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<!DOCTYPE html>
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<meta charset="utf-8">
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<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>
</ul>
<br><ul class="navigation-links"><li>Next:&nbsp;<a href="a_l.html">Literature&emsp;<small>[a.l]</small></a></li></ul>
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@ -1619,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -1689,6 +1689,8 @@ Wiley, 1986 [2nd edition]
</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>
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@ -1702,7 +1704,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -1606,6 +1606,8 @@ Table of contents
<li><a href="c_m.html">Mathematics</a><span class="headline-id">c.m</span></li>
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@ -1619,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<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>
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build/c_m.html
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@ -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>
</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>
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@ -1625,7 +1627,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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build/c_m_cs.html
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<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>
</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>
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@ -1621,7 +1623,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<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>
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build/c_m_cs_cyl.html
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<!DOCTYPE html>
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<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">
<h6 id="c_m_cs_cyl_grad"><a href="#c_m_cs_cyl_grad">Gradient</a></h6>
<div class="outline-text-6" id="text-c_m_cs_cyl_grad">
<div class="eqlabel" id="orgbec1efd">
<div class="eqlabel" id="org03db374">
<p>
<a id="cylgrad"></a><a href="./c_m_cs_cyl.html#cylgrad"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org21e7bea">
<div class="alteqlabels" id="orgb61510b">
<ul class="org-ul">
<li>Gr4(1.79)</li>
</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">
<h6 id="c_m_cs_cyl_div"><a href="#c_m_cs_cyl_div">Divergence</a></h6>
<div class="outline-text-6" id="text-c_m_cs_cyl_div">
<div class="eqlabel" id="org59ba5e0">
<div class="eqlabel" id="orgd9435d5">
<p>
<a id="cyl_div"></a><a href="./c_m_cs_cyl.html#cyl_div"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgd4f5fe9">
<div class="alteqlabels" id="org5c1c30c">
<ul class="org-ul">
<li>Gr4(2.21)</li>
</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">
<h6 id="c_m_cs_cyl_curl"><a href="#c_m_cs_cyl_curl">Curl</a></h6>
<div class="outline-text-6" id="text-c_m_cs_cyl_curl">
<div class="eqlabel" id="orgc77e287">
<div class="eqlabel" id="org695c029">
<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">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org808a8dd">
<div class="alteqlabels" id="orge332907">
<ul class="org-ul">
<li>Gr4(2.21)</li>
</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>
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@ -1753,7 +1755,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<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>
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build/c_m_cs_hyp.html
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@ -1601,6 +1601,8 @@ Table of contents
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<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>
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@ -1614,7 +1616,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<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>
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build/c_m_cs_sph.html
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@ -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>
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@ -1727,7 +1729,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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build/c_m_dc.html
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@ -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>
</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>
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@ -1624,7 +1626,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<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>
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build/c_m_dc_curl.html
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<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>
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@ -1626,7 +1628,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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36
build/c_m_dc_d2.html
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@ -1599,9 +1599,9 @@ Table of contents
<div class="outline-text-5" id="text-c_m_dc_d2">
</div>
<div id="outline-container-orgd5309ec" class="outline-6">
<h6 id="orgd5309ec"><a href="#orgd5309ec">Divergence of gradient</a></h6>
<div class="outline-text-6" id="text-orgd5309ec">
<div id="outline-container-orgbe5f73d" class="outline-6">
<h6 id="orgbe5f73d"><a href="#orgbe5f73d">Divergence of gradient</a></h6>
<div class="outline-text-6" id="text-orgbe5f73d">
<p>
\({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
@ -1610,36 +1610,36 @@ given by the Laplacian of the corresponding vector elements.
</div>
</div>
<div id="outline-container-org3f8f01e" class="outline-6">
<h6 id="org3f8f01e"><a href="#org3f8f01e">Curl of a gradient</a></h6>
<div class="outline-text-6" id="text-org3f8f01e">
<div id="outline-container-orgee1ca6a" class="outline-6">
<h6 id="orgee1ca6a"><a href="#orgee1ca6a">Curl of a gradient</a></h6>
<div class="outline-text-6" id="text-orgee1ca6a">
<p>
This always vanishes.
</p>
</div>
</div>
<div id="outline-container-org86a1e1f" class="outline-6">
<h6 id="org86a1e1f"><a href="#org86a1e1f">Gradient of the divergence</a></h6>
<div class="outline-text-6" id="text-org86a1e1f">
<div id="outline-container-org2789da2" class="outline-6">
<h6 id="org2789da2"><a href="#org2789da2">Gradient of the divergence</a></h6>
<div class="outline-text-6" id="text-org2789da2">
<p>
\({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.
</p>
</div>
</div>
<div id="outline-container-orgdca9d9a" class="outline-6">
<h6 id="orgdca9d9a"><a href="#orgdca9d9a">Divergence of a curl</a></h6>
<div class="outline-text-6" id="text-orgdca9d9a">
<div id="outline-container-org62ec566" class="outline-6">
<h6 id="org62ec566"><a href="#org62ec566">Divergence of a curl</a></h6>
<div class="outline-text-6" id="text-org62ec566">
<p>
This always vanishes.
</p>
</div>
</div>
<div id="outline-container-org613cb02" class="outline-6">
<h6 id="org613cb02"><a href="#org613cb02">Curl of curl</a></h6>
<div class="outline-text-6" id="text-org613cb02">
<div id="outline-container-org5cd2844" class="outline-6">
<h6 id="org5cd2844"><a href="#org5cd2844">Curl of curl</a></h6>
<div class="outline-text-6" id="text-org5cd2844">
<p>
\[
{\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>
<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>
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@ -1664,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p>
</div>

