jscaux
/
pqed_html
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Update 2022-02-14 06:33

This commit is contained in:
Jean-Sébastien 2022-02-14 06:33:37 +01:00
parent f8446c1405
commit 4cfe8cef59
204 changed files with 1839 additions and 941 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

6
build/a.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1606,6 +1606,8 @@ Table of contents
<li><a href="a_l.html">Literature</a><span class="headline-id">a.l</span></li> <li><a href="a_l.html">Literature</a><span class="headline-id">a.l</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Next:&nbsp;<a href="a_l.html">Literature&emsp;<small>[a.l]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1619,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/a_l.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1689,6 +1689,8 @@ Wiley, 1986 [2nd edition]
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="a.html">Appendices&emsp;<small>[a]</small></a></li><li>Next:&nbsp;<a href="c.html">Compendium&emsp;<small>[c]</small></a></li><li>Up:&nbsp;<a href="a.html">Appendices&emsp;<small>[a]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1702,7 +1704,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1606,6 +1606,8 @@ Table of contents
<li><a href="c_m.html">Mathematics</a><span class="headline-id">c.m</span></li> <li><a href="c_m.html">Mathematics</a><span class="headline-id">c.m</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Next:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1619,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1612,6 +1612,8 @@ Table of contents
<li><a href="c_m_uf.html">Useful Formulas</a><span class="headline-id">c.m.uf</span></li> <li><a href="c_m_uf.html">Useful Formulas</a><span class="headline-id">c.m.uf</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c.html">Compendium&emsp;<small>[c]</small></a></li><li>Next:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li><li>Up:&nbsp;<a href="c.html">Compendium&emsp;<small>[c]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1625,7 +1627,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_cs.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1608,6 +1608,8 @@ Table of contents
<li><a href="c_m_cs_hyp.html">Hyperbolic Coordinates</a><span class="headline-id">c.m.cs.hyp</span></li> <li><a href="c_m_cs_hyp.html">Hyperbolic Coordinates</a><span class="headline-id">c.m.cs.hyp</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_ip.html">Integration by Parts&emsp;<small>[c.m.ic.ip]</small></a></li><li>Next:&nbsp;<a href="c_m_cs_sph.html">Spherical Coordinates&emsp;<small>[c.m.cs.sph]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1621,7 +1623,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

18
build/c_m_cs_cyl.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1643,14 +1643,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
<div id="outline-container-c_m_cs_cyl_grad" class="outline-6"> <div id="outline-container-c_m_cs_cyl_grad" class="outline-6">
<h6 id="c_m_cs_cyl_grad"><a href="#c_m_cs_cyl_grad">Gradient</a></h6> <h6 id="c_m_cs_cyl_grad"><a href="#c_m_cs_cyl_grad">Gradient</a></h6>
<div class="outline-text-6" id="text-c_m_cs_cyl_grad"> <div class="outline-text-6" id="text-c_m_cs_cyl_grad">
<div class="eqlabel" id="orgbec1efd"> <div class="eqlabel" id="org03db374">
<p> <p>
<a id="cylgrad"></a><a href="./c_m_cs_cyl.html#cylgrad"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16"> <a id="cylgrad"></a><a href="./c_m_cs_cyl.html#cylgrad"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org21e7bea"> <div class="alteqlabels" id="orgb61510b">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr4(1.79)</li> <li>Gr4(1.79)</li>
</ul> </ul>
@ -1671,14 +1671,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
<div id="outline-container-c_m_cs_cyl_div" class="outline-6"> <div id="outline-container-c_m_cs_cyl_div" class="outline-6">
<h6 id="c_m_cs_cyl_div"><a href="#c_m_cs_cyl_div">Divergence</a></h6> <h6 id="c_m_cs_cyl_div"><a href="#c_m_cs_cyl_div">Divergence</a></h6>
<div class="outline-text-6" id="text-c_m_cs_cyl_div"> <div class="outline-text-6" id="text-c_m_cs_cyl_div">
<div class="eqlabel" id="org59ba5e0"> <div class="eqlabel" id="orgd9435d5">
<p> <p>
<a id="cyl_div"></a><a href="./c_m_cs_cyl.html#cyl_div"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16"> <a id="cyl_div"></a><a href="./c_m_cs_cyl.html#cyl_div"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgd4f5fe9"> <div class="alteqlabels" id="org5c1c30c">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr4(2.21)</li> <li>Gr4(2.21)</li>
</ul> </ul>
@ -1699,14 +1699,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
<div id="outline-container-c_m_cs_cyl_curl" class="outline-6"> <div id="outline-container-c_m_cs_cyl_curl" class="outline-6">
<h6 id="c_m_cs_cyl_curl"><a href="#c_m_cs_cyl_curl">Curl</a></h6> <h6 id="c_m_cs_cyl_curl"><a href="#c_m_cs_cyl_curl">Curl</a></h6>
<div class="outline-text-6" id="text-c_m_cs_cyl_curl"> <div class="outline-text-6" id="text-c_m_cs_cyl_curl">
<div class="eqlabel" id="orgc77e287"> <div class="eqlabel" id="org695c029">
<p> <p>
<a id="cyl_curl"></a><a href="./c_m_cs_cyl.html#cyl_curl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16"> <a id="cyl_curl"></a><a href="./c_m_cs_cyl.html#cyl_curl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org808a8dd"> <div class="alteqlabels" id="orge332907">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr4(2.21)</li> <li>Gr4(2.21)</li>
</ul> </ul>
@ -1740,6 +1740,8 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_cs_sph.html">Spherical Coordinates&emsp;<small>[c.m.cs.sph]</small></a></li><li>Next:&nbsp;<a href="c_m_cs_hyp.html">Hyperbolic Coordinates&emsp;<small>[c.m.cs.hyp]</small></a></li><li>Up:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1753,7 +1755,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_cs_hyp.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1601,6 +1601,8 @@ Table of contents
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_cs_cyl.html">Cylindrical Coordinates&emsp;<small>[c.m.cs.cyl]</small></a></li><li>Next:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li><li>Up:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1614,7 +1616,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_cs_sph.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1714,6 +1714,8 @@ Infinitesimal surface element: depends on situation.
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li><li>Next:&nbsp;<a href="c_m_cs_cyl.html">Cylindrical Coordinates&emsp;<small>[c.m.cs.cyl]</small></a></li><li>Up:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1727,7 +1729,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_dc.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1611,6 +1611,8 @@ Table of contents
<li><a href="c_m_dc_d2.html">Second Derivatives</a><span class="headline-id">c.m.dc.d2</span></li> <li><a href="c_m_dc_d2.html">Second Derivatives</a><span class="headline-id">c.m.dc.d2</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va_pds.html">Position, Displacement and Separation Vectors&emsp;<small>[c.m.va.pds]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_g.html">Gradient&emsp;<small>[c.m.dc.g]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1624,7 +1626,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_dc_curl.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1613,6 +1613,8 @@ v_x &amp; v_y &amp; v_z \end{array} \right| \nonumber \\
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_div.html">The Divergence&emsp;<small>[c.m.dc.div]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_pr.html">Product Rules&emsp;<small>[c.m.dc.pr]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1626,7 +1628,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

36
build/c_m_dc_d2.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1599,9 +1599,9 @@ Table of contents
<div class="outline-text-5" id="text-c_m_dc_d2"> <div class="outline-text-5" id="text-c_m_dc_d2">
</div> </div>
<div id="outline-container-orgd5309ec" class="outline-6"> <div id="outline-container-orgbe5f73d" class="outline-6">
<h6 id="orgd5309ec"><a href="#orgd5309ec">Divergence of gradient</a></h6> <h6 id="orgbe5f73d"><a href="#orgbe5f73d">Divergence of gradient</a></h6>
<div class="outline-text-6" id="text-orgd5309ec"> <div class="outline-text-6" id="text-orgbe5f73d">
<p> <p>
\({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\). \({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
@ -1610,36 +1610,36 @@ given by the Laplacian of the corresponding vector elements.
</div> </div>
</div> </div>
<div id="outline-container-org3f8f01e" class="outline-6"> <div id="outline-container-orgee1ca6a" class="outline-6">
<h6 id="org3f8f01e"><a href="#org3f8f01e">Curl of a gradient</a></h6> <h6 id="orgee1ca6a"><a href="#orgee1ca6a">Curl of a gradient</a></h6>
<div class="outline-text-6" id="text-org3f8f01e"> <div class="outline-text-6" id="text-orgee1ca6a">
<p> <p>
This always vanishes. This always vanishes.
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-org86a1e1f" class="outline-6"> <div id="outline-container-org2789da2" class="outline-6">
<h6 id="org86a1e1f"><a href="#org86a1e1f">Gradient of the divergence</a></h6> <h6 id="org2789da2"><a href="#org2789da2">Gradient of the divergence</a></h6>
<div class="outline-text-6" id="text-org86a1e1f"> <div class="outline-text-6" id="text-org2789da2">
<p> <p>
\({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name. \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-orgdca9d9a" class="outline-6"> <div id="outline-container-org62ec566" class="outline-6">
<h6 id="orgdca9d9a"><a href="#orgdca9d9a">Divergence of a curl</a></h6> <h6 id="org62ec566"><a href="#org62ec566">Divergence of a curl</a></h6>
<div class="outline-text-6" id="text-orgdca9d9a"> <div class="outline-text-6" id="text-org62ec566">
<p> <p>
This always vanishes. This always vanishes.
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-org613cb02" class="outline-6"> <div id="outline-container-org5cd2844" class="outline-6">
<h6 id="org613cb02"><a href="#org613cb02">Curl of curl</a></h6> <h6 id="org5cd2844"><a href="#org5cd2844">Curl of curl</a></h6>
<div class="outline-text-6" id="text-org613cb02"> <div class="outline-text-6" id="text-org5cd2844">
<p> <p>
\[ \[
{\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v} {\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v}
@ -1651,6 +1651,8 @@ This always vanishes.
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_pr.html">Product Rules&emsp;<small>[c.m.dc.pr]</small></a></li><li>Next:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1664,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_dc_del.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1612,6 +1612,8 @@ The <b>del/grad/nabla operator</b> is defined as
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_g.html">Gradient&emsp;<small>[c.m.dc.g]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_div.html">The Divergence&emsp;<small>[c.m.dc.div]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1625,7 +1627,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_dc_div.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1609,6 +1609,8 @@ Table of contents
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_del.html">The \({\boldsymbol \nabla}\) Operator&emsp;<small>[c.m.dc.del]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_curl.html">The Curl&emsp;<small>[c.m.dc.curl]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1622,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_dc_g.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1633,6 +1633,8 @@ is a vector called the <b>gradient</b> of \(T\).
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_del.html">The \({\boldsymbol \nabla}\) Operator&emsp;<small>[c.m.dc.del]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1646,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_dc_pr.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1606,6 +1606,8 @@ Six product rules (on inside front cover of Gr).
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_curl.html">The Curl&emsp;<small>[c.m.dc.curl]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_d2.html">Second Derivatives&emsp;<small>[c.m.dc.d2]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1619,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_dd.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1608,6 +1608,8 @@ Table of contents
<li><a href="c_m_dd_3d.html">The Three-Dimensional Delta Function</a><span class="headline-id">c.m.dd.3d</span></li> <li><a href="c_m_dd_3d.html">The Three-Dimensional Delta Function</a><span class="headline-id">c.m.dd.3d</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_cs_hyp.html">Hyperbolic Coordinates&emsp;<small>[c.m.cs.hyp]</small></a></li><li>Next:&nbsp;<a href="c_m_dd_div.html">The Divergence of \(\hat{\bf r}/r^2\)&emsp;<small>[c.m.dd.div]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1621,7 +1623,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_dd_1d.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1620,6 +1620,8 @@ Consequences: for any smooth differentiable function \(f(x)\),
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dd_div.html">The Divergence of \(\hat{\bf r}/r^2\)&emsp;<small>[c.m.dd.div]</small></a></li><li>Next:&nbsp;<a href="c_m_dd_3d.html">The Three-Dimensional Delta Function&emsp;<small>[c.m.dd.3d]</small></a></li><li>Up:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1633,7 +1635,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

