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Update 2022-03-07 20:40

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Jean-Sébastien 2022-03-07 20:40:36 +01:00
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194 changed files with 1487 additions and 5980 deletions
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12
build/a.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1645,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

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build/a_l.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1728,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1645,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

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build/c_m.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1651,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_cs.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1647,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

24
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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1667,14 +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="org3f7fe1a"> <div class="eqlabel" id="orgfd5fb6e">
<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="orgcf66eee"> <div class="alteqlabels" id="org16d29dc">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr4(1.79)</li> <li>Gr4(1.79)</li>
</ul> </ul>
@ -1695,14 +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="org77a88dd"> <div class="eqlabel" id="org1dfbb35">
<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="orga1ce1b4"> <div class="alteqlabels" id="org035dafc">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr4(2.21)</li> <li>Gr4(2.21)</li>
</ul> </ul>
@ -1723,14 +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="orgb7e3d3b"> <div class="eqlabel" id="org747dbea">
<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="orgd068b29"> <div class="alteqlabels" id="orgcd134fe">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr4(2.21)</li> <li>Gr4(2.21)</li>
</ul> </ul>
@ -1779,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1640,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

44
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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1631,14 +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="org3e4cc27"> <div class="eqlabel" id="org348038f">
<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="org6cfcb56"> <div class="alteqlabels" id="org5f5a1a3">
</div> </div>
@ -1663,14 +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="org115ed32"> <div class="eqlabel" id="orgbdd14bc">
<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="org965d0cb"> <div class="alteqlabels" id="orga250579">
</div> </div>
@ -1693,14 +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="orgdb194fb"> <div class="eqlabel" id="org6494f4b">
<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="org96ce6ff"> <div class="alteqlabels" id="orgaafdc77">
</div> </div>
@ -1716,14 +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="org40e244c"> <div class="eqlabel" id="orga40a4ab">
<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="org6796383"> <div class="alteqlabels" id="orgc7eb689">
</div> </div>
@ -1744,14 +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="org7c9e5d8"> <div class="eqlabel" id="orgb45a655">
<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="org94fd9f5"> <div class="alteqlabels" id="org5dfccd6">
</div> </div>
@ -1768,14 +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="org25fdd4e"> <div class="eqlabel" id="orgdca9c80">
<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="org3d0214d"> <div class="alteqlabels" id="org52febbe">
</div> </div>
@ -1792,14 +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="org082892a"> <div class="eqlabel" id="org26cc7dd">
<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="orge37528b"> <div class="alteqlabels" id="orgeafd92e">
</div> </div>
@ -1818,14 +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="orgf48e07a"> <div class="eqlabel" id="org9c1e636">
<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="orge21efb8"> <div class="alteqlabels" id="org83d937d">
</div> </div>
@ -1859,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1650,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1652,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

46
build/c_m_dc_d2.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1623,9 +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-org3814a3d" class="outline-6"> <div id="outline-container-orgc12bd11" class="outline-6">
<h6 id="org3814a3d"><a href="#org3814a3d">Divergence of gradient</a></h6> <h6 id="orgc12bd11"><a href="#orgc12bd11">Divergence of gradient</a></h6>
<div class="outline-text-6" id="text-org3814a3d"> <div class="outline-text-6" id="text-orgc12bd11">
<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
@ -1634,44 +1626,44 @@ given by the Laplacian of the corresponding vector elements.
</div> </div>
</div> </div>
<div id="outline-container-orgc8aa2ed" class="outline-6"> <div id="outline-container-org0ea7ae4" class="outline-6">
<h6 id="orgc8aa2ed"><a href="#orgc8aa2ed">Curl of a gradient</a></h6> <h6 id="org0ea7ae4"><a href="#org0ea7ae4">Curl of a gradient</a></h6>
<div class="outline-text-6" id="text-orgc8aa2ed"> <div class="outline-text-6" id="text-org0ea7ae4">
<p> <p>
This always vanishes. This always vanishes.
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-orge3b0dcb" class="outline-6"> <div id="outline-container-org78d85e1" class="outline-6">
<h6 id="orge3b0dcb"><a href="#orge3b0dcb">Gradient of the divergence</a></h6> <h6 id="org78d85e1"><a href="#org78d85e1">Gradient of the divergence</a></h6>
<div class="outline-text-6" id="text-orge3b0dcb"> <div class="outline-text-6" id="text-org78d85e1">
<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-org1f5474a" class="outline-6"> <div id="outline-container-orge7f86bf" class="outline-6">
<h6 id="org1f5474a"><a href="#org1f5474a">Divergence of a curl</a></h6> <h6 id="orge7f86bf"><a href="#orge7f86bf">Divergence of a curl</a></h6>
<div class="outline-text-6" id="text-org1f5474a"> <div class="outline-text-6" id="text-orge7f86bf">
<p> <p>
This always vanishes. This always vanishes.
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-org4b8cb79" class="outline-6"> <div id="outline-container-orgbb8199b" class="outline-6">
<h6 id="org4b8cb79"><a href="#org4b8cb79">Curl of curl</a></h6> <h6 id="orgbb8199b"><a href="#orgbb8199b">Curl of curl</a></h6>
<div class="outline-text-6" id="text-org4b8cb79"> <div class="outline-text-6" id="text-orgbb8199b">
<div class="eqlabel" id="orgad52ac5"> <div class="eqlabel" id="orga59fc73">
<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="orgde36146"> <div class="alteqlabels" id="org384a104">
</div> </div>
@ -1702,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1651,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_dc_div.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1648,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_dc_g.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1672,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

36
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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1630,14 +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="org4bcb61a"> <div class="eqlabel" id="orged10256">
<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="org8c15975"> <div class="alteqlabels" id="orgfc8793a">
<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>
@ -1657,14 +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="org7784802"> <div class="eqlabel" id="org4b4bdf3">
<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="orgb3682de"> <div class="alteqlabels" id="org02913cc">
<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>
@ -1684,14 +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="org96b1b94"> <div class="eqlabel" id="org1df402a">
<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="org29d8167"> <div class="alteqlabels" id="orgfa5050d">
<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>
@ -1711,14 +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="orgfe2b08f"> <div class="eqlabel" id="org5048e1e">
<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="orgdce994d"> <div class="alteqlabels" id="orgd39dff6">
<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>
@ -1738,14 +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="orgbb8798b"> <div class="eqlabel" id="orgbd31452">
<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="org93d7ea5"> <div class="alteqlabels" id="org825aae4">
<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>
@ -1765,14 +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="orgd9fde5c"> <div class="eqlabel" id="org9c6b6a4">
<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="orgb1d2b5b"> <div class="alteqlabels" id="orgb030476">
<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>
@ -1813,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1647,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1659,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

24
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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1644,14 +1636,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
More generally, More generally,
</p> </p>
<div class="eqlabel" id="orgf5d25ef"> <div class="eqlabel" id="org492d0c0">
<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="org6fba030"> <div class="alteqlabels" id="org9b8bed8">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (1.100)</li> <li>Gr (1.100)</li>
</ul> </ul>
@ -1670,14 +1662,14 @@ More generally,
Since Since
</p> </p>
<div class="eqlabel" id="org9588ee0"> <div class="eqlabel" id="orga3b7a91">
<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="orgb0b4434"> <div class="alteqlabels" id="org98ac626">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (1.101)</li> <li>Gr (1.101)</li>
</ul> </ul>
@ -1693,14 +1685,14 @@ Since
<p> <p>
we have that we have that
</p> </p>
<div class="eqlabel" id="orgd223c1e"> <div class="eqlabel" id="org596ba00">
<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="org2ae8bc7"> <div class="alteqlabels" id="org620c397">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (1.102)</li> <li>Gr (1.102)</li>
</ul> </ul>
@ -1732,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_dd_div.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1665,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_ic.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1650,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_ic_ftc.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1648,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_ic_ftg.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1657,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_ic_gauss.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1651,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_ic_ip.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1658,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

30
build/c_m_ic_lsv.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1623,9 +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-org2486563" class="outline-6"> <div id="outline-container-org7fab499" class="outline-6">
<h6 id="org2486563"><a href="#org2486563">Line Integrals</a></h6> <h6 id="org7fab499"><a href="#org7fab499">Line Integrals</a></h6>
<div class="outline-text-6" id="text-org2486563"> <div class="outline-text-6" id="text-org7fab499">
<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}
@ -1654,9 +1646,9 @@ Integral over a closed loop:
</div> </div>
</div> </div>
<div id="outline-container-orgf44a1b8" class="outline-6"> <div id="outline-container-org543b5ed" class="outline-6">
<h6 id="orgf44a1b8"><a href="#orgf44a1b8">Surface Integrals</a></h6> <h6 id="org543b5ed"><a href="#org543b5ed">Surface Integrals</a></h6>
<div class="outline-text-6" id="text-orgf44a1b8"> <div class="outline-text-6" id="text-org543b5ed">
<p> <p>
\[ \[
\int_{\cal S} {\bf v} \cdot d{\bf a} \int_{\cal S} {\bf v} \cdot d{\bf a}
@ -1676,9 +1668,9 @@ Over a closed surface:
</div> </div>
</div> </div>
<div id="outline-container-org7c26dcf" class="outline-6"> <div id="outline-container-org015d5b1" class="outline-6">
<h6 id="org7c26dcf"><a href="#org7c26dcf">Volume Integrals</a></h6> <h6 id="org015d5b1"><a href="#org015d5b1">Volume Integrals</a></h6>
<div class="outline-text-6" id="text-org7c26dcf"> <div class="outline-text-6" id="text-org015d5b1">
<p> <p>
\[ \[
\int_{\cal V} T d\tau \int_{\cal V} T d\tau
@ -1719,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

16
build/c_m_ic_stokes.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1621,14 +1613,14 @@ Table of contents
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a><span class="headline-id">c.m.ic.stokes</span></h5> </svg></a><span class="headline-id">c.m.ic.stokes</span></h5>
<div class="outline-text-5" id="text-c_m_ic_stokes"> <div class="outline-text-5" id="text-c_m_ic_stokes">
<div class="eqlabel" id="org603ea63"> <div class="eqlabel" id="org39cf0b7">
<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">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.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="org95b24de"> <div class="alteqlabels" id="orga45987e">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (1.57)</li> <li>Gr (1.57)</li>
</ul> </ul>
@ -1673,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_uf.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1647,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_uf_cyl.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1660,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_uf_sph.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1660,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_uf_vi.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1663,7 +1655,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_va.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1649,7 +1641,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_va_cp.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1678,7 +1670,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_va_n.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1673,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_va_pds.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1707,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_va_sp.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1673,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_va_tp.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1720,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_vf.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1646,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_vf_helm.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1665,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/c_m_vf_pot.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1692,7 +1684,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

16
build/d.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1622,8 +1614,8 @@ Table of contents
</svg></a><span class="headline-id">d</span></h2> </svg></a><span class="headline-id">d</span></h2>
<div class="outline-text-2" id="text-d"> <div class="outline-text-2" id="text-d">
<details class="objectives" id="orge784719"> <details class="objectives" id="orgcc61564">
<summary id="orgfb0d046"> <summary id="org585bc00">
Objectives Objectives
</summary> </summary>
@ -1687,7 +1679,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/d_emd.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1667,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/d_emd_ce.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1662,7 +1654,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/d_emd_emw.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1665,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/d_emf.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1662,7 +1654,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

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build/d_ems.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1684,7 +1676,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/d_ems_ca.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1667,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/d_ems_ms.html
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1664,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/d_emsm.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1667,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/d_emsm_msm.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1665,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/d_m.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1675,7 +1667,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/d_red.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1669,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

