second identity</b> or <b>Green's theorem</b> <b>Green's second identity</b> or <b>Green's theorem</b>
</p> </p>
<div class="eqlabel" id="orgee2e293"> <div class="eqlabel" id="org964d072">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org20f4498"> <div class="alteqlabels" id="orge29c408">
<ul class="org-ul"> <ul class="org-ul">
<li>J (1.35)</li> <li>J (1.35)</li>
</ul> </ul>
@@ -1718,7 +1718,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>
build/ems_ca_fe_uP.html
+11 -11
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@@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1635,14 +1635,14 @@ This yields
The first term on the left-hand side vanishes since \(\Phi\) satisfies Laplace. 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 The right-hand side can be made to vanish if \(\Phi\) obeys either
</p> </p>
<div class="eqlabel" id="org13e35fc"> <div class="eqlabel" id="org0acb722">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgccd338e"> <div class="alteqlabels" id="org03f8c83">
</div> </div>
@@ -1654,14 +1654,14 @@ The right-hand side can be made to vanish if \(\Phi\) obeys either
<p> <p>
or or
</p> </p>
<div class="eqlabel" id="org68968dd"> <div class="eqlabel" id="org3d1a359">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org062fea2"> <div class="alteqlabels" id="orga2c0bab">
</div> </div>
@@ -1689,19 +1689,19 @@ additive constant.
<p> <p>
We can thus finally state the We can thus finally state the
</p> </p>
<div class="eqlabel" id="org154b9ea"> <div class="eqlabel" id="orgf4dd440">
<p> <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"> <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="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</p> </p>
<div class="alteqlabels" id="org43b26fb"> <div class="alteqlabels" id="org786cfb7">
</div> </div>
</div> </div>
<div class="core div" id="orgc742981"> <div class="core div" id="org9b96393">
<p> <p>
<b>Uniqueness Theorem</b> <b>Uniqueness Theorem</b>
</p> </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 and corollaries and that things are
</p> </p>
<div class="info div" id="org2adbb79"> <div class="info div" id="org333667e">
<p> <p>
<b>Comment/warning</b>: <b>uniqueness theorem on uniqueness theorems</b> <br> <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 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>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-15 Tue 08:10 --> <!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary> </summary>
<ul> <ul>
<li> <li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span> <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li> </li>
@@ -1645,7 +1645,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-15 Tue 08:10</p> <p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

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