31
build/c_m_dd_3d.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1620,14 +1620,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
More generally, More generally,
</p> </p>
<div class="eqlabel" id="org16a4f22"> <div class="eqlabel" id="org28e4fae">
<p> <p>
<a id="divdel"></a><a href="./c_m_dd_3d.html#divdel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16"> <a id="divdel"></a><a href="./c_m_dd_3d.html#divdel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgf96ad98"> <div class="alteqlabels" id="org3ed5b35">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (1.100)</li> <li>Gr (1.100)</li>
</ul> </ul>
@ -1646,14 +1646,14 @@ More generally,
Since Since
</p> </p>
<div class="eqlabel" id="org71f92c6"> <div class="eqlabel" id="org51c5e82">
<p> <p>
<a id="div1or"></a><a href="./c_m_dd_3d.html#div1or"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16"> <a id="div1or"></a><a href="./c_m_dd_3d.html#div1or"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="org39f00df"> <div class="alteqlabels" id="org725ab8d">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (1.101)</li> <li>Gr (1.101)</li>
</ul> </ul>
@ -1669,15 +1669,32 @@ Since
<p> <p>
we have that we have that
</p> </p>
<div class="eqlabel" id="orgbcd1aff">
<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="orgc217127">
<ul class="org-ul">
<li>Gr (1.102)</li>
</ul>
</div>
</div>
\begin{equation} \begin{equation}
{\boldsymbol \nabla}^2 \left( \frac{1}{|{\bf r} - {\bf r}'|} \right) = -4\pi \delta^{(3)} ({\bf r} - {\bf r}') {\boldsymbol \nabla}^2 \left( \frac{1}{|{\bf r} - {\bf r}'|} \right) = -4\pi \delta^{(3)} ({\bf r} - {\bf r}')
\label{Gr(1.102)} \tag{Lap1or}\label{Lap1or}
\end{equation} \end{equation}
</div> </div>
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dd_1d.html">The One-Dimensional Dirac Delta Function&emsp;<small>[c.m.dd.1d]</small></a></li><li>Next:&nbsp;<a href="c_m_vf.html">Vector Fields&emsp;<small>[c.m.vf]</small></a></li><li>Up:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1691,7 +1708,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_dd_div.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1626,6 +1626,8 @@ Problem: in Gr(1.84), we've divided by zero when \(r = 0\).
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li><li>Next:&nbsp;<a href="c_m_dd_1d.html">The One-Dimensional Dirac Delta Function&emsp;<small>[c.m.dd.1d]</small></a></li><li>Up:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1639,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_ic.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1611,6 +1611,8 @@ Table of contents
<li><a href="c_m_ic_ip.html">Integration by Parts</a><span class="headline-id">c.m.ic.ip</span></li> <li><a href="c_m_ic_ip.html">Integration by Parts</a><span class="headline-id">c.m.ic.ip</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_d2.html">Second Derivatives&emsp;<small>[c.m.dc.d2]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_lsv.html">Line, Surface and Volume Integrals&emsp;<small>[c.m.ic.lsv]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1624,7 +1626,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_ic_ftc.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1609,6 +1609,8 @@ Table of contents
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_lsv.html">Line, Surface and Volume Integrals&emsp;<small>[c.m.ic.lsv]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_ftg.html">The Fundamental Theorem for Gradients&emsp;<small>[c.m.ic.ftg]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1622,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_ic_ftg.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1618,6 +1618,8 @@ Table of contents
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_ftc.html">The Fundamental Theorem of Calculus&emsp;<small>[c.m.ic.ftc]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_gauss.html">Gauss' Theorem&emsp;<small>[c.m.ic.gauss]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1631,7 +1633,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_ic_gauss.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1612,6 +1612,8 @@ This is know either as <b>Gauss' theorem</b>, <b>Green's theorem</b> or the <b>d
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_ftg.html">The Fundamental Theorem for Gradients&emsp;<small>[c.m.ic.ftg]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_stokes.html">Stokes' Theorem&emsp;<small>[c.m.ic.stokes]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1625,7 +1627,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_ic_ip.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1619,6 +1619,8 @@ or in other words
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_stokes.html">Stokes' Theorem&emsp;<small>[c.m.ic.stokes]</small></a></li><li>Next:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1632,7 +1634,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

24
build/c_m_ic_lsv.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1599,9 +1599,9 @@ Table of contents
<div class="outline-text-5" id="text-c_m_ic_lsv"> <div class="outline-text-5" id="text-c_m_ic_lsv">
</div> </div>
<div id="outline-container-org3f3c8c7" class="outline-6"> <div id="outline-container-orgdc409a4" class="outline-6">
<h6 id="org3f3c8c7"><a href="#org3f3c8c7">Line Integrals</a></h6> <h6 id="orgdc409a4"><a href="#orgdc409a4">Line Integrals</a></h6>
<div class="outline-text-6" id="text-org3f3c8c7"> <div class="outline-text-6" id="text-orgdc409a4">
<p> <p>
\[ \[
{\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l} {\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l}
@ -1630,9 +1630,9 @@ Integral over a closed loop:
</div> </div>
</div> </div>
<div id="outline-container-orgbda984c" class="outline-6"> <div id="outline-container-org465c39d" class="outline-6">
<h6 id="orgbda984c"><a href="#orgbda984c">Surface Integrals</a></h6> <h6 id="org465c39d"><a href="#org465c39d">Surface Integrals</a></h6>
<div class="outline-text-6" id="text-orgbda984c"> <div class="outline-text-6" id="text-org465c39d">
<p> <p>
\[ \[
\int_{\cal S} {\bf v} \cdot d{\bf a} \int_{\cal S} {\bf v} \cdot d{\bf a}
@ -1652,9 +1652,9 @@ Over a closed surface:
</div> </div>
</div> </div>
<div id="outline-container-orgc3891db" class="outline-6"> <div id="outline-container-org794c5b9" class="outline-6">
<h6 id="orgc3891db"><a href="#orgc3891db">Volume Integrals</a></h6> <h6 id="org794c5b9"><a href="#org794c5b9">Volume Integrals</a></h6>
<div class="outline-text-6" id="text-orgc3891db"> <div class="outline-text-6" id="text-org794c5b9">
<p> <p>
\[ \[
\int_{\cal V} T d\tau \int_{\cal V} T d\tau
@ -1680,6 +1680,8 @@ d\tau = dx dy dz
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_ftc.html">The Fundamental Theorem of Calculus&emsp;<small>[c.m.ic.ftc]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1693,7 +1695,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

10
build/c_m_ic_stokes.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1597,14 +1597,14 @@ Table of contents
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a><span class="headline-id">c.m.ic.stokes</span></h5> </svg></a><span class="headline-id">c.m.ic.stokes</span></h5>
<div class="outline-text-5" id="text-c_m_ic_stokes"> <div class="outline-text-5" id="text-c_m_ic_stokes">
<div class="eqlabel" id="org9333ada"> <div class="eqlabel" id="org54009e0">
<p> <p>
<a id="Stokes"></a><a href="./c_m_ic_stokes.html#Stokes"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16"> <a id="Stokes"></a><a href="./c_m_ic_stokes.html#Stokes"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/> <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a> </svg></a>
</p> </p>
<div class="alteqlabels" id="orgf5a9283"> <div class="alteqlabels" id="org8b7df95">
<ul class="org-ul"> <ul class="org-ul">
<li>Gr (1.57)</li> <li>Gr (1.57)</li>
</ul> </ul>
@ -1634,6 +1634,8 @@ the boundary shrinks to a point.
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_gauss.html">Gauss' Theorem&emsp;<small>[c.m.ic.gauss]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_ip.html">Integration by Parts&emsp;<small>[c.m.ic.ip]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1647,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_uf.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1608,6 +1608,8 @@ Table of contents
<li><a href="c_m_uf_vi.html">Vector identities</a><span class="headline-id">c.m.uf.vi</span></li> <li><a href="c_m_uf_vi.html">Vector identities</a><span class="headline-id">c.m.uf.vi</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_vf_pot.html">Potentials&emsp;<small>[c.m.vf.pot]</small></a></li><li>Next:&nbsp;<a href="c_m_uf_cyl.html">Cylindrical coordinates&emsp;<small>[c.m.uf.cyl]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1621,7 +1623,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_uf_cyl.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1621,6 +1621,8 @@ Table of contents
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_uf.html">Useful Formulas&emsp;<small>[c.m.uf]</small></a></li><li>Next:&nbsp;<a href="c_m_uf_sph.html">Spherical coordinates&emsp;<small>[c.m.uf.sph]</small></a></li><li>Up:&nbsp;<a href="c_m_uf.html">Useful Formulas&emsp;<small>[c.m.uf]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1634,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_uf_sph.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1621,6 +1621,8 @@ Table of contents
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_uf_cyl.html">Cylindrical coordinates&emsp;<small>[c.m.uf.cyl]</small></a></li><li>Next:&nbsp;<a href="c_m_uf_vi.html">Vector identities&emsp;<small>[c.m.uf.vi]</small></a></li><li>Up:&nbsp;<a href="c_m_uf.html">Useful Formulas&emsp;<small>[c.m.uf]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1634,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_uf_vi.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1624,6 +1624,8 @@ Table of contents
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_uf_sph.html">Spherical coordinates&emsp;<small>[c.m.uf.sph]</small></a></li><li>Up:&nbsp;<a href="c_m_uf.html">Useful Formulas&emsp;<small>[c.m.uf]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1637,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_va.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1610,6 +1610,8 @@ Table of contents
<li><a href="c_m_va_pds.html">Position, Displacement and Separation Vectors</a><span class="headline-id">c.m.va.pds</span></li> <li><a href="c_m_va_pds.html">Position, Displacement and Separation Vectors</a><span class="headline-id">c.m.va.pds</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li><li>Next:&nbsp;<a href="c_m_va_n.html">Notation and algebraic properties&emsp;<small>[c.m.va.n]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1623,7 +1625,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_va_cp.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1639,6 +1639,8 @@ this relation making plain that
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va_sp.html">Scalar product&emsp;<small>[c.m.va.sp]</small></a></li><li>Next:&nbsp;<a href="c_m_va_tp.html">Triple Products&emsp;<small>[c.m.va.tp]</small></a></li><li>Up:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1652,7 +1654,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_va_n.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1634,6 +1634,8 @@ Component notation (in \({\mathbb R}^3\) with cartesian coordinates):
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li><li>Next:&nbsp;<a href="c_m_va_sp.html">Scalar product&emsp;<small>[c.m.va.sp]</small></a></li><li>Up:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1647,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_va_pds.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1668,6 +1668,8 @@ Separation vector: for two position vectors \({\bf r}_a, {\bf r}_b\),
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va_tp.html">Triple Products&emsp;<small>[c.m.va.tp]</small></a></li><li>Next:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li><li>Up:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1681,7 +1683,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_va_sp.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1634,6 +1634,8 @@ In a general coordinate system with metric \(g\),
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va_n.html">Notation and algebraic properties&emsp;<small>[c.m.va.n]</small></a></li><li>Next:&nbsp;<a href="c_m_va_cp.html">Cross product&emsp;<small>[c.m.va.cp]</small></a></li><li>Up:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1647,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_va_tp.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1681,6 +1681,8 @@ All higher vector products can be reduced to combinations of single vector produ
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va_cp.html">Cross product&emsp;<small>[c.m.va.cp]</small></a></li><li>Next:&nbsp;<a href="c_m_va_pds.html">Position, Displacement and Separation Vectors&emsp;<small>[c.m.va.pds]</small></a></li><li>Up:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1694,7 +1696,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_vf.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1607,6 +1607,8 @@ Table of contents
<li><a href="c_m_vf_pot.html">Potentials</a><span class="headline-id">c.m.vf.pot</span></li> <li><a href="c_m_vf_pot.html">Potentials</a><span class="headline-id">c.m.vf.pot</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dd_3d.html">The Three-Dimensional Delta Function&emsp;<small>[c.m.dd.3d]</small></a></li><li>Next:&nbsp;<a href="c_m_vf_helm.html">The Helmholtz Theorem&emsp;<small>[c.m.vf.helm]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1620,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_vf_helm.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1626,6 +1626,8 @@ where
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_vf.html">Vector Fields&emsp;<small>[c.m.vf]</small></a></li><li>Next:&nbsp;<a href="c_m_vf_pot.html">Potentials&emsp;<small>[c.m.vf.pot]</small></a></li><li>Up:&nbsp;<a href="c_m_vf.html">Vector Fields&emsp;<small>[c.m.vf]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1639,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/c_m_vf_pot.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1653,6 +1653,8 @@ The following conditions are equivalent:
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_vf_helm.html">The Helmholtz Theorem&emsp;<small>[c.m.vf.helm]</small></a></li><li>Next:&nbsp;<a href="c_m_uf.html">Useful Formulas&emsp;<small>[c.m.uf]</small></a></li><li>Up:&nbsp;<a href="c_m_vf.html">Vector Fields&emsp;<small>[c.m.vf]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1666,7 +1668,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