20
build/emd.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1622,8 +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="org2aba3cc"> <details class="prereq" id="org417b126">
<summary id="org30ca6a6"> <summary id="org6bf52b1">
Prerequisites Prerequisites
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1632,8 +1624,8 @@ Prerequisites
</ul> </ul>
</details> </details>
<details class="objectives" id="org98af4b0"> <details class="objectives" id="org93939b2">
<summary id="org78e61da"> <summary id="org78ba2fd">
Objectives Objectives
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1674,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/emd_Fl.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1648,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

34
build/emd_Fl_Fl.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1652,18 +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="orgdfb0aad"> <div class="core div" id="org9cc8d48">
<p> <p>
<b>Electromotive force (or electromotance)</b>, <b>Electromotive force (or electromotance)</b>,
</p> </p>
<div class="eqlabel" id="org64afdaf"> <div class="eqlabel" id="orgb9f75a5">
<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="orgb3cda90"> <div class="alteqlabels" id="orgcb4fab4">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.9)</li> <li>Gr (7.9)</li>
</ul> </ul>
@ -1688,14 +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="orgec7b520"> <div class="eqlabel" id="orgb7faf93">
<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="orgf70f495"> <div class="alteqlabels" id="orgffe5f68">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.14)</li> <li>Gr (7.14)</li>
</ul> </ul>
@ -1710,18 +1702,18 @@ to the rate of change of the magnetic flux,
\] \]
so we obtain so we obtain
</p> </p>
<div class="core div" id="orgdfedc05"> <div class="core div" id="org80fe686">
<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="orge87df83"> <div class="eqlabel" id="orgfa5af5c">
<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="org7b32fe7"> <div class="alteqlabels" id="orga84de5d">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.15)</li> <li>Gr (7.15)</li>
</ul> </ul>
@ -1745,15 +1737,15 @@ for any loop (on a wire or not). Using Stokes' theorem,
\] \]
we obtain we obtain
</p> </p>
<div class="core div" id="orgafa0d15"> <div class="core div" id="org2c503e3">
<div class="eqlabel" id="org13d3c14"> <div class="eqlabel" id="org9023002">
<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="orgd662a28"> <div class="alteqlabels" id="orgb7f3817">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.16)</li> <li>Gr (7.16)</li>
</ul> </ul>
@ -1798,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1647,14 +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="org87af1ba"> <div class="eqlabel" id="org78f90c5">
<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="org95d3f8b"> <div class="alteqlabels" id="org0277370">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.31)</li> <li>Gr (7.31)</li>
</ul> </ul>
@ -1689,15 +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="org247bd37"> <div class="core div" id="org9994672">
<div class="eqlabel" id="orgdf4c394"> <div class="eqlabel" id="org2cf22c6">
<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="orge069b86"> <div class="alteqlabels" id="orgde5daac">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.34)</li> <li>Gr (7.34)</li>
</ul> </ul>
@ -1726,7 +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="org03fb4e8"> <div class="example div" id="orgd2854e0">
<p> <p>
<b>Example: energy in coaxial cable</b> <b>Example: energy in coaxial cable</b>
</p> </p>
@ -1776,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1649,14 +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="org197212f"> <div class="eqlabel" id="org766e684">
<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="org4709e6a"> <div class="alteqlabels" id="orga92dd51">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.22)</li> <li>Gr (7.22)</li>
</ul> </ul>
@ -1673,14 +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="orgb952a44"> <div class="eqlabel" id="org577027c">
<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="org53f3037"> <div class="alteqlabels" id="org21aff84">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.23)</li> <li>Gr (7.23)</li>
</ul> </ul>
@ -1695,7 +1687,7 @@ M_{12} = M_{21}
\] \]
</p> </p>
<div class="example div" id="org6347791"> <div class="example div" id="org9334fe0">
<p> <p>
<b>Example: solenoid in solenoid</b> <b>Example: solenoid in solenoid</b>
</p> </p>
@ -1743,14 +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="org667fec5"> <div class="eqlabel" id="org4807c27">
<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="org28a1a02"> <div class="alteqlabels" id="orga19e4d1">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.25)</li> <li>Gr (7.25)</li>
</ul> </ul>
@ -1773,7 +1765,7 @@ Inductance: measured in <b>henries</b> (\(H\)). \(H = V s/A\).
</p> </p>
<div class="example div" id="org00e72a3"> <div class="example div" id="org940f5ab">
<p> <p>
<b>Example: self-inductance of toroidal coil</b> <b>Example: self-inductance of toroidal coil</b>
</p> </p>
@ -1812,7 +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="org958ea6c"> <div class="example div" id="orgbeea66c">
<p> <p>
<b>Example: circuit</b> <b>Example: circuit</b>
</p> </p>
@ -1861,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

42
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1645,7 +1637,7 @@ law in integral form:
<div class="example div" id="org045463b"> <div class="example div" id="org56facb7">
<p> <p>
<b>Example: loop with time-dependent flux</b> <b>Example: loop with time-dependent flux</b>
</p> </p>
@ -1661,10 +1653,10 @@ through a horizontal circular region of radius \(R\).
<p> <p>
<b>Solution</b>: <b>Solution</b>:
amperian loop of radius \(s\), apply Faraday: amperian loop of radius \(r\), apply Faraday:
\[ \[
\oint {\bf E} \cdot d{\bf l} = E (2\pi s) = -\frac{d\Phi}{dt} = -\pi s^2 \frac{dB}{dt} \oint {\bf E} \cdot d{\bf l} = E (2\pi r) = -\frac{d\Phi}{dt} = -\pi r^2 \frac{dB}{dt}
\Rightarrow {\bf E} = -\frac{s}{2} \frac{dB}{dt} \hat{\boldsymbol \varphi}. \Rightarrow {\bf E} = -\frac{r}{2} \frac{dB}{dt} \hat{\boldsymbol \varphi}.
\] \]
Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz. Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
</p> </p>
@ -1672,7 +1664,7 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
</div> </div>
<div class="example div" id="org047ea10"> <div class="example div" id="org367fcfc">
<p> <p>
<b>Example: wheel with charged rim traversed by flux</b> <b>Example: wheel with charged rim traversed by flux</b>
</p> </p>
@ -1713,7 +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="org733cbdd"> <div class="example div" id="orgb2a276c">
<p> <p>
<b>Example: field from wire with time-dependent current</b> <b>Example: field from wire with time-dependent current</b>
</p> </p>
@ -1723,31 +1715,31 @@ Consider an infinitely long straight wire which carries current \(I(t)\).
</p> </p>
<p> <p>
<b>Task</b>: find the induced \({\bf E}\) field as a function of distance \(s\) from wire. <b>Task</b>: find the induced \({\bf E}\) field as a function of distance \(r\) from wire.
</p> </p>
<p> <p>
<b>Solution</b>: assuming we can use the quasistatic approximation, the <b>Solution</b>: assuming we can use the quasistatic approximation, the
magnetic field is \(B = \frac{\mu_0 I}{2\pi s}\) magnetic field is \(B = \frac{\mu_0 I}{2\pi r}\)
and circles the wire. Like \({\bf B}\) field of solenoid, \({\bf E}\) runs parallel and circles the wire. Like \({\bf B}\) field of solenoid, \({\bf E}\) runs parallel
to wire. Amperian loop with sides at distances \(s_0\) and \(s\): to wire. Amperian loop with sides at distances \(r_0\) and \(r\):
\[ \[
\oint {\bf E} \cdot d{\bf l} = E(s_0)l - E(s)l = -\frac{d}{dt} \int {\bf B} \cdot d{\bf a} \oint {\bf E} \cdot d{\bf l} = E(r_0)l - E(r)l = -\frac{d}{dt} \int {\bf B} \cdot d{\bf a}
= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \int_{s_0}^s \frac{ds'}{s'} = -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \int_{r_0}^s \frac{dr'}{r'}
= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \ln(s/s_0). = -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \ln(r/r_0).
\] \]
So: So:
\[ \[
{\bf E} (s) = \left[ \frac{\mu_0}{2\pi} \frac{dI}{dt} \ln s + K \right] \hat{\bf x} {\bf E} (r) = \left[ \frac{\mu_0}{2\pi} \frac{dI}{dt} \ln r + K \right] \hat{\bf x}
\label{Gr(7.19)} \label{Gr(7.19)}
\] \]
where \(K\) is a constant (depends on the history of \(I(t)\)). where \(K\) is a constant (depends on the history of \(I(t)\)).
</p> </p>
<p> <p>
<b>N.B.</b>: this can't be true always, since it blows up as \(s \rightarrow \infty\). <b>N.B.</b>: this can't be true always, since it blows up as \(r \rightarrow \infty\).
Reason: in this case, we've overstepped the quasistatic limit. We need Reason: in this case, we've overstepped the quasistatic limit. We need
\(s \ll c\tau\) where \(\tau\) is a typical time scale for change of \(I(t)\). \(r \ll c\tau\) where \(\tau\) is a typical time scale for change of \(I(t)\).
</p> </p>
</div> </div>
@ -1772,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1648,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1624,18 +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="org40e4789"> <div class="core div" id="orgce899b3">
<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="org3e8ad9b"> <div class="eqlabel" id="org6517c73">
<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="orgc606bec"> <div class="alteqlabels" id="orgfa86c68">
</div> </div>
@ -1652,7 +1644,7 @@ Full set of equations for the electromagnetic field:
<p> <p>
Complement: Complement:
</p> </p>
<div class="core div" id="org28daec0"> <div class="core div" id="orgba4da4a">
<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>
\[ \[
@ -1675,15 +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="org4bb3e78"> <div class="core div" id="org8fce2a6">
<div class="eqlabel" id="org05437e5"> <div class="eqlabel" id="org384b902">
<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="org61742b2"> <div class="alteqlabels" id="orgb8b5d0a">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.42)</li> <li>Gr (7.42)</li>
</ul> </ul>
@ -1721,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1630,15 +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="orgdf7b3f9"> <div class="core div" id="org76622f8">
<div class="eqlabel" id="org5844c7b"> <div class="eqlabel" id="org47af899">
<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="org1e192cc"> <div class="alteqlabels" id="org4b2df9a">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.36)</li> <li>Gr (7.36)</li>
</ul> </ul>
@ -1667,18 +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="orgb28580f"> <div class="core div" id="org1ac2c47">
<p> <p>
<b>Displacement current</b> <b>Displacement current</b>
</p> </p>
<div class="eqlabel" id="org4a72dcc"> <div class="eqlabel" id="orgd74a8cb">
<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="org57849e1"> <div class="alteqlabels" id="org44de33e">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.37)</li> <li>Gr (7.37)</li>
</ul> </ul>
@ -1732,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1638,14 +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="org764ac3e"> <div class="eqlabel" id="org83e3419">
<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="org0f4f12d"> <div class="alteqlabels" id="org20c79ee">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (7.35)</li> <li>Gr (7.35)</li>
</ul> </ul>
@ -1690,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1671,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

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build/emd_ce.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1622,8 +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="orgca631bc"> <details class="prereq" id="org36d2595">
<summary id="org2ad112f"> <summary id="orgac28d59">
Prerequisites Prerequisites
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1631,8 +1623,8 @@ Prerequisites
</ul> </ul>
</details> </details>
<details class="objectives" id="orgbdb48d8"> <details class="objectives" id="org84c31d2">
<summary id="org495ffbc"> <summary id="org5891649">
Objectives Objectives
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1670,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1624,18 +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="org6e52344"> <div class="main div" id="org7d69a07">
<p> <p>
<b>Angular momentum of EM fields</b> <b>Angular momentum of EM fields</b>
</p> </p>
<div class="eqlabel" id="org9b249c7"> <div class="eqlabel" id="org0855fcd">
<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="org9913b19"> <div class="alteqlabels" id="org434669a">
</div> </div>
@ -1669,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1643,7 +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="orgf1c2b2a"> <div class="core div" id="org75f48b5">
<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>
\[ \[
@ -1683,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