6
build/c_m_dc_del.html
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<title>Pre-Quantum Electrodynamics</title>
@ -1612,6 +1612,8 @@ The <b>del/grad/nabla operator</b> is defined as
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@ -1625,7 +1627,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -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>
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@ -1622,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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<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>
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@ -1646,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -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>
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@ -1619,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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<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>
</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>
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@ -1621,7 +1623,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -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>
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@ -1633,7 +1635,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p>
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31
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<title>Pre-Quantum Electrodynamics</title>
@ -1620,14 +1620,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
More generally,
</p>
<div class="eqlabel" id="org16a4f22">
<div class="eqlabel" id="org28e4fae">
<p>
<a id="divdel"></a><a href="./c_m_dd_3d.html#divdel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgf96ad98">
<div class="alteqlabels" id="org3ed5b35">
<ul class="org-ul">
<li>Gr (1.100)</li>
</ul>
@ -1646,14 +1646,14 @@ More generally,
Since
</p>
<div class="eqlabel" id="org71f92c6">
<div class="eqlabel" id="org51c5e82">
<p>
<a id="div1or"></a><a href="./c_m_dd_3d.html#div1or"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="org39f00df">
<div class="alteqlabels" id="org725ab8d">
<ul class="org-ul">
<li>Gr (1.101)</li>
</ul>
@ -1669,15 +1669,32 @@ Since
<p>
we have that
</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}
{\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}
</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>
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@ -1691,7 +1708,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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<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>
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@ -1639,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -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>
</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>
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@ -1624,7 +1626,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -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>
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@ -1622,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<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>
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@ -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>
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@ -1631,7 +1633,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p>
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@ -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>
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@ -1625,7 +1627,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<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>
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@ -1619,6 +1619,8 @@ or in other words
</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>
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@ -1632,7 +1634,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<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>
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@ -1599,9 +1599,9 @@ Table of contents
<div class="outline-text-5" id="text-c_m_ic_lsv">
</div>
<div id="outline-container-org3f3c8c7" class="outline-6">
<h6 id="org3f3c8c7"><a href="#org3f3c8c7">Line Integrals</a></h6>
<div class="outline-text-6" id="text-org3f3c8c7">
<div id="outline-container-orgdc409a4" class="outline-6">
<h6 id="orgdc409a4"><a href="#orgdc409a4">Line Integrals</a></h6>
<div class="outline-text-6" id="text-orgdc409a4">
<p>
\[
{\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 id="outline-container-orgbda984c" class="outline-6">
<h6 id="orgbda984c"><a href="#orgbda984c">Surface Integrals</a></h6>
<div class="outline-text-6" id="text-orgbda984c">
<div id="outline-container-org465c39d" class="outline-6">
<h6 id="org465c39d"><a href="#org465c39d">Surface Integrals</a></h6>
<div class="outline-text-6" id="text-org465c39d">
<p>
\[
\int_{\cal S} {\bf v} \cdot d{\bf a}
@ -1652,9 +1652,9 @@ Over a closed surface:
</div>
</div>
<div id="outline-container-orgc3891db" class="outline-6">
<h6 id="orgc3891db"><a href="#orgc3891db">Volume Integrals</a></h6>
<div class="outline-text-6" id="text-orgc3891db">
<div id="outline-container-org794c5b9" class="outline-6">
<h6 id="org794c5b9"><a href="#org794c5b9">Volume Integrals</a></h6>
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<p>
\[
\int_{\cal V} T d\tau
@ -1680,6 +1680,8 @@ d\tau = dx dy dz
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@ -1597,14 +1597,14 @@ Table of contents
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@ -1634,6 +1634,8 @@ the boundary shrinks to a point.
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@ -1608,6 +1608,8 @@ Table of contents
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@ -1621,6 +1621,8 @@ Table of contents
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@ -1621,6 +1621,8 @@ Table of contents
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@ -1624,6 +1624,8 @@ Table of contents
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@ -1610,6 +1610,8 @@ Table of contents
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@ -1639,6 +1639,8 @@ this relation making plain that
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@ -1634,6 +1634,8 @@ Component notation (in \({\mathbb R}^3\) with cartesian coordinates):
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@ -1668,6 +1668,8 @@ Separation vector: for two position vectors \({\bf r}_a, {\bf r}_b\),
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@ -1634,6 +1634,8 @@ In a general coordinate system with metric \(g\),
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@ -1681,6 +1681,8 @@ All higher vector products can be reduced to combinations of single vector produ
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@ -1607,6 +1607,8 @@ Table of contents
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@ -1626,6 +1626,8 @@ where
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@ -1653,6 +1653,8 @@ The following conditions are equivalent:
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@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">d</span></h2>
<div class="outline-text-2" id="text-d">
<details class="objectives" id="orge321671">
<summary id="orgc02d8c9">
<details class="objectives" id="orgece4d0a">
<summary id="org51b47eb">
Objectives
</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>
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<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>
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<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 id="postamble" class="status">
<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>
</div>