10
build/d.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">d</span></h2> </svg></a><span class="headline-id">d</span></h2>
<div class="outline-text-2" id="text-d"> <div class="outline-text-2" id="text-d">
<details class="objectives" id="orge321671"> <details class="objectives" id="orgece4d0a">
<summary id="orgc02d8c9"> <summary id="org51b47eb">
Objectives Objectives
</summary> </summary>
@ -1648,6 +1648,8 @@ I have little doubt you'll pass the course.
<li><a href="d_red.html">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span></li> <li><a href="d_red.html">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Next:&nbsp;<a href="d_m.html">Diagnostics: Mathematical Preliminaries&emsp;<small>[d.m]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1661,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/d_emd.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1628,6 +1628,8 @@ As a strict minimum, you should be able to:
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emsm_msm.html">Diagnostics: Magnetostatics in Matter&emsp;<small>[d.emsm.msm]</small></a></li><li>Next:&nbsp;<a href="d_emd_ce.html">Diagnostics: Conservation Laws&emsp;<small>[d.emd.ce]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1641,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/d_emd_ce.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1623,6 +1623,8 @@ As a strict minimum, you should be able to:
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emd.html">Diagnostics: Electromagnetodynamics&emsp;<small>[d.emd]</small></a></li><li>Next:&nbsp;<a href="d_emd_emw.html">Diagnostics: Electromagnetic Waves&emsp;<small>[d.emd.emw]</small></a></li><li>Up:&nbsp;<a href="d_emd.html">Diagnostics: Electromagnetodynamics&emsp;<small>[d.emd]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1636,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/d_emd_emw.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1626,6 +1626,8 @@ As a strict minimum, you should be able to:
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emd_ce.html">Diagnostics: Conservation Laws&emsp;<small>[d.emd.ce]</small></a></li><li>Next:&nbsp;<a href="d_emf.html">Diagnostics: Potentials, Gauges and Fields&emsp;<small>[d.emf]</small></a></li><li>Up:&nbsp;<a href="d_emd.html">Diagnostics: Electromagnetodynamics&emsp;<small>[d.emd]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1639,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/d_emf.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1623,6 +1623,8 @@ As a strict minimum, you should be able to:
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emd_emw.html">Diagnostics: Electromagnetic Waves&emsp;<small>[d.emd.emw]</small></a></li><li>Next:&nbsp;<a href="d_red.html">Diagnostics: Relativistic Electrodynamics&emsp;<small>[d.red]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1636,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/d_ems.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1645,6 +1645,8 @@ As a strict minimum, you should be able to:
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_m.html">Diagnostics: Mathematical Preliminaries&emsp;<small>[d.m]</small></a></li><li>Next:&nbsp;<a href="d_ems_ca.html">Diagnostics: Calculating or Approximating the Electostatic Potential&emsp;<small>[d.ems.ca]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1658,7 +1660,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/d_ems_ca.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1628,6 +1628,8 @@ a point source charge \(q\) at \({\bf r}_s\) with an infinite grounded conductin
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_ems.html">Diagnostics: Electromagnetostatics&emsp;<small>[d.ems]</small></a></li><li>Next:&nbsp;<a href="d_emsm.html">Diagnostics: Electromagnetostatics in Matter&emsp;<small>[d.emsm]</small></a></li><li>Up:&nbsp;<a href="d_ems.html">Diagnostics: Electromagnetostatics&emsp;<small>[d.ems]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1641,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/d_ems_ms.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1625,6 +1625,8 @@ As a strict minimum, you should be able to:
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emsm.html">Diagnostics: Electromagnetostatics in Matter&emsp;<small>[d.emsm]</small></a></li><li>Next:&nbsp;<a href="d_emsm_msm.html">Diagnostics: Magnetostatics in Matter&emsp;<small>[d.emsm.msm]</small></a></li><li>Up:&nbsp;<a href="d_ems.html">Diagnostics: Electromagnetostatics&emsp;<small>[d.ems]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1638,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/d_emsm.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1628,6 +1628,8 @@ As a strict minimum, you should be able to:
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_ems_ca.html">Diagnostics: Calculating or Approximating the Electostatic Potential&emsp;<small>[d.ems.ca]</small></a></li><li>Next:&nbsp;<a href="d_ems_ms.html">Diagnostics: Magnetostatics&emsp;<small>[d.ems.ms]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1641,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/d_emsm_msm.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1626,6 +1626,8 @@ As a strict minimum, you should be able to:
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_ems_ms.html">Diagnostics: Magnetostatics&emsp;<small>[d.ems.ms]</small></a></li><li>Next:&nbsp;<a href="d_emd.html">Diagnostics: Electromagnetodynamics&emsp;<small>[d.emd]</small></a></li><li>Up:&nbsp;<a href="d_emsm.html">Diagnostics: Electromagnetostatics in Matter&emsp;<small>[d.emsm]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1639,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/d_m.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1636,6 +1636,8 @@ Things you should be able to do (ideally: from scratch, on a blank sheet of pape
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li><li>Next:&nbsp;<a href="d_ems.html">Diagnostics: Electromagnetostatics&emsp;<small>[d.ems]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1649,7 +1651,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/d_red.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1630,6 +1630,8 @@ As a strict minimum, you should be able to:
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emf.html">Diagnostics: Potentials, Gauges and Fields&emsp;<small>[d.emf]</small></a></li><li>Next:&nbsp;<a href="a.html">Appendices&emsp;<small>[a]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1643,7 +1645,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

14
build/emd.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">emd</span></h2> </svg></a><span class="headline-id">emd</span></h2>
<div class="outline-text-2" id="text-emd"> <div class="outline-text-2" id="text-emd">
<details class="prereq" id="org09895e9"> <details class="prereq" id="org8c17216">
<summary id="org4a0bf93"> <summary id="org07b9266">
Prerequisites Prerequisites
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1608,8 +1608,8 @@ Prerequisites
</ul> </ul>
</details> </details>
<details class="objectives" id="orgef3f98e"> <details class="objectives" id="org1cf61f0">
<summary id="org1810cca"> <summary id="org2c97d99">
Objectives Objectives
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1635,6 +1635,8 @@ Objectives
<li><a href="emd_emw.html">Electromagnetic waves in vacuum</a><span class="headline-id">emd.emw</span></li> <li><a href="emd_emw.html">Electromagnetic waves in vacuum</a><span class="headline-id">emd.emw</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Next:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1648,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emd_Fl.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1609,6 +1609,8 @@ Table of contents
<li><a href="emd_Fl_e.html">Energy in Magnetic Fields</a><span class="headline-id">emd.Fl.e</span></li> <li><a href="emd_Fl_e.html">Energy in Magnetic Fields</a><span class="headline-id">emd.Fl.e</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd.html">Electromagnetodynamics&emsp;<small>[emd]</small></a></li><li>Next:&nbsp;<a href="emd_Fl_Fl.html">Faraday's Law&emsp;<small>[emd.Fl.Fl]</small></a></li><li>Up:&nbsp;<a href="emd.html">Electromagnetodynamics&emsp;<small>[emd]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1622,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/emd_Fl_Fl.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1638,7 +1638,7 @@ Empirically: the changing magnetic field induces an electric current around
the circuit. This current is really driven by an electric field having a component the circuit. This current is really driven by an electric field having a component
along the wire. The line integral of this field is called the along the wire. The line integral of this field is called the
</p> </p>
<div class="core div" id="orgee21897"> <div class="core div" id="org60c9879">
<p> <p>
<b>Electromotive force (or electromotance)</b>, <b>Electromotive force (or electromotance)</b>,
\[ \[
@ -1660,7 +1660,7 @@ to the rate of change of the magnetic flux,
\] \]
so we obtain so we obtain
</p> </p>
<div class="core div" id="orgd1f18df"> <div class="core div" id="org02e64db">
<p> <p>
<b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>) <b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
\[ \[
@ -1678,7 +1678,7 @@ for any loop (on a wire or not). Using Stokes' theorem,
\] \]
we obtain we obtain
</p> </p>
<div class="core div" id="org04bf1f1"> <div class="core div" id="org3c09004">
<p> <p>
<b>Faraday's law</b> (differential form) <b>Faraday's law</b> (differential form)
\[ \[
@ -1700,6 +1700,8 @@ to an opposing counter-reaction.
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li><li>Next:&nbsp;<a href="emd_Fl_ief.html">The Induced Electric Field&emsp;<small>[emd.Fl.ief]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1713,7 +1715,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

10
build/emd_Fl_e.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1657,7 +1657,7 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold
\] \]
We can integrate over all space: after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with We can integrate over all space: after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with
</p> </p>
<div class="core div" id="org183e158"> <div class="core div" id="org0ecd8ee">
<p> <p>
\[ \[
W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2 W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2
@ -1678,7 +1678,7 @@ W_{mag} = \frac{1}{2} \int d\tau ({\bf A} \cdot {\bf J}) = \frac{1}{2\mu_0} \int
\hspace{2cm} \mbox{(7.31 and 7.34)} \hspace{2cm} \mbox{(7.31 and 7.34)}
\end{align} \end{align}
<div class="example div" id="org5a2dc32"> <div class="example div" id="org46b2091">
<p> <p>
\paragraph{Example 7.13:} coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\). \paragraph{Example 7.13:} coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\).
Find energy stored in section of length \(l\). Find energy stored in section of length \(l\).
@ -1702,6 +1702,8 @@ Note: gives easy way to find inductance, since \(W = \frac{1}{2} L I^2\).
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_i.html">Inductance&emsp;<small>[emd.Fl.i]</small></a></li><li>Next:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1715,7 +1717,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/emd_Fl_i.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1637,7 +1637,7 @@ M_{12} = M_{21}
\] \]
</p> </p>
<div class="example div" id="orgb4beaba"> <div class="example div" id="orgc3b71d6">
<p> <p>
\paragraph{Example 7.10:} \paragraph{Example 7.10:}
short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) lies concentrically inside short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) lies concentrically inside
@ -1687,7 +1687,7 @@ Inductance: measured in {\bf henries} (\(H\)). \(H = V s/A\).
</p> </p>
<div class="example div" id="org24edf9c"> <div class="example div" id="orgd00f4d4">
<p> <p>
\paragraph{Example 7.11:} find self-inductance of toroidal coil with \paragraph{Example 7.11:} find self-inductance of toroidal coil with
rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\)) rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\))
@ -1714,7 +1714,7 @@ Total flux: \(N\) times this, so self-inductance is
Inductance (like capacitance) is intrinsically positive. Use Lenz law. Think of {\bf back EMF}. Inductance (like capacitance) is intrinsically positive. Use Lenz law. Think of {\bf back EMF}.
</p> </p>
<div class="example div" id="org0b14b9f"> <div class="example div" id="org7ae308f">
<p> <p>
\paragraph{Example 7.12:} circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\). \paragraph{Example 7.12:} circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\).
What is the current ? What is the current ?
@ -1738,6 +1738,8 @@ where \(\tau \equiv L/R\) is the {\bf time constant} of the circuit.
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_ief.html">The Induced Electric Field&emsp;<small>[emd.Fl.ief]</small></a></li><li>Next:&nbsp;<a href="emd_Fl_e.html">Energy in Magnetic Fields&emsp;<small>[emd.Fl.e]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1751,7 +1753,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/emd_Fl_ief.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1621,7 +1621,7 @@ law in integral form:
<div class="example div" id="org7f7b579"> <div class="example div" id="orgaba3d08">
<p> <p>
{\bf Example 7.7:} {\bf Example 7.7:}
\({\bf B}(t)\) points up in circular region of radius \(R\). What is the induced \({\bf E}(t)\) ? \({\bf B}(t)\) points up in circular region of radius \(R\). What is the induced \({\bf E}(t)\) ?
@ -1637,7 +1637,7 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
</div> </div>
<div class="example div" id="orgd8d7c07"> <div class="example div" id="org655e44f">
<p> <p>
{\bf Example 7.8:} wheel or radius \(b\) with line charge \(\lambda\) on the rim. {\bf Example 7.8:} wheel or radius \(b\) with line charge \(\lambda\) on the rim.
Uniform magnetic field \({\bf B}_0\) in central region up to \(a &lt; b\), Uniform magnetic field \({\bf B}_0\) in central region up to \(a &lt; b\),
@ -1671,7 +1671,7 @@ called the {\bf quasistatic} approximation, and works provided we deal with
'slow enough' phenomena. 'slow enough' phenomena.
</p> </p>
<div class="example div" id="org1188389"> <div class="example div" id="org0cc8dcd">
<p> <p>
{\bf Example 7.9:} infinitely long straight wire carries \(I(t)\). Find {\bf Example 7.9:} infinitely long straight wire carries \(I(t)\). Find
induced \({\bf E}\) field as a function of distance \(s\) from wire. induced \({\bf E}\) field as a function of distance \(s\) from wire.
@ -1704,6 +1704,8 @@ Reason: in this case, we've overstepped the quasistatic limit. We need
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_Fl.html">Faraday's Law&emsp;<small>[emd.Fl.Fl]</small></a></li><li>Next:&nbsp;<a href="emd_Fl_i.html">Inductance&emsp;<small>[emd.Fl.i]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1717,7 +1719,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emd_Me.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1609,6 +1609,8 @@ Table of contents
<li><a href="emd_Me_mc.html">Magnetic Charge</a><span class="headline-id">emd.Me.mc</span></li> <li><a href="emd_Me_mc.html">Magnetic Charge</a><span class="headline-id">emd.Me.mc</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_e.html">Energy in Magnetic Fields&emsp;<small>[emd.Fl.e]</small></a></li><li>Next:&nbsp;<a href="emd_Me_ebM.html">Electrodynamics Before Maxwell&emsp;<small>[emd.Me.ebM]</small></a></li><li>Up:&nbsp;<a href="emd.html">Electromagnetodynamics&emsp;<small>[emd]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1622,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/emd_Me_Me.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1600,7 +1600,7 @@ Table of contents
<p> <p>
Full set of equations for the electromagnetic field: Full set of equations for the electromagnetic field:
</p> </p>
<div class="core div" id="org8953ea9"> <div class="core div" id="org02575df">
<p> <p>
{\bf Maxwell's equations} {\it (in vacuum)} {\bf Maxwell's equations} {\it (in vacuum)}
</p> </p>
@ -1616,7 +1616,7 @@ Full set of equations for the electromagnetic field:
<p> <p>
Complement: Complement:
</p> </p>
<div class="core div" id="org04de767"> <div class="core div" id="org67bee1c">
<p> <p>
{\bf Force law} {\bf Force law}
\[ \[
@ -1640,7 +1640,7 @@ take divergence of \((iv)\).
<p> <p>
Better way of writing: all fields on left, all sources on right, Better way of writing: all fields on left, all sources on right,
</p> </p>
<div class="core div" id="org8800b57"> <div class="core div" id="org95e5ef3">
\begin{align} \begin{align}
(i) &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0}, (i) &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0},
&amp;(iii) {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} = 0, \\ &amp;(iii) {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} = 0, \\
@ -1655,6 +1655,8 @@ Better way of writing: all fields on left, all sources on right,
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Me_dc.html">Maxwell's Correction to Ampère's Law; the Displacement Current&emsp;<small>[emd.Me.dc]</small></a></li><li>Next:&nbsp;<a href="emd_Me_mc.html">Magnetic Charge&emsp;<small>[emd.Me.mc]</small></a></li><li>Up:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1668,7 +1670,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