24
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1636,18 +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="orgecd6647"> <div class="main div" id="org60dbc44">
<p> <p>
<b>Momentum density in the EM fields</b> <b>Momentum density in the EM fields</b>
</p> </p>
<div class="eqlabel" id="orga1d9e2c"> <div class="eqlabel" id="org7808fcd">
<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">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.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="orgf73a06d"> <div class="alteqlabels" id="org1473c1c">
</div> </div>
@ -1663,18 +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="org8cc6bb8"> <div class="main div" id="orgf7ee289">
<p> <p>
<b>Continuity equation for EM momentum</b> <b>Continuity equation for EM momentum</b>
</p> </p>
<div class="eqlabel" id="orgcad090e"> <div class="eqlabel" id="orgd7db956">
<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">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.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="org69cf28d"> <div class="alteqlabels" id="orge8fbc29">
</div> </div>
@ -1707,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

32
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1675,18 +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="orga7d370d"> <div class="main div" id="orgaa8145a">
<p> <p>
<b>Maxwell stress tensor</b> <b>Maxwell stress tensor</b>
</p> </p>
<div class="eqlabel" id="orge5429c3"> <div class="eqlabel" id="org0e3a6cc">
<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="org0306495"> <div class="alteqlabels" id="org4826577">
</div> </div>
@ -1699,7 +1691,7 @@ T_{ij} \equiv \varepsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2\right)
</div> </div>
<p> <p>
The element \(T_{ij}\) represents the force per unit area in the $i$th direction acting on a surface element oriented in the $j$th direction. Diagonal elements are pressures, off-diagonal elements are shears. The element \(T_{ij}\) represents the force per unit area in the \(i\) direction acting on a surface element oriented in the \(j\) direction. Diagonal elements are pressures, off-diagonal elements are shears.
</p> </p>
@ -1707,18 +1699,18 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
<p> <p>
We then obtain the We then obtain the
</p> </p>
<div class="main div" id="org4781d10"> <div class="main div" id="org66e8efe">
<p> <p>
<b>EM force per unit volume</b> <b>EM force per unit volume</b>
</p> </p>
<div class="eqlabel" id="orgd9f91b7"> <div class="eqlabel" id="org49902c2">
<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="org7d4a06b"> <div class="alteqlabels" id="orge0c52b1">
</div> </div>
@ -1734,18 +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="orgea18677"> <div class="main div" id="org887bc5d">
<p> <p>
<b>Total force on charges in volume</b> <b>Total force on charges in volume</b>
</p> </p>
<div class="eqlabel" id="orgef98657"> <div class="eqlabel" id="orgd12201e">
<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="org4c9c200"> <div class="alteqlabels" id="org0dc7e2f">
</div> </div>
@ -1778,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

52
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1645,14 +1637,14 @@ 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="org0197499"> <div class="eqlabel" id="orgc9958b2">
<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="org9aacea3"> <div class="alteqlabels" id="orgf9bbccb">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (8.6)</li> <li>Gr (8.6)</li>
</ul> </ul>
@ -1665,7 +1657,7 @@ at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\). Thus,
\frac{dW}{dt} = \int_{\cal V} d\tau ~ {\bf E} \cdot {\bf J} \frac{dW}{dt} = \int_{\cal V} d\tau ~ {\bf E} \cdot {\bf J}
\tag{dWdt_intEJ}\label{dWdt_intEJ} \tag{dWdt_intEJ}\label{dWdt_intEJ}
\] \]
The integrand is the work done per unit time, per unit volume, {\it i.e.} the power delivered per unit volume. The integrand is the work done per unit time, per unit volume, <i>i.e.</i> the power delivered per unit volume.
In terms of fields alone: use Ampère-Maxwell: In terms of fields alone: use Ampère-Maxwell:
\[ \[
{\bf E} \cdot {\bf J} = \frac{1}{\mu_0} {\bf E} \cdot ({\boldsymbol \nabla} \times {\bf B}) - \varepsilon_0 {\bf E} \cdot \frac{\partial {\bf E}}{\partial t} {\bf E} \cdot {\bf J} = \frac{1}{\mu_0} {\bf E} \cdot ({\boldsymbol \nabla} \times {\bf B}) - \varepsilon_0 {\bf E} \cdot \frac{\partial {\bf E}}{\partial t}
@ -1692,18 +1684,18 @@ 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="orgf118f4f"> <div class="main div" id="orgead023b">
<p> <p>
<b>Poynting's theorem</b> <b>Poynting's theorem</b>
</p> </p>
<div class="eqlabel" id="org1b7cdac"> <div class="eqlabel" id="org1b1ef48">
<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="orgd7b6cac"> <div class="alteqlabels" id="org9c51283">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (8.9)</li> <li>Gr (8.9)</li>
</ul> </ul>
@ -1729,18 +1721,18 @@ 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="orgf3198a5"> <div class="core div" id="orgafa4bdd">
<p> <p>
<b>Poynting vector</b> <b>Poynting vector</b>
</p> </p>
<div class="eqlabel" id="org8725431"> <div class="eqlabel" id="org0aaf227">
<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="org6f2879d"> <div class="alteqlabels" id="org05edf23">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (8.10)</li> <li>Gr (8.10)</li>
</ul> </ul>
@ -1759,18 +1751,18 @@ 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="org3a4bb91"> <div class="core div" id="orgf7b3c73">
<p> <p>
<b>Poynting's theorem</b> (integral form) <b>Poynting's theorem</b> (integral form)
</p> </p>
<div class="eqlabel" id="orgbf2cc63"> <div class="eqlabel" id="org610ce5e">
<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="org5f484e3"> <div class="alteqlabels" id="org462a7a9">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (8.11)</li> <li>Gr (8.11)</li>
</ul> </ul>
@ -1789,18 +1781,18 @@ 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="orgbc7eb16"> <div class="core div" id="org49394fc">
<p> <p>
<b>Energy in electromagnetic fields</b> <b>Energy in electromagnetic fields</b>
</p> </p>
<div class="eqlabel" id="org89872c3"> <div class="eqlabel" id="org2fd18f3">
<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="org67613ca"> <div class="alteqlabels" id="orge0e6bd0">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (8.5)</li> <li>Gr (8.5)</li>
</ul> </ul>
@ -1829,18 +1821,18 @@ Then,
\] \]
so we get the so we get the
</p> </p>
<div class="core div" id="org487db23"> <div class="core div" id="orgb7f6aa8">
<p> <p>
<b>Poynting theorem</b> (differential form) <b>Poynting theorem</b> (differential form)
</p> </p>
<div class="eqlabel" id="org9593699"> <div class="eqlabel" id="orgc43f9ba">
<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="org129becb"> <div class="alteqlabels" id="org50e67b6">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (8.14)</li> <li>Gr (8.14)</li>
</ul> </ul>
@ -1863,7 +1855,7 @@ and has a similar for to the continuity equation
<div class="example div" id="orgd9e0ab5"> <div class="example div" id="org76ab6ea">
<p> <p>
<b>Example: Joule heating</b> <b>Example: Joule heating</b>
</p> </p>
@ -1895,7 +1887,7 @@ and points radially inwards. Energy per unit time passing surface of wire:
\[ \[
\int d{\bf a} \cdot {\bf S} = S (2\pi a L) = -V I \int d{\bf a} \cdot {\bf S} = S (2\pi a L) = -V I
\] \]
where the minus sign means energy is flowing {\it in} (the wire heats up), where the minus sign means energy is flowing <i>in</i> (the wire heats up),
and the value is as expected. and the value is as expected.
</p> </p>
@ -1920,7 +1912,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

20
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1622,8 +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="orgd522e84"> <details class="prereq" id="org42aca70">
<summary id="org3c8d630"> <summary id="org3fdd75c">
Prerequisites Prerequisites
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1632,8 +1624,8 @@ Prerequisites
</ul> </ul>
</details> </details>
<details class="objectives" id="org98124eb"> <details class="objectives" id="org8aeb437">
<summary id="org6546918"> <summary id="org16d834d">
Objectives Objectives
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1674,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

50
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1652,12 +1644,30 @@ so for a monochromatic EM plan wave,
\] \]
or more succinctly: or more succinctly:
</p> </p>
<div class="main div" id="org0e45759"> <div class="main div" id="orgee78283">
<p>
<b>Poynting vector of a monochromatic EM wave</b>
</p>
<div class="eqlabel" id="orgdedd25b">
<p>
<a id="Poynting_mpw"></a><a href="./emd_emw_ep.html#Poynting_mpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgdfa9b5e">
<ul class="org-ul">
<li>Gr (9.57)</li>
</ul>
</div>
</div>
<p> <p>
{\bf Poynting vector of a monochromatic EM wave}
\[ \[
{\boldsymbol S} = c u ~\hat{\boldsymbol k} {\boldsymbol S} = c u ~\hat{\boldsymbol k}
\] \tag{Poynting_mpw}\label{Poyting_mpw}
\]
</p> </p>
</div> </div>
@ -1668,12 +1678,12 @@ 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="org4b55ff9"> <div class="main div" id="org108cc57">
<p> <p>
{\bf Momentum density of a monochromatic EM wave} <b>Momentum density of a monochromatic EM wave</b>
\[ \[
{\boldsymbol g} = \frac{1}{c^2} {\boldsymbol S} = \frac{u}{c} ~\hat{\boldsymbol k} {\boldsymbol g} = \frac{1}{c^2} {\boldsymbol S} = \frac{u}{c} ~\hat{\boldsymbol k}
\] \]
</p> </p>
</div> </div>
@ -1688,14 +1698,14 @@ Time averages: integrating over a (integer number of) cycle(s), we have
</p> </p>
<p> <p>
The average power per unit time per unit area transported by an EM wave is called the {\bf Intensity} The average power per unit time per unit area transported by an EM wave is called the <b>Intensity</b>
\[ \[
I \equiv \langle S \rangle = \frac{c\varepsilon_0}{2} E_0^2 I \equiv \langle S \rangle = \frac{c\varepsilon_0}{2} E_0^2
\] \]
</p> </p>
<p> <p>
The {\it radiation pressure} is the momentum transfer per unit area per unit of time The <b>radiation pressure</b> is the momentum transfer per unit area per unit of time
\[ \[
P = \frac{1}{A}\frac{\Delta p}{\Delta t} = \frac{\langle g \rangle A c \Delta t}{A \Delta t} = \frac{\varepsilon_0}{2} E_0^2 = \frac{I}{c}. P = \frac{1}{A}\frac{\Delta p}{\Delta t} = \frac{\langle g \rangle A c \Delta t}{A \Delta t} = \frac{\varepsilon_0}{2} E_0^2 = \frac{I}{c}.
\] \]
@ -1719,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