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@ -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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

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@ -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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

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<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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

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build/d_ems.html
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<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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

6
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<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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

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@ -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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

6
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<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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

6
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<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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

6
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<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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

6
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@ -1,7 +1,7 @@
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<title>Pre-Quantum Electrodynamics</title>
@ -1630,6 +1630,8 @@ As a strict minimum, you should be able to:
</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>
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<hr>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

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<meta charset="utf-8">
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<title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">emd</span></h2>
<div class="outline-text-2" id="text-emd">
<details class="prereq" id="org09895e9">
<summary id="org4a0bf93">
<details class="prereq" id="org8c17216">
<summary id="org07b9266">
Prerequisites
</summary>
<ul class="org-ul">
@ -1608,8 +1608,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="orgef3f98e">
<summary id="org1810cca">
<details class="objectives" id="org1cf61f0">
<summary id="org2c97d99">
Objectives
</summary>
<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>
</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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

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<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>
</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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

12
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<head>
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<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
along the wire. The line integral of this field is called the
</p>
<div class="core div" id="orgee21897">
<div class="core div" id="org60c9879">
<p>
<b>Electromotive force (or electromotance)</b>,
\[
@ -1660,7 +1660,7 @@ to the rate of change of the magnetic flux,
\]
so we obtain
</p>
<div class="core div" id="orgd1f18df">
<div class="core div" id="org02e64db">
<p>
<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
</p>
<div class="core div" id="org04bf1f1">
<div class="core div" id="org3c09004">
<p>
<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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

