Update 2022-03-07 20:40
This commit is contained in:
Jean-Sébastien
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build/a.html
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-03-02 Wed 15:45 -->
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<!-- 2022-03-07 Mon 20:38 -->
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<title>Pre-Quantum Electrodynamics</title>
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@ -1098,14 +1098,6 @@ Table of contents
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<li>
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<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
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</li>
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<li>
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<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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</li>
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<li>
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<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
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</li>
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</ul>
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@ -1645,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-03-02 Wed 15:45</p>
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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<p class="validation"></p>
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</div>
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@ -1,7 +1,7 @@
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-03-02 Wed 15:45 -->
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<!-- 2022-03-07 Mon 20:38 -->
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<title>Pre-Quantum Electrodynamics</title>
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@ -1098,14 +1098,6 @@ Table of contents
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<li>
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||||
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
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</li>
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<li>
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<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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</li>
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||||
<li>
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<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
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</li>
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</ul>
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@ -1728,7 +1720,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-03-02 Wed 15:45</p>
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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<p class="validation"></p>
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</div>
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12
build/c.html
12
build/c.html
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-03-02 Wed 15:45 -->
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<!-- 2022-03-07 Mon 20:38 -->
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<title>Pre-Quantum Electrodynamics</title>
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@ -1098,14 +1098,6 @@ Table of contents
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<li>
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<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
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||||
</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>
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</li>
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||||
<li>
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<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
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||||
</li>
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||||
</ul>
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@ -1645,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-03-02 Wed 15:45</p>
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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<p class="validation"></p>
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</div>
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@ -1,7 +1,7 @@
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-03-02 Wed 15:45 -->
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<!-- 2022-03-07 Mon 20:38 -->
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<title>Pre-Quantum Electrodynamics</title>
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@ -1098,14 +1098,6 @@ Table of contents
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<li>
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<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
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||||
</li>
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||||
<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>
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</li>
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<li>
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<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
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</li>
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</ul>
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@ -1651,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-03-02 Wed 15:45</p>
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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<p class="validation"></p>
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</div>
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@ -1,7 +1,7 @@
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<!DOCTYPE html>
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<html lang="en">
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||||
<head>
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||||
<!-- 2022-03-02 Wed 15:45 -->
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<!-- 2022-03-07 Mon 20:38 -->
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||||
<meta charset="utf-8">
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||||
<meta name="viewport" content="width=device-width, initial-scale=1">
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<title>Pre-Quantum Electrodynamics</title>
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|
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@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
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|
||||
</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>
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</li>
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||||
</ul>
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@ -1647,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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|||
</div>
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<div id="postamble" class="status">
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-03-02 Wed 15:45</p>
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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<p class="validation"></p>
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</div>
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@ -1,7 +1,7 @@
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<!DOCTYPE html>
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<html lang="en">
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||||
<head>
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||||
<!-- 2022-03-02 Wed 15:45 -->
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<!-- 2022-03-07 Mon 20:38 -->
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1">
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||||
<title>Pre-Quantum Electrodynamics</title>
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@ -1098,14 +1098,6 @@ Table of contents
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|||
<li>
|
||||
<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>
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||||
</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>
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</li>
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</ul>
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@ -1667,14 +1659,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
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<div id="outline-container-c_m_cs_cyl_grad" class="outline-6">
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<h6 id="c_m_cs_cyl_grad"><a href="#c_m_cs_cyl_grad">Gradient</a></h6>
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<div class="outline-text-6" id="text-c_m_cs_cyl_grad">
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<div class="eqlabel" id="org3f7fe1a">
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<div class="eqlabel" id="orgfd5fb6e">
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<p>
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<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">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="orgcf66eee">
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<div class="alteqlabels" id="org16d29dc">
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<ul class="org-ul">
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<li>Gr4(1.79)</li>
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</ul>
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@ -1695,14 +1687,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
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<div id="outline-container-c_m_cs_cyl_div" class="outline-6">
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<h6 id="c_m_cs_cyl_div"><a href="#c_m_cs_cyl_div">Divergence</a></h6>
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<div class="outline-text-6" id="text-c_m_cs_cyl_div">
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<div class="eqlabel" id="org77a88dd">
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<div class="eqlabel" id="org1dfbb35">
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<p>
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<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">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="orga1ce1b4">
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<div class="alteqlabels" id="org035dafc">
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<ul class="org-ul">
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<li>Gr4(2.21)</li>
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</ul>
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@ -1723,14 +1715,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
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<div id="outline-container-c_m_cs_cyl_curl" class="outline-6">
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<h6 id="c_m_cs_cyl_curl"><a href="#c_m_cs_cyl_curl">Curl</a></h6>
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<div class="outline-text-6" id="text-c_m_cs_cyl_curl">
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<div class="eqlabel" id="orgb7e3d3b">
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<div class="eqlabel" id="org747dbea">
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<p>
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<a id="cyl_curl"></a><a href="./c_m_cs_cyl.html#cyl_curl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="orgd068b29">
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<div class="alteqlabels" id="orgcd134fe">
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<ul class="org-ul">
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<li>Gr4(2.21)</li>
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</ul>
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@ -1779,7 +1771,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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|||
</div>
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||||
<div id="postamble" class="status">
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||||
<p class="author">Author: Jean-Sébastien Caux</p>
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<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>
|
||||
</div>
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@ -1,7 +1,7 @@
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<!DOCTYPE html>
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||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
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||||
</li>
|
||||
|
||||
</ul>
|
||||
|
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@ -1640,7 +1632,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
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@ -1,7 +1,7 @@
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<!DOCTYPE html>
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||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
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</li>
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</ul>
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@ -1631,14 +1623,14 @@ which \(r\) is the distance from the chosen origin,
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The usual Cartesian coordinates relate to spherical coordinates
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according to
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</p>
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<div class="eqlabel" id="org3e4cc27">
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<div class="eqlabel" id="org348038f">
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<p>
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<a id="sph_xyz"></a><a href="./c_m_cs_sph.html#sph_xyz"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org6cfcb56">
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<div class="alteqlabels" id="org5f5a1a3">
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</div>
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@ -1663,14 +1655,14 @@ A generic vector can be expressed as
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where the explicit relation between spherical and
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Cartesian unit vectors is
|
||||
</p>
|
||||
<div class="eqlabel" id="org115ed32">
|
||||
<div class="eqlabel" id="orgbdd14bc">
|
||||
<p>
|
||||
<a id="sph_uv"></a><a href="./c_m_cs_sph.html#sph_uv"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org965d0cb">
|
||||
<div class="alteqlabels" id="orga250579">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1693,14 +1685,14 @@ and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).
|
|||
<p>
|
||||
An infinitesimal displacement \(d{\bf l}\) can be written as
|
||||
</p>
|
||||
<div class="eqlabel" id="orgdb194fb">
|
||||
<div class="eqlabel" id="org6494f4b">
|
||||
<p>
|
||||
<a id="sph_dl"></a><a href="./c_m_cs_sph.html#sph_dl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org96ce6ff">
|
||||
<div class="alteqlabels" id="orgaafdc77">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1716,14 +1708,14 @@ d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin
|
|||
<p>
|
||||
Infinitesimal volume element:
|
||||
</p>
|
||||
<div class="eqlabel" id="org40e244c">
|
||||
<div class="eqlabel" id="orga40a4ab">
|
||||
<p>
|
||||
<a id="sph_dtau"></a><a href="./c_m_cs_sph.html#sph_dtau"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org6796383">
|
||||
<div class="alteqlabels" id="orgc7eb689">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1744,14 +1736,14 @@ Infinitesimal surface element: depends on situation.
|
|||
<div id="outline-container-c_m_cs_sph_grad" class="outline-6">
|
||||
<h6 id="c_m_cs_sph_grad"><a href="#c_m_cs_sph_grad">Gradient</a></h6>
|
||||
<div class="outline-text-6" id="text-c_m_cs_sph_grad">
|
||||
<div class="eqlabel" id="org7c9e5d8">
|
||||
<div class="eqlabel" id="orgb45a655">
|
||||
<p>
|
||||
<a id="sph_grad"></a><a href="./c_m_cs_sph.html#sph_grad"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org94fd9f5">
|
||||
<div class="alteqlabels" id="org5dfccd6">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1768,14 +1760,14 @@ Infinitesimal surface element: depends on situation.
|
|||
<div id="outline-container-c_m_cs_sph_div" class="outline-6">
|
||||
<h6 id="c_m_cs_sph_div"><a href="#c_m_cs_sph_div">Divergence</a></h6>
|
||||
<div class="outline-text-6" id="text-c_m_cs_sph_div">
|
||||
<div class="eqlabel" id="org25fdd4e">
|
||||
<div class="eqlabel" id="orgdca9c80">
|
||||
<p>
|
||||
<a id="sph_div"></a><a href="./c_m_cs_sph.html#sph_div"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org3d0214d">
|
||||
<div class="alteqlabels" id="org52febbe">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1792,14 +1784,14 @@ Infinitesimal surface element: depends on situation.
|
|||
<div id="outline-container-c_m_cs_sph_curl" class="outline-6">
|
||||
<h6 id="c_m_cs_sph_curl"><a href="#c_m_cs_sph_curl">Curl</a></h6>
|
||||
<div class="outline-text-6" id="text-c_m_cs_sph_curl">
|
||||
<div class="eqlabel" id="org082892a">
|
||||
<div class="eqlabel" id="org26cc7dd">
|
||||
<p>
|
||||
<a id="sph_curl"></a><a href="./c_m_cs_sph.html#sph_curl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orge37528b">
|
||||
<div class="alteqlabels" id="orgeafd92e">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1818,14 +1810,14 @@ Infinitesimal surface element: depends on situation.
|
|||
<div id="outline-container-c_m_cs_sph_lap" class="outline-6">
|
||||
<h6 id="c_m_cs_sph_lap"><a href="#c_m_cs_sph_lap">Laplacian</a></h6>
|
||||
<div class="outline-text-6" id="text-c_m_cs_sph_lap">
|
||||
<div class="eqlabel" id="orgf48e07a">
|
||||
<div class="eqlabel" id="org9c1e636">
|
||||
<p>
|
||||
<a id="sph_Lap"></a><a href="./c_m_cs_sph.html#sph_Lap"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orge21efb8">
|
||||
<div class="alteqlabels" id="org83d937d">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1859,7 +1851,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1650,7 +1642,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1652,7 +1644,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1623,9 +1615,9 @@ Table of contents
|
|||
<div class="outline-text-5" id="text-c_m_dc_d2">
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3814a3d" class="outline-6">
|
||||
<h6 id="org3814a3d"><a href="#org3814a3d">Divergence of gradient</a></h6>
|
||||
<div class="outline-text-6" id="text-org3814a3d">
|
||||
<div id="outline-container-orgc12bd11" class="outline-6">
|
||||
<h6 id="orgc12bd11"><a href="#orgc12bd11">Divergence of gradient</a></h6>
|
||||
<div class="outline-text-6" id="text-orgc12bd11">
|
||||
<p>
|
||||
\({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
|
||||
The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
|
||||
|
|
@ -1634,44 +1626,44 @@ given by the Laplacian of the corresponding vector elements.