73
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1622,7 +1614,7 @@ Table of contents
</svg></a><span class="headline-id">emd.emw.mpw</span></h4> </svg></a><span class="headline-id">emd.emw.mpw</span></h4>
<div class="outline-text-4" id="text-emd_emw_mpw"> <div class="outline-text-4" id="text-emd_emw_mpw">
<p> <p>
A {\it monochromatic} wave is one having a single frequency in its temporal dependence. Say that the propagation direction is \(\hat{\boldsymbol z}\): we'd then have A <b>monochromatic</b> wave is one having a single frequency in its temporal dependence. Say that the propagation direction is \(\hat{\boldsymbol z}\): we'd then have
\[ \[
{\bf E} (z, t) = {\bf E}_0 e^{i(k z - \omega t)}, \hspace{1cm} {\bf E} (z, t) = {\bf E}_0 e^{i(k z - \omega t)}, \hspace{1cm}
{\bf B} (z,t) = {\bf B}_0 e^{i(k z - \omega t)} {\bf B} (z,t) = {\bf B}_0 e^{i(k z - \omega t)}
@ -1632,7 +1624,7 @@ Maxwell's equations impose constraints. Since \({\boldsymbol \nabla} \cdot {\bf
(E_0)_z = 0 = (B_0)_z (E_0)_z = 0 = (B_0)_z
\label{Gr(9.44)} \label{Gr(9.44)}
\] \]
so {\bf electromagnetic waves are transverse}. so <b>electromagnetic waves are transverse</b>.
</p> </p>
<p> <p>
@ -1643,9 +1635,26 @@ From Faraday: \({\boldsymbol \nabla} \times {\bf E} = -\partial {\bf B}/\partia
\label{Gr(9.46)} \label{Gr(9.46)}
\] \]
so \({\bf E}\) and \({\bf B}\) are mutually perpendicular, and so \({\bf E}\) and \({\bf B}\) are mutually perpendicular, and
</p>
<div class="eqlabel" id="org4747373">
<p>
<a id="EBmpw"></a><a href="./emd_emw_mpw.html#EBmpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org2ea2f4c">
<ul class="org-ul">
<li>Gr (9.47)</li>
</ul>
</div>
</div>
<p>
\[ \[
B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0. B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0.
\label{Gr(9.47)} \tag{EBmpw}\label{EBmpw}
\] \]
</p> </p>
@ -1653,19 +1662,37 @@ 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="org53e84bf"> <div class="core div" id="orga666428">
<p>
<b>E and B fields for a monochromatic EM plane wave</b>
</p>
<div class="eqlabel" id="orge12acff">
<p>
<a id="mpw"></a><a href="./emd_emw_mpw.html#mpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgd889027">
<ul class="org-ul">
<li>Gr (9.49)</li>
</ul>
</div>
</div>
<p> <p>
{\bf E and B fields for a monochromatic EM plane wave}
\[ \[
{\boldsymbol E} ({\boldsymbol r},t ) = E_0 e^{i ({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol n}, {\boldsymbol E} ({\boldsymbol r},t ) = E_0 e^{i ({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol n},
\hspace{10mm} \hspace{10mm}
{\boldsymbol B} ({\boldsymbol r}, t) = \frac{E_0}{c} e^{i({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol k} \times \hat{\boldsymbol n} {\boldsymbol B} ({\boldsymbol r}, t) = \frac{E_0}{c} e^{i({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol k} \times \hat{\boldsymbol n}
= \frac{1}{c} ~\hat{\boldsymbol k} \times {\boldsymbol E} ({\boldsymbol r}, t) = \frac{1}{c} ~\hat{\boldsymbol k} \times {\boldsymbol E} ({\boldsymbol r}, t)
\] \tag{mpw}\label{mpw}
with the transversality condition \]
with the <b>transversality condition</b>
\[ \[
\hat{\boldsymbol k} \cdot \hat{\boldsymbol n} = 0 \hat{\boldsymbol k} \cdot \hat{\boldsymbol n} = 0
\] \]
</p> </p>
</div> </div>
@ -1697,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1622,7 +1614,6 @@ Table of contents
</svg></a><span class="headline-id">emd.emw.we</span></h4> </svg></a><span class="headline-id">emd.emw.we</span></h4>
<div class="outline-text-4" id="text-emd_emw_we"> <div class="outline-text-4" id="text-emd_emw_we">
<p> <p>
\subsubsection*{The wave equation for \({\bf E}\) and \({\bf B}\)}
Take Maxwell's equations in vacuum: Take Maxwell's equations in vacuum:
</p> </p>
\begin{align} \begin{align}
@ -1650,15 +1641,32 @@ 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="orgb3cd985"> <div class="core div" id="orgc9c5d44">
<p>
<b>Wave equations for electric and magnetic fields in vacuum</b>
</p>
<div class="eqlabel" id="orgb29f135">
<p>
<a id="WaveEq"></a><a href="./emd_emw_we.html#WaveEq"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orge5c16c5">
<ul class="org-ul">
<li>Gr (9.41)</li>
</ul>
</div>
</div>
<p> <p>
{\bf Wave equations for electric and magnetic fields in vacuum}
\[ \[
{\boldsymbol \nabla}^2 {\bf E} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf E}}{\partial t^2}, {\boldsymbol \nabla}^2 {\bf E} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf E}}{\partial t^2},
\hspace{1cm} \hspace{1cm}
{\boldsymbol \nabla}^2 {\bf B} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf B}}{\partial t^2}. {\boldsymbol \nabla}^2 {\bf B} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf B}}{\partial t^2}.
\label{Gr(9.41)} \tag{WaveEq}\label{WaveEq}
\] \]
</p> </p>
</div> </div>
@ -1681,7 +1689,7 @@ That is, a form
{\bf E} ({\bf r},t) = {\bf E}_0 e^{i ({\bf k} \cdot {\bf r} - \omega t)}, \hspace{1cm} {\bf E} ({\bf r},t) = {\bf E}_0 e^{i ({\bf k} \cdot {\bf r} - \omega t)}, \hspace{1cm}
{\bf B} ({\bf r},t) = {\bf B}_0 e^{i ({\bf k} \cdot {\bf r} - \omega t)}, {\bf B} ({\bf r},t) = {\bf B}_0 e^{i ({\bf k} \cdot {\bf r} - \omega t)},
\] \]
solves (\ref{Gr(9.41)}) for \(\omega = c |{\bf k}|\). solves <a href="./emd_emw_we.html#WaveEq">WaveEq</a> for \(\omega = c |{\bf k}|\).
Here and under, we use complex exponentials for convenience, remembering that Here and under, we use complex exponentials for convenience, remembering that
the actual electric and magnetic fields are given by the real part. the actual electric and magnetic fields are given by the real part.
</p> </p>
@ -1706,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1646,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1646,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
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<!DOCTYPE html> <!DOCTYPE html>
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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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1637,12 +1629,12 @@ free charges and currents.
</p> </p>
<p> <p>
From static case: electric polarization \({\bf P}\) produces bound charge density (\ref{Gr(4.12)}) From static case: electric polarization \({\bf P}\) produces bound charge density <a href="./emsm_esm_po.html#rhob">rhob</a>
\[ \[
\rho_b = -{\boldsymbol \nabla} \cdot {\bf P} \rho_b = -{\boldsymbol \nabla} \cdot {\bf P}
\label{Gr(7.46)} \label{Gr(7.46)}
\] \]
and magnetization \({\bf M}\) produces bound current density (\ref{Gr(6.13)}) and magnetization \({\bf M}\) produces bound current density <a href="./emsm_msm_fmo_bc.html#JbcurlM">JbcurlM</a>
\[ \[
{\bf J}_b = {\boldsymbol \nabla} \times {\bf M} {\bf J}_b = {\boldsymbol \nabla} \times {\bf M}
\label{Gr(7.47)} \label{Gr(7.47)}
@ -1657,13 +1649,30 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
\] \]
We therefore have the We therefore have the
</p> </p>
<div class="core div" id="org0421a72"> <div class="core div" id="orgf1835ae">
<p>
<b>Polarization current density</b>
</p>
<div class="eqlabel" id="orgfad2da8">
<p>
<a id="Jp"></a><a href="./emdm_Me_Mem.html#Jp"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgf15b4cb">
<ul class="org-ul">
<li>Gr (7.48)</li>
</ul>
</div>
</div>
<p> <p>
{\bf Polarization current density}
\[ \[
{\bf J}_p = \frac{\partial {\bf P}}{\partial t} {\bf J}_p = \frac{\partial {\bf P}}{\partial t}
\label{Gr(7.48)} \tag{Jp}\label{Jp}
\] \]
</p> </p>
</div> </div>
@ -1675,10 +1684,9 @@ 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="org642846e"> <aside id="orgadb90c7">
<p> <p>
Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current 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.
\({\boldsymbol J}_b\) but this is not the convention used here.
</p> </p>
</aside> </aside>
<p> <p>
@ -1696,31 +1704,56 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
<p> <p>
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,
{\it free} and {\it bound}: <b>free</b> and <b>bound</b>:
</p> </p>
<div class="main div" id="orgba471ef"> <div class="main div" id="orgbe78924">
<div class="eqlabel" id="org4679dd8">
<p>
<a id="rhofb"></a><a href="./emdm_Me_Mem.html#rhofb"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgb7e7ed8">
<ul class="org-ul">
<li>Gr (7.49)</li>
</ul>
</div>
</div>
<p> <p>
\[ \[
\rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P} \rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
\label{Gr(7.49)} \tag{rhofb}\label{rhofb}
\] \]
</p> </p>
</div> </div>
<p> <p>
and current can be separated into three parts, {\it free}, {\it bound} and and current can be separated into three parts, <b>free</b>, <b>bound</b> and
{\it polarization}: <b>polarization</b>:
</p> </p>
<div class="main div" id="org2ffd81b"> <div class="main div" id="org1c506a7">
<div class="eqlabel" id="orgbf0c17d">
<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">
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org33aea7e">
<ul class="org-ul">
<li>Gr (7.50)</li>
</ul>
</div>
</div>
<p> <p>
\[ \[
{\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M} {\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol \nabla} \times {\bf M} + \frac{\partial {\bf P}}{\partial t}.
</p> \tag{Jfbp}\label{Jfbp}
<ul class="org-ul">
<li>\frac{∂ {\bf P}}{∂ t}.</li>
</ul>
<p>
\label{Gr(7.50)}
\] \]
</p> </p>
@ -1735,24 +1768,19 @@ Gauss's law: can be rewritten
\] \]
where (as in static case) where (as in static case)
</p> </p>
<div class="core div" id="org88bb3c5"> <div class="core div" id="org501f375">
<p> <p>
\[ \[
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P} {\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
\label{Gr(7.52)} \label{Gr(7.52)}
\] \]
</p> </p>
</div> </div>
<p> <p>
Ampère's law including Maxwell's term: Ampère's law including Maxwell's term:
\[ \[
{\boldsymbol ∇} × {\bf B} = μ_0 \left( {\bf J}_f + {\boldsymbol ∇} × {\bf M} {\boldsymbol \nabla} \times {\bf B} = \mu_0 \left( {\bf J}_f + {\boldsymbol \nabla} \times {\bf M} + \frac{\partial {\bf P}}{\partial t} \right) + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t},
</p>
<ul class="org-ul">
<li>\frac{∂ {\bf P}}{∂ t} \right) + μ_0 ε_0 \frac{∂ {\bf E}}{∂ t},</li>
</ul>
<p>
\] \]
or or
\[ \[
@ -1761,12 +1789,12 @@ or
\] \]
where as before where as before
</p> </p>
<div class="core div" id="org90cea20"> <div class="core div" id="orgfaac9ca">
<p> <p>
\[ \[
{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M} {\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
\label{Gr(7.54)} \label{Gr(7.54)}
\] \]
</p> </p>
</div> </div>
@ -1779,21 +1807,36 @@ 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="orgd6526ab"> <div class="core div" id="org4a1df55">
<p> <p>
{\bf Maxwell's equations {\it (in matter)}} <b>Maxwell's equations</b> <i>(in matter)</i>
</p> </p>
<div class="eqlabel" id="orga6eb31a">
<p>
<a id="Max_mat"></a><a href="./emdm_Me_Mem.html#Max_mat"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org154a0ec">
<ul class="org-ul">
<li>Gr (7.55)</li>
</ul>
</div>
</div>
\begin{align} \begin{align}
(i)~~ &amp;{\boldsymbol \nabla} \cdot {\bf D} = \rho_f, \nonumber \\ (i)~~ &amp;{\boldsymbol \nabla} \cdot {\bf D} = \rho_f, \nonumber \\
(ii)~~ &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\ (ii)~~ &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\
(iii)~~ &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\ (iii)~~ &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\
(iv)~~ &amp;{\boldsymbol \nabla} \times {\bf H} = {\bf J}_f + \frac{\partial {\bf D}}{\partial t}. (iv)~~ &amp;{\boldsymbol \nabla} \times {\bf H} = {\bf J}_f + \frac{\partial {\bf D}}{\partial t}.
\label{Gr(7.55)} \tag{Max_mat}\label{Max_mat}
\end{align} \end{align}
</div> </div>
<p> <p>
Last term: {\bf displacement current}, Last term: <b>displacement current</b>,
\[ \[
{\bf J}_d = \frac{\partial {\bf D}}{\partial t} {\bf J}_d = \frac{\partial {\bf D}}{\partial t}
\label{Gr(7.58)} \label{Gr(7.58)}
@ -1801,23 +1844,37 @@ Last term: {\bf displacement current},
</p> </p>
<p> <p>
Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and \({\bf H}\) This must all be complemented by the <b>constitutive relations</b> giving \({\bf D}\) and \({\bf H}\)
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="orgd345cd6"> <div class="main div" id="orge60b08f">
<div class="eqlabel" id="org88284f8">
<p>
<a id="consrel"></a><a href="./emdm_Me_Mem.html#consrel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org21479eb">
<ul class="org-ul">
<li>Gr (7.56,7.57)</li>
</ul>
</div>
</div>
<p> <p>
\[ \[
{\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm} {\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
{\bf M} = \chi_m {\bf H} {\bf M} = \chi_m {\bf H}
\label{Gr(7.56)} \]
\]
so so
\[ \[
{\bf D} = \varepsilon {\bf E}, \hspace{1cm} {\bf D} = \varepsilon {\bf E}, \hspace{1cm}
{\bf H} = \frac{1}{\mu} {\bf B}, {\bf H} = \frac{1}{\mu} {\bf B},
\label{Gr(7.57)} \tag{consrel}\label{consrel}
\] \]
where \(\varepsilon \equiv \varepsilon_0(1 + \chi_e)\) and \(\mu \equiv \mu_0 (1 + \chi_m)\). where \(\varepsilon \equiv \varepsilon_0(1 + \chi_e)\) and \(\mu \equiv \mu_0 (1 + \chi_m)\).
</p> </p>
@ -1842,7 +1899,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