10
build/emd_Me_dc.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1607,7 +1607,7 @@ the continuity equation as
\] \]
The extra term would thus be eliminated if we were to put The extra term would thus be eliminated if we were to put
</p> </p>
<div class="core div" id="orgba15335"> <div class="core div" id="org2b447e1">
<p> <p>
\[ \[
{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}
@ -1631,7 +1631,7 @@ Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation
<p> <p>
Maxwell baptized this term the Maxwell baptized this term the
</p> </p>
<div class="core div" id="org79a35ce"> <div class="core div" id="org10e929a">
<p> <p>
{\bf Displacement current} {\bf Displacement current}
\[ \[
@ -1664,6 +1664,8 @@ Flat surface: OK, \(E = 0\) and \(I_{\mbox{enc}} = I\). Balloon surface: \(I
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Me_ebM.html">Electrodynamics Before Maxwell&emsp;<small>[emd.Me.ebM]</small></a></li><li>Next:&nbsp;<a href="emd_Me_Me.html">Maxwell's Equations&emsp;<small>[emd.Me.Me]</small></a></li><li>Up:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1677,7 +1679,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emd_Me_ebM.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1634,6 +1634,8 @@ So for non-steady currents, the 'current enclosed by a loop' is ill-defined.
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li><li>Next:&nbsp;<a href="emd_Me_dc.html">Maxwell's Correction to Ampère's Law; the Displacement Current&emsp;<small>[emd.Me.dc]</small></a></li><li>Up:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1647,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emd_Me_mc.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1632,6 +1632,8 @@ Maxwell's equations {\bf beg} for magnetic charges. But we've never found any!
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Me_Me.html">Maxwell's Equations&emsp;<small>[emd.Me.Me]</small></a></li><li>Next:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li><li>Up:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1645,7 +1647,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

14
build/emd_ce.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">emd.ce</span></h3> </svg></a><span class="headline-id">emd.ce</span></h3>
<div class="outline-text-3" id="text-emd_ce"> <div class="outline-text-3" id="text-emd_ce">
<details class="prereq" id="orga7642d4"> <details class="prereq" id="orgeb45464">
<summary id="org89bea54"> <summary id="org2f2d9c5">
Prerequisites Prerequisites
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1607,8 +1607,8 @@ Prerequisites
</ul> </ul>
</details> </details>
<details class="objectives" id="orgf85a819"> <details class="objectives" id="org9e5041f">
<summary id="orgb7f774b"> <summary id="orgbcd06e1">
Objectives Objectives
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1631,6 +1631,8 @@ Objectives
<li><a href="emd_ce_amom.html">Angular Momentum</a><span class="headline-id">emd.ce.amom</span></li> <li><a href="emd_ce_amom.html">Angular Momentum</a><span class="headline-id">emd.ce.amom</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Me_mc.html">Magnetic Charge&emsp;<small>[emd.Me.mc]</small></a></li><li>Next:&nbsp;<a href="emd_ce_ce.html">The Continuity Equation&emsp;<small>[emd.ce.ce]</small></a></li><li>Up:&nbsp;<a href="emd.html">Electromagnetodynamics&emsp;<small>[emd]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1644,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

8
build/emd_ce_amom.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1600,7 +1600,7 @@ Table of contents
<p> <p>
The angular momentum of EM fields is directly given by The angular momentum of EM fields is directly given by
</p> </p>
<div class="main div" id="orgbedab67"> <div class="main div" id="org76c33ae">
<p> <p>
{\bf Angular momentum of EM fields} {\bf Angular momentum of EM fields}
\[ \[
@ -1615,6 +1615,8 @@ The angular momentum of EM fields is directly given by
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_mom.html">Momentum&emsp;<small>[emd.ce.mom]</small></a></li><li>Next:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1628,7 +1630,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

8
build/emd_ce_ce.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1619,7 +1619,7 @@ This means that
\] \]
Since this is true for any volume, we have (re)derived the Since this is true for any volume, we have (re)derived the
</p> </p>
<div class="core div" id="org8328070"> <div class="core div" id="org9d47204">
<p> <p>
{\bf Continuity equation} {\bf Continuity equation}
\[ \[
@ -1645,6 +1645,8 @@ imposes a functional constraint on these sources: not {\it any} \(\rho\) and
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li><li>Next:&nbsp;<a href="emd_ce_poy.html">Poynting's Theorem; the Poynting Vector&emsp;<small>[emd.ce.poy]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1658,7 +1660,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

10
build/emd_ce_mom.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1612,7 +1612,7 @@ in which the first integral can be interpreted as the momentum stored in the EM
<p> <p>
This is thus simply a conservation law for momentum, with This is thus simply a conservation law for momentum, with
</p> </p>
<div class="main div" id="orgfb9b180"> <div class="main div" id="org75dce9f">
<p> <p>
{\bf Momentum density in the EM fields} {\bf Momentum density in the EM fields}
\[ \[
@ -1624,7 +1624,7 @@ This is thus simply a conservation law for momentum, with
<p> <p>
In a region in which the mechanical momentum is not changing due to external influences, we then have the In a region in which the mechanical momentum is not changing due to external influences, we then have the
</p> </p>
<div class="main div" id="org03df6a3"> <div class="main div" id="orgeb01a5d">
<p> <p>
{\bf Continuity equation for EM momentum} {\bf Continuity equation for EM momentum}
\[ \[
@ -1638,6 +1638,8 @@ In a region in which the mechanical momentum is not changing due to external inf
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_mst.html">Maxwell's Stress Tensor&emsp;<small>[emd.ce.mst]</small></a></li><li>Next:&nbsp;<a href="emd_ce_amom.html">Angular Momentum&emsp;<small>[emd.ce.amom]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1651,7 +1653,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

12
build/emd_ce_mst.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1654,7 +1654,7 @@ and similarly for \({\boldsymbol B}\). We thus get
<p> <p>
This expression can be greatly simplified by introducing the This expression can be greatly simplified by introducing the
</p> </p>
<div class="main div" id="org7f141f8"> <div class="main div" id="orgcd90456">
<p> <p>
{\bf Maxwell stress tensor} {\bf Maxwell stress tensor}
\[ \[
@ -1677,7 +1677,7 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
<p> <p>
We then obtain We then obtain
</p> </p>
<div class="main div" id="org6e156bd"> <div class="main div" id="orgbba46d7">
<p> <p>
{\bf EM force per unit volume} {\bf EM force per unit volume}
\[ \[
@ -1689,7 +1689,7 @@ We then obtain
<p> <p>
where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
</p> </p>
<div class="main div" id="org7c98547"> <div class="main div" id="org9e182ba">
<p> <p>
{\bf Total force on charges in volume} {\bf Total force on charges in volume}
\[ \[
@ -1703,6 +1703,8 @@ where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_poy.html">Poynting's Theorem; the Poynting Vector&emsp;<small>[emd.ce.poy]</small></a></li><li>Next:&nbsp;<a href="emd_ce_mom.html">Momentum&emsp;<small>[emd.ce.mom]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1716,7 +1718,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

18
build/emd_ce_poy.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1666,7 +1666,7 @@ so we get
Substituting this in \ref{Gr(8.6)} and using the divergence theorem, Substituting this in \ref{Gr(8.6)} and using the divergence theorem,
we obtain we obtain
</p> </p>
<div class="main div" id="orgab4b619"> <div class="main div" id="org1ad9e6b">
<p> <p>
{\bf Poynting's theorem} {\bf Poynting's theorem}
\[ \[
@ -1691,7 +1691,7 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
<p> <p>
Energy per unit time, per unit area carried by EM fields: Energy per unit time, per unit area carried by EM fields:
</p> </p>
<div class="core div" id="org9cfe7aa"> <div class="core div" id="org97f8866">
<p> <p>
{\bf Poynting vector} {\bf Poynting vector}
\[ \[
@ -1704,7 +1704,7 @@ Energy per unit time, per unit area carried by EM fields:
<p> <p>
We can thus express Poynting's theorem more compactly: We can thus express Poynting's theorem more compactly:
</p> </p>
<div class="core div" id="orgdeee59b"> <div class="core div" id="orgc096fa3">
<p> <p>
{\bf Poynting's theorem} {\bf Poynting's theorem}
\[ \[
@ -1717,7 +1717,7 @@ We can thus express Poynting's theorem more compactly:
<p> <p>
where we have defined the total where we have defined the total
</p> </p>
<div class="core div" id="org57db6a8"> <div class="core div" id="org56f314c">
<p> <p>
{\bf Energy in electromagnetic fields} {\bf Energy in electromagnetic fields}
\[ \[
@ -1740,7 +1740,7 @@ Then,
\] \]
so we get the so we get the
</p> </p>
<div class="core div" id="org65ae443"> <div class="core div" id="orgd477ba8">
<p> <p>
{\bf Poynting theorem (differential form)} {\bf Poynting theorem (differential form)}
\[ \[
@ -1757,7 +1757,7 @@ and has a similar for to the continuity equation
<div class="example div" id="org0e3381a"> <div class="example div" id="orgf36d646">
<p> <p>
\paragraph{Example 8.1} Current in a wire: Joule heating. Energy per unit time delivered to wire: from Poynting. \paragraph{Example 8.1} Current in a wire: Joule heating. Energy per unit time delivered to wire: from Poynting.
Assuming that the field is uniform, the electric field parallel to the wire is Assuming that the field is uniform, the electric field parallel to the wire is
@ -1787,6 +1787,8 @@ and the value is as expected.
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_ce.html">The Continuity Equation&emsp;<small>[emd.ce.ce]</small></a></li><li>Next:&nbsp;<a href="emd_ce_mst.html">Maxwell's Stress Tensor&emsp;<small>[emd.ce.mst]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1800,7 +1802,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

14
build/emd_emw.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">emd.emw</span></h3> </svg></a><span class="headline-id">emd.emw</span></h3>
<div class="outline-text-3" id="text-emd_emw"> <div class="outline-text-3" id="text-emd_emw">
<details class="prereq" id="org3bd5d73"> <details class="prereq" id="org976b39e">
<summary id="orgec57498"> <summary id="org4571516">
Prerequisites Prerequisites
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1608,8 +1608,8 @@ Prerequisites
</ul> </ul>
</details> </details>
<details class="objectives" id="org7eaef42"> <details class="objectives" id="orgc2083f7">
<summary id="org71ec3b5"> <summary id="org74289a2">
Objectives Objectives
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1635,6 +1635,8 @@ Objectives
<li><a href="emd_emw_ep.html">Energy and Momentum</a><span class="headline-id">emd.emw.ep</span></li> <li><a href="emd_emw_ep.html">Energy and Momentum</a><span class="headline-id">emd.emw.ep</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_amom.html">Angular Momentum&emsp;<small>[emd.ce.amom]</small></a></li><li>Next:&nbsp;<a href="emd_emw_we.html">The Wave Equation&emsp;<small>[emd.emw.we]</small></a></li><li>Up:&nbsp;<a href="emd.html">Electromagnetodynamics&emsp;<small>[emd]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1648,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

10
build/emd_emw_ep.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1628,7 +1628,7 @@ so for a monochromatic EM plan wave,
\] \]
or more succinctly: or more succinctly:
</p> </p>
<div class="main div" id="org7992b89"> <div class="main div" id="org58ea3a9">
<p> <p>
{\bf Poynting vector of a monochromatic EM wave} {\bf Poynting vector of a monochromatic EM wave}
\[ \[
@ -1644,7 +1644,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
<p> <p>
Similary, we get the Similary, we get the
</p> </p>
<div class="main div" id="org0141dca"> <div class="main div" id="org9272ec4">
<p> <p>
{\bf Momentum density of a monochromatic EM wave} {\bf Momentum density of a monochromatic EM wave}
\[ \[
@ -1680,6 +1680,8 @@ The {\it radiation pressure} is the momentum transfer per unit area per unit of
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_emw_mpw.html">Monochromatic Plane Waves&emsp;<small>[emd.emw.mpw]</small></a></li><li>Next:&nbsp;<a href="emdm.html">Electromagnetodynamics in Matter&emsp;<small>[emdm]</small></a></li><li>Up:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1693,7 +1695,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