|
|||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc8aa2ed" class="outline-6">
|
||||
<h6 id="orgc8aa2ed"><a href="#orgc8aa2ed">Curl of a gradient</a></h6>
|
||||
<div class="outline-text-6" id="text-orgc8aa2ed">
|
||||
<div id="outline-container-org0ea7ae4" class="outline-6">
|
||||
<h6 id="org0ea7ae4"><a href="#org0ea7ae4">Curl of a gradient</a></h6>
|
||||
<div class="outline-text-6" id="text-org0ea7ae4">
|
||||
<p>
|
||||
This always vanishes.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge3b0dcb" class="outline-6">
|
||||
<h6 id="orge3b0dcb"><a href="#orge3b0dcb">Gradient of the divergence</a></h6>
|
||||
<div class="outline-text-6" id="text-orge3b0dcb">
|
||||
<div id="outline-container-org78d85e1" class="outline-6">
|
||||
<h6 id="org78d85e1"><a href="#org78d85e1">Gradient of the divergence</a></h6>
|
||||
<div class="outline-text-6" id="text-org78d85e1">
|
||||
<p>
|
||||
\({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1f5474a" class="outline-6">
|
||||
<h6 id="org1f5474a"><a href="#org1f5474a">Divergence of a curl</a></h6>
|
||||
<div class="outline-text-6" id="text-org1f5474a">
|
||||
<div id="outline-container-orge7f86bf" class="outline-6">
|
||||
<h6 id="orge7f86bf"><a href="#orge7f86bf">Divergence of a curl</a></h6>
|
||||
<div class="outline-text-6" id="text-orge7f86bf">
|
||||
<p>
|
||||
This always vanishes.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4b8cb79" class="outline-6">
|
||||
<h6 id="org4b8cb79"><a href="#org4b8cb79">Curl of curl</a></h6>
|
||||
<div class="outline-text-6" id="text-org4b8cb79">
|
||||
<div class="eqlabel" id="orgad52ac5">
|
||||
<div id="outline-container-orgbb8199b" class="outline-6">
|
||||
<h6 id="orgbb8199b"><a href="#orgbb8199b">Curl of curl</a></h6>
|
||||
<div class="outline-text-6" id="text-orgbb8199b">
|
||||
<div class="eqlabel" id="orga59fc73">
|
||||
<p>
|
||||
<a id="curlcurl"></a><a href="./c_m_dc_d2.html#curlcurl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orgde36146">
|
||||
<div class="alteqlabels" id="org384a104">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1702,7 +1694,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1651,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1648,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1672,7 +1664,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1630,14 +1622,14 @@ explicited as follows:
|
|||
<p>
|
||||
<b>Gradient of a product</b>:
|
||||
</p>
|
||||
<div class="eqlabel" id="org4bcb61a">
|
||||
<div class="eqlabel" id="orged10256">
|
||||
<p>
|
||||
<a id="grad_prod"></a><a href="./c_m_dc_pr.html#grad_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org8c15975">
|
||||
<div class="alteqlabels" id="orgfc8793a">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (3)</li>
|
||||
<li>W (1-111)</li>
|
||||
|
|
@ -1657,14 +1649,14 @@ explicited as follows:
|
|||
<p>
|
||||
<b>Gradient of a scalar product</b>:
|
||||
</p>
|
||||
<div class="eqlabel" id="org7784802">
|
||||
<div class="eqlabel" id="org4b4bdf3">
|
||||
<p>
|
||||
<a id="grad_sprod"></a><a href="./c_m_dc_pr.html#grad_sprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orgb3682de">
|
||||
<div class="alteqlabels" id="org02913cc">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (4)</li>
|
||||
<li>W (1-112)</li>
|
||||
|
|
@ -1684,14 +1676,14 @@ explicited as follows:
|
|||
<p>
|
||||
<b>Divergence of a product</b>:
|
||||
</p>
|
||||
<div class="eqlabel" id="org96b1b94">
|
||||
<div class="eqlabel" id="org1df402a">
|
||||
<p>
|
||||
<a id="div_prod"></a><a href="./c_m_dc_pr.html#div_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org29d8167">
|
||||
<div class="alteqlabels" id="orgfa5050d">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (5)</li>
|
||||
<li>W (1-115)</li>
|
||||
|
|
@ -1711,14 +1703,14 @@ explicited as follows:
|
|||
<p>
|
||||
<b>Divergence of a cross product</b>:
|
||||
</p>
|
||||
<div class="eqlabel" id="orgfe2b08f">
|
||||
<div class="eqlabel" id="org5048e1e">
|
||||
<p>
|
||||
<a id="div_xprod"></a><a href="./c_m_dc_pr.html#div_xprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orgdce994d">
|
||||
<div class="alteqlabels" id="orgd39dff6">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (6)</li>
|
||||
<li>W (1-116)</li>
|
||||
|
|
@ -1738,14 +1730,14 @@ explicited as follows:
|
|||
<p>
|
||||
<b>Curl of a product</b>:
|
||||
</p>
|
||||
<div class="eqlabel" id="orgbb8798b">
|
||||
<div class="eqlabel" id="orgbd31452">
|
||||
<p>
|
||||
<a id="curl_prod"></a><a href="./c_m_dc_pr.html#curl_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org93d7ea5">
|
||||
<div class="alteqlabels" id="org825aae4">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7)</li>
|
||||
<li>W (1-118)</li>
|
||||
|
|
@ -1765,14 +1757,14 @@ explicited as follows:
|
|||
<p>
|
||||
<b>Curl of a cross product</b>:
|
||||
</p>
|
||||
<div class="eqlabel" id="orgd9fde5c">
|
||||
<div class="eqlabel" id="org9c6b6a4">
|
||||
<p>
|
||||
<a id="curl_xprod"></a><a href="./c_m_dc_pr.html#curl_xprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orgb1d2b5b">
|
||||
<div class="alteqlabels" id="orgb030476">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (8)</li>
|
||||
<li>W (1-119)</li>
|
||||
|
|
@ -1813,7 +1805,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1647,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1659,7 +1651,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1644,14 +1636,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
|
|||
More generally,
|
||||
</p>
|
||||
|
||||
<div class="eqlabel" id="orgf5d25ef">
|
||||
<div class="eqlabel" id="org492d0c0">
|
||||
<p>
|
||||
<a id="divdel"></a><a href="./c_m_dd_3d.html#divdel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org6fba030">
|
||||
<div class="alteqlabels" id="org9b8bed8">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (1.100)</li>
|
||||
</ul>
|
||||
|
|
@ -1670,14 +1662,14 @@ More generally,
|
|||
Since
|
||||
</p>
|
||||
|
||||
<div class="eqlabel" id="org9588ee0">
|
||||
<div class="eqlabel" id="orga3b7a91">
|
||||
<p>
|
||||
<a id="div1or"></a><a href="./c_m_dd_3d.html#div1or"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orgb0b4434">
|
||||
<div class="alteqlabels" id="org98ac626">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (1.101)</li>
|
||||
</ul>
|
||||
|
|
@ -1693,14 +1685,14 @@ Since
|
|||
<p>
|
||||
we have that
|
||||
</p>
|
||||
<div class="eqlabel" id="orgd223c1e">
|
||||
<div class="eqlabel" id="org596ba00">
|
||||
<p>
|
||||
<a id="Lap1or"></a><a href="./c_m_dd_3d.html#Lap1or"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org2ae8bc7">
|
||||
<div class="alteqlabels" id="org620c397">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (1.102)</li>
|
||||
</ul>
|
||||
|
|
@ -1732,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1665,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1650,7 +1642,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1648,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1657,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1651,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1658,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1623,9 +1615,9 @@ Table of contents
|
|||
<div class="outline-text-5" id="text-c_m_ic_lsv">
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2486563" class="outline-6">
|
||||
<h6 id="org2486563"><a href="#org2486563">Line Integrals</a></h6>
|
||||
<div class="outline-text-6" id="text-org2486563">
|
||||
<div id="outline-container-org7fab499" class="outline-6">
|
||||
<h6 id="org7fab499"><a href="#org7fab499">Line Integrals</a></h6>
|
||||
<div class="outline-text-6" id="text-org7fab499">
|
||||
<p>
|
||||
\[
|
||||
{\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 id="outline-container-orgf44a1b8" class="outline-6">
|
||||
<h6 id="orgf44a1b8"><a href="#orgf44a1b8">Surface Integrals</a></h6>
|
||||
<div class="outline-text-6" id="text-orgf44a1b8">
|
||||
<div id="outline-container-org543b5ed" class="outline-6">
|
||||
<h6 id="org543b5ed"><a href="#org543b5ed">Surface Integrals</a></h6>
|
||||
<div class="outline-text-6" id="text-org543b5ed">
|
||||
<p>
|
||||
\[
|
||||
\int_{\cal S} {\bf v} \cdot d{\bf a}
|
||||
|
|
@ -1676,9 +1668,9 @@ Over a closed surface:
|
|||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7c26dcf" class="outline-6">
|
||||
<h6 id="org7c26dcf"><a href="#org7c26dcf">Volume Integrals</a></h6>
|
||||
<div class="outline-text-6" id="text-org7c26dcf">
|
||||
<div id="outline-container-org015d5b1" class="outline-6">
|
||||
<h6 id="org015d5b1"><a href="#org015d5b1">Volume Integrals</a></h6>
|
||||
<div class="outline-text-6" id="text-org015d5b1">
|
||||
<p>
|
||||
\[
|
||||
\int_{\cal V} T d\tau
|
||||
|
|
@ -1719,7 +1711,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</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"/>
|
||||
</svg></a><span class="headline-id">c.m.ic.stokes</span></h5>
|
||||
<div class="outline-text-5" id="text-c_m_ic_stokes">
|
||||
<div class="eqlabel" id="org603ea63">
|
||||
<div class="eqlabel" id="org39cf0b7">
|
||||
<p>
|
||||
<a id="Stokes"></a><a href="./c_m_ic_stokes.html#Stokes"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org95b24de">
|
||||
<div class="alteqlabels" id="orga45987e">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (1.57)</li>
|
||||
</ul>
|
||||
|
|
@ -1673,7 +1665,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1647,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1660,7 +1652,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1660,7 +1652,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1663,7 +1655,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1649,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1678,7 +1670,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1673,7 +1665,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1707,7 +1699,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1673,7 +1665,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1720,7 +1712,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1646,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1665,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1692,7 +1684,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
16
build/d.html
16
build/d.html
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1622,8 +1614,8 @@ Table of contents
|
|||
</svg></a><span class="headline-id">d</span></h2>
|
||||
|
||||
<div class="outline-text-2" id="text-d">
|
||||
<details class="objectives" id="orge784719">
|
||||
<summary id="orgfb0d046">
|
||||
<details class="objectives" id="orgcc61564">
|
||||
<summary id="org585bc00">
|
||||
Objectives
|
||||
</summary>
|
||||
|
||||
|
|
@ -1687,7 +1679,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<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>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1667,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1662,7 +1654,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1665,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1662,7 +1654,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1684,7 +1676,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1667,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1664,7 +1656,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1667,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1665,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1675,7 +1667,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1669,7 +1661,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1622,8 +1614,8 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emd</span></h2>