157
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1624,74 +1616,177 @@ Table of contents
<p> <p>
Discontinuities between different media, deduced from Discontinuities between different media, deduced from
</p> </p>
<div class="core div" id="org48bc400"> <div class="core div" id="org2152f07">
<p> <p>
{\bf Maxwell's equations {\it (in matter)}, integral form} <b>Maxwell's equations</b> <i>(in matter)</i>, <i>integral form</i>
</p> </p>
<div class="eqlabel" id="org896f0bf">
<p>
<a id="Max_mat_int"></a><a href="./emdm_Me_bc.html#Max_mat_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org37cc7e4">
</div>
</div>
\begin{align} \begin{align}
(i)~~ &amp;\oint_{\cal S} {\bf D} \cdot d{\bf a} = Q_{f_{enc}}, \nonumber \\ (i)~~ &amp;\oint_{\cal S} {\bf D} \cdot d{\bf a} = Q_{f_{enc}}, \nonumber \\
(ii)~~ &amp;\oint_{\cal S} {\bf B} \cdot d{\bf a} = 0 \nonumber \\ (ii)~~ &amp;\oint_{\cal S} {\bf B} \cdot d{\bf a} = 0 \nonumber \\
(iii)~~ &amp;\oint_{\cal P} {\bf E} \cdot d{\bf l} = -\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}, \nonumber \\ (iii)~~ &amp;\oint_{\cal P} {\bf E} \cdot d{\bf l} = -\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}, \nonumber \\
(iv)~~ &amp;\oint_{\cal P} {\bf H} \cdot d{\bf l} = I_{f_{enc}} + \frac{d}{dt} \int_{\cal S} {\bf D} \cdot d{\bf a}. (iv)~~ &amp;\oint_{\cal P} {\bf H} \cdot d{\bf l} = I_{f_{enc}} + \frac{d}{dt} \int_{\cal S} {\bf D} \cdot d{\bf a}.
\tag{Max_mat_int}\label{Max_mat_int}
\end{align} \end{align}
</div> </div>
<p> <p>
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>
<div class="main div" id="orgaee2676">
<div class="eqlabel" id="org35b4f05">
<p>
<a id="Ddisc"></a><a href="./emdm_Me_bc.html#Ddisc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgc9c66ee">
<ul class="org-ul">
<li>Gr (7.60)</li>
</ul>
</div>
</div>
<p>
\[ \[
\boxed{
D^{\perp}_1 - D^{\perp}_2 = \sigma_f D^{\perp}_1 - D^{\perp}_2 = \sigma_f
} \tag{Ddisc}\label{Ddisc}
\label{Gr(7.59)}
\] \]
Same reasoning applied to \((ii)\) gives </p>
</div>
<p>
Same reasoning applied to \((ii)\) gives <a href="./ems_ms_vp_mbc.html#Bdisc">Bdisc</a>
</p>
<div class="main div" id="org0c24b54">
<div class="eqlabel" id="org262505f">
<div class="alteqlabels" id="org560c237">
</div>
</div>
<p>
\[ \[
\boxed{
B^{\perp}_1 - B^{\perp}_2 = 0 B^{\perp}_1 - B^{\perp}_2 = 0
}
\label{Gr(7.60)}
\] \]
</p>
</div>
<p>
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>
<div class="main div" id="org2c8267d">
<p>
\[ \[
\boxed{
{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0 {\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0
}
\label{Gr(7.61)}
\] \]
</p>
</div>
<p>
Similarly, \((iv)\) implies \({\bf H}_1 \cdot {\bf l} - {\bf H}_2 \cdot {\bf l} = I_{f_{enc}}\). Similarly, \((iv)\) implies \({\bf H}_1 \cdot {\bf l} - {\bf H}_2 \cdot {\bf l} = I_{f_{enc}}\).
No volume current can contribute, but a surface current can. Can write 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 and thus
</p>
<div class="main div" id="orge9d0c89">
<div class="eqlabel" id="org15b0fbc">
<p>
<a id="Hdisc"></a><a href="./emsm_msm_H_A.html#Hdisc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a><a href="./emdm_Me_bc.html#Hdisc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgc1f1499">
<ul class="org-ul">
<li>Gr (7.63)</li>
</ul>
</div>
</div>
<p>
\[ \[
\boxed{
{\bf H}_1^{\parallel} - {\bf H}_2^{\parallel} = {\bf K}_f \times \hat{\bf n} {\bf H}_1^{\parallel} - {\bf H}_2^{\parallel} = {\bf K}_f \times \hat{\bf n}
} \tag{Hdisc}\label{Hdisc}
\label{Gr(7.62)}
\] \]
</p>
</div>
<p>
These are the general boundary conditions for electrodynamics. These are the general boundary conditions for electrodynamics.
</p> </p>
<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="org8b0c892">
<div class="eqlabel" id="org2d4783e">
<p>
<a id="disc_lm"></a><a href="./emdm_Me_bc.html#disc_lm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orga65ee45">
<ul class="org-ul">
<li>Gr (7.64)</li>
</ul>
</div>
</div>
\begin{align} \begin{align}
(i)~~ &amp;\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = \sigma_f, \nonumber \\ (i)~~ &amp;\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = \sigma_f, \nonumber \\
(ii)~~ &amp;B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\ (ii)~~ &amp;B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\
(iii)~~ &amp;{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\ (iii)~~ &amp;{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\
(iv)~~ &amp;\frac{1}{\mu_1} {\bf B}_1^{\parallel} - \frac{1}{\mu_2} {\bf B}_2^{\parallel} = {\bf K}_f \times \hat{\bf n}. (iv)~~ &amp;\frac{1}{\mu_1} {\bf B}_1^{\parallel} - \frac{1}{\mu_2} {\bf B}_2^{\parallel} = {\bf K}_f \times \hat{\bf n}.
\label{Gr(7.63)} \tag{disc_lm}\label{disc_lm}
\end{align} \end{align}
</div>
<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="org656d25d">
<p>
<a id="disc_nfc"></a><a href="./emdm_Me_bc.html#disc_nfc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org856682c">
<ul class="org-ul">
<li>Gr (7.64)</li>
</ul>
</div>
</div>
\begin{align} \begin{align}
(i)~~ &amp;\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = 0, \nonumber \\ (i)~~ &amp;\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = 0, \nonumber \\
(ii)~~ &amp;B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\ (ii)~~ &amp;B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\
(iii)~~ &amp;{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\ (iii)~~ &amp;{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\
(iv)~~ &amp;\frac{1}{\mu_1} {\bf B}_1^{\parallel} - \frac{1}{\mu_2} {\bf B}_2^{\parallel} = 0. (iv)~~ &amp;\frac{1}{\mu_1} {\bf B}_1^{\parallel} - \frac{1}{\mu_2} {\bf B}_2^{\parallel} = 0.
\label{Gr(7.64)} \tag{disc_nfc}\label{disc_nfc}
\end{align} \end{align}
<p> <p>
These are basis of theory of reflection and refraction. These are basis of theory of reflection and refraction.
@ -1715,7 +1810,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1649,7 +1641,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1615,7 +1607,7 @@ Table of contents
</ul> </ul>
</details> </details>
</nav> </nav>
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li>Absorption and Dispersion</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_Ba.html">Brewster's Angle&emsp;<small>[emdm.emwm.refl.Ba]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad_c.html">EM Waves in Conductors&emsp;<small>[emdm.emwm.ad.c]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul> <ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li>Absorption and Dispersion</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_oi.html">Oblique Incidence&emsp;<small>[emdm.emwm.refl.oi]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad_c.html">EM Waves in Conductors&emsp;<small>[emdm.emwm.ad.c]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
<h4 id="emdm_emwm_ad">Absorption and Dispersion<a class="headline-permalink" href="./emdm_emwm_ad.html#emdm_emwm_ad"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16"> <h4 id="emdm_emwm_ad">Absorption and Dispersion<a class="headline-permalink" href="./emdm_emwm_ad.html#emdm_emwm_ad"><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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@ -1630,7 +1622,7 @@ Table of contents
<li><a href="emdm_emwm_ad_c.html">EM Waves in Conductors</a><span class="headline-id">emdm.emwm.ad.c</span></li> <li><a href="emdm_emwm_ad_c.html">EM Waves in Conductors</a><span class="headline-id">emdm.emwm.ad.c</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_Ba.html">Brewster's Angle&emsp;<small>[emdm.emwm.refl.Ba]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad_c.html">EM Waves in Conductors&emsp;<small>[emdm.emwm.ad.c]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul> <br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_oi.html">Oblique Incidence&emsp;<small>[emdm.emwm.refl.oi]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad_c.html">EM Waves in Conductors&emsp;<small>[emdm.emwm.ad.c]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
<br> <br>
<hr> <hr>
<div class="license"> <div class="license">
@ -1645,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1628,13 +1620,13 @@ We now consider EM waves inside a bulk conductor and will consider nonvanishing
\] \]
which means that the Maxwell equations reduce to which means that the Maxwell equations reduce to
</p> </p>
\begin{align} \begin{align*}
{\boldsymbol \nabla} \cdot {\boldsymbol E} &amp;= \frac{\rho_f}{\varepsilon}, {\boldsymbol \nabla} \cdot {\boldsymbol E} &amp;= \frac{\rho_f}{\varepsilon},
\hspace{10mm} &amp; \hspace{10mm} &amp;
{\boldsymbol \nabla} \cdot {\boldsymbol B} &amp;= 0, \nonumber\\ {\boldsymbol \nabla} \cdot {\boldsymbol B} &amp;= 0, \nonumber\\
{\boldsymbol \nabla} \times {\boldsymbol E} &amp;= -\frac{\partial {\boldsymbol B}}{\partial t} &amp; {\boldsymbol \nabla} \times {\boldsymbol E} &amp;= -\frac{\partial {\boldsymbol B}}{\partial t} &amp;
{\boldsymbol \nabla} \times {\boldsymbol B} &amp;= \mu \sigma {\boldsymbol E} + \mu \varepsilon \frac{\partial {\boldsymbol E}}{\partial t}. {\boldsymbol \nabla} \times {\boldsymbol B} &amp;= \mu \sigma {\boldsymbol E} + \mu \varepsilon \frac{\partial {\boldsymbol E}}{\partial t}.
\end{align} \end{align*}
<p> <p>
Putting together the continuity equation for free charge Putting together the continuity equation for free charge
\[ \[
@ -1660,7 +1652,7 @@ After the free charge has dissipated, we have
{\boldsymbol \nabla} \times {\boldsymbol B} &amp;= \mu \sigma {\boldsymbol E} + \mu \varepsilon \frac{\partial {\boldsymbol E}}{\partial t} {\boldsymbol \nabla} \times {\boldsymbol B} &amp;= \mu \sigma {\boldsymbol E} + \mu \varepsilon \frac{\partial {\boldsymbol E}}{\partial t}
\end{align} \end{align}
<p> <p>
Applying ${\boldsymbol ∇} × $ to the curl equations gives the modified wave equations Applying \({\boldsymbol \nabla} \times\) to the curl equations gives the modified wave equations
\[ \[
{\boldsymbol \nabla}^2 {\boldsymbol E} = \mu \varepsilon \frac{\partial^2 {\boldsymbol E}}{\partial t^2} + \mu \sigma \frac{\partial {\boldsymbol E}}{\partial t}, \hspace{10mm} {\boldsymbol \nabla}^2 {\boldsymbol E} = \mu \varepsilon \frac{\partial^2 {\boldsymbol E}}{\partial t^2} + \mu \sigma \frac{\partial {\boldsymbol E}}{\partial t}, \hspace{10mm}
{\boldsymbol \nabla}^2 {\boldsymbol B} = \mu \varepsilon \frac{\partial^2 {\boldsymbol B}}{\partial t^2} + \mu \sigma \frac{\partial {\boldsymbol B}}{\partial t} {\boldsymbol \nabla}^2 {\boldsymbol B} = \mu \varepsilon \frac{\partial^2 {\boldsymbol B}}{\partial t^2} + \mu \sigma \frac{\partial {\boldsymbol B}}{\partial t}
@ -1684,7 +1676,7 @@ The wave can thus be written (letting \(\hat{\boldsymbol k}\) represent the dire
{\boldsymbol E} ({\boldsymbol r}, t) = {\boldsymbol E}_0 e^{-{\boldsymbol \kappa} \cdot {\boldsymbol r}} e^{i ({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} {\boldsymbol E} ({\boldsymbol r}, t) = {\boldsymbol E}_0 e^{-{\boldsymbol \kappa} \cdot {\boldsymbol r}} e^{i ({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)}
\] \]
with a similar solution for \({\boldsymbol B}\). with a similar solution for \({\boldsymbol B}\).
The quantity \(d = \frac{1}{\kappa}\) is known as the {\bf skin depth}. The quantity \(d = \frac{1}{\kappa}\) is known as the <b>skin depth</b>.
</p> </p>
<p> <p>
@ -1741,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
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<html lang="en"> <html lang="en">
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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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1624,13 +1616,13 @@ Table of contents
<p> <p>
In matter regions without free charge and free current: Maxwell's equations are In matter regions without free charge and free current: Maxwell's equations are
</p> </p>
\begin{align}
\begin{align*}
(i)~~ &amp;{\boldsymbol \nabla} \cdot {\bf D} = 0, \nonumber \\ (i)~~ &amp;{\boldsymbol \nabla} \cdot {\bf D} = 0, \nonumber \\
(ii)~~ &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\ (ii)~~ &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\
(iii)~~ &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\ (iii)~~ &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\
(iv)~~ &amp;{\boldsymbol \nabla} \times {\bf H} = \frac{\partial {\bf D}}{\partial t}. (iv)~~ &amp;{\boldsymbol \nabla} \times {\bf H} = \frac{\partial {\bf D}}{\partial t}.
\label{Gr(9.65)} \end{align*}
\end{align}
<p> <p>
For linear medium: For linear medium:
\[ \[
@ -1640,13 +1632,31 @@ For linear medium:
\] \]
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="orga20c833">
<div class="eqlabel" id="org28a8d44">
<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">
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org79eef6d">
<ul class="org-ul">
<li>Gr (9.65)</li>
</ul>
</div>
</div>
\begin{align} \begin{align}
(i)~~ &amp;{\boldsymbol \nabla} \cdot {\bf E} = 0, \nonumber \\ (i)~~ &amp;{\boldsymbol \nabla} \cdot {\bf E} = 0, \nonumber \\
(ii)~~ &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\ (ii)~~ &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\
(iii)~~ &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\ (iii)~~ &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\
(iv)~~ &amp;{\boldsymbol \nabla} \times {\bf B} = \mu \varepsilon \frac{\partial {\bf E}}{\partial t}. (iv)~~ &amp;{\boldsymbol \nabla} \times {\bf B} = \mu \varepsilon \frac{\partial {\bf E}}{\partial t}.
\label{Gr(9.65)} \tag{Max_lh}\label{Max_lh}
\end{align} \end{align}
</div>
<p> <p>
These are the same equations as in vacuum, except for the substitution of \(\mu_0 \varepsilon_0\) by \(\mu \varepsilon\). These are the same equations as in vacuum, except for the substitution of \(\mu_0 \varepsilon_0\) by \(\mu \varepsilon\).
</p> </p>
@ -1659,13 +1669,30 @@ 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="orgd0bce92"> <div class="main div" id="orga25a79e">
<p>
<b>Index of refraction</b>
</p>
<div class="eqlabel" id="orgc9df98d">
<p>
<a id="n"></a><a href="./emdm_emwm_plm.html#n"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org574afc6">
<ul class="org-ul">
<li>Gr (9.69)</li>
</ul>
</div>
</div>
<p> <p>
{\bf Index of refraction}
\[ \[
n \equiv \sqrt{\frac{\mu \varepsilon}{\mu_0 \varepsilon_0}} n \equiv \sqrt{\frac{\mu \varepsilon}{\mu_0 \varepsilon_0}}
\label{Gr(9.69)} \tag{n}\label{n}
\] \]
</p> </p>
</div> </div>
@ -1675,7 +1702,7 @@ Fact: for most materials, \(\mu\) is very close to \(\mu_0\), so
n \simeq \sqrt{\varepsilon_r} n \simeq \sqrt{\varepsilon_r}
\label{Gr(9.70)} \label{Gr(9.70)}
\] \]
with \(\varepsilon_r\) being the dielectric constant \ref{Gr(4.34)}. with \(\varepsilon_r\) being the dielectric constant <a href="./emsm_esm_ld_sp.html#epsr">epsr</a>.
</p> </p>
<p> <p>