8
build/emd_emw_mpw.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1629,7 +1629,7 @@ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0.
Generalizing to propagation in the direction of an arbitrary wavevector Generalizing to propagation in the direction of an arbitrary wavevector
\({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
</p> </p>
<div class="core div" id="org0336f47"> <div class="core div" id="org19a2b1c">
<p> <p>
{\bf E and B fields for a monochromatic EM plane wave} {\bf E and B fields for a monochromatic EM plane wave}
\[ \[
@ -1658,6 +1658,8 @@ or if you prefer explicit real parts (adding a possible phase shift \(\delta\)):
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_emw_we.html">The Wave Equation&emsp;<small>[emd.emw.we]</small></a></li><li>Next:&nbsp;<a href="emd_emw_ep.html">Energy and Momentum&emsp;<small>[emd.emw.ep]</small></a></li><li>Up:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1671,7 +1673,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

8
build/emd_emw_we.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1626,7 +1626,7 @@ These take the form of coupled first-order partial differential equations for \(
Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\), Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
we get the we get the
</p> </p>
<div class="core div" id="org832615b"> <div class="core div" id="org3cd266d">
<p> <p>
{\bf Wave equations for electric and magnetic fields in vacuum} {\bf Wave equations for electric and magnetic fields in vacuum}
\[ \[
@ -1667,6 +1667,8 @@ the actual electric and magnetic fields are given by the real part.
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li><li>Next:&nbsp;<a href="emd_emw_mpw.html">Monochromatic Plane Waves&emsp;<small>[emd.emw.mpw]</small></a></li><li>Up:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1680,7 +1682,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emdm.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1607,6 +1607,8 @@ Table of contents
<li><a href="emdm_emwm.html">Electromagnetic Waves in Matter</a><span class="headline-id">emdm.emwm</span></li> <li><a href="emdm_emwm.html">Electromagnetic Waves in Matter</a><span class="headline-id">emdm.emwm</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Next:&nbsp;<a href="emdm_Me.html">Maxwell's Equations in Matter&emsp;<small>[emdm.Me]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1620,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emdm_Me.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1607,6 +1607,8 @@ Table of contents
<li><a href="emdm_Me_bc.html">Boundary Conditions</a><span class="headline-id">emdm.Me.bc</span></li> <li><a href="emdm_Me_bc.html">Boundary Conditions</a><span class="headline-id">emdm.Me.bc</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm.html">Electromagnetodynamics in Matter&emsp;<small>[emdm]</small></a></li><li>Next:&nbsp;<a href="emdm_Me_Mem.html">Maxwell's Equations in Matter&emsp;<small>[emdm.Me.Mem]</small></a></li><li>Up:&nbsp;<a href="emdm.html">Electromagnetodynamics in Matter&emsp;<small>[emdm]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1620,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

22
build/emdm_Me_Mem.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1633,7 +1633,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
\] \]
We therefore have the We therefore have the
</p> </p>
<div class="core div" id="orga19bc43"> <div class="core div" id="orgc16990f">
<p> <p>
{\bf Polarization current density} {\bf Polarization current density}
\[ \[
@ -1651,7 +1651,7 @@ the polarization current is the result of linear motion of charge when
polarization changes). We can check consistency with the continuity equation polarization changes). We can check consistency with the continuity equation
associated to the conservation of bound charges: associated to the conservation of bound charges:
</p> </p>
<aside id="org7492618"> <aside id="orgbe14b68">
<p> <p>
Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current
\({\boldsymbol J}_b\) but this is not the convention used here. \({\boldsymbol J}_b\) but this is not the convention used here.
@ -1674,7 +1674,7 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
In view of this: total charge density can be separated into 2 parts, In view of this: total charge density can be separated into 2 parts,
{\it free} and {\it bound}: {\it free} and {\it bound}:
</p> </p>
<div class="main div" id="org6458291"> <div class="main div" id="org27840ee">
<p> <p>
\[ \[
\rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P} \rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
@ -1687,7 +1687,7 @@ In view of this: total charge density can be separated into 2 parts,
and current can be separated into three parts, {\it free}, {\it bound} and and current can be separated into three parts, {\it free}, {\it bound} and
{\it polarization}: {\it polarization}:
</p> </p>
<div class="main div" id="org79300fd"> <div class="main div" id="orge8444d0">
<p> <p>
\[ \[
{\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M} {\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
@ -1711,7 +1711,7 @@ Gauss's law: can be rewritten
\] \]
where (as in static case) where (as in static case)
</p> </p>
<div class="core div" id="org0cf7843"> <div class="core div" id="org3e52dc6">
<p> <p>
\[ \[
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P} {\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
@ -1737,7 +1737,7 @@ or
\] \]
where as before where as before
</p> </p>
<div class="core div" id="org4d58c35"> <div class="core div" id="org5550f7c">
<p> <p>
\[ \[
{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M} {\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
@ -1755,7 +1755,7 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
<p> <p>
In terms of free charges and currents, we thus get In terms of free charges and currents, we thus get
</p> </p>
<div class="core div" id="org4ca5e56"> <div class="core div" id="orge1d07ac">
<p> <p>
{\bf Maxwell's equations {\it (in matter)}} {\bf Maxwell's equations {\it (in matter)}}
</p> </p>
@ -1781,7 +1781,7 @@ Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and
in terms of \({\bf E}\) and \({\bf B}\). in terms of \({\bf E}\) and \({\bf B}\).
For the restricted case of linear media: For the restricted case of linear media:
</p> </p>
<div class="main div" id="orgac29d47"> <div class="main div" id="org83b4dbd">
<p> <p>
\[ \[
{\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm} {\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
@ -1803,6 +1803,8 @@ where \(\varepsilon \equiv \varepsilon_0(1 + \chi_e)\) and \(\mu \equiv \mu_0 (1
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_Me.html">Maxwell's Equations in Matter&emsp;<small>[emdm.Me]</small></a></li><li>Next:&nbsp;<a href="emdm_Me_bc.html">Boundary Conditions&emsp;<small>[emdm.Me.bc]</small></a></li><li>Up:&nbsp;<a href="emdm_Me.html">Maxwell's Equations in Matter&emsp;<small>[emdm.Me]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1816,7 +1818,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

8
build/emdm_Me_bc.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1600,7 +1600,7 @@ Table of contents
<p> <p>
Discontinuities between different media, deduced from Discontinuities between different media, deduced from
</p> </p>
<div class="core div" id="org72eb54c"> <div class="core div" id="org55029ff">
<p> <p>
{\bf Maxwell's equations {\it (in matter)}, integral form} {\bf Maxwell's equations {\it (in matter)}, integral form}
</p> </p>
@ -1676,6 +1676,8 @@ These are basis of theory of reflection and refraction.
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_Me_Mem.html">Maxwell's Equations in Matter&emsp;<small>[emdm.Me.Mem]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li><li>Up:&nbsp;<a href="emdm_Me.html">Maxwell's Equations in Matter&emsp;<small>[emdm.Me]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1689,7 +1691,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emdm_emwm.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1610,6 +1610,8 @@ Table of contents
<li><a href="emdm_emwm_wg.html">Waveguides</a><span class="headline-id">emdm.emwm.wg</span></li> <li><a href="emdm_emwm_wg.html">Waveguides</a><span class="headline-id">emdm.emwm.wg</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_Me_bc.html">Boundary Conditions&emsp;<small>[emdm.Me.bc]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_plm.html">Propagation in Linear Media&emsp;<small>[emdm.emwm.plm]</small></a></li><li>Up:&nbsp;<a href="emdm.html">Electromagnetodynamics in Matter&emsp;<small>[emdm]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1623,7 +1625,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emdm_emwm_ad.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1606,6 +1606,8 @@ Table of contents
<li><a href="emdm_emwm_ad_c.html">EM Waves in Conductors</a><span class="headline-id">emdm.emwm.ad.c</span></li> <li><a href="emdm_emwm_ad_c.html">EM Waves in Conductors</a><span class="headline-id">emdm.emwm.ad.c</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_Ba.html">Brewster's Angle&emsp;<small>[emdm.emwm.refl.Ba]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad_c.html">EM Waves in Conductors&emsp;<small>[emdm.emwm.ad.c]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1619,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emdm_emwm_ad_c.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1702,6 +1702,8 @@ so the final form of the fields is
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_ad.html">Absorption and Dispersion&emsp;<small>[emdm.emwm.ad]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_wg.html">Waveguides&emsp;<small>[emdm.emwm.wg]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_ad.html">Absorption and Dispersion&emsp;<small>[emdm.emwm.ad]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1715,7 +1717,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

8
build/emdm_emwm_plm.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1635,7 +1635,7 @@ v = \frac{1}{\sqrt{\mu \varepsilon}} = \frac{c}{n}
\] \]
where the index of refraction of the material is defined as where the index of refraction of the material is defined as
</p> </p>
<div class="main div" id="orgc7bc968"> <div class="main div" id="orga0618f6">
<p> <p>
{\bf Index of refraction} {\bf Index of refraction}
\[ \[
@ -1677,6 +1677,8 @@ I = \frac{1}{2} \varepsilon v E_0^2
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refr.html">Refraction&emsp;<small>[emdm.emwm.refr]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1690,7 +1692,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emdm_emwm_refl.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1621,6 +1621,8 @@ Detailed study: starts from boundary conditions \ref{Gr(7.64)},
<li><a href="emdm_emwm_refl_Ba.html">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</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> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refr.html">Refraction&emsp;<small>[emdm.emwm.refr]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1634,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emdm_emwm_refl_Ba.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1601,6 +1601,8 @@ Table of contents
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_Fe.html">Fresnel's Equations&emsp;<small>[emdm.emwm.refl.Fe]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad.html">Absorption and Dispersion&emsp;<small>[emdm.emwm.ad]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1614,7 +1616,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emdm_emwm_refl_Fe.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1602,6 +1602,8 @@ Table of contents
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_oi.html">Oblique Incidence&emsp;<small>[emdm.emwm.refl.oi]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_Ba.html">Brewster's Angle&emsp;<small>[emdm.emwm.refl.Ba]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1615,7 +1617,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emdm_emwm_refl_ni.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1693,6 +1693,8 @@ R + T = 1.
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_oi.html">Oblique Incidence&emsp;<small>[emdm.emwm.refl.oi]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1706,7 +1708,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

14
build/emdm_emwm_refl_oi.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1635,7 +1635,7 @@ These forms for incident, reflected and transmitted wave can be substituted in t
<p> <p>
From now on we will orient the axes so that \({\boldsymbol k}_I\) lies in the \(xz\) plane. This means that \({\boldsymbol k}_R\) and \({\boldsymbol k}_T\) also lie in that plane. This is the From now on we will orient the axes so that \({\boldsymbol k}_I\) lies in the \(xz\) plane. This means that \({\boldsymbol k}_R\) and \({\boldsymbol k}_T\) also lie in that plane. This is the
</p> </p>
<div class="core div" id="orgb0a9710"> <div class="core div" id="orge8258d6">
<p> <p>
{\bf First law of reflection:} {\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. the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
@ -1650,7 +1650,7 @@ Specializing (\ref{eq:RTObliquek}) to our notations, we have
with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
and the angle of refraction (\(\theta_T\)) obey the following laws: and the angle of refraction (\(\theta_T\)) obey the following laws:
</p> </p>
<div class="core div" id="orga98e40b"> <div class="core div" id="org823940c">
<p> <p>
{\bf Law of reflection} {\bf Law of reflection}
\[ \[
@ -1708,7 +1708,7 @@ while the third equation becomes
\] \]
Writing everything in terms of the incident amplitude, we get Writing everything in terms of the incident amplitude, we get
</p> </p>
<div class="main div" id="org521f546"> <div class="main div" id="orgd20a172">
<p> <p>
{\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)} {\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)}
\[ \[
@ -1728,7 +1728,7 @@ Amplitudes for transmitted and reflected wave: depend on angle of incidence:
Behaviour: for \(\theta_I = 0\) we recover (\ref{Gr(9.82)}). Behaviour: for \(\theta_I = 0\) we recover (\ref{Gr(9.82)}).
For grazing waves \(\theta_I \rightarrow \pi/2\) we have that \(\alpha \rightarrow \infty\) and the wave is totally reflected. The most interesting angle is the one at which \(\alpha = \beta\) and the reflected wave has zero amplitude. This is known as For grazing waves \(\theta_I \rightarrow \pi/2\) we have that \(\alpha \rightarrow \infty\) and the wave is totally reflected. The most interesting angle is the one at which \(\alpha = \beta\) and the reflected wave has zero amplitude. This is known as
</p> </p>
<div class="main div" id="org51d503c"> <div class="main div" id="orgd033d51">
<p> <p>
{\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}} {\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}}
\[ \[
@ -1763,6 +1763,8 @@ Of course, we get \(R + T = 1\) as expected.
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_Fe.html">Fresnel's Equations&emsp;<small>[emdm.emwm.refl.Fe]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1776,7 +1778,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emdm_emwm_refr.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1602,6 +1602,8 @@ Table of contents
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_plm.html">Propagation in Linear Media&emsp;<small>[emdm.emwm.plm]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1615,7 +1617,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emdm_emwm_wg.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1608,6 +1608,8 @@ Table of contents
<li><a href="emdm_emwm_wg_c.html">Coaxial Lines</a><span class="headline-id">emdm.emwm.wg.c</span></li> <li><a href="emdm_emwm_wg_c.html">Coaxial Lines</a><span class="headline-id">emdm.emwm.wg.c</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_ad_c.html">EM Waves in Conductors&emsp;<small>[emdm.emwm.ad.c]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_wg_gw.html">Guided waves&emsp;<small>[emdm.emwm.wg.gw]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1621,7 +1623,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emdm_emwm_wg_c.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1628,6 +1628,8 @@ where \(A\) is a constant amplitude. Substituting and taking the real part,
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_wg_r.html">Rectangular Waveguides&emsp;<small>[emdm.emwm.wg.r]</small></a></li><li>Next:&nbsp;<a href="emf.html">Electromagnetic Fields&emsp;<small>[emf]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_wg.html">Waveguides&emsp;<small>[emdm.emwm.wg]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1641,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emdm_emwm_wg_gw.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1652,6 +1652,8 @@ with an identical equation for \(B_z\). If \(E_z = 0\) the waves are called {\bf
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_wg.html">Waveguides&emsp;<small>[emdm.emwm.wg]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_wg_r.html">Rectangular Waveguides&emsp;<small>[emdm.emwm.wg.r]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_wg.html">Waveguides&emsp;<small>[emdm.emwm.wg]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1665,7 +1667,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emdm_emwm_wg_r.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1661,6 +1661,8 @@ which is {\it greater} than \(c\). The energy of the wave however propagates at
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_wg_gw.html">Guided waves&emsp;<small>[emdm.emwm.wg.gw]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_wg_c.html">Coaxial Lines&emsp;<small>[emdm.emwm.wg.c]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_wg.html">Waveguides&emsp;<small>[emdm.emwm.wg]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1674,7 +1676,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