|
||||
|
||||
<div class="outline-text-2" id="text-emd">
|
||||
<details class="prereq" id="org2aba3cc">
|
||||
<summary id="org30ca6a6">
|
||||
<details class="prereq" id="org417b126">
|
||||
<summary id="org6bf52b1">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1632,8 +1624,8 @@ Prerequisites
|
|||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="org98af4b0">
|
||||
<summary id="org78e61da">
|
||||
<details class="objectives" id="org93939b2">
|
||||
<summary id="org78ba2fd">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1674,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1648,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</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
|
||||
along the wire. The line integral of this field is called the
|
||||
</p>
|
||||
<div class="core div" id="orgdfb0aad">
|
||||
<div class="core div" id="org9cc8d48">
|
||||
<p>
|
||||
<b>Electromotive force (or electromotance)</b>,
|
||||
</p>
|
||||
<div class="eqlabel" id="org64afdaf">
|
||||
<div class="eqlabel" id="orgb9f75a5">
|
||||
<p>
|
||||
<a id="elmofo"></a><a href="./emd_Fl_Fl.html#elmofo"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orgb3cda90">
|
||||
<div class="alteqlabels" id="orgcb4fab4">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.9)</li>
|
||||
</ul>
|
||||
|
|
@ -1688,14 +1680,14 @@ The precise statement associated to Faraday's observations
|
|||
is that the electromotive force is proportional
|
||||
to the rate of change of the magnetic flux,
|
||||
</p>
|
||||
<div class="eqlabel" id="orgec7b520">
|
||||
<div class="eqlabel" id="orgb7faf93">
|
||||
<p>
|
||||
<a id="Fl_flux"></a><a href="./emd_Fl_Fl.html#Fl_flux"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orgf70f495">
|
||||
<div class="alteqlabels" id="orgffe5f68">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.14)</li>
|
||||
</ul>
|
||||
|
|
@ -1710,18 +1702,18 @@ to the rate of change of the magnetic flux,
|
|||
\]
|
||||
so we obtain
|
||||
</p>
|
||||
<div class="core div" id="orgdfedc05">
|
||||
<div class="core div" id="org80fe686">
|
||||
<p>
|
||||
<b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
|
||||
</p>
|
||||
<div class="eqlabel" id="orge87df83">
|
||||
<div class="eqlabel" id="orgfa5af5c">
|
||||
<p>
|
||||
<a id="Fl_int"></a><a href="./emd_Fl_Fl.html#Fl_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org7b32fe7">
|
||||
<div class="alteqlabels" id="orga84de5d">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.15)</li>
|
||||
</ul>
|
||||
|
|
@ -1745,15 +1737,15 @@ for any loop (on a wire or not). Using Stokes' theorem,
|
|||
\]
|
||||
we obtain
|
||||
</p>
|
||||
<div class="core div" id="orgafa0d15">
|
||||
<div class="eqlabel" id="org13d3c14">
|
||||
<div class="core div" id="org2c503e3">
|
||||
<div class="eqlabel" id="org9023002">
|
||||
<p>
|
||||
<a id="Fl"></a><a href="./emd_Fl_Fl.html#Fl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orgd662a28">
|
||||
<div class="alteqlabels" id="orgb7f3817">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.16)</li>
|
||||
</ul>
|
||||
|
|
@ -1798,7 +1790,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</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:
|
||||
</p>
|
||||
<div class="eqlabel" id="org87af1ba">
|
||||
<div class="eqlabel" id="org78f90c5">
|
||||
<p>
|
||||
<a id="W_intAJ"></a><a href="./emd_Fl_e.html#W_intAJ"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org95d3f8b">
|
||||
<div class="alteqlabels" id="org0277370">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.31)</li>
|
||||
</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
|
||||
</p>
|
||||
<div class="core div" id="org247bd37">
|
||||
<div class="eqlabel" id="orgdf4c394">
|
||||
<div class="core div" id="org9994672">
|
||||
<div class="eqlabel" id="org2cf22c6">
|
||||
<p>
|
||||
<a id="W_intBsq"></a><a href="./emd_Fl_e.html#W_intBsq"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orge069b86">
|
||||
<div class="alteqlabels" id="orgde5daac">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.34)</li>
|
||||
</ul>
|
||||
|
|
@ -1726,7 +1718,7 @@ W_{mag} &= \frac{1}{2} \int d\tau ~({\bf A} \cdot {\bf J}) &= \frac{1}{2
|
|||
which are equations <a href="./ems_es_e.html#W_vcd">W_vcd</a>, <a href="./ems_es_e.html#W_intEsq">W_intEsq</a>, <a href="./emd_Fl_e.html#W_intAJ">W_intAJ</a> and <a href="./emd_Fl_e.html#W_intBsq">W_intBsq</a>.
|
||||
</p>
|
||||
|
||||
<div class="example div" id="org03fb4e8">
|
||||
<div class="example div" id="orgd2854e0">
|
||||
<p>
|
||||
<b>Example: energy in coaxial cable</b>
|
||||
</p>
|
||||
|
|
@ -1776,7 +1768,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1649,14 +1641,14 @@ so
|
|||
\]
|
||||
and we can write the mutual inductance as the <b>Neumann formula</b>,
|
||||
</p>
|
||||
<div class="eqlabel" id="org197212f">
|
||||
<div class="eqlabel" id="org766e684">
|
||||
<p>
|
||||
<a id="Newmann_M"></a><a href="./emd_Fl_i.html#Newmann_M"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org4709e6a">
|
||||
<div class="alteqlabels" id="orga92dd51">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.22)</li>
|
||||
</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:
|
||||
first, \(M_{21}\) is purely geometrical. Second,
|
||||
</p>
|
||||
<div class="eqlabel" id="orgb952a44">
|
||||
<div class="eqlabel" id="org577027c">
|
||||
<p>
|
||||
<a id="Msym"></a><a href="./emd_Fl_i.html#Msym"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org53f3037">
|
||||
<div class="alteqlabels" id="org21aff84">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.23)</li>
|
||||
</ul>
|
||||
|
|
@ -1695,7 +1687,7 @@ M_{12} = M_{21}
|
|||
\]
|
||||
</p>
|
||||
|
||||
<div class="example div" id="org6347791">
|
||||
<div class="example div" id="org9334fe0">
|
||||
<p>
|
||||
<b>Example: solenoid in solenoid</b>
|
||||
</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:
|
||||
</p>
|
||||
<div class="eqlabel" id="org667fec5">
|
||||
<div class="eqlabel" id="org4807c27">
|
||||
<p>
|
||||
<a id="PLI"></a><a href="./emd_Fl_i.html#PLI"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org28a1a02">
|
||||
<div class="alteqlabels" id="orga19e4d1">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.25)</li>
|
||||
</ul>
|
||||
|
|
@ -1773,7 +1765,7 @@ Inductance: measured in <b>henries</b> (\(H\)). \(H = V s/A\).
|
|||
</p>
|
||||
|
||||
|
||||
<div class="example div" id="org00e72a3">
|
||||
<div class="example div" id="org940f5ab">
|
||||
<p>
|
||||
<b>Example: self-inductance of toroidal coil</b>
|
||||
</p>
|
||||
|
|
@ -1812,7 +1804,7 @@ Inductance (like capacitance) is intrinsically positive. Use Lenz law.
|
|||
Think of <i>back EMF</i>.
|
||||
</p>
|
||||
|
||||
<div class="example div" id="org958ea6c">
|
||||
<div class="example div" id="orgbeea66c">
|
||||
<p>
|
||||
<b>Example: circuit</b>
|
||||
</p>
|
||||
|
|
@ -1861,7 +1853,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1645,7 +1637,7 @@ law in integral form:
|
|||
|
||||
|
||||
|
||||
<div class="example div" id="org045463b">
|
||||
<div class="example div" id="org56facb7">
|
||||
<p>
|
||||
<b>Example: loop with time-dependent flux</b>
|
||||
</p>
|
||||
|
|
@ -1661,10 +1653,10 @@ through a horizontal circular region of radius \(R\).
|
|||
|
||||
<p>
|
||||
<b>Solution</b>:
|
||||
amperian loop of radius \(s\), apply Faraday:
|
||||
amperian loop of radius \(r\), apply Faraday:
|
||||
\[
|
||||
\oint {\bf E} \cdot d{\bf l} = E (2\pi s) = -\frac{d\Phi}{dt} = -\pi s^2 \frac{dB}{dt}
|
||||
\Rightarrow {\bf E} = -\frac{s}{2} \frac{dB}{dt} \hat{\boldsymbol \varphi}.
|
||||
\oint {\bf E} \cdot d{\bf l} = E (2\pi r) = -\frac{d\Phi}{dt} = -\pi r^2 \frac{dB}{dt}
|
||||
\Rightarrow {\bf E} = -\frac{r}{2} \frac{dB}{dt} \hat{\boldsymbol \varphi}.
|
||||
\]
|
||||
Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
|
||||
</p>
|
||||
|
|
@ -1672,7 +1664,7 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
|
|||
</div>
|
||||
|
||||
|
||||
<div class="example div" id="org047ea10">
|
||||
<div class="example div" id="org367fcfc">
|
||||
<p>
|
||||
<b>Example: wheel with charged rim traversed by flux</b>
|
||||
</p>
|
||||
|
|
@ -1713,7 +1705,7 @@ called the <b>quasistatic</b> approximation, and works provided we deal with
|
|||
<i>slow enough</i> phenomena.
|
||||
</p>
|
||||
|
||||
<div class="example div" id="org733cbdd">
|
||||
<div class="example div" id="orgb2a276c">
|
||||
<p>
|
||||
<b>Example: field from wire with time-dependent current</b>
|
||||
</p>
|
||||
|
|
@ -1723,31 +1715,31 @@ Consider an infinitely long straight wire which carries current \(I(t)\).
|
|||
</p>
|
||||
|
||||
<p>
|
||||
<b>Task</b>: find the induced \({\bf E}\) field as a function of distance \(s\) from wire.
|
||||
<b>Task</b>: find the induced \({\bf E}\) field as a function of distance \(r\) from wire.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
<b>Solution</b>: assuming we can use the quasistatic approximation, the
|
||||
magnetic field is \(B = \frac{\mu_0 I}{2\pi s}\)
|
||||
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
|
||||
to wire. Amperian loop with sides at distances \(s_0\) and \(s\):
|
||||
to wire. Amperian loop with sides at distances \(r_0\) and \(r\):
|
||||
\[
|
||||
\oint {\bf E} \cdot d{\bf l} = E(s_0)l - E(s)l = -\frac{d}{dt} \int {\bf B} \cdot d{\bf a}
|
||||
= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \int_{s_0}^s \frac{ds'}{s'}
|
||||
= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \ln(s/s_0).
|
||||
\oint {\bf E} \cdot d{\bf l} = E(r_0)l - E(r)l = -\frac{d}{dt} \int {\bf B} \cdot d{\bf a}
|
||||
= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \int_{r_0}^s \frac{dr'}{r'}
|
||||
= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \ln(r/r_0).
|
||||
\]
|
||||
So:
|
||||
\[
|
||||
{\bf E} (s) = \left[ \frac{\mu_0}{2\pi} \frac{dI}{dt} \ln s + K \right] \hat{\bf x}
|
||||
{\bf E} (r) = \left[ \frac{\mu_0}{2\pi} \frac{dI}{dt} \ln r + K \right] \hat{\bf x}
|
||||
\label{Gr(7.19)}
|
||||
\]
|
||||
where \(K\) is a constant (depends on the history of \(I(t)\)).
|
||||
</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
|
||||
\(s \ll c\tau\) 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>
|
||||
|
||||
</div>
|
||||
|
|
@ -1772,7 +1764,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1648,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1624,18 +1616,18 @@ Table of contents
|
|||
<p>
|
||||
Full set of equations for the electromagnetic field:
|
||||
</p>
|
||||
<div class="core div" id="org40e4789">
|
||||
<div class="core div" id="orgce899b3">
|
||||
<p>
|
||||
<b>Maxwell's equations</b> <i>(in vacuum)</i>
|
||||
</p>
|
||||
<div class="eqlabel" id="org3e8ad9b">
|
||||
<div class="eqlabel" id="org6517c73">
|
||||
<p>
|
||||
<a id="Max_vac"></a><a href="./emd_Me_Me.html#Max_vac"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orgc606bec">
|
||||
<div class="alteqlabels" id="orgfa86c68">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1652,7 +1644,7 @@ Full set of equations for the electromagnetic field:
|
|||
<p>
|
||||
Complement:
|
||||
</p>
|
||||
<div class="core div" id="org28daec0">
|
||||
<div class="core div" id="orgba4da4a">
|
||||
<p>
|
||||
Force law <a href="./ems_ms_lf_pc.html#LorFo">LorFo</a>
|
||||
\[
|
||||
|
|
@ -1675,15 +1667,15 @@ take divergence of \((iv)\) and use \((i)\).