@ -1716,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
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<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1625,15 +1617,14 @@ Table of contents
<p> <p>
Interesting question: what happens to an EM wave as it passes from one medium to another? Interesting question: what happens to an EM wave as it passes from one medium to another?
Incident wave: produces reflected and transmitted waves. Incident wave: produces reflected and transmitted waves.
Detailed study: starts from boundary conditions \ref{Gr(7.64)}, Detailed study: starts from boundary conditions <a href="./emdm_Me_bc.html#disc_nfc">disc_nfc</a>,
</p> </p>
\begin{align} \begin{align*}
(i)~~ &amp;\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = 0, \nonumber \\ (i)~~ &amp;\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = 0, \nonumber \\
(ii)~~ &amp;B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\ (ii)~~ &amp;B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\
(iii)~~ &amp;{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\ (iii)~~ &amp;{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\
(iv)~~ &amp;\frac{1}{\mu_1} {\bf B}_1^{\parallel} - \frac{1}{\mu_2} {\bf B}_2^{\parallel} = 0. (iv)~~ &amp;\frac{1}{\mu_1} {\bf B}_1^{\parallel} - \frac{1}{\mu_2} {\bf B}_2^{\parallel} = 0.
\label{eq:EMBdryCondAtMediumInterface} \end{align*}
\end{align}
</div> </div>
@ -1641,8 +1632,6 @@ Detailed study: starts from boundary conditions \ref{Gr(7.64)},
<ul class="child-links-list"> <ul class="child-links-list">
<li><a href="emdm_emwm_refl_ni.html">Normal Incidence</a><span class="headline-id">emdm.emwm.refl.ni</span></li> <li><a href="emdm_emwm_refl_ni.html">Normal Incidence</a><span class="headline-id">emdm.emwm.refl.ni</span></li>
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<li><a href="emdm_emwm_refl_Fe.html">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span></li>
<li><a href="emdm_emwm_refl_Ba.html">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span></li>
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@ -1660,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1629,7 +1621,7 @@ in \(x\) direction approaches surface:
{\boldsymbol B}_I (z,t) = \frac{1}{v_1} E_{0_I} e^{i (k_1 z - \omega t)} \hat{\boldsymbol y} {\boldsymbol B}_I (z,t) = \frac{1}{v_1} E_{0_I} e^{i (k_1 z - \omega t)} \hat{\boldsymbol y}
\label{Gr(9.75)} \label{Gr(9.75)}
\] \]
(in which we have used \(B_0 = \frac{1}{v} E_0\), \ref{Gr(9.47)}). (in which we have used \(B_0 = \frac{1}{v} E_0\), <a href="./emd_emw_mpw.html#EBmpw">EBmpw</a>.
</p> </p>
<p> <p>
@ -1644,19 +1636,19 @@ Reflected wave:
<p> <p>
For the transmitted wave, we put For the transmitted wave, we put
\[ </p>
{\boldsymbol E}_T (z,t) = E_{0_T} e^{i (k_2 z - \omega t)} \hat{\boldsymbol x}, \hspace{1cm}
{\boldsymbol B}_T (z,t) = \frac{1}{v_2} E_{0_T} e^{i (k_2 z - \omega t)} \hat{\boldsymbol y}. \begin{align*}
\label{Gr(9.75)} {\boldsymbol E}_T (z,t) &amp;= E_{0_T} e^{i (k_2 z - \omega t)} \hat{\boldsymbol x}, \\
\] {\boldsymbol B}_T (z,t) &amp;= \frac{1}{v_2} E_{0_T} e^{i (k_2 z - \omega t)} \hat{\boldsymbol y}.
\end{align*}
<p>
Our boundary is by choice of coordinate system at \(z = 0\). Our setup calls for solving the boundary conditions <a href="./emdm_Me_bc.html#disc_nfc">disc_nfc</a> with \({\boldsymbol E}_I + {\boldsymbol E}_R\) and \({\boldsymbol B}_I + {\boldsymbol B}_R\) on one side, and \({\boldsymbol E}_T\) and \({\boldsymbol B}_T\) on the other.
</p> </p>
<p> <p>
Our boundary is by choice of coordinate system at \(z = 0\). Our setup calls for solving the boundary conditions \ref{eq:EMBdryCondAtMediumInterface} with \({\boldsymbol E}_I + {\boldsymbol E}_R\) and \({\boldsymbol B}_I + {\boldsymbol B}_R\) on one side, and \({\boldsymbol E}_T\) and \({\boldsymbol B}_T\) on the other. At normal incidence, there are no perpendicular components of the fields relative to the surface, so <a href="./emdm_Me_bc.html#disc_nfc">disc_nfc</a> (i) and (ii) are obeyed. (iii) means that
</p>
<p>
At normal incidence, there are no perpendicular components of the fields relative to the surface, so \ref{eq:EMBdryCondAtMediumInterface} (i) and (ii) are obeyed. (iii) means that
\[ \[
E_{0_I} + E_{0_R} = E_{0_T} E_{0_I} + E_{0_R} = E_{0_T}
\label{Gr(9.78)} \label{Gr(9.78)}
@ -1677,10 +1669,27 @@ E_{0_I} - E_{0_R} = \beta E_{0_T}, \hspace{1cm}
<p> <p>
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>
<div class="eqlabel" id="org653a14a">
<p>
<a id="ERT"></a><a href="./emdm_emwm_refl_ni.html#ERT"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org5690572">
<ul class="org-ul">
<li>Gr (9.82)</li>
</ul>
</div>
</div>
<p>
\[ \[
E_{0_R} = \frac{1 - \beta}{1 + \beta} E_{0_I}, \hspace{10mm} E_{0_R} = \frac{1 - \beta}{1 + \beta} E_{0_I}, \hspace{10mm}
E_{0_T} = \frac{2}{1 + \beta} E_{0_I}. E_{0_T} = \frac{2}{1 + \beta} E_{0_I}.
\label{Gr(9.82)} \tag{ERT}\label{ERT}
\] \]
</p> </p>
@ -1701,12 +1710,14 @@ R \equiv \frac{I_R}{I_I} = \frac{E^2_{0_R}}{E^2_{0_I}} = \left( \frac{1 - \beta}
\label{Gr(9.86)} \label{Gr(9.86)}
\] \]
while the transmitted intensity is while the transmitted intensity is
\[ </p>
\begin{equation}
T \equiv \frac{I_T}{I_I} = \frac{v_2 \varepsilon_2}{v_1 \varepsilon_1} \frac{E^2_{0_T}}{E^2_{0_I}} T \equiv \frac{I_T}{I_I} = \frac{v_2 \varepsilon_2}{v_1 \varepsilon_1} \frac{E^2_{0_T}}{E^2_{0_I}}
= \sqrt{\frac{\mu_1 \varepsilon_2}{\mu_2 \varepsilon_1}} \frac{E^2_{0_T}}{E^2_{0_I}} = \frac{4 \beta}{(1 + \beta)^2}. = \sqrt{\frac{\mu_1 \varepsilon_2}{\mu_2 \varepsilon_1}} \frac{E^2_{0_T}}{E^2_{0_I}} = \frac{4 \beta}{(1 + \beta)^2}.
\label{Gr(9.86)} \end{equation}
\] <p>
We thus have that the {\bf reflection} and {\bf transmission coefficients} satisfy We thus have that the <b>reflection</b> and <b>transmission coefficients</b> satisfy
\[ \[
R + T = 1. R + T = 1.
\label{Gr(9.88)} \label{Gr(9.88)}
@ -1732,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li class="toc-currentpage"> <li class="toc-currentpage">
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1615,7 +1607,7 @@ Table of contents
</ul> </ul>
</details> </details>
</nav> </nav>
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm_refl.html">Reflection and Transmission</a></li><li>Oblique Incidence</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_Fe.html">Fresnel's Equations&emsp;<small>[emdm.emwm.refl.Fe]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul><div id="outline-container-emdm_emwm_refl_oi" class="outline-5"> <ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm_refl.html">Reflection and Transmission</a></li><li>Oblique Incidence</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad.html">Absorption and Dispersion&emsp;<small>[emdm.emwm.ad]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul><div id="outline-container-emdm_emwm_refl_oi" class="outline-5">
<h5 id="emdm_emwm_refl_oi">Oblique Incidence<a class="headline-permalink" href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16"> <h5 id="emdm_emwm_refl_oi">Oblique Incidence<a class="headline-permalink" href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.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"/>
@ -1642,14 +1634,28 @@ Transmitted wave:
{\boldsymbol B}_T ({\boldsymbol r},t) = \frac{1}{v_2} \hat{\boldsymbol k}_T \times {\boldsymbol E}_{T} ({\boldsymbol r}, t). {\boldsymbol B}_T ({\boldsymbol r},t) = \frac{1}{v_2} \hat{\boldsymbol k}_T \times {\boldsymbol E}_{T} ({\boldsymbol r}, t).
\] \]
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>
<div class="eqlabel" id="org862531f">
<p>
<a id="RTobliquek"></a><a href="./emdm_emwm_refl_oi.html#RTobliquek"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org4435847">
</div>
</div>
<p>
\[ \[
k_I v_1 = k_R v_1 = k_T v_2 ~~\longrightarrow~~ k_I = k_R = \frac{v_2}{v_1} k_T = \frac{n_1}{n_2} k_T k_I v_1 = k_R v_1 = k_T v_2 ~~\longrightarrow~~ k_I = k_R = \frac{v_2}{v_1} k_T = \frac{n_1}{n_2} k_T
\label{eq:RTObliquek} \tag{RTobliquek}\label{RTobliquek}
\] \]
</p> </p>
<p> <p>
These forms for incident, reflected and transmitted wave can be substituted in the boundary conditions (\ref{eq:EMBdryCondAtMediumInterface}). Since these must be valid for any \(x\) and \(y\) (on the interface at \(z=0\)), we must have that the \(x\) and \(y\) components of the wavevectors coincide for all the waves: These forms for incident, reflected and transmitted wave can be substituted in the boundary conditions <a href="./emdm_Me_bc.html#disc_nfc">disc_nfc</a>. Since these must be valid for any \(x\) and \(y\) (on the interface at \(z=0\)), we must have that the \(x\) and \(y\) components of the wavevectors coincide for all the waves:
\[ \[
k_{I_x} = k_{R_x} = k_{T_x}, \hspace{10mm} k_{I_x} = k_{R_x} = k_{T_x}, \hspace{10mm}
k_{I_y} = k_{R_y} = k_{T_y} k_{I_y} = k_{R_y} = k_{T_y}
@ -1659,29 +1665,29 @@ 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="orge681e56"> <div class="core div" id="org6f173e3">
<p> <p>
{\bf First law of reflection:} <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.
</p> </p>
</div> </div>
<p> <p>
Specializing (\ref{eq:RTObliquek}) to our notations, we have Specializing <a href="./emdm_emwm_refl_oi.html#RTobliquek">RTobliquek</a> to our notations, we have
\[ \[
k_I \sin \theta_I = k_R \sin \theta_R = k_T \sin \theta_T k_I \sin \theta_I = k_R \sin \theta_R = k_T \sin \theta_T
\] \]
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="orgfbf5032"> <div class="core div" id="orgbf9459c">
<p> <p>
{\bf Law of reflection} <b>Law of reflection</b>
\[ \[
\theta_I = \theta_R \theta_I = \theta_R
\] \]
{\bf Law of refraction (Snell's law)} <b>Law of refraction (Snell's law)</b>
\[ \[
n_1 \sin \theta_I = n_2 \sin \theta_T n_1 \sin \theta_I = n_2 \sin \theta_T
\] \]
</p> </p>
@ -1691,13 +1697,25 @@ 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="org8d76e61">
<p>
<a id="EBRT"></a><a href="./emdm_emwm_refl_oi.html#EBRT"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org2c4d03c">
</div>
</div>
\begin{align} \begin{align}
\varepsilon_1 \left({\boldsymbol E}_{0_I} + {\boldsymbol E}_{0_R} \right)_z &amp;= \varepsilon_2 \left({\boldsymbol E}_{0_T} \right)_z, \varepsilon_1 \left({\boldsymbol E}_{0_I} + {\boldsymbol E}_{0_R} \right)_z &amp;= \varepsilon_2 \left({\boldsymbol E}_{0_T} \right)_z,
&amp; \hspace{10mm} &amp; \hspace{10mm}
\left({\boldsymbol B}_{0_I} +{\boldsymbol B}_{0_R} \right)_z &amp;= \left({\boldsymbol B}_{0_T}\right)_z, \nonumber \\ \left({\boldsymbol B}_{0_I} +{\boldsymbol B}_{0_R} \right)_z &amp;= \left({\boldsymbol B}_{0_T}\right)_z, \nonumber \\
\left( {\boldsymbol E}_{0_I} + {\boldsymbol E}_{0_R} \right)_{x,y} &amp;= \left({\boldsymbol E}_{0_T}\right)_{x,y}, \left( {\boldsymbol E}_{0_I} + {\boldsymbol E}_{0_R} \right)_{x,y} &amp;= \left({\boldsymbol E}_{0_T}\right)_{x,y},
&amp; \frac{1}{\mu_1} \left({\boldsymbol B}_{0_I} + {\boldsymbol B}_{0_R} \right)_{x,y} &amp;= \frac{1}{\mu_2} \left({\boldsymbol B}_{0_T}\right)_{x,y}. &amp; \frac{1}{\mu_1} \left({\boldsymbol B}_{0_I} + {\boldsymbol B}_{0_R} \right)_{x,y} &amp;= \frac{1}{\mu_2} \left({\boldsymbol B}_{0_T}\right)_{x,y}.
\label{eq:EMBdryCondAtMediumInterface:amp} \tag{EBRT}\label{EBRT}
\end{align} \end{align}
<p> <p>
@ -1709,8 +1727,8 @@ The two cases of polarization parallel and perpendicular to the plane of inciden
</p> </p>
<p> <p>
\paragraph{Polarization in plane of incidence:} <b>Polarization in plane of incidence</b>:
in this case the first equation of (\ref{eq:EMBdryCondAtMediumInterface:amp}) gives in this case the first equation of <a href="./emdm_emwm_refl_oi.html#EBRT">EBRT</a> gives
\[ \[
\varepsilon_1 \left(-E_{0_I} \sin \theta_I + E_{0_R} \sin \theta_R \right) = -\varepsilon_2 E_{0_T} \sin \theta_T. \varepsilon_1 \left(-E_{0_I} \sin \theta_I + E_{0_R} \sin \theta_R \right) = -\varepsilon_2 E_{0_T} \sin \theta_T.
\] \]
@ -1732,13 +1750,28 @@ 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="orgcf2803c"> <div class="main div" id="org3866053">
<p>
<b>Fresnel's equations for reflection and transmission amplitudes (parallel case)</b>
</p>
<div class="eqlabel" id="org31e8ee3">
<p>
<a id="Fresnel"></a><a href="./emdm_emwm_refl_oi.html#Fresnel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org1f91c78">
</div>
</div>
<p> <p>
{\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)}
\[ \[
E_{0_R} = \frac{\alpha - \beta}{\alpha + \beta} E_{0_I}, E_{0_R} = \frac{\alpha - \beta}{\alpha + \beta} E_{0_I},
\hspace{10mm} \hspace{10mm}
E_{0_T} = \frac{2}{\alpha + \beta} E_{0_I} E_{0_T} = \frac{2}{\alpha + \beta} E_{0_I}
\tag{Fresnel}\label{Fresnel}
\] \]
</p> </p>
@ -1749,15 +1782,28 @@ Amplitudes for transmitted and reflected wave: depend on angle of incidence:
\[ \[
\alpha = \frac{\sqrt{1 - \sin^2 \theta_T}}{\cos \theta_I} = \frac{\left[1 - \left(\frac{n_1}{n_2}\right)^2 \sin^2 \theta_I\right]^{1/2}}{\cos \theta_I} \alpha = \frac{\sqrt{1 - \sin^2 \theta_T}}{\cos \theta_I} = \frac{\left[1 - \left(\frac{n_1}{n_2}\right)^2 \sin^2 \theta_I\right]^{1/2}}{\cos \theta_I}
\] \]
Behaviour: for \(\theta_I = 0\) we recover (\ref{Gr(9.82)}). 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
</p> </p>
<div class="main div" id="org7503b43"> <div class="main div" id="org328f2a5">
<div class="eqlabel" id="org3b9cec2">
<p> <p>
{\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}} <a id="Brewster"></a><a href="./emdm_emwm_refl_oi.html#Brewster"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
\[ <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
\theta_B = \arcsin \left[ \frac{1 - \beta^2}{(n_1/n_2)^2 - \beta^2} \right]^{1/2} <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
\] </svg></a>
</p>
<div class="alteqlabels" id="orgf448df4">
</div>
</div>
<p>
<b>Brewster's angle</b> <i>(at which the reflected wave amplitude vanishes)</i>
\[
\theta_B = \arcsin \left[ \frac{1 - \beta^2}{(n_1/n_2)^2 - \beta^2} \right]^{1/2}
\tag{Brewster}\label{Brewster}
\]
</p> </p>
</div> </div>
@ -1785,9 +1831,7 @@ Of course, we get \(R + T = 1\) as expected.
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad.html">Absorption and Dispersion&emsp;<small>[emdm.emwm.ad]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_Fe.html">Fresnel's Equations&emsp;<small>[emdm.emwm.refl.Fe]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul>
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@ -1802,7 +1846,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
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<!DOCTYPE html> <!DOCTYPE html>
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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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1641,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1647,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1667,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