14
build/emf.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">emf</span></h2> </svg></a><span class="headline-id">emf</span></h2>
<div class="outline-text-2" id="text-emf"> <div class="outline-text-2" id="text-emf">
<details class="prereq" id="orgc2e1e28"> <details class="prereq" id="orgd0eb59c">
<summary id="orge525c1b"> <summary id="org6131f4a">
Prerequisites Prerequisites
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1607,8 +1607,8 @@ Prerequisites
</ul> </ul>
</details> </details>
<details class="objectives" id="orgcd01d6b"> <details class="objectives" id="org1db29ae">
<summary id="org510a3e2"> <summary id="orgab2baad">
Objectives Objectives
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1629,6 +1629,8 @@ Objectives
<li><a href="emf_g.html">Gauge Freedom and Choices</a><span class="headline-id">emf.g</span></li> <li><a href="emf_g.html">Gauge Freedom and Choices</a><span class="headline-id">emf.g</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Next:&nbsp;<a href="emf_svp.html">Scalar and Vector Potentials&emsp;<small>[emf.svp]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1642,7 +1644,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emf_g.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1626,6 +1626,8 @@ implementing a <b>gauge transformation</b>.
<li><a href="emf_g_Lg.html">Lorenz Gauge; d'Alembertian; Inhomogeneous Maxwell Equations</a><span class="headline-id">emf.g.Lg</span></li> <li><a href="emf_g_Lg.html">Lorenz Gauge; d'Alembertian; Inhomogeneous Maxwell Equations</a><span class="headline-id">emf.g.Lg</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emf_svp.html">Scalar and Vector Potentials&emsp;<small>[emf.svp]</small></a></li><li>Next:&nbsp;<a href="emf_g_Cg.html">Coulomb Gauge&emsp;<small>[emf.g.Cg]</small></a></li><li>Up:&nbsp;<a href="emf.html">Electromagnetic Fields&emsp;<small>[emf]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1639,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/emf_g_Cg.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1625,6 +1625,8 @@ Although Gauss's law looks nice in the Coulomb gauge, Amp{\`e}re-Maxwell does no
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emf_g.html">Gauge Freedom and Choices&emsp;<small>[emf.g]</small></a></li><li>Next:&nbsp;<a href="emf_g_Lg.html">Lorenz Gauge; d'Alembertian; Inhomogeneous Maxwell Equations&emsp;<small>[emf.g.Lg]</small></a></li><li>Up:&nbsp;<a href="emf_g.html">Gauge Freedom and Choices&emsp;<small>[emf.g]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1638,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

10
build/emf_g_Lg.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1613,7 +1613,7 @@ while the equation for \(V\) becomes
\] \]
These can be written compactly upon introducing a new operator: the These can be written compactly upon introducing a new operator: the
</p> </p>
<div class="core div" id="org9685743"> <div class="core div" id="orgb7b775a">
<p> <p>
{\bf d'Alembertian operator} {\bf d'Alembertian operator}
\[ \[
@ -1626,7 +1626,7 @@ These can be written compactly upon introducing a new operator: the
<p> <p>
so we get the so we get the
</p> </p>
<div class="core div" id="org8c1d592"> <div class="core div" id="org1d4f993">
<p> <p>
{\bf Inhomogeneous Maxwell equations (Lorenz gauge)} {\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
\[ \[
@ -1661,6 +1661,8 @@ we have by direct inspection
</div> </div>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emf_g_Cg.html">Coulomb Gauge&emsp;<small>[emf.g.Cg]</small></a></li><li>Next:&nbsp;<a href="red.html">Relativistic Electrodynamics&emsp;<small>[red]</small></a></li><li>Up:&nbsp;<a href="emf_g.html">Gauge Freedom and Choices&emsp;<small>[emf.g]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1674,7 +1676,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

14
build/emf_svp.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1614,7 +1614,7 @@ Useful strategy: represent fields in terms of potentials.
<p> <p>
Easiest: Easiest:
</p> </p>
<div class="core div" id="org449388e"> <div class="core div" id="orgea17c1a">
<p> <p>
\[ \[
{\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A} {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
@ -1630,7 +1630,7 @@ Putting this into Faraday's law gives
\] \]
so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
</p> </p>
<div class="core div" id="org8d2221f"> <div class="core div" id="orge40ddf5">
<p> <p>
\[ \[
{\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t} {\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}
@ -1643,7 +1643,7 @@ so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \
<p> <p>
Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting (\ref{eq:E_from_Potentials}) into Gauss's law gives Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting (\ref{eq:E_from_Potentials}) into Gauss's law gives
</p> </p>
<div class="main div" id="orgf979da5"> <div class="main div" id="org73cca63">
<p> <p>
\[ \[
{\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0} {\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
@ -1659,7 +1659,7 @@ whereas Amp{\`ere}-Maxwell becomes
\] \]
which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\), which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
</p> </p>
<div class="main div" id="orgc475199"> <div class="main div" id="org0170d34">
<p> <p>
\[ \[
\left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right) \left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
@ -1680,6 +1680,8 @@ which becomes after simple rearrangement and use of the identity \({\boldsymbol
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emf.html">Electromagnetic Fields&emsp;<small>[emf]</small></a></li><li>Next:&nbsp;<a href="emf_g.html">Gauge Freedom and Choices&emsp;<small>[emf.g]</small></a></li><li>Up:&nbsp;<a href="emf.html">Electromagnetic Fields&emsp;<small>[emf]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1693,7 +1695,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

6
build/ems.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1609,6 +1609,8 @@ Table of contents
<li><a href="ems_ms.html">Magnetostatics</a><span class="headline-id">ems.ms</span></li> <li><a href="ems_ms.html">Magnetostatics</a><span class="headline-id">ems.ms</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Next:&nbsp;<a href="ems_es.html">Electrostatics&emsp;<small>[ems.es]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1622,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

14
build/ems_ca.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">ems.ca</span></h3> </svg></a><span class="headline-id">ems.ca</span></h3>
<div class="outline-text-3" id="text-ems_ca"> <div class="outline-text-3" id="text-ems_ca">
<details class="prereq" id="orga055812"> <details class="prereq" id="org0bf319c">
<summary id="org2cb689b"> <summary id="orga92f3c7">
Prerequisites Prerequisites
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1607,8 +1607,8 @@ Prerequisites
</ul> </ul>
</details> </details>
<details class="objectives" id="orgb006f6c"> <details class="objectives" id="org87cd363">
<summary id="org037b124"> <summary id="org64f736b">
Objectives Objectives
</summary> </summary>
<ul class="org-ul"> <ul class="org-ul">
@ -1631,6 +1631,8 @@ Objectives
<li><a href="ems_ca_me.html">The Multipole Expansion</a><span class="headline-id">ems.ca.me</span></li> <li><a href="ems_ca_me.html">The Multipole Expansion</a><span class="headline-id">ems.ca.me</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_c_cap.html">Capacitors&emsp;<small>[ems.es.c.cap]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li><li>Up:&nbsp;<a href="ems.html">Electromagnetostatics&emsp;<small>[ems]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1644,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

57
build/ems_ca_fe.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1603,7 +1603,11 @@ A generic configuration of static charges coupled via the Coulomb interaction
defines an electrostatic problem, whose solution is in principle obtained defines an electrostatic problem, whose solution is in principle obtained
from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a> from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
</p> </p>
<div class="main div" id="org27034ee"> <div class="main div" id="org98cdfd8">
<p>
</p>
\begin{equation*} \begin{equation*}
{\bf E} ({\bf r}) = \frac{1}{4\pi\varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') \frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3} {\bf E} ({\bf r}) = \frac{1}{4\pi\varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') \frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3}
\end{equation*} \end{equation*}
@ -1611,44 +1615,53 @@ from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_v
</div> </div>
<p> <p>
or (often simpler) by calculating the electrostatic potential, using either the or (often simpler) by calculating the electrostatic potential, using either the
explicit construction (\ref{eq:V_from_rho}) explicit construction <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
</p> </p>
<div class="main div" id="org1620b08"> <div class="main div" id="org12e97ac">
<p>
</p>
<p> <p>
\[ \[
V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|}. \phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|}.
\tag{\ref{eq:V_from_rho}} \]
\]
</p> </p>
</div> </div>
<p> <p>
Alternately, we have also seen that the two fundamental equations for the Alternately, we have also seen that the two fundamental equations for the
electrostatic field, Gauss's law (\ref{Gr(2.14)}) and the no-perpetual-machine (vanishing curl) electrostatic field, Gauss's law <a href="./ems_es_ef_Gl.html#Gl_d">Gl_d</a> and the vanishing curl
condition (\ref{Gr(2.20)}) can be expressed as the single condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as the single
'local' (differential) condition (Poisson's equation) (\ref{eq:Poisson}) <i>local</i> differential condition (Poisson's equation)
<a href="./ems_es_ep_PL.html#Poi">🐟</a>
</p>
<div class="core div" id="orgcacf446">
<p>
</p> </p>
<div class="core div" id="org1361b41">
<p> <p>
\[ \[
{\boldsymbol \nabla}^2 V = -\frac{\rho}{\varepsilon_0}. {\boldsymbol \nabla}^2 \phi = -\frac{\rho}{\varepsilon_0}.
\tag{\ref{eq:Poisson}} \]
\]
</p> </p>
</div> </div>
<p> <p>
In the specific case where the charge density vanishes, we fall back onto the simpler 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>
Laplace equation
</p> </p>
<div class="core div" id="org2305746"> <div class="core div" id="orgea7b7e8">
<p>
</p>
<p> <p>
\[ \[
{\boldsymbol \nabla}^2 V = 0 {\boldsymbol \nabla}^2 \phi = 0
\tag{\ref{eq:Laplace}} \]
\]
</p> </p>
</div> </div>
@ -1663,6 +1676,8 @@ Laplace equation
<li><a href="ems_ca_fe_uP.html">Uniqueness of Solution to Poisson's Equation</a><span class="headline-id">ems.ca.fe.uP</span></li> <li><a href="ems_ca_fe_uP.html">Uniqueness of Solution to Poisson's Equation</a><span class="headline-id">ems.ca.fe.uP</span></li>
</ul> </ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca.html">Calculating or Approximating the Electrostatic Potential&emsp;<small>[ems.ca]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe_L.html">The Laplace Equation&emsp;<small>[ems.ca.fe.L]</small></a></li><li>Up:&nbsp;<a href="ems_ca.html">Calculating or Approximating the Electrostatic Potential&emsp;<small>[ems.ca]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1676,7 +1691,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