|
|||
<p>
|
||||
Better way of writing: all fields on left, all sources on right,
|
||||
</p>
|
||||
<div class="core div" id="org4bb3e78">
|
||||
<div class="eqlabel" id="org05437e5">
|
||||
<div class="core div" id="org8fce2a6">
|
||||
<div class="eqlabel" id="org384b902">
|
||||
<p>
|
||||
<a id="Max_vac_s"></a><a href="./emd_Me_Me.html#Max_vac_s"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org61742b2">
|
||||
<div class="alteqlabels" id="orgb8b5d0a">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.42)</li>
|
||||
</ul>
|
||||
|
|
@ -1721,7 +1713,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</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
|
||||
</p>
|
||||
<div class="core div" id="orgdf7b3f9">
|
||||
<div class="eqlabel" id="org5844c7b">
|
||||
<div class="core div" id="org76622f8">
|
||||
<div class="eqlabel" id="org47af899">
|
||||
<p>
|
||||
<a id="AmpMax"></a><a href="./emd_Me_dc.html#AmpMax"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org1e192cc">
|
||||
<div class="alteqlabels" id="org4b2df9a">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.36)</li>
|
||||
</ul>
|
||||
|
|
@ -1667,18 +1659,18 @@ Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation
|
|||
<p>
|
||||
Maxwell baptized this term the
|
||||
</p>
|
||||
<div class="core div" id="orgb28580f">
|
||||
<div class="core div" id="org1ac2c47">
|
||||
<p>
|
||||
<b>Displacement current</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org4a72dcc">
|
||||
<div class="eqlabel" id="orgd74a8cb">
|
||||
<p>
|
||||
<a id="Jd"></a><a href="./emd_Me_dc.html#Jd"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org57849e1">
|
||||
<div class="alteqlabels" id="org44de33e">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.37)</li>
|
||||
</ul>
|
||||
|
|
@ -1732,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1638,14 +1630,14 @@ Fatal inconsistency: div of curl must always vanish. Check on \((iii)\):
|
|||
\]
|
||||
But: try same with \((iv)\):
|
||||
</p>
|
||||
<div class="eqlabel" id="org764ac3e">
|
||||
<div class="eqlabel" id="org83e3419">
|
||||
<p>
|
||||
<a id="divcurlB"></a><a href="./emd_Me_ebM.html#divcurlB"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org0f4f12d">
|
||||
<div class="alteqlabels" id="org20c79ee">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (7.35)</li>
|
||||
</ul>
|
||||
|
|
@ -1690,7 +1682,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1671,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1622,8 +1614,8 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emd.ce</span></h3>
|
||||
|
||||
<div class="outline-text-3" id="text-emd_ce">
|
||||
<details class="prereq" id="orgca631bc">
|
||||
<summary id="org2ad112f">
|
||||
<details class="prereq" id="org36d2595">
|
||||
<summary id="orgac28d59">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1631,8 +1623,8 @@ Prerequisites
|
|||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="orgbdb48d8">
|
||||
<summary id="org495ffbc">
|
||||
<details class="objectives" id="org84c31d2">
|
||||
<summary id="org5891649">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1670,7 +1662,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1624,18 +1616,18 @@ Table of contents
|
|||
<p>
|
||||
The angular momentum of EM fields is directly given by
|
||||
</p>
|
||||
<div class="main div" id="org6e52344">
|
||||
<div class="main div" id="org7d69a07">
|
||||
<p>
|
||||
<b>Angular momentum of EM fields</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org9b249c7">
|
||||
<div class="eqlabel" id="org0855fcd">
|
||||
<p>
|
||||
<a id="lrxg"></a><a href="./emd_ce_amom.html#lrxg"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org9913b19">
|
||||
<div class="alteqlabels" id="org434669a">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1669,7 +1661,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1643,7 +1635,7 @@ This means that
|
|||
\]
|
||||
Since this is true for any volume, we have (re)derived the
|
||||
</p>
|
||||
<div class="core div" id="orgf1c2b2a">
|
||||
<div class="core div" id="org75f48b5">
|
||||
<p>
|
||||
<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 id="postamble" class="status">
|
||||
<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>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1636,18 +1628,18 @@ in which the first integral can be interpreted as the momentum stored in the EM
|
|||
<p>
|
||||
This is thus simply a conservation law for momentum, with
|
||||
</p>
|
||||
<div class="main div" id="orgecd6647">
|
||||
<div class="main div" id="org60dbc44">
|
||||
<p>
|
||||
<b>Momentum density in the EM fields</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="orga1d9e2c">
|
||||
<div class="eqlabel" id="org7808fcd">
|
||||
<p>
|
||||
<a id="gExB"></a><a href="./emd_ce_mom.html#gExB"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orgf73a06d">
|
||||
<div class="alteqlabels" id="org1473c1c">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1663,18 +1655,18 @@ This is thus simply a conservation law for momentum, with
|
|||
<p>
|
||||
In a region in which the mechanical momentum is not changing due to external influences, we then have the
|
||||
</p>
|
||||
<div class="main div" id="org8cc6bb8">
|
||||
<div class="main div" id="orgf7ee289">
|
||||
<p>
|
||||
<b>Continuity equation for EM momentum</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="orgcad090e">
|
||||
<div class="eqlabel" id="orgd7db956">
|
||||
<p>
|
||||
<a id="contg"></a><a href="./emd_ce_mom.html#contg"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org69cf28d">
|
||||
<div class="alteqlabels" id="orge8fbc29">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1707,7 +1699,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1675,18 +1667,18 @@ and similarly for \({\boldsymbol B}\). We thus get
|
|||
<p>
|
||||
This expression can be greatly simplified by introducing the
|
||||
</p>
|
||||
<div class="main div" id="orga7d370d">
|
||||
<div class="main div" id="orgaa8145a">
|
||||
<p>
|
||||
<b>Maxwell stress tensor</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="orge5429c3">
|
||||
<div class="eqlabel" id="org0e3a6cc">
|
||||
<p>
|
||||
<a id="MaxST"></a><a href="./emd_ce_mst.html#MaxST"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org0306495">
|
||||
<div class="alteqlabels" id="org4826577">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1699,7 +1691,7 @@ T_{ij} \equiv \varepsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2\right)
|
|||
|
||||
</div>
|
||||
<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>
|
||||
|
||||
|
||||
|
|
@ -1707,18 +1699,18 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
|
|||
<p>
|
||||
We then obtain the
|
||||
</p>
|
||||
<div class="main div" id="org4781d10">
|
||||
<div class="main div" id="org66e8efe">
|
||||
<p>
|
||||
<b>EM force per unit volume</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="orgd9f91b7">
|
||||
<div class="eqlabel" id="org49902c2">
|
||||
<p>
|
||||
<a id="fT"></a><a href="./emd_ce_mst.html#fT"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org7d4a06b">
|
||||
<div class="alteqlabels" id="orge0c52b1">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1734,18 +1726,18 @@ We then obtain the
|
|||
<p>
|
||||
where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
|
||||
</p>
|
||||
<div class="main div" id="orgea18677">
|
||||
<div class="main div" id="org887bc5d">
|
||||
<p>
|
||||
<b>Total force on charges in volume</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="orgef98657">
|
||||
<div class="eqlabel" id="orgd12201e">
|
||||
<p>
|
||||
<a id="totFo"></a><a href="./emd_ce_mst.html#totFo"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org4c9c200">
|
||||
<div class="alteqlabels" id="org0dc7e2f">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1778,7 +1770,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</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
|
||||
at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\). Thus,
|
||||
</p>
|
||||
<div class="eqlabel" id="org0197499">
|
||||
<div class="eqlabel" id="orgc9958b2">
|
||||
<p>
|
||||
<a id="dWdt_intEJ"></a><a href="./emd_ce_poy.html#dWdt_intEJ"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org9aacea3">
|
||||
<div class="alteqlabels" id="orgf9bbccb">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (8.6)</li>
|
||||
</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}
|
||||
\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:
|
||||
\[
|
||||
{\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,
|
||||
we obtain
|
||||
</p>
|
||||
<div class="main div" id="orgf118f4f">
|
||||
<div class="main div" id="orgead023b">
|
||||
<p>
|
||||
<b>Poynting's theorem</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org1b7cdac">
|
||||
<div class="eqlabel" id="org1b1ef48">
|
||||
<p>
|
||||
<a id="👉Thm"></a><a href="./emd_ce_poy.html#👉Thm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orgd7b6cac">
|
||||
<div class="alteqlabels" id="org9c51283">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (8.9)</li>
|
||||
</ul>
|
||||
|
|
@ -1729,18 +1721,18 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
|
|||
<p>
|
||||
Energy per unit time, per unit area carried by EM fields: given by the
|
||||
</p>
|
||||
<div class="core div" id="orgf3198a5">
|
||||
<div class="core div" id="orgafa4bdd">
|
||||
<p>
|
||||
<b>Poynting vector</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org8725431">
|
||||
<div class="eqlabel" id="org0aaf227">
|
||||
<p>
|
||||
<a id="PoyntingVec"></a><a href="./emd_ce_poy.html#PoyntingVec"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org6f2879d">
|
||||
<div class="alteqlabels" id="org05edf23">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (8.10)</li>
|
||||
</ul>
|
||||
|
|
@ -1759,18 +1751,18 @@ Energy per unit time, per unit area carried by EM fields: given by the
|
|||
<p>
|
||||
We can thus express Poynting's theorem more compactly:
|
||||
</p>
|
||||
<div class="core div" id="org3a4bb91">
|
||||
<div class="core div" id="orgf7b3c73">
|
||||
<p>
|
||||
<b>Poynting's theorem</b> (integral form)
|
||||
</p>
|
||||
<div class="eqlabel" id="orgbf2cc63">
|
||||
<div class="eqlabel" id="org610ce5e">
|
||||
<p>
|
||||
<a id="PoyntingThm_int"></a><a href="./emd_ce_poy.html#PoyntingThm_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org5f484e3">
|
||||
<div class="alteqlabels" id="org462a7a9">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (8.11)</li>
|
||||
</ul>
|
||||
|
|
@ -1789,18 +1781,18 @@ We can thus express Poynting's theorem more compactly:
|
|||
<p>
|
||||
where we have defined the total
|
||||
</p>
|
||||
<div class="core div" id="orgbc7eb16">
|
||||
<div class="core div" id="org49394fc">
|
||||
<p>
|
||||
<b>Energy in electromagnetic fields</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org89872c3">
|
||||
<div class="eqlabel" id="org2fd18f3">
|
||||
<p>
|
||||
<a id="Uem"></a><a href="./emd_ce_poy.html#Uem"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org67613ca">
|
||||
<div class="alteqlabels" id="orge0e6bd0">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (8.5)</li>
|
||||
</ul>
|
||||
|
|
@ -1829,18 +1821,18 @@ Then,
|
|||
\]
|
||||
so we get the
|
||||
</p>
|
||||
<div class="core div" id="org487db23">
|
||||
<div class="core div" id="orgb7f6aa8">
|
||||
<p>
|
||||
<b>Poynting theorem</b> (differential form)
|
||||
</p>
|
||||
<div class="eqlabel" id="org9593699">
|
||||
<div class="eqlabel" id="orgc43f9ba">
|
||||
<p>
|
||||
<a id="PoyntingThm"></a><a href="./emd_ce_poy.html#PoyntingThm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org129becb">
|
||||
<div class="alteqlabels" id="org50e67b6">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (8.14)</li>
|
||||
</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>
|
||||
<b>Example: Joule heating</b>
|
||||
</p>
|
||||
|
|
@ -1895,7 +1887,7 @@ and points radially inwards. Energy per unit time passing surface of wire:
|
|||
\[
|
||||
\int d{\bf a} \cdot {\bf S} = S (2\pi a L) = -V I
|
||||
\]
|
||||
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.