10
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
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<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
</p>
<div class="core div" id="org183e158">
<div class="core div" id="org0ecd8ee">
<p>
\[
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)}
\end{align}
<div class="example div" id="org5a2dc32">
<div class="example div" id="org46b2091">
<p>
\paragraph{Example 7.13:} coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\).
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>
<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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

12
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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
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<meta charset="utf-8">
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<title>Pre-Quantum Electrodynamics</title>
@ -1637,7 +1637,7 @@ M_{12} = M_{21}
\]
</p>
<div class="example div" id="orgb4beaba">
<div class="example div" id="orgc3b71d6">
<p>
\paragraph{Example 7.10:}
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>
<div class="example div" id="org24edf9c">
<div class="example div" id="orgd00f4d4">
<p>
\paragraph{Example 7.11:} find self-inductance of toroidal coil with
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}.
</p>
<div class="example div" id="org0b14b9f">
<div class="example div" id="org7ae308f">
<p>
\paragraph{Example 7.12:} circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\).
What is the current ?
@ -1738,6 +1738,8 @@ where \(\tau \equiv L/R\) is the {\bf time constant} of the circuit.
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@ -1751,7 +1753,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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<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>
{\bf Example 7.7:}
\({\bf B}(t)\) 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 class="example div" id="orgd8d7c07">
<div class="example div" id="org655e44f">
<p>
{\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 \(a &lt; b\),
@ -1671,7 +1671,7 @@ called the {\bf quasistatic} approximation, and works provided we deal with
'slow enough' phenomena.
</p>
<div class="example div" id="org1188389">
<div class="example div" id="org0cc8dcd">
<p>
{\bf Example 7.9:} infinitely long straight wire carries \(I(t)\). Find
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>
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@ -1717,7 +1719,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -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>
</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>
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<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 id="postamble" class="status">
<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>
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@ -1600,7 +1600,7 @@ Table of contents
<p>
Full set of equations for the electromagnetic field:
</p>
<div class="core div" id="org8953ea9">
<div class="core div" id="org02575df">
<p>
{\bf Maxwell's equations} {\it (in vacuum)}
</p>
@ -1616,7 +1616,7 @@ Full set of equations for the electromagnetic field:
<p>
Complement:
</p>
<div class="core div" id="org04de767">
<div class="core div" id="org67bee1c">
<p>
{\bf Force law}
\[
@ -1640,7 +1640,7 @@ take divergence of \((iv)\).
<p>
Better way of writing: all fields on left, all sources on right,
</p>
<div class="core div" id="org8800b57">
<div class="core div" id="org95e5ef3">
\begin{align}
(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, \\
@ -1655,6 +1655,8 @@ Better way of writing: all fields on left, all sources on right,
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@ -1668,7 +1670,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -1607,7 +1607,7 @@ the continuity equation as
\]
The extra term would thus be eliminated if we were to put
</p>
<div class="core div" id="orgba15335">
<div class="core div" id="org2b447e1">
<p>
\[
{\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>
Maxwell baptized this term the
</p>
<div class="core div" id="org79a35ce">
<div class="core div" id="org10e929a">
<p>
{\bf Displacement current}
\[
@ -1664,6 +1664,8 @@ Flat surface: OK, \(E = 0\) and \(I_{\mbox{enc}} = I\). Balloon surface: \(I
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@ -1677,7 +1679,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<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>
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@ -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>
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@ -1647,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -1632,6 +1632,8 @@ Maxwell's equations {\bf beg} for magnetic charges. But we've never found any!
</div>
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@ -1645,7 +1647,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">emd.ce</span></h3>
<div class="outline-text-3" id="text-emd_ce">
<details class="prereq" id="orga7642d4">
<summary id="org89bea54">
<details class="prereq" id="orgeb45464">
<summary id="org2f2d9c5">
Prerequisites
</summary>
<ul class="org-ul">
@ -1607,8 +1607,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="orgf85a819">