386
build/ems_ca_fe_L.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,10 @@ Table of contents
</svg></a><span class="headline-id">ems.ca.fe.L</span></h5> </svg></a><span class="headline-id">ems.ca.fe.L</span></h5>
<div class="outline-text-5" id="text-ems_ca_fe_L"> <div class="outline-text-5" id="text-ems_ca_fe_L">
<p> <p>
Of course, the simplest situation is to start by looking at the region of space In regions of space where there is no charge density,
where there is no charge density. The potential then solves Laplace's equation. How can it possibly look ? the potential must solve Laplace's equation.
Let us discuss how solutions to this equation look,
in increasingly complicated situations.
</p> </p>
</div> </div>
@ -1608,18 +1610,85 @@ where there is no charge density. The potential then solves Laplace's equation.
<h6 id="ems_ca_fe_L_1d"><a href="#ems_ca_fe_L_1d">The Laplace Equation in One Dimension</a></h6> <h6 id="ems_ca_fe_L_1d"><a href="#ems_ca_fe_L_1d">The Laplace Equation in One Dimension</a></h6>
<div class="outline-text-6" id="text-ems_ca_fe_L_1d"> <div class="outline-text-6" id="text-ems_ca_fe_L_1d">
<p> <p>
In one dimension, the potential is a single-variable
function \(\phi (x)\) and the Laplace equation reads
</p>
<div class="eqlabel" id="org8b9eaa8">
<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="org1a79944">
</div>
</div>
<p>
\[ \[
\frac{d^2 V(x)}{dx^2} = 0 \Longrightarrow V(x) = mx + b \frac{d^2 \phi(x)}{dx^2} = 0.
\label{Gr(3.6)} \tag{Lap_1d}\label{Lap_1d}
\] \]
Properties:
\paragraph{1.} \(V(x)\) is the average of \(V(x + a)\) and \(V(x - a)\) for any \(a\).
\paragraph{2.} Solutions to Laplace's equation have no local maxima or minima.
</p> </p>
<p> <p>
Boundary conditions: always work: two end values, one end value + same end derivative value. The solution to this
Not always: one end value + derivative value at other end, two end derivative values. </p>
<div class="eqlabel" id="orgd6d6f8b">
<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="org135f906">
<ul class="org-ul">
<li>Gr (3.6)</li>
</ul>
</div>
</div>
<p>
\[
\phi(x) = a x + b
\tag{Lap_1d_sol}\label{Lap_1d_sol}
\]
</p>
<p>
Properties:
</p>
<ul class="org-ul">
<li>
<b>Balance</b>: \(\phi(x)\) is the average of \(\phi(x + dx)\) and \(\phi(x - dx)\) for any \(dx\) (with \(x \pm dx\) still being in
the region where Laplace is satisfied, of course).</li>
<li>
<b>No extrema</b>: \(\phi(x)\) has no local extrema. Max/min
values must occur at boundaries.</li>
</ul>
<p>
In a particular problem, to fix the solution (said
otherwise: to fix the parameters \(a\) and \(b\) in <a href="./ems_ca_fe_L.html#Lap_1d_sol">Lap_1d_sol</a>), we need to appeal to boundary
conditions. Concretely, for a finite segment,
a solution exists and is unique if one is
provided with any of these possibilities:
</p>
<ul class="org-ul">
<li>\(\phi\) at both boundaries</li>
<li>\(\phi\) and \(\frac{d\phi}{dx}\) at one boundary</li>
<li>\(\phi\) at one boundary, \(\frac{d\phi}{dx}\) at the other.</li>
</ul>
<p>
Specifying \(\frac{d\phi}{dx}\) at both boundaries
provides insufficient information, since you get
an inconsistency if the derivatives don't match.
</p> </p>
</div> </div>
</div> </div>
@ -1629,16 +1698,45 @@ Not always: one end value + derivative value at other end, two end derivative v
<h6 id="ems_ca_fe_L_2d"><a href="#ems_ca_fe_L_2d">The Laplace Equation in Two Dimensions</a></h6> <h6 id="ems_ca_fe_L_2d"><a href="#ems_ca_fe_L_2d">The Laplace Equation in Two Dimensions</a></h6>
<div class="outline-text-6" id="text-ems_ca_fe_L_2d"> <div class="outline-text-6" id="text-ems_ca_fe_L_2d">
<p> <p>
\[ In two dimensions, the potential becomes a function
\frac{d^2 V}{dx^2} + \frac{d^2 V}{dy^2} = 0. of two variables (here: \(x\) and \(y\)), so Laplace's
\] equation now reads
Properties:
\paragraph{1.} The value of \(V(x,y)\) equals the average value around the point:
\[
V(x,y) = \frac{1}{2\pi R} \oint V dl
\]
\paragraph{2.} \(V\) has no local maxima or minima. All extrema occur at the boundaries.
</p> </p>
<div class="eqlabel" id="orgb8aa28d">
<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="org01054fc">
</div>
</div>
\begin{equation*}
\frac{\partial^2 \phi (x,y)}{\partial x^2}
+ \frac{\partial^2 \phi (x,y)}{\partial y^2} = 0.
\tag{Lap_2d}\label{Lap_2d}
\end{equation*}
<p>
Properties:
</p>
<ul class="org-ul">
<li>
<b>Balance</b>: \(\phi(x,y)\) equals the average value around the point:</li>
</ul>
<p>
\[
\phi(x,y) = \frac{1}{2\pi R} \oint dl ~\phi
\]
</p>
<ul class="org-ul">
<li>
<b>No extrema</b>: \(\phi\) has no local maxima or minima. All extrema occur at the boundaries.</li>
</ul>
</div> </div>
</div> </div>
@ -1647,42 +1745,197 @@ V(x,y) = \frac{1}{2\pi R} \oint V dl
<h6 id="ems_ca_fe_L_3d"><a href="#ems_ca_fe_L_3d">The Laplace Equation in Three Dimensions</a></h6> <h6 id="ems_ca_fe_L_3d"><a href="#ems_ca_fe_L_3d">The Laplace Equation in Three Dimensions</a></h6>
<div class="outline-text-6" id="text-ems_ca_fe_L_3d"> <div class="outline-text-6" id="text-ems_ca_fe_L_3d">
<p> <p>
\[ In three dimensions, we will write the potential
{\boldsymbol \nabla}^2 V = 0 as a function of a 3-dimensional vector, \(\phi({\bf r})\).
\] The Laplace equation is (we repeat)
Properties:
\paragraph{1.} \(V({\bf r})\) is the average value of \(V\) over any spherical surface
centered at \({\bf r}\):
\[
V({\bf r}) = \frac{1}{4\pi R^2} \oint V da
\]
\paragraph{2.} \(V\) can have no local maxima or minima. All extrema occur at the boundaries.
</p> </p>
<p> <p>
\paragraph{Another way of seeing this} is to write the second derivatives as
\[ \[
\frac{\partial^2 V({\bf r})}{\partial x^2} = f_x ({\bf r}), \hspace{5mm} {\boldsymbol \nabla}^2 \phi ({\bf r}) = 0
\frac{\partial^2 V({\bf r})}{\partial y^2} = f_y ({\bf r}), \hspace{5mm} \]
\frac{\partial^2 V({\bf r})}{\partial z^2} = f_z ({\bf r}), \hspace{5mm} </p>
<p>
<b>Theorem</b>: if \(\phi\) satisfies Laplace, then its value at
a point equals its value averaged over any sphere
\(S_{\bf r}\) centered on this point,
\[
\phi({\bf r}) = \frac{1}{4\pi R^2} \oint_{S_{\bf r}} da' ~\phi ({\bf r}')
\]
</p>
<details id="org8c78d21">
<summary id="org6bdc443">
<strong>Physicist's proof</strong>
</summary>
<p>
Consider a sphere of radius \(R\) centered at the origin
carrying charge \(q\) spread with a uniform surface charge density over its surface. Bring in a point charge \(q'\) from
infinity up to a distance \(R'\) (with \(R' &gt; R\)) from the center
of the sphere.
</p>
<p>
We know that the field created by the sphere coincides
with that of a point charge \(q\) at the origin.
Since the potential at \({\bf r = 0}\) created by the charge
\(q'\) at \({\bf r'}\) is simply \(\phi_{q', {\bf r}'} (0) = \frac{q'}{4\pi \varepsilon_0 R'}\),
the work
required to bring the \(q'\) charge into position is thus
simply \(W = q \times \phi_{q', {\bf r}'} (0) = \frac{q q'}{4 \pi \varepsilon_0 R'}\) by <a href="./ems_es_efo_e.html#Wab">Wab</a>.
</p>
<p>
We can however proceed the other way: fixing \(q'\) in place,
and then bringing the charged sphere into position;
the work (energy) has to coincide with our previous result.
But this energy is now given by the integral of the
potential \(\phi_{q', {\bf r'}}\)
created by \(q'\) (sitting at \({\bf r'}\)) over the sphere
times the surface charge density on the sphere,
namely
</p>
<p>
\[
W = \oint_{S_R} da ~\sigma ~\phi_{q', {\bf r}'} ({\bf r})
\]
</p>
<p>
But \(\sigma = q/4\pi R^2\) and is a constant over the
sphere, so \(W = q \times \frac{1}{4\pi R^2} \oint da ~\phi_{q', {\bf r}'} ({\bf r})\).
</p>
<p>
Equating this with the previous results shows that
</p>
<p>
\[
\phi_{q', {\bf r'}} (0) = \frac{1}{4\pi R^2} \oint_{S_R} da ~\phi_{q', {\bf r}'} ({\bf r})
\]
</p>
<p>
namely that for the potential created by a single point
charge \(q'\) at \(R'\),
the value at a point (here the origin)
coincides with the value averaged over a sphere
or an arbitrary radius \(R\) centered on the same point.
</p>
<p>
By the principle of superposition, this works for an
arbitrary distribution of charges outside the sphere,
proving the theorem.
</p>
</details>
<details id="orgff2611b">
<summary id="org89a3b1b">
<strong>Formal proof</strong>
</summary>
<p>
Consider a function \(f({\bf r})\) and its average over
a ball of radius \(R\) centered on \({\bf r}\):
</p>
<p>
\[
f_{S_R} ({\bf r}) \equiv \frac{1}{4\pi R^2}\oint_{S_R} da' ~ f ({\bf r} + {\bf r}')
\]
</p>
<p>
For convenience we will hereafter put \({\bf r} = 0\).
In spherical coordinates, we have \(da' = R^2 sin \theta d\theta d\phi \equiv R^2 d\Omega\).
Differentiating with respect to \(R\),
</p>
<p>
\[
\frac{d}{dR} f_{S_R} = \frac{1}{4\pi} \oint_{S_R} d\Omega ~\left.\frac{\partial f}{\partial r}\right|_{r=R}
\]
</p>
<p>
with \(f\) differentiated with respect to the radial coordiate.
We can rewrite this by noting that \(R^2 d\Omega \hat{\bf r}\)
is the normal differential surface area \(d{\bf a}\), while
\(\left.\frac{\partial f}{\partial r}\right|_{r=R}\) is the radial component of the gradient
of \(f\) in spherical coordinates. Thus,
</p>
<p>
\[
\frac{d}{dR} f_{S_R} = \frac{1}{4\pi R^2} \oint_{S_R} d{\bf a} \cdot ~\nabla f
\]
</p>
<p>
Invoking the divergence theorem and using the definition
of the Laplacian operator \(\nabla^2 = \nabla \cdot \nabla\),
we get the following general
</p>
<p>
<b>Theorem</b>:
</p>
<p>
\[
\frac{d}{dR} f_{S_R} = \frac{1}{4\pi R^2} \int_{V_R} d\tau ~\nabla^2 f
\]
</p>
<p>
For the electrostatic potential away from charges, we have
\[
\nabla^2 \phi = 0 ~\rightarrow \frac{d}{dR} \phi_{S_R} = 0
\]
namely the ball average is independent of the ball size.
Since the value at the center is simply the average for
an infinitesimally small ball, we get the result announced above.
</p>
</details>
<p>
<b>Theorem (Earnshaw, mathematical versoin)</b>: \(\phi\) has no local extrema except at the boundaries.
</p>
<p>
<b>Proof</b>: write the second derivatives as
</p>
<p>
\[
\frac{\partial^2 \phi({\bf r})}{\partial x^2} = f_x ({\bf r}), \hspace{5mm}
\frac{\partial^2 \phi({\bf r})}{\partial y^2} = f_y ({\bf r}), \hspace{5mm}
\frac{\partial^2 \phi({\bf r})}{\partial z^2} = f_z ({\bf r}), \hspace{5mm}
f_x + f_y + f_z = 0. f_x + f_y + f_z = 0.
\] \]
The \(f_a ({\bf r})\) represent the three components of the curvature of \(V({\bf r})\). </p>
An extremum of \(V\) at \({\bf r}_e\) would be characterized by \({\boldsymbol \nabla} V |_{{\bf r}_e} \cdot \delta{\bf r} = 0\)
<p>
The \(f_a ({\bf r})\) represent the three components of the curvature of \(\phi({\bf r})\).
An extremum of \(\phi\) at \({\bf r}_e\) would be characterized by \({\boldsymbol \nabla} \phi |_{{\bf r}_e} \cdot \delta{\bf r} = 0\)
for any infinitesimal displacement \(\delta{\bf r}\) around the extremum point. For a local for any infinitesimal displacement \(\delta{\bf r}\) around the extremum point. For a local
minimum, the second derivative form should be greater than zero, \(\sum_{i,j} \frac{\partial^2 V}{\partial r_i \partial r_j} \delta r_i \delta r_j &gt; 0\) minimum, the second derivative form should be greater than zero, \(\sum_{i,j} \frac{\partial^2 \phi}{\partial r_i \partial r_j} \delta r_i \delta r_j &gt; 0\)
for any displacement vector. Choosing alternately displacements along the three axes, for any displacement vector. Choosing alternately displacements along the three axes,
the form becomes \(f_x (\delta x)^2\), \(f_y (\delta y)^2\) or \(f_z (\delta z)^2\). Since the squared displacements the form becomes \(f_x (\delta x)^2\), \(f_y (\delta y)^2\) or \(f_z (\delta z)^2\). Since the squared displacements
are necessarily positive, we thus require \(f_x &gt; 0\), \(f_y &gt; 0\) and \(f_z &gt; 0\). This is impossible in view are necessarily positive, we thus require \(f_x &gt; 0\), \(f_y &gt; 0\) and \(f_z &gt; 0\). This is impossible in view
of the \(f_x + f_y + f_z = 0\) condition above. of the \(f_x + f_y + f_z = 0\) condition above.
</p> </p>
<div class="info div" id="orge499be0"> <div class="info div" id="org950ce9d">
<p> <p>
<b>Earnshaw's theorem</b> <br> <b>Earnshaw's theorem (physical version)</b> <br>
Since solutions to Laplace's equation have no local minimum, It is impossible to find a static distribution of charges which generates an electrostatic field
it is impossible to find a static distribution of charges which generates an electrostatic field displaying a stable equilibrium position in empty space.
with a stable equilibrium position for a test charge.
</p> </p>
</div> </div>
@ -1692,47 +1945,56 @@ Going back to Poisson's equation, we can make a few comments:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>representation (\ref{eq:Poisson}) highlights the 'local' nature of the coupling between electrostatic fields and charges: fields are 'created' where the charges 'sit'. This is also seen by looking at the integrand of (\ref{eq:V_from_rho}). If electrostatics was nonlocal, a modified representation like (\ref{eq:V_from_rho}) would still exist, but not a local differential one like (\ref{eq:Poisson}).</li> <li>representation <a href="./ems_es_ep_PL.html#Poi">🐟</a> highlights the 'local' nature of the coupling between electrostatic fields and charges: fields are 'created' where the charges 'sit'. This is also seen by looking at the integrand of <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>. If electrostatics was nonlocal, a modified representation like <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a> would still exist, but not a local differential one like Poisson's equation.</li>
<li>as written, representations (\ref{eq:E_from_rho}) and (\ref{eq:V_from_rho}) require the knowledge of the charge density distribution \(\rho({\bf r})\) throughout space to determine the potential at any given point.</li> <li>as written, representations <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a> and <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a> require the knowledge of the charge density distribution \(\rho({\bf r})\) throughout space to determine the potential at any given point.</li>
<li>(\ref{eq:Poisson}), being purely local, might allow to determine the potential at a specified point, provided we know the charge density distribution around this specified point, and at some set of other reference points (to make the solution unique).</li> <li>Poisson's equation <a href="./ems_es_ep_PL.html#Poi">🐟</a>, being purely local, might allow to determine the potential at a specified point, provided we know the charge density distribution around this specified point, and at some set of other reference points (to make the solution unique).</li>
</ul> </ul>
<p> <p>
We therefore want to ask the question: <b>under what conditions can an electrostatic problem be fully We therefore want to ask the question: <i>under what conditions can an electrostatic problem be fully
defined by solving Poisson's equation ?</b> We start by mentioning some cases, and interpreting them thereafter. defined by solving Poisson's equation ?</i>
</p>
<p>
We start by mentioning some cases, and interpreting them thereafter.
</p>
<p>
<b>Charge density is known throughout space</b>: in this case,
the electrostatic potential is uniquely determined
by Poisson's equation, which
is explicitly solved by <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>.
One can eplicitly verify this:
</p> </p>
<p> <p>
<b>First case</b>: the electrostatic potential is uniquely determined
if \(\rho({\bf r})\) is given throughout all space. In this case, Poisson's equation
is explicitly solved by (\ref{eq:V_from_rho}). Explicit check:
\[ \[
{\boldsymbol \nabla}^2 V ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') {\boldsymbol \nabla}^2 \frac{1}{|{\bf r} - {\bf r}'|} {\boldsymbol \nabla}^2 V ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') {\boldsymbol \nabla}^2 \frac{1}{|{\bf r} - {\bf r}'|}
= \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' (-4\pi) \delta ({\bf r} - {\bf r}') = -\frac{\rho ({\bf r})}{\varepsilon_0}. = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' (-4\pi) \delta ({\bf r} - {\bf r}') = -\frac{\rho ({\bf r})}{\varepsilon_0}.
\] \]
where we have used (\ref{Gr(1.102)}), and the fact that the delta function is always resolved since we </p>
integrate over all space. Note: it is implicitly assumed that the integral in (\ref{eq:V_from_rho})
<p>
where we have used <a href="./c_m_dd_3d.html#Lap1or">Lap1or</a>, and the fact that the delta function is always resolved since we
integrate over all space. Note: it is implicitly assumed that the integral in <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
converges, <i>i.e.</i> that the charge density \(\rho({\bf r})\) is sufficiently well-behaved (does not converges, <i>i.e.</i> that the charge density \(\rho({\bf r})\) is sufficiently well-behaved (does not
become singular). become singular).
</p> </p>
<p> <p>
<b>First case (corollary)</b>: the electrostatic potential is uniquely determined <b>Charge density in closed volume, and boundary surface charge density are known</b>: the electrostatic potential is uniquely determined
in a certain volume \({\cal V}\) bounded by boundary \({\cal S}\), provided the charge density in a certain volume \({\cal V}\) bounded by boundary \({\cal S}\), provided the charge density
\(\rho ({\bf x})\) is given everywhere within \({\cal V}\), vanishes outside of \({\cal V}\), \(\rho ({\bf x})\) is given everywhere within \({\cal V}\), vanishes outside of \({\cal V}\),
and the value of the surface charge density \(\sigma\) is given everywhere on the boundary \({\cal S}\). and the value of the surface charge density \(\sigma\) is given everywhere on the boundary \({\cal S}\).
Of course, \({\cal S}\) need not be a connected surface. Of course, \({\cal S}\) need not be a connected surface.
</p> </p>
<p>
This is obvious: we know where all the charges are, so this is really the same as the first case.
</p>
<p> <p>
<b>Second case</b>: the electrostatic potential is uniquely determined <b>Charge density in closed volume, and potential at boundary are known</b>: the electrostatic potential is uniquely determined
in a certain volume \({\cal V}\) bounded by boundary \({\cal S}\), provided the charge density in a certain volume \({\cal V}\) bounded by boundary \({\cal S}\), provided the charge density
\(\rho ({\bf x})\) is given everywhere within \({\cal V}\), and the value of \(V\) is given everywhere on the \(\rho ({\bf x})\) is given everywhere within \({\cal V}\), and the value of \(V\) is given everywhere on the
boundary \({\cal S}\). Of course, \({\cal S}\) need not be a connected surface. boundary \({\cal S}\). Of course, \({\cal S}\) need not be a connected surface.
@ -1748,14 +2010,20 @@ to obtain \(V\) within \({\cal V}\).
Given a solution \(V_1 ({\bf r})\), we can easily show that it is unique. Suppose there was another solution Given a solution \(V_1 ({\bf r})\), we can easily show that it is unique. Suppose there was another solution
\(V_2 ({\bf r})\). Look at the difference, \(U \equiv V_1 - V_2\). In the bulk, \(U\) obeys the Laplace \(V_2 ({\bf r})\). Look at the difference, \(U \equiv V_1 - V_2\). In the bulk, \(U\) obeys the Laplace
equation equation
</p>
<p>
\[ \[
{\boldsymbol \nabla}^2 U = {\boldsymbol \nabla}^2 V_1 - {\boldsymbol \nabla}^2 V_2 = -\frac{\rho}{\varepsilon_0} + \frac{\rho}{\varepsilon_0} = 0. {\boldsymbol \nabla}^2 U = {\boldsymbol \nabla}^2 V_1 - {\boldsymbol \nabla}^2 V_2 = -\frac{\rho}{\varepsilon_0} + \frac{\rho}{\varepsilon_0} = 0.
\] \]
</p>
<p>
Moreover, \(U ({\bf r}) = 0\) for \({\bf r} \in {\cal S}\). Since solutions to the Laplace equation take Moreover, \(U ({\bf r}) = 0\) for \({\bf r} \in {\cal S}\). Since solutions to the Laplace equation take
their maximal and minimal value on the boundary, we must have \(U = 0\) \(\forall {\bf r} \in {\cal V}\) their maximal and minimal value on the boundary, we must have \(U = 0\) \(\forall {\bf r} \in {\cal V}\)
(Griffiths' proof).
</p> </p>
<p> <p>
This all feels a bit amateurish and not very systematic. Can we be more precise and general? What kinds of boundary information do we really need to specify the solution uniquely ? This all feels a bit amateurish and not very systematic. Can we be more precise and general? What kinds of boundary information do we really need to specify the solution uniquely ?
</p> </p>
@ -1765,6 +2033,8 @@ This all feels a bit amateurish and not very systematic. Can we be more precise
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe_g.html">Green's Identities&emsp;<small>[ems.ca.fe.g]</small></a></li><li>Up:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1778,7 +2048,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