|
||||
</p>
|
||||
|
||||
|
|
@ -1920,7 +1912,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<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>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1622,8 +1614,8 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emd.emw</span></h3>
|
||||
|
||||
<div class="outline-text-3" id="text-emd_emw">
|
||||
<details class="prereq" id="orgd522e84">
|
||||
<summary id="org3c8d630">
|
||||
<details class="prereq" id="org42aca70">
|
||||
<summary id="org3fdd75c">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1632,8 +1624,8 @@ Prerequisites
|
|||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="org98124eb">
|
||||
<summary id="org6546918">
|
||||
<details class="objectives" id="org8aeb437">
|
||||
<summary id="org16d834d">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1674,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1652,12 +1644,30 @@ so for a monochromatic EM plan wave,
|
|||
\]
|
||||
or more succinctly:
|
||||
</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>
|
||||
{\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>
|
||||
|
||||
</div>
|
||||
|
|
@ -1668,12 +1678,12 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
|
|||
<p>
|
||||
Similary, we get the
|
||||
</p>
|
||||
<div class="main div" id="org4b55ff9">
|
||||
<div class="main div" id="org108cc57">
|
||||
<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>
|
||||
|
||||
</div>
|
||||
|
|
@ -1688,14 +1698,14 @@ Time averages: integrating over a (integer number of) cycle(s), we have
|
|||
</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>
|
||||
\[
|
||||
I \equiv \langle S \rangle = \frac{c\varepsilon_0}{2} E_0^2
|
||||
\]
|
||||
</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
|
||||
\[
|
||||
P = \frac{1}{A}\frac{\Delta p}{\Delta t} = \frac{\langle g \rangle A c \Delta t}{A \Delta t} = \frac{\varepsilon_0}{2} E_0^2 = \frac{I}{c}.
|
||||
\]
|
||||
|
|
@ -1719,7 +1729,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1622,7 +1614,7 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emd.emw.mpw</span></h4>
|
||||
<div class="outline-text-4" id="text-emd_emw_mpw">
|
||||
<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 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
|
||||
\label{Gr(9.44)}
|
||||
\]
|
||||
so {\bf electromagnetic waves are transverse}.
|
||||
so <b>electromagnetic waves are transverse</b>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
|
|
@ -1643,9 +1635,26 @@ From Faraday: \({\boldsymbol \nabla} \times {\bf E} = -\partial {\bf B}/\partia
|
|||
\label{Gr(9.46)}
|
||||
\]
|
||||
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>
|
||||
\[
|
||||
B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0.
|
||||
\label{Gr(9.47)}
|
||||
\tag{EBmpw}\label{EBmpw}
|
||||
\]
|
||||
</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
|
||||
\({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
|
||||
</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>
|
||||
{\bf E and B fields for a monochromatic EM plane wave}
|
||||
\[
|
||||
{\boldsymbol E} ({\boldsymbol r},t ) = E_0 e^{i ({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol n},
|
||||
\hspace{10mm}
|
||||
{\boldsymbol B} ({\boldsymbol r}, t) = \frac{E_0}{c} e^{i({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol k} \times \hat{\boldsymbol n}
|
||||
= \frac{1}{c} ~\hat{\boldsymbol k} \times {\boldsymbol E} ({\boldsymbol r}, t)
|
||||
\]
|
||||
with the transversality condition
|
||||
{\boldsymbol E} ({\boldsymbol r},t ) = E_0 e^{i ({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol n},
|
||||
\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}
|
||||
= \frac{1}{c} ~\hat{\boldsymbol k} \times {\boldsymbol E} ({\boldsymbol r}, t)
|
||||
\tag{mpw}\label{mpw}
|
||||
\]
|
||||
with the <b>transversality condition</b>
|
||||
\[
|
||||
\hat{\boldsymbol k} \cdot \hat{\boldsymbol n} = 0
|
||||
\]
|
||||
\hat{\boldsymbol k} \cdot \hat{\boldsymbol n} = 0
|
||||
\]
|
||||
</p>
|
||||
|
||||
</div>
|
||||
|
|
@ -1697,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1622,7 +1614,6 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emd.emw.we</span></h4>
|
||||
<div class="outline-text-4" id="text-emd_emw_we">
|
||||
<p>
|
||||
\subsubsection*{The wave equation for \({\bf E}\) and \({\bf B}\)}
|
||||
Take Maxwell's equations in vacuum:
|
||||
</p>
|
||||
\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 \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
|
||||
we get the
|
||||
</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>
|
||||
{\bf Wave equations for electric and magnetic fields in vacuum}
|
||||
\[
|
||||
{\boldsymbol \nabla}^2 {\bf E} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf E}}{\partial t^2},
|
||||
\hspace{1cm}
|
||||
{\boldsymbol \nabla}^2 {\bf B} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf B}}{\partial t^2}.
|
||||
\label{Gr(9.41)}
|
||||
\]
|
||||
{\boldsymbol \nabla}^2 {\bf E} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf E}}{\partial t^2},
|
||||
\hspace{1cm}
|
||||
{\boldsymbol \nabla}^2 {\bf B} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf B}}{\partial t^2}.
|
||||
\tag{WaveEq}\label{WaveEq}
|
||||
\]
|
||||
</p>
|
||||
|
||||
</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 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
|
||||
the actual electric and magnetic fields are given by the real part.
|
||||
</p>
|
||||
|
|
@ -1706,7 +1714,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1646,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1646,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1637,12 +1629,12 @@ free charges and currents.
|
|||
</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>
|
||||
\[
|
||||
\rho_b = -{\boldsymbol \nabla} \cdot {\bf P}
|
||||
\label{Gr(7.46)}
|
||||
\]
|
||||
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>
|
||||
\[
|
||||
{\bf J}_b = {\boldsymbol \nabla} \times {\bf M}
|
||||
\label{Gr(7.47)}
|
||||
|
|
@ -1657,13 +1649,30 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
|
|||
\]
|
||||
We therefore have the
|
||||
</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>
|
||||
{\bf Polarization current density}
|
||||
\[
|
||||
{\bf J}_p = \frac{\partial {\bf P}}{\partial t}
|
||||
\label{Gr(7.48)}
|
||||
\]
|
||||
{\bf J}_p = \frac{\partial {\bf P}}{\partial t}
|
||||
\tag{Jp}\label{Jp}
|
||||
\]
|
||||
</p>
|
||||
|
||||
</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
|
||||
associated to the conservation of bound charges:
|
||||
</p>
|
||||
<aside id="org642846e">
|
||||
<aside id="orgadb90c7">
|
||||
<p>
|
||||
Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current
|
||||
\({\boldsymbol J}_b\) but this is not the convention used here.
|
||||
Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current \({\boldsymbol J}_b\) but this is not the common convention.
|
||||
</p>
|
||||
</aside>
|
||||
<p>
|
||||
|
|
@ -1696,31 +1704,56 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
|
|||
|
||||
<p>
|
||||
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>
|
||||
<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>
|
||||
\[
|
||||
\rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
|
||||
\label{Gr(7.49)}
|
||||
\]
|
||||
\rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
|
||||
\tag{rhofb}\label{rhofb}
|
||||
\]
|
||||
</p>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
and current can be separated into three parts, {\it free}, {\it bound} and
|
||||
{\it polarization}:
|
||||
and current can be separated into three parts, <b>free</b>, <b>bound</b> and
|
||||
<b>polarization</b>:
|
||||
</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">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 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>
|
||||
\[
|
||||
{\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\frac{∂ {\bf P}}{∂ t}.</li>
|
||||
</ul>
|
||||
<p>
|
||||
\label{Gr(7.50)}
|
||||
{\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol \nabla} \times {\bf M} + \frac{\partial {\bf P}}{\partial t}.
|
||||
\tag{Jfbp}\label{Jfbp}
|
||||
\]
|
||||
</p>
|
||||
|
||||
|
|
@ -1735,24 +1768,19 @@ Gauss's law: can be rewritten
|
|||
\]
|
||||
where (as in static case)
|
||||
</p>
|
||||
<div class="core div" id="org88bb3c5">
|
||||
<div class="core div" id="org501f375">
|
||||
<p>
|
||||
\[
|
||||
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
|
||||
\label{Gr(7.52)}
|
||||
\]
|
||||
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
|
||||
\label{Gr(7.52)}
|
||||
\]
|
||||
</p>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
Ampère's law including Maxwell's term:
|
||||
\[
|
||||
{\boldsymbol ∇} × {\bf B} = μ_0 \left( {\bf J}_f + {\boldsymbol ∇} × {\bf M}
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\frac{∂ {\bf P}}{∂ t} \right) + μ_0 ε_0 \frac{∂ {\bf E}}{∂ t},</li>
|
||||
</ul>
|
||||
<p>
|
||||
{\boldsymbol \nabla} \times {\bf B} = \mu_0 \left( {\bf J}_f + {\boldsymbol \nabla} \times {\bf M} + \frac{\partial {\bf P}}{\partial t} \right) + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t},
|
||||
\]
|
||||
or
|
||||
\[
|
||||
|
|
@ -1761,12 +1789,12 @@ or
|
|||
\]
|
||||
where as before
|
||||
</p>
|
||||
<div class="core div" id="org90cea20">
|
||||
<div class="core div" id="orgfaac9ca">
|
||||
<p>
|
||||
\[
|
||||
{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
|
||||
\label{Gr(7.54)}
|
||||
\]
|
||||
{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
|
||||
\label{Gr(7.54)}
|
||||
\]
|
||||
</p>
|
||||
|
||||
</div>
|
||||
|
|
@ -1779,21 +1807,36 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
|
|||
<p>
|
||||
In terms of free charges and currents, we thus get
|
||||
</p>
|
||||
<div class="core div" id="orgd6526ab">
|
||||
<div class="core div" id="org4a1df55">
|
||||
<p>
|
||||
{\bf Maxwell's equations {\it (in matter)}}
|
||||
<b>Maxwell's equations</b> <i>(in matter)</i>
|
||||
</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}
|
||||
(i)~~ &{\boldsymbol \nabla} \cdot {\bf D} = \rho_f, \nonumber \\
|
||||
(ii)~~ &{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\
|
||||
(iii)~~ &{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\
|
||||
(iv)~~ &{\boldsymbol \nabla} \times {\bf H} = {\bf J}_f + \frac{\partial {\bf D}}{\partial t}.
|
||||
\label{Gr(7.55)}
|
||||
(i)~~ &{\boldsymbol \nabla} \cdot {\bf D} = \rho_f, \nonumber \\
|
||||
(ii)~~ &{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\
|
||||
(iii)~~ &{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\
|
||||
(iv)~~ &{\boldsymbol \nabla} \times {\bf H} = {\bf J}_f + \frac{\partial {\bf D}}{\partial t}.
|
||||
\tag{Max_mat}\label{Max_mat}
|
||||
\end{align}
|
||||
|
||||
</div>
|
||||
<p>
|
||||
Last term: {\bf displacement current},
|
||||
Last term: <b>displacement current</b>,
|
||||
\[
|
||||
{\bf J}_d = \frac{\partial {\bf D}}{\partial t}
|
||||
\label{Gr(7.58)}
|
||||
|
|
@ -1801,23 +1844,37 @@ Last term: {\bf displacement current},
|
|||
</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 \({\bf B}\).