14
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1668,7 +1660,7 @@ Putting these back into Maxwell (i) and (ii) gives the decoupled equations
\[ \[
\left[ \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \left(\frac{\omega}{c}\right)^2 - k^2 \right] E_z = 0, \left[ \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \left(\frac{\omega}{c}\right)^2 - k^2 \right] E_z = 0,
\] \]
with an identical equation for \(B_z\). If \(E_z = 0\) the waves are called {\bf TE} waves (for {\it transverse electric}), and if \(B_z = 0\) they are called {\bf TM} (for {\it transverse magnetic}) waves. If \(E_z = 0 = B_z\) they are called {\bf TEM waves}. The latter cannot occur in a hollow waveguide (simple proof: Gauss + Faraday). with an identical equation for \(B_z\). If \(E_z = 0\) the waves are called <b>TE</b> waves (for <i>transverse electric</i>), and if \(B_z = 0\) they are called <b>TM</b> (for <i>transverse magnetic</i>}) waves. If \(E_z = 0 = B_z\) they are called <b>TEM</b> waves. The latter cannot occur in a hollow waveguide (simple proof: Gauss + Faraday).
</p> </p>
</div> </div>
</div> </div>
@ -1691,7 +1683,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

17
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<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1662,7 +1654,7 @@ In view of this, if the frequency is low enough, namely
\[ \[
\omega &lt; c\pi \sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2} \equiv \omega_{mn} \omega &lt; c\pi \sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2} \equiv \omega_{mn}
\] \]
then the wavenumber is imaginary and the travelling wave is exponentially attenuated. This frequency is called the {\bf cutoff frequency} for this mode. then the wavenumber is imaginary and the travelling wave is exponentially attenuated. This frequency is called the <b>cutoff frequency</b> for this mode.
</p> </p>
<p> <p>
@ -1674,7 +1666,8 @@ and wave velocity
\[ \[
v = \frac{\omega}{k} = \frac{c}{\sqrt{1 - \left(\frac{\omega_{mn}}{\omega}\right)^2}} v = \frac{\omega}{k} = \frac{c}{\sqrt{1 - \left(\frac{\omega_{mn}}{\omega}\right)^2}}
\] \]
which is {\it greater} than \(c\). The energy of the wave however propagates at the {\bf group velocity} which is <i>greater</i> than \(c\). The energy of the wave however propagates at the
<b>group velocity</b>
\[ \[
v_g = \frac{d\omega}{dk} = c \sqrt{1 - \left(\frac{\omega_{mn}}{\omega}\right)^2}. v_g = \frac{d\omega}{dk} = c \sqrt{1 - \left(\frac{\omega_{mn}}{\omega}\right)^2}.
\] \]
@ -1700,7 +1693,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1622,8 +1614,8 @@ Table of contents
</svg></a><span class="headline-id">emf</span></h2> </svg></a><span class="headline-id">emf</span></h2>
<div class="outline-text-2" id="text-emf"> <div class="outline-text-2" id="text-emf">
<details class="prereq" id="orgb93b9c3"> <details class="prereq" id="org8e30f8b">
<summary id="org621593e"> <summary id="org2c7b815">
Prerequisites Prerequisites
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1631,8 +1623,8 @@ Prerequisites
</ul> </ul>
</details> </details>
<details class="objectives" id="org179e7b9"> <details class="objectives" id="org5818573">
<summary id="orgb47e5d6"> <summary id="orgda754dd">
Objectives Objectives
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1668,7 +1660,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1665,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1664,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</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>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1637,7 +1629,7 @@ while the equation for \(V\) becomes
\] \]
These can be written compactly upon introducing a new operator: the These can be written compactly upon introducing a new operator: the
</p> </p>
<div class="core div" id="org242c17e"> <div class="core div" id="org0aaab77">
<p> <p>
{\bf d'Alembertian operator} {\bf d'Alembertian operator}
\[ \[
@ -1650,7 +1642,7 @@ 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="orgb00ab8a"> <div class="core div" id="orgc8e8b42">
<p> <p>
{\bf Inhomogeneous Maxwell equations (Lorenz gauge)} {\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
\[ \[
@ -1700,7 +1692,7 @@ 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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