54
build/ems_ca_fe_g.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html> <!DOCTYPE html>
<html lang="en"> <html lang="en">
<head> <head>
<!-- 2022-02-10 Thu 08:32 --> <!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8"> <meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title> <title>Pre-Quantum Electrodynamics</title>
@ -1597,13 +1597,13 @@ Table of contents
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/> <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a><span class="headline-id">ems.ca.fe.g</span></h5> </svg></a><span class="headline-id">ems.ca.fe.g</span></h5>
<div class="outline-text-5" id="text-ems_ca_fe_g"> <div class="outline-text-5" id="text-ems_ca_fe_g">
<div class="info div" id="org5bc5392"> <div class="info div" id="orgc7ef55c">
<p> <p>
<b>George Green</b> <b>George Green</b>
</p> </p>
<aside id="orgb80e217"> <aside id="org5a19053">
<p> <p>
See a \href{short biography of George Green on Wikipedia}{https://en.wikipedia.org/wiki/George\_Green\_(mathematician)}. See a <a href="https://en.wikipedia.org/wiki/George%5C_Green%5C_(mathematician)">short bio on wikipedia</a>
</p> </p>
</aside> </aside>
<p> <p>
@ -1632,19 +1632,53 @@ and
\[ \[
\phi {\boldsymbol \nabla} \psi \cdot {\bf n} = \phi \frac{\partial \psi}{\partial n}. \phi {\boldsymbol \nabla} \psi \cdot {\bf n} = \phi \frac{\partial \psi}{\partial n}.
\] \]
Substituting this in the divergence theorem gives {\bf Green's first identity} Substituting this in the divergence theorem gives <b>Green's first identity</b>
</p>
<div class="eqlabel" id="org713a4b7">
<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="orgfb4fc39">
<ul class="org-ul">
<li>J (1.34)</li>
</ul>
</div>
</div>
<p>
\[ \[
\int_{\cal V} d\tau ~(\phi {\boldsymbol \nabla}^2 \psi + {\boldsymbol \nabla} \phi \cdot {\boldsymbol \nabla} \psi) = \oint_{\cal S} da ~\phi \frac{\partial \psi}{\partial n}. \int_{\cal V} d\tau ~(\phi {\boldsymbol \nabla}^2 \psi + {\boldsymbol \nabla} \phi \cdot {\boldsymbol \nabla} \psi) = \oint_{\cal S} da ~\phi \frac{\partial \psi}{\partial n}.
\label{eq:GreensFirstIdentity} \tag{Green1}\label{Green1}
\] \]
This first identity will prove crucial in the argument that follows. This first identity will prove crucial in the argument that follows.
As an aside for now, for completeness, if we do the same thing again but with \(\phi\) and \(\psi\) As an aside for now, for completeness, if we do the same thing again but with \(\phi\) and \(\psi\)
interchanged, and subtract the result, we obtain another useful result known as interchanged, and subtract the result, we obtain another useful result known as
{\bf Green's second identity} or {\bf Green's theorem} <b>Green's second identity</b> or <b>Green's theorem</b>
</p>
<div class="eqlabel" id="org6298bab">
<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="org29adc56">
<ul class="org-ul">
<li>J (1.35)</li>
</ul>
</div>
</div>
<p>
\[ \[
\int_{\cal V} d\tau (\phi {\boldsymbol \nabla}^2 \psi - \psi {\boldsymbol \nabla}^2 \phi) \int_{\cal V} d\tau (\phi {\boldsymbol \nabla}^2 \psi - \psi {\boldsymbol \nabla}^2 \phi)
= \oint_{\cal S} da \left(\phi \frac{\partial \psi}{\partial n} - \psi \frac{\partial \phi}{\partial n} \right). = \oint_{\cal S} da \left(\phi \frac{\partial \psi}{\partial n} - \psi \frac{\partial \phi}{\partial n} \right).
\label{eq:GreensTheorem} \tag{Green2}\label{Green2}
\] \]
</p> </p>
</div> </div>
@ -1653,6 +1687,8 @@ interchanged, and subtract the result, we obtain another useful result known as
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca_fe_L.html">The Laplace Equation&emsp;<small>[ems.ca.fe.L]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe_uP.html">Uniqueness of Solution to Poisson's Equation&emsp;<small>[ems.ca.fe.uP]</small></a></li><li>Up:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li></ul>
<br>
<hr> <hr>
<div class="license"> <div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/" <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@ -1666,7 +1702,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p> <p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p> <p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p> <p class="validation"></p>
</div> </div>

Some files were not shown because too many files have changed in this diff Show More