|
||||
For the restricted case of linear media:
|
||||
</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>
|
||||
\[
|
||||
{\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
|
||||
{\bf M} = \chi_m {\bf H}
|
||||
\label{Gr(7.56)}
|
||||
\]
|
||||
{\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
|
||||
{\bf M} = \chi_m {\bf H}
|
||||
\]
|
||||
so
|
||||
\[
|
||||
{\bf D} = \varepsilon {\bf E}, \hspace{1cm}
|
||||
{\bf H} = \frac{1}{\mu} {\bf B},
|
||||
\label{Gr(7.57)}
|
||||
\]
|
||||
{\bf D} = \varepsilon {\bf E}, \hspace{1cm}
|
||||
{\bf H} = \frac{1}{\mu} {\bf B},
|
||||
\tag{consrel}\label{consrel}
|
||||
\]
|
||||
where \(\varepsilon \equiv \varepsilon_0(1 + \chi_e)\) and \(\mu \equiv \mu_0 (1 + \chi_m)\).
|
||||
</p>
|
||||
|
||||
|
|
@ -1842,7 +1899,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<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>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1624,74 +1616,177 @@ Table of contents
|
|||
<p>
|
||||
Discontinuities between different media, deduced from
|
||||
</p>
|
||||
<div class="core div" id="org48bc400">
|
||||
<div class="core div" id="org2152f07">
|
||||
<p>
|
||||
{\bf Maxwell's equations {\it (in matter)}, integral form}
|
||||
<b>Maxwell's equations</b> <i>(in matter)</i>, <i>integral form</i>
|
||||
</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}
|
||||
(i)~~ &\oint_{\cal S} {\bf D} \cdot d{\bf a} = Q_{f_{enc}}, \nonumber \\
|
||||
(ii)~~ &\oint_{\cal S} {\bf B} \cdot d{\bf a} = 0 \nonumber \\
|
||||
(iii)~~ &\oint_{\cal P} {\bf E} \cdot d{\bf l} = -\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}, \nonumber \\
|
||||
(iv)~~ &\oint_{\cal P} {\bf H} \cdot d{\bf l} = I_{f_{enc}} + \frac{d}{dt} \int_{\cal S} {\bf D} \cdot d{\bf a}.
|
||||
(i)~~ &\oint_{\cal S} {\bf D} \cdot d{\bf a} = Q_{f_{enc}}, \nonumber \\
|
||||
(ii)~~ &\oint_{\cal S} {\bf B} \cdot d{\bf a} = 0 \nonumber \\
|
||||
(iii)~~ &\oint_{\cal P} {\bf E} \cdot d{\bf l} = -\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}, \nonumber \\
|
||||
(iv)~~ &\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}
|
||||
|
||||
</div>
|
||||
<p>
|
||||
Applying \((i)\) to wafer-thin Gaussian pillbox straddling boundary between 2 materials:
|
||||
\({\bf D}_1 \cdot {\bf a} - {\bf D}_2 \cdot {\bf a} = \sigma_f a\) so
|
||||
</p>
|
||||
<div class="main div" id="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>
|
||||
\[
|
||||
\boxed{
|
||||
D^{\perp}_1 - D^{\perp}_2 = \sigma_f
|
||||
}
|
||||
\label{Gr(7.59)}
|
||||
\tag{Ddisc}\label{Ddisc}
|
||||
\]
|
||||
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>
|
||||
\[
|
||||
\boxed{
|
||||
B^{\perp}_1 - B^{\perp}_2 = 0
|
||||
}
|
||||
\label{Gr(7.60)}
|
||||
\]
|
||||
</p>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
For \((iii)\): Amperian loop straddling surface: \({\bf E}_1 \cdot {\bf l} - {\bf E}_2 \cdot {\bf l} =
|
||||
-\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}\). Limit of small loop: flux vanishes, therefore
|
||||
</p>
|
||||
<div class="main div" id="org2c8267d">
|
||||
<p>
|
||||
\[
|
||||
\boxed{
|
||||
{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0
|
||||
}
|
||||
\label{Gr(7.61)}
|
||||
\]
|
||||
</p>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
Similarly, \((iv)\) implies \({\bf H}_1 \cdot {\bf l} - {\bf H}_2 \cdot {\bf l} = I_{f_{enc}}\).
|
||||
No volume current can contribute, but a surface current can. Can write
|
||||
\(I_{f_{enc}} = {\bf K}_f \cdot (\hat{\bf n} \times {\bf l}) = ({\bf K}_f \times \hat{\bf n}) \cdot {\bf l}\)
|
||||
and thus
|
||||
</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>
|
||||
\[
|
||||
\boxed{
|
||||
{\bf H}_1^{\parallel} - {\bf H}_2^{\parallel} = {\bf K}_f \times \hat{\bf n}
|
||||
}
|
||||
\label{Gr(7.62)}
|
||||
\tag{Hdisc}\label{Hdisc}
|
||||
\]
|
||||
</p>
|
||||
|
||||
</div>
|
||||
<p>
|
||||
These are the general boundary conditions for electrodynamics.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
In case of linear media: can be expressed in terms of \({\bf E}\) and \({\bf B}\) alone:
|
||||
</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}
|
||||
(i)~~ &\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = \sigma_f, \nonumber \\
|
||||
(ii)~~ &B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\
|
||||
(iii)~~ &{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\
|
||||
(iv)~~ &\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}
|
||||
|
||||
</div>
|
||||
<p>
|
||||
If there is no free charge and no free current at boundary:
|
||||
</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}
|
||||
(i)~~ &\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = 0, \nonumber \\
|
||||
(ii)~~ &B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\
|
||||
(iii)~~ &{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\
|
||||
(iv)~~ &\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}
|
||||
<p>
|
||||
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 id="postamble" class="status">
|
||||
<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>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1649,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1615,7 +1607,7 @@ Table of contents
|
|||
</ul>
|
||||
</details>
|
||||
</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: <a href="emdm_emwm_refl_Ba.html">Brewster's Angle <small>[emdm.emwm.refl.Ba]</small></a></li><li>Next: <a href="emdm_emwm_ad_c.html">EM Waves in Conductors <small>[emdm.emwm.ad.c]</small></a></li><li>Up: <a href="emdm_emwm.html">Electromagnetic Waves in Matter <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: <a href="emdm_emwm_refl_oi.html">Oblique Incidence <small>[emdm.emwm.refl.oi]</small></a></li><li>Next: <a href="emdm_emwm_ad_c.html">EM Waves in Conductors <small>[emdm.emwm.ad.c]</small></a></li><li>Up: <a href="emdm_emwm.html">Electromagnetic Waves in Matter <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">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
|
|
@ -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>
|
||||
</ul>
|
||||
|
||||
<br><ul class="navigation-links"><li>Prev: <a href="emdm_emwm_refl_Ba.html">Brewster's Angle <small>[emdm.emwm.refl.Ba]</small></a></li><li>Next: <a href="emdm_emwm_ad_c.html">EM Waves in Conductors <small>[emdm.emwm.ad.c]</small></a></li><li>Up: <a href="emdm_emwm.html">Electromagnetic Waves in Matter <small>[emdm.emwm]</small></a></li></ul>
|
||||
<br><ul class="navigation-links"><li>Prev: <a href="emdm_emwm_refl_oi.html">Oblique Incidence <small>[emdm.emwm.refl.oi]</small></a></li><li>Next: <a href="emdm_emwm_ad_c.html">EM Waves in Conductors <small>[emdm.emwm.ad.c]</small></a></li><li>Up: <a href="emdm_emwm.html">Electromagnetic Waves in Matter <small>[emdm.emwm]</small></a></li></ul>
|
||||
<br>
|
||||
<hr>
|
||||
<div class="license">
|
||||
|
|
@ -1645,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</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
|
||||
</p>
|
||||
\begin{align}
|
||||
\begin{align*}
|
||||
{\boldsymbol \nabla} \cdot {\boldsymbol E} &= \frac{\rho_f}{\varepsilon},
|
||||
\hspace{10mm} &
|
||||
{\boldsymbol \nabla} \cdot {\boldsymbol B} &= 0, \nonumber\\
|
||||
{\boldsymbol \nabla} \times {\boldsymbol E} &= -\frac{\partial {\boldsymbol B}}{\partial t} &
|
||||
{\boldsymbol \nabla} \times {\boldsymbol B} &= \mu \sigma {\boldsymbol E} + \mu \varepsilon \frac{\partial {\boldsymbol E}}{\partial t}.
|
||||
\end{align}
|
||||
\end{align*}
|
||||
<p>
|
||||
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} &= \mu \sigma {\boldsymbol E} + \mu \varepsilon \frac{\partial {\boldsymbol E}}{\partial t}
|
||||
\end{align}
|
||||
<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 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)}
|
||||
\]
|
||||
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>
|
||||
|
|
@ -1741,7 +1733,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1624,13 +1616,13 @@ Table of contents
|
|||
<p>
|
||||
In matter regions without free charge and free current: Maxwell's equations are
|
||||
</p>
|
||||
\begin{align}
|
||||
|
||||
\begin{align*}
|
||||
(i)~~ &{\boldsymbol \nabla} \cdot {\bf D} = 0, \nonumber \\
|
||||
(ii)~~ &{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\
|
||||
(iii)~~ &{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\
|
||||
(iv)~~ &{\boldsymbol \nabla} \times {\bf H} = \frac{\partial {\bf D}}{\partial t}.
|
||||
\label{Gr(9.65)}
|
||||
\end{align}
|
||||
\end{align*}
|
||||
<p>
|
||||
For linear medium:
|
||||
\[
|
||||
|
|
@ -1640,13 +1632,31 @@ For linear medium:
|
|||
\]
|
||||
If the medium is homogeneous (no spatial dependence of \(\varepsilon\) or \(\mu\)),
|
||||
</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">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 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}
|
||||
(i)~~ &{\boldsymbol \nabla} \cdot {\bf E} = 0, \nonumber \\
|
||||
(ii)~~ &{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\
|
||||
(iii)~~ &{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\
|
||||
(iv)~~ &{\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}
|
||||
|
||||
</div>
|
||||
<p>
|
||||
These are the same equations as in vacuum, except for the substitution of \(\mu_0 \varepsilon_0\) by \(\mu \varepsilon\).
|
||||
</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
|
||||
</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>
|
||||
{\bf Index of refraction}
|
||||
\[
|
||||
n \equiv \sqrt{\frac{\mu \varepsilon}{\mu_0 \varepsilon_0}}
|
||||
\label{Gr(9.69)}
|
||||
\]
|
||||
n \equiv \sqrt{\frac{\mu \varepsilon}{\mu_0 \varepsilon_0}}
|
||||
\tag{n}\label{n}
|
||||
\]
|
||||
</p>
|
||||
|
||||
</div>
|
||||
|
|
@ -1675,7 +1702,7 @@ Fact: for most materials, \(\mu\) is very close to \(\mu_0\), so
|
|||
n \simeq \sqrt{\varepsilon_r}
|
||||
\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>
|
||||
|
|
@ -1716,7 +1743,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1625,15 +1617,14 @@ Table of contents
|
|||
<p>
|
||||
Interesting question: what happens to an EM wave as it passes from one medium to another?