<summary id="orgb7f774b">
<details class="objectives" id="org9e5041f">
<summary id="orgbcd06e1">
Objectives
</summary>
<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>
</ul>
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@ -1644,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<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>
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@ -1600,7 +1600,7 @@ Table of contents
<p>
The angular momentum of EM fields is directly given by
</p>
<div class="main div" id="orgbedab67">
<div class="main div" id="org76c33ae">
<p>
{\bf Angular momentum of EM fields}
\[
@ -1615,6 +1615,8 @@ The angular momentum of EM fields is directly given by
</div>
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@ -1628,7 +1630,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -1619,7 +1619,7 @@ This means that
\]
Since this is true for any volume, we have (re)derived the
</p>
<div class="core div" id="org8328070">
<div class="core div" id="org9d47204">
<p>
{\bf Continuity equation}
\[
@ -1645,6 +1645,8 @@ imposes a functional constraint on these sources: not {\it any} \(\rho\) and
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@ -1658,7 +1660,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -1612,7 +1612,7 @@ in which the first integral can be interpreted as the momentum stored in the EM
<p>
This is thus simply a conservation law for momentum, with
</p>
<div class="main div" id="orgfb9b180">
<div class="main div" id="org75dce9f">
<p>
{\bf Momentum density in the EM fields}
\[
@ -1624,7 +1624,7 @@ This is thus simply a conservation law for momentum, with
<p>
In a region in which the mechanical momentum is not changing due to external influences, we then have the
</p>
<div class="main div" id="org03df6a3">
<div class="main div" id="orgeb01a5d">
<p>
{\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
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@ -1651,7 +1653,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
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@ -1654,7 +1654,7 @@ and similarly for \({\boldsymbol B}\). We thus get
<p>
This expression can be greatly simplified by introducing the
</p>
<div class="main div" id="org7f141f8">
<div class="main div" id="orgcd90456">
<p>
{\bf Maxwell stress tensor}
\[
@ -1677,7 +1677,7 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
<p>
We then obtain
</p>
<div class="main div" id="org6e156bd">
<div class="main div" id="orgbba46d7">
<p>
{\bf EM force per unit volume}
\[
@ -1689,7 +1689,7 @@ We then obtain
<p>
where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
</p>
<div class="main div" id="org7c98547">
<div class="main div" id="org9e182ba">
<p>
{\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>
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@ -1716,7 +1718,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -1666,7 +1666,7 @@ so we get
Substituting this in \ref{Gr(8.6)} and using the divergence theorem,
we obtain
</p>
<div class="main div" id="orgab4b619">
<div class="main div" id="org1ad9e6b">
<p>
{\bf Poynting's theorem}
\[
@ -1691,7 +1691,7 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
<p>
Energy per unit time, per unit area carried by EM fields:
</p>
<div class="core div" id="org9cfe7aa">
<div class="core div" id="org97f8866">
<p>
{\bf Poynting vector}
\[
@ -1704,7 +1704,7 @@ Energy per unit time, per unit area carried by EM fields:
<p>
We can thus express Poynting's theorem more compactly:
</p>
<div class="core div" id="orgdeee59b">
<div class="core div" id="orgc096fa3">
<p>
{\bf Poynting's theorem}
\[
@ -1717,7 +1717,7 @@ We can thus express Poynting's theorem more compactly:
<p>
where we have defined the total
</p>
<div class="core div" id="org57db6a8">
<div class="core div" id="org56f314c">
<p>
{\bf Energy in electromagnetic fields}
\[
@ -1740,7 +1740,7 @@ Then,
\]
so we get the
</p>
<div class="core div" id="org65ae443">
<div class="core div" id="orgd477ba8">
<p>
{\bf Poynting theorem (differential form)}
\[
@ -1757,7 +1757,7 @@ and has a similar for to the continuity equation
<div class="example div" id="org0e3381a">
<div class="example div" id="orgf36d646">
<p>
\paragraph{Example 8.1} Current in a wire: Joule heating. Energy per unit time delivered to wire: from Poynting.
Assuming that the field is uniform, the electric field parallel to the wire is
@ -1787,6 +1787,8 @@ and the value is as expected.
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<p class="author">Author: Jean-Sébastien Caux</p>
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@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">emd.emw</span></h3>
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<summary id="org4571516">
Prerequisites
</summary>
<ul class="org-ul">
@ -1608,8 +1608,8 @@ Prerequisites
</ul>
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Objectives
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@ -1635,6 +1635,8 @@ Objectives
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@ -1628,7 +1628,7 @@ so for a monochromatic EM plan wave,
\]
or more succinctly:
</p>
<div class="main div" id="org7992b89">
<div class="main div" id="org58ea3a9">
<p>
{\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>
Similary, we get the
</p>
<div class="main div" id="org0141dca">
<div class="main div" id="org9272ec4">