20
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1638,7 +1630,7 @@ Useful strategy: represent fields in terms of potentials.
<p> <p>
Easiest: Easiest:
</p> </p>
<div class="core div" id="orgf88fb96"> <div class="core div" id="org5bf4193">
<p> <p>
\[ \[
{\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A} {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
@ -1654,7 +1646,7 @@ Putting this into Faraday's law gives
\] \]
so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
</p> </p>
<div class="core div" id="org2f23671"> <div class="core div" id="orgffce8dc">
<p> <p>
\[ \[
{\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t} {\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}
@ -1667,7 +1659,7 @@ so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \
<p> <p>
Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting (\ref{eq:E_from_Potentials}) into Gauss's law gives Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting (\ref{eq:E_from_Potentials}) into Gauss's law gives
</p> </p>
<div class="main div" id="org5127f9c"> <div class="main div" id="org26743e8">
<p> <p>
\[ \[
{\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0} {\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
@ -1683,7 +1675,7 @@ whereas Amp{\`ere}-Maxwell becomes
\] \]
which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\), which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
</p> </p>
<div class="main div" id="org3f2c1f0"> <div class="main div" id="orga86c8c2">
<p> <p>
\[ \[
\left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right) \left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
@ -1719,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1648,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

20
build/ems_ca.html
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1626,8 +1618,8 @@ Table of contents
<li>Gr 3</li> <li>Gr 3</li>
</ul> </ul>
<details class="prereq" id="org3b926ab"> <details class="prereq" id="org192cbf5">
<summary id="org2707d3f"> <summary id="org78425f6">
Prerequisites Prerequisites
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1635,8 +1627,8 @@ Prerequisites
</ul> </ul>
</details> </details>
<details class="objectives" id="orgfae12c2"> <details class="objectives" id="orge457a7e">
<summary id="org18515f0"> <summary id="orge350b5c">
Objectives Objectives
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1674,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

20
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1632,7 +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="orgbb90bd9"> <div class="main div" id="orgb661984">
<p> <p>
</p> </p>
@ -1646,7 +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="org9f1e898"> <div class="main div" id="org5d15b9f">
<p> <p>
</p> </p>
@ -1666,7 +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="org7c70185"> <div class="core div" id="org7aef84c">
<p> <p>
</p> </p>
@ -1682,7 +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="org1935b19"> <div class="core div" id="org3f412f2">
<p> <p>
</p> </p>
@ -1720,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

46
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@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1638,14 +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="org84cc03f"> <div class="eqlabel" id="org5c6cddc">
<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="org61fa4ec"> <div class="alteqlabels" id="org5f00f86">
</div> </div>
@ -1660,14 +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="orgb8c4067"> <div class="eqlabel" id="orgc4fd451">
<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="org6b2acba"> <div class="alteqlabels" id="org1f2c6cb">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (3.6)</li> <li>Gr (3.6)</li>
</ul> </ul>
@ -1726,14 +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="orgf3e1009"> <div class="eqlabel" id="orge167f78">
<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="org696fc74"> <div class="alteqlabels" id="org9aae751">
</div> </div>
@ -1786,14 +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="org9f81e63"> <div class="eqlabel" id="orge400754">
<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="org66bdc2f"> <div class="alteqlabels" id="orgf37e10c">
</div> </div>
@ -1805,8 +1797,8 @@ a point equals its value averaged over a sphere
\] \]
</p> </p>
<details id="org0f27c64"> <details id="org4bb0c5f">
<summary id="org881e712"> <summary id="orgaf332f4">
<strong>Physicist's proof</strong> <strong>Physicist's proof</strong>
</summary> </summary>
<p> <p>
@ -1868,8 +1860,8 @@ proving the theorem.
</p> </p>
</details> </details>
<details id="org0dbbb13"> <details id="orgfce7dc3">
<summary id="orgf2b7c76"> <summary id="org452b0d4">
<strong>Formal proof</strong> <strong>Formal proof</strong>
</summary> </summary>
@ -1919,14 +1911,14 @@ we get the following general
<p> <p>
<b>Theorem</b>: <b>Theorem</b>:
</p> </p>
<div class="eqlabel" id="org230e535"> <div class="eqlabel" id="orgfba2553">
<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="org37e5ede"> <div class="alteqlabels" id="orgc5c6de6">
</div> </div>
@ -1979,19 +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="org8692599"> <div class="eqlabel" id="org8465450">
<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="orgcae1a27"> <div class="alteqlabels" id="org0fbdf78">
</div> </div>
</div> </div>
<div class="info div" id="orgab38e5b"> <div class="info div" id="org9dd741d">
<p> <p>
<b>Earnshaw's theorem (physical version)</b> <br> <b>Earnshaw's theorem (physical version)</b> <br>
</p> </p>
@ -2110,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

24
build/ems_ca_fe_g.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-03-02 Wed 15:45 --> <!-- 2022-03-07 Mon 20:38 -->
<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>
@ -1098,14 +1098,6 @@ Table of contents
<li> <li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span> <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li> </li>
</ul> </ul>
@ -1621,11 +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="org5424abd"> <div class="info div" id="orga6b382e">
<p> <p>
<b>George Green</b> <b>George Green</b>
</p> </p>
<aside id="org7013f8e"> <aside id="org8e26888">
<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>
@ -1658,14 +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="org17f1f87"> <div class="eqlabel" id="orgeb35436">
<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="org9716c8b"> <div class="alteqlabels" id="org6a05402">
<ul class="org-ul"> <ul class="org-ul">
<li>J (1.34)</li> <li>J (1.34)</li>
</ul> </ul>
@ -1683,14 +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="orgdecb68b"> <div class="eqlabel" id="org3cd71e0">
<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="org1abae59"> <div class="alteqlabels" id="org2a9c8de">
<ul class="org-ul"> <ul class="org-ul">
<li>J (1.35)</li> <li>J (1.35)</li>
</ul> </ul>
@ -1726,7 +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: 2022-03-02 Wed 15:45</p> <p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

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