|
||||
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>
|
||||
\begin{align}
|
||||
\begin{align*}
|
||||
(i)~~ &\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = 0, \nonumber \\
|
||||
(ii)~~ &B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\
|
||||
(iii)~~ &{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\
|
||||
(iv)~~ &\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>
|
||||
|
||||
|
||||
|
|
@ -1641,8 +1632,6 @@ Detailed study: starts from boundary conditions \ref{Gr(7.64)},
|
|||
<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_oi.html">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span></li>
|
||||
<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>
|
||||
</ul>
|
||||
|
||||
<br><ul class="navigation-links"><li>Prev: <a href="emdm_emwm_refr.html">Refraction <small>[emdm.emwm.refr]</small></a></li><li>Next: <a href="emdm_emwm_refl_ni.html">Normal Incidence <small>[emdm.emwm.refl.ni]</small></a></li><li>Up: <a href="emdm_emwm.html">Electromagnetic Waves in Matter <small>[emdm.emwm]</small></a></li></ul>
|
||||
|
|
@ -1660,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
File diff suppressed because it is too large
Load Diff
File diff suppressed because it is too large
Load Diff
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</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}
|
||||
\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>
|
||||
|
|
@ -1644,19 +1636,19 @@ Reflected wave:
|
|||
|
||||
<p>
|
||||
For the transmitted wave, we put
|
||||
\[
|
||||
{\boldsymbol E}_T (z,t) = E_{0_T} e^{i (k_2 z - \omega t)} \hat{\boldsymbol x}, \hspace{1cm}
|
||||
{\boldsymbol B}_T (z,t) = \frac{1}{v_2} E_{0_T} e^{i (k_2 z - \omega t)} \hat{\boldsymbol y}.
|
||||
\label{Gr(9.75)}
|
||||
\]
|
||||
</p>
|
||||
|
||||
\begin{align*}
|
||||
{\boldsymbol E}_T (z,t) &= E_{0_T} e^{i (k_2 z - \omega t)} \hat{\boldsymbol x}, \\
|
||||
{\boldsymbol B}_T (z,t) &= \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>
|
||||
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.
|
||||
</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
|
||||
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
|
||||
\[
|
||||
E_{0_I} + E_{0_R} = E_{0_T}
|
||||
\label{Gr(9.78)}
|
||||
|
|
@ -1677,10 +1669,27 @@ E_{0_I} - E_{0_R} = \beta E_{0_T}, \hspace{1cm}
|
|||
<p>
|
||||
Solving these coupled equations, we can write
|
||||
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>
|
||||
\[
|
||||
E_{0_R} = \frac{1 - \beta}{1 + \beta} E_{0_I}, \hspace{10mm}
|
||||
E_{0_T} = \frac{2}{1 + \beta} E_{0_I}.
|
||||
\label{Gr(9.82)}
|
||||
\tag{ERT}\label{ERT}
|
||||
\]
|
||||
</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)}
|
||||
\]
|
||||
while the transmitted intensity is
|
||||
\[
|
||||
</p>
|
||||
|
||||
\begin{equation}
|
||||
T \equiv \frac{I_T}{I_I} = \frac{v_2 \varepsilon_2}{v_1 \varepsilon_1} \frac{E^2_{0_T}}{E^2_{0_I}}
|
||||
= \sqrt{\frac{\mu_1 \varepsilon_2}{\mu_2 \varepsilon_1}} \frac{E^2_{0_T}}{E^2_{0_I}} = \frac{4 \beta}{(1 + \beta)^2}.
|
||||
\label{Gr(9.86)}
|
||||
\]
|
||||
We thus have that the {\bf reflection} and {\bf transmission coefficients} satisfy
|
||||
\end{equation}
|
||||
<p>
|
||||
We thus have that the <b>reflection</b> and <b>transmission coefficients</b> satisfy
|
||||
\[
|
||||
R + T = 1.
|
||||
\label{Gr(9.88)}
|
||||
|
|
@ -1732,7 +1743,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<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>
|
||||
|
||||
</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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1615,7 +1607,7 @@ Table of contents
|
|||
</ul>
|
||||
</details>
|
||||
</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: <a href="emdm_emwm_refl_ni.html">Normal Incidence <small>[emdm.emwm.refl.ni]</small></a></li><li>Next: <a href="emdm_emwm_refl_Fe.html">Fresnel's Equations <small>[emdm.emwm.refl.Fe]</small></a></li><li>Up: <a href="emdm_emwm_refl.html">Reflection and Transmission <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: <a href="emdm_emwm_refl_ni.html">Normal Incidence <small>[emdm.emwm.refl.ni]</small></a></li><li>Next: <a href="emdm_emwm_ad.html">Absorption and Dispersion <small>[emdm.emwm.ad]</small></a></li><li>Up: <a href="emdm_emwm_refl.html">Reflection and Transmission <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">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 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).
|
||||
\]
|
||||
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>
|
||||
\[
|
||||
k_I v_1 = k_R v_1 = k_T v_2 ~~\longrightarrow~~ k_I = k_R = \frac{v_2}{v_1} k_T = \frac{n_1}{n_2} k_T
|
||||
\label{eq:RTObliquek}
|
||||
\tag{RTobliquek}\label{RTobliquek}
|
||||
\]
|
||||
</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_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>
|
||||
From now on we will orient the axes so that \({\boldsymbol k}_I\) lies in the \(xz\) plane. This means that \({\boldsymbol k}_R\) and \({\boldsymbol k}_T\) also lie in that plane. This is the
|
||||
</p>
|
||||
<div class="core div" id="orge681e56">
|
||||
<div class="core div" id="org6f173e3">
|
||||
<p>
|
||||
{\bf First law of reflection:}
|
||||
the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
|
||||
<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.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
<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
|
||||
\[
|
||||
k_I \sin \theta_I = k_R \sin \theta_R = k_T \sin \theta_T
|
||||
\]
|
||||
with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
|
||||
and the angle of refraction (\(\theta_T\)) obey the following laws:
|
||||
</p>
|
||||
<div class="core div" id="orgfbf5032">
|
||||
<div class="core div" id="orgbf9459c">
|
||||
<p>
|
||||
{\bf Law of reflection}
|
||||
\[
|
||||
<b>Law of reflection</b>
|
||||
\[
|
||||
\theta_I = \theta_R
|
||||
\]
|
||||
{\bf Law of refraction (Snell's law)}
|
||||
\[
|
||||
<b>Law of refraction (Snell's law)</b>
|
||||
\[
|
||||
n_1 \sin \theta_I = n_2 \sin \theta_T
|
||||
\]
|
||||
</p>
|
||||
|
|
@ -1691,13 +1697,25 @@ and the angle of refraction (\(\theta_T\)) obey the following laws:
|
|||
<p>
|
||||
This takes care of the spatially-dependent exponential factors in the boundary conditions. The coefficients must further obey
|
||||
</p>
|
||||
<div class="eqlabel" id="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}
|
||||
\varepsilon_1 \left({\boldsymbol E}_{0_I} + {\boldsymbol E}_{0_R} \right)_z &= \varepsilon_2 \left({\boldsymbol E}_{0_T} \right)_z,
|
||||
& \hspace{10mm}
|
||||
\left({\boldsymbol B}_{0_I} +{\boldsymbol B}_{0_R} \right)_z &= \left({\boldsymbol B}_{0_T}\right)_z, \nonumber \\
|
||||
\left( {\boldsymbol E}_{0_I} + {\boldsymbol E}_{0_R} \right)_{x,y} &= \left({\boldsymbol E}_{0_T}\right)_{x,y},
|
||||
& \frac{1}{\mu_1} \left({\boldsymbol B}_{0_I} + {\boldsymbol B}_{0_R} \right)_{x,y} &= \frac{1}{\mu_2} \left({\boldsymbol B}_{0_T}\right)_{x,y}.
|
||||
\label{eq:EMBdryCondAtMediumInterface:amp}
|
||||
\tag{EBRT}\label{EBRT}
|
||||
\end{align}
|
||||
|
||||
<p>
|
||||
|
|
@ -1709,8 +1727,8 @@ The two cases of polarization parallel and perpendicular to the plane of inciden
|
|||
</p>
|
||||
|
||||
<p>
|
||||
\paragraph{Polarization in plane of incidence:}
|
||||
in this case the first equation of (\ref{eq:EMBdryCondAtMediumInterface:amp}) gives
|
||||
<b>Polarization in plane of incidence</b>:
|
||||
in this case the first equation of <a href="./emdm_emwm_refl_oi.html#EBRT">EBRT</a> gives
|
||||
\[
|
||||
\varepsilon_1 \left(-E_{0_I} \sin \theta_I + E_{0_R} \sin \theta_R \right) = -\varepsilon_2 E_{0_T} \sin \theta_T.
|
||||
\]
|
||||
|
|
@ -1732,13 +1750,28 @@ while the third equation becomes
|
|||
\]
|
||||
Writing everything in terms of the incident amplitude, we get
|
||||
</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>
|
||||
{\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)}
|
||||
\[
|
||||
E_{0_R} = \frac{\alpha - \beta}{\alpha + \beta} E_{0_I},
|
||||
\hspace{10mm}
|
||||
E_{0_T} = \frac{2}{\alpha + \beta} E_{0_I}
|
||||
\tag{Fresnel}\label{Fresnel}
|
||||
\]
|
||||
</p>
|
||||
|
||||
|
|
@ -1749,15 +1782,28 @@ Amplitudes for transmitted and reflected wave: depend on angle of incidence:
|
|||
\[
|
||||
\alpha = \frac{\sqrt{1 - \sin^2 \theta_T}}{\cos \theta_I} = \frac{\left[1 - \left(\frac{n_1}{n_2}\right)^2 \sin^2 \theta_I\right]^{1/2}}{\cos \theta_I}
|
||||
\]
|
||||
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
|
||||
</p>
|
||||
<div class="main div" id="org7503b43">
|
||||
<div class="main div" id="org328f2a5">
|
||||
<div class="eqlabel" id="org3b9cec2">
|
||||
<p>
|
||||
{\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}}
|
||||
\[
|
||||
\theta_B = \arcsin \left[ \frac{1 - \beta^2}{(n_1/n_2)^2 - \beta^2} \right]^{1/2}
|
||||
\]
|
||||
<a id="Brewster"></a><a href="./emdm_emwm_refl_oi.html#Brewster"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="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>
|
||||
|
||||
</div>
|
||||
|
|
@ -1785,9 +1831,7 @@ Of course, we get \(R + T = 1\) as expected.
|
|||
</div>
|
||||
|
||||
|
||||
|
||||
|
||||
<br><ul class="navigation-links"><li>Prev: <a href="emdm_emwm_refl_ni.html">Normal Incidence <small>[emdm.emwm.refl.ni]</small></a></li><li>Next: <a href="emdm_emwm_refl_Fe.html">Fresnel's Equations <small>[emdm.emwm.refl.Fe]</small></a></li><li>Up: <a href="emdm_emwm_refl.html">Reflection and Transmission <small>[emdm.emwm.refl]</small></a></li></ul>
|
||||
<br><ul class="navigation-links"><li>Prev: <a href="emdm_emwm_refl_ni.html">Normal Incidence <small>[emdm.emwm.refl.ni]</small></a></li><li>Next: <a href="emdm_emwm_ad.html">Absorption and Dispersion <small>[emdm.emwm.ad]</small></a></li><li>Up: <a href="emdm_emwm_refl.html">Reflection and Transmission <small>[emdm.emwm.refl]</small></a></li></ul>
|
||||
<br>
|
||||
<hr>
|
||||
<div class="license">
|
||||
|
|
@ -1802,7 +1846,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1641,7 +1633,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1647,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1667,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1668,7 +1660,7 @@ Putting these back into Maxwell (i) and (ii) gives the decoupled equations
|
|||
\[
|
||||
\left[ \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \left(\frac{\omega}{c}\right)^2 - k^2 \right] E_z = 0,
|
||||
\]
|
||||
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>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -1691,7 +1683,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1662,7 +1654,7 @@ In view of this, if the frequency is low enough, namely
|
|||
\[
|
||||
\omega < c\pi \sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2} \equiv \omega_{mn}
|
||||
\]
|
||||
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>
|
||||
|
|
@ -1674,7 +1666,8 @@ and wave velocity
|
|||
\[
|
||||
v = \frac{\omega}{k} = \frac{c}{\sqrt{1 - \left(\frac{\omega_{mn}}{\omega}\right)^2}}
|
||||
\]
|
||||
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>
|
||||
\[
|
||||
v_g = \frac{d\omega}{dk} = c \sqrt{1 - \left(\frac{\omega_{mn}}{\omega}\right)^2}.
|
||||
\]
|
||||
|
|
@ -1700,7 +1693,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1622,8 +1614,8 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emf</span></h2>
|
||||
|
||||
<div class="outline-text-2" id="text-emf">
|
||||
<details class="prereq" id="orgb93b9c3">
|
||||
<summary id="org621593e">
|
||||
<details class="prereq" id="org8e30f8b">
|
||||
<summary id="org2c7b815">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1631,8 +1623,8 @@ Prerequisites
|
|||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="org179e7b9">
|
||||
<summary id="orgb47e5d6">
|
||||
<details class="objectives" id="org5818573">
|
||||
<summary id="orgda754dd">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1668,7 +1660,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1665,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1664,7 +1656,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1637,7 +1629,7 @@ while the equation for \(V\) becomes
|
|||
\]
|
||||
These can be written compactly upon introducing a new operator: the
|
||||
</p>
|
||||
<div class="core div" id="org242c17e">
|
||||
<div class="core div" id="org0aaab77">
|
||||
<p>
|
||||
{\bf d'Alembertian operator}
|
||||
\[
|
||||
|
|
@ -1650,7 +1642,7 @@ These can be written compactly upon introducing a new operator: the
|
|||
<p>
|
||||
so we get the
|
||||
</p>
|
||||
<div class="core div" id="orgb00ab8a">
|
||||
<div class="core div" id="orgc8e8b42">
|
||||
<p>
|
||||
{\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
|
||||
\[
|
||||
|
|
@ -1700,7 +1692,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1638,7 +1630,7 @@ Useful strategy: represent fields in terms of potentials.
|
|||
<p>
|
||||
Easiest:
|
||||
</p>
|
||||
<div class="core div" id="orgf88fb96">
|
||||
<div class="core div" id="org5bf4193">
|
||||
<p>
|
||||
\[
|
||||
{\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
|
||||
|
|
@ -1654,7 +1646,7 @@ Putting this into Faraday's law gives
|
|||
\]
|
||||
so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
|
||||
</p>
|
||||
<div class="core div" id="org2f23671">
|
||||
<div class="core div" id="orgffce8dc">
|
||||
<p>
|
||||
\[
|
||||
{\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>
|
||||
Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting (\ref{eq:E_from_Potentials}) into Gauss's law gives
|
||||
</p>
|
||||
<div class="main div" id="org5127f9c">
|
||||
<div class="main div" id="org26743e8">
|
||||
<p>
|
||||
\[
|
||||
{\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
|
||||
|
|
@ -1683,7 +1675,7 @@ whereas Amp{\`ere}-Maxwell becomes
|
|||
\]
|
||||
which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
|
||||
</p>
|
||||
<div class="main div" id="org3f2c1f0">
|
||||
<div class="main div" id="orga86c8c2">
|
||||
<p>
|
||||
\[
|
||||
\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 id="postamble" class="status">
|
||||
<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>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1648,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1626,8 +1618,8 @@ Table of contents
|
|||
<li>Gr 3</li>
|
||||
</ul>
|
||||
|
||||
<details class="prereq" id="org3b926ab">
|
||||
<summary id="org2707d3f">
|
||||
<details class="prereq" id="org192cbf5">
|
||||
<summary id="org78425f6">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1635,8 +1627,8 @@ Prerequisites
|
|||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="orgfae12c2">
|
||||
<summary id="org18515f0">
|
||||
<details class="objectives" id="orge457a7e">
|
||||
<summary id="orge350b5c">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1674,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</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
|
||||
from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
|
||||
</p>
|
||||
<div class="main div" id="orgbb90bd9">
|
||||
<div class="main div" id="orgb661984">
|
||||
<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
|
||||
explicit construction <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
|
||||
</p>
|
||||
<div class="main div" id="org9f1e898">
|
||||
<div class="main div" id="org5d15b9f">
|
||||
<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>
|
||||
</p>
|
||||
|
||||
<div class="core div" id="org7c70185">
|
||||
<div class="core div" id="org7aef84c">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
|
|
@ -1682,7 +1674,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
|
|||
<p>
|
||||
In the specific case where the charge density vanishes, we fall back onto the simpler Laplace equation <a href="./ems_es_ep_PL.html#Lap">Lap</a>
|
||||
</p>
|
||||
<div class="core div" id="org1935b19">
|
||||
<div class="core div" id="org3f412f2">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
|
|
@ -1720,7 +1712,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</ul>
|
||||
|
|
@ -1638,14 +1630,14 @@ In one dimension, the potential is a single-variable
|
|||
function \(\phi (x)\) and the Laplace equation reads
|
||||
</p>
|
||||
|
||||
<div class="eqlabel" id="org84cc03f">
|
||||
<div class="eqlabel" id="org5c6cddc">
|
||||
<p>
|
||||
<a id="Lap_1d"></a><a href="./ems_ca_fe_L.html#Lap_1d"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org61fa4ec">
|
||||
<div class="alteqlabels" id="org5f00f86">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1660,14 +1652,14 @@ function \(\phi (x)\) and the Laplace equation reads
|
|||
<p>
|
||||
The solution to this is
|
||||
</p>
|
||||
<div class="eqlabel" id="orgb8c4067">
|
||||
<div class="eqlabel" id="orgc4fd451">
|
||||
<p>
|
||||
<a id="Lap_1d_sol"></a><a href="./ems_ca_fe_L.html#Lap_1d_sol"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org6b2acba">
|
||||
<div class="alteqlabels" id="org1f2c6cb">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (3.6)</li>
|
||||
</ul>
|
||||
|
|
@ -1726,14 +1718,14 @@ In two dimensions, the potential becomes a function
|
|||
of two variables (here: \(x\) and \(y\)), so Laplace's
|
||||
equation now reads
|
||||
</p>
|
||||
<div class="eqlabel" id="orgf3e1009">
|
||||
<div class="eqlabel" id="orge167f78">
|
||||
<p>
|
||||
<a id="Lap_2d"></a><a href="./ems_ca_fe_L.html#Lap_2d"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org696fc74">
|
||||
<div class="alteqlabels" id="org9aae751">
|
||||
|
||||
</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
|
||||
(and of course not containing any charges),
|
||||
</p>
|
||||
<div class="eqlabel" id="org9f81e63">
|
||||
<div class="eqlabel" id="orge400754">
|
||||
<p>
|
||||
<a id="p_ball_avg"></a><a href="./ems_ca_fe_L.html#p_ball_avg"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org66bdc2f">
|
||||
<div class="alteqlabels" id="orgf37e10c">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1805,8 +1797,8 @@ a point equals its value averaged over a sphere
|
|||
\]
|
||||
</p>
|
||||
|
||||
<details id="org0f27c64">
|
||||
<summary id="org881e712">
|
||||
<details id="org4bb0c5f">
|
||||
<summary id="orgaf332f4">
|
||||
<strong>Physicist's proof</strong>
|
||||
</summary>
|
||||
<p>
|
||||
|
|
@ -1868,8 +1860,8 @@ proving the theorem.
|
|||
</p>
|
||||
</details>
|
||||
|
||||
<details id="org0dbbb13">
|
||||
<summary id="orgf2b7c76">
|
||||
<details id="orgfce7dc3">
|
||||
<summary id="org452b0d4">
|
||||
<strong>Formal proof</strong>
|
||||
</summary>
|
||||
|
||||
|
|
@ -1919,14 +1911,14 @@ we get the following general
|
|||
<p>
|
||||
<b>Theorem</b>:
|
||||
</p>
|
||||
<div class="eqlabel" id="org230e535">
|
||||
<div class="eqlabel" id="orgfba2553">
|
||||
<p>
|
||||
<a id="dfdR_intLap"></a><a href="./ems_ca_fe_L.html#dfdR_intLap"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org37e5ede">
|
||||
<div class="alteqlabels" id="orgc5c6de6">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1979,19 +1971,19 @@ are necessarily positive, we thus require \(f_x > 0\), \(f_y > 0\) and \(f
|
|||
of the \(f_x + f_y + f_z = 0\) condition above.
|
||||
</p>
|
||||
|
||||
<div class="eqlabel" id="org8692599">
|
||||
<div class="eqlabel" id="org8465450">
|
||||
<p>
|
||||
<a id="Earnshaw"></a><a href="./ems_ca_fe_L.html#Earnshaw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orgcae1a27">
|
||||
<div class="alteqlabels" id="org0fbdf78">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="info div" id="orgab38e5b">
|
||||
<div class="info div" id="org9dd741d">
|
||||
<p>
|
||||
<b>Earnshaw's theorem (physical version)</b> <br>
|
||||
</p>
|
||||
|
|
@ -2110,7 +2102,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-03-02 Wed 15:45 -->
|
||||
<!-- 2022-03-07 Mon 20:38 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1098,14 +1098,6 @@ Table of contents
|
|||
<li>
|
||||
<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>
|
||||
|
||||
</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"/>
|
||||
</svg></a><span class="headline-id">ems.ca.fe.g</span></h5>
|
||||
<div class="outline-text-5" id="text-ems_ca_fe_g">
|
||||
<div class="info div" id="org5424abd">
|
||||
<div class="info div" id="orga6b382e">
|
||||
<p>
|
||||
<b>George Green</b>
|
||||
</p>
|
||||
<aside id="org7013f8e">
|
||||
<aside id="org8e26888">
|
||||
<p>
|
||||
See a <a href="https://en.wikipedia.org/wiki/George%5C_Green%5C_(mathematician)">short bio on wikipedia</a>
|
||||
</p>
|
||||
|
|
@ -1658,14 +1650,14 @@ and
|
|||
\]
|
||||
Substituting this in the divergence theorem gives <b>Green's first identity</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org17f1f87">
|
||||
<div class="eqlabel" id="orgeb35436">
|
||||
<p>
|
||||
<a id="Green1"></a><a href="./ems_ca_fe_g.html#Green1"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org9716c8b">
|
||||
<div class="alteqlabels" id="org6a05402">
|
||||
<ul class="org-ul">
|
||||
<li>J (1.34)</li>
|
||||
</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
|
||||
<b>Green's second identity</b> or <b>Green's theorem</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="orgdecb68b">
|
||||
<div class="eqlabel" id="org3cd71e0">
|
||||
<p>
|
||||
<a id="Green2"></a><a href="./ems_ca_fe_g.html#Green2"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
|
||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org1abae59">
|
||||
<div class="alteqlabels" id="org2a9c8de">
|
||||
<ul class="org-ul">
|
||||
<li>J (1.35)</li>
|
||||
</ul>
|
||||
|
|
@ -1726,7 +1718,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-03-02 Wed 15:45</p>
|
||||
<p class="date">Created: 2022-03-07 Mon 20:38</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
Some files were not shown because too many files have changed in this diff Show More
Loading…
Reference in New Issue