<p>
{\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>
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@ -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
\({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
</p>
<div class="core div" id="org0336f47">
<div class="core div" id="org19a2b1c">
<p>
{\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 \(\delta\)):
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@ -1626,7 +1626,7 @@ These take the form of coupled first-order partial differential equations for \(
Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
we get the
</p>
<div class="core div" id="org832615b">
<div class="core div" id="org3cd266d">
<p>
{\bf Wave equations for electric and magnetic fields in vacuum}
\[
@ -1667,6 +1667,8 @@ the actual electric and magnetic fields are given by the real part.
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@ -1633,7 +1633,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
\]
We therefore have the
</p>
<div class="core div" id="orga19bc43">
<div class="core div" id="orgc16990f">
<p>
{\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
associated to the conservation of bound charges:
</p>
<aside id="org7492618">
<aside id="orgbe14b68">
<p>
Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current
\({\boldsymbol J}_b\) but this is not the convention used here.
@ -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,
{\it free} and {\it bound}:
</p>
<div class="main div" id="org6458291">
<div class="main div" id="org27840ee">
<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
{\it polarization}:
</p>
<div class="main div" id="org79300fd">
<div class="main div" id="orge8444d0">
<p>
\[
{\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
@ -1711,7 +1711,7 @@ Gauss's law: can be rewritten
\]
where (as in static case)
</p>
<div class="core div" id="org0cf7843">
<div class="core div" id="org3e52dc6">
<p>
\[
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
@ -1737,7 +1737,7 @@ or
\]
where as before
</p>
<div class="core div" id="org4d58c35">
<div class="core div" id="org5550f7c">
<p>
\[
{\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>
In terms of free charges and currents, we thus get
</p>
<div class="core div" id="org4ca5e56">
<div class="core div" id="orge1d07ac">
<p>
{\bf Maxwell's equations {\it (in matter)}}
</p>
@ -1781,7 +1781,7 @@ Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and
in terms of \({\bf E}\) and \({\bf B}\).
For the restricted case of linear media:
</p>
<div class="main div" id="orgac29d47">
<div class="main div" id="org83b4dbd">
<p>
\[
{\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
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@ -1600,7 +1600,7 @@ Table of contents
<p>
Discontinuities between different media, deduced from
</p>
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<div class="core div" id="org55029ff">
<p>
{\bf Maxwell's equations {\it (in matter)}, integral form}
</p>
@ -1676,6 +1676,8 @@ These are basis of theory of reflection and refraction.
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@ -1610,6 +1610,8 @@ Table of contents
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@ -1606,6 +1606,8 @@ Table of contents
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@ -1702,6 +1702,8 @@ so the final form of the fields is
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@ -1635,7 +1635,7 @@ v = \frac{1}{\sqrt{\mu \varepsilon}} = \frac{c}{n}
\]
where the index of refraction of the material is defined as
</p>
<div class="main div" id="orgc7bc968">
<div class="main div" id="orga0618f6">
<p>
{\bf Index of refraction}
\[
@ -1677,6 +1677,8 @@ I = \frac{1}{2} \varepsilon v E_0^2
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@ -1690,7 +1692,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<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>
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@ -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>
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@ -1601,6 +1601,8 @@ Table of contents
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<p class="author">Author: Jean-Sébastien Caux</p>
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@ -1602,6 +1602,8 @@ Table of contents
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@ -1615,7 +1617,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<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>
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@ -1693,6 +1693,8 @@ R + T = 1.
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@ -1706,7 +1708,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -1635,7 +1635,7 @@ These forms for incident, reflected and transmitted wave can be substituted in t
<p>
From now on we will orient the axes so that \({\boldsymbol k}_I\) lies in the \(xz\) plane. This means that \({\boldsymbol k}_R\) and \({\boldsymbol k}_T\) also lie in that plane. This is the
</p>
<div class="core div" id="orgb0a9710">
<div class="core div" id="orge8258d6">
<p>
{\bf First law of reflection:}
the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
@ -1650,7 +1650,7 @@ Specializing (\ref{eq:RTObliquek}) to our notations, we have
with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
and the angle of refraction (\(\theta_T\)) obey the following laws:
</p>
<div class="core div" id="orga98e40b">
<div class="core div" id="org823940c">
<p>
{\bf Law of reflection}
\[
@ -1708,7 +1708,7 @@ while the third equation becomes
\]
Writing everything in terms of the incident amplitude, we get
</p>
<div class="main div" id="org521f546">
<div class="main div" id="orgd20a172">
<p>
{\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)}).
For grazing waves \(\theta_I \rightarrow \pi/2\) we have that \(\alpha \rightarrow \infty\) and the wave is totally reflected. The most interesting angle is the one at which \(\alpha = \beta\) and the reflected wave has zero amplitude. This is known as
</p>
<div class="main div" id="org51d503c">
<div class="main div" id="orgd033d51">
<p>
{\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.
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@ -1776,7 +1778,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -1602,6 +1602,8 @@ Table of contents
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<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>
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@ -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>
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<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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6
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@ -1628,6 +1628,8 @@ where \(A\) is a constant amplitude. Substituting and taking the real part,
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@ -1652,6 +1652,8 @@ with an identical equation for \(B_z\). If \(E_z = 0\) the waves are called {\bf
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@ -1661,6 +1661,8 @@ which is {\it greater} than \(c\). The energy of the wave however propagates at
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@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">emf</span></h2>
<div class="outline-text-2" id="text-emf">
<details class="prereq" id="orgc2e1e28">
<summary id="orge525c1b">
<details class="prereq" id="orgd0eb59c">
<summary id="org6131f4a">
Prerequisites
</summary>
<ul class="org-ul">
@ -1607,8 +1607,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="orgcd01d6b">
<summary id="org510a3e2">
<details class="objectives" id="org1db29ae">
<summary id="orgab2baad">
Objectives
</summary>
<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>
</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>
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@ -1642,7 +1644,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
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@ -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>
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@ -1625,6 +1625,8 @@ Although Gauss's law looks nice in the Coulomb gauge, Amp{\`e}re-Maxwell does no
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@ -1613,7 +1613,7 @@ while the equation for \(V\) becomes
\]
These can be written compactly upon introducing a new operator: the
</p>
<div class="core div" id="org9685743">
<div class="core div" id="orgb7b775a">
<p>
{\bf d'Alembertian operator}
\[
@ -1626,7 +1626,7 @@ These can be written compactly upon introducing a new operator: the
<p>
so we get the
</p>
<div class="core div" id="org8c1d592">
<div class="core div" id="org1d4f993">
<p>
{\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
\[
@ -1661,6 +1661,8 @@ we have by direct inspection
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@ -1614,7 +1614,7 @@ Useful strategy: represent fields in terms of potentials.
<p>
Easiest:
</p>
<div class="core div" id="org449388e">
<div class="core div" id="orgea17c1a">
<p>
\[
{\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
@ -1630,7 +1630,7 @@ Putting this into Faraday's law gives
\]
so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
</p>
<div class="core div" id="org8d2221f">
<div class="core div" id="orge40ddf5">
<p>
\[
{\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>
Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting (\ref{eq:E_from_Potentials}) into Gauss's law gives
</p>
<div class="main div" id="orgf979da5">
<div class="main div" id="org73cca63">
<p>
\[
{\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
@ -1659,7 +1659,7 @@ whereas Amp{\`ere}-Maxwell becomes
\]
which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
</p>
<div class="main div" id="orgc475199">
<div class="main div" id="org0170d34">
<p>
\[
\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>
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<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 id="postamble" class="status">
<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>
</div>

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@ -1,7 +1,7 @@
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<meta charset="utf-8">
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<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>
</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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

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@ -1,7 +1,7 @@
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<html lang="en">
<head>
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,8 @@ Table of contents
</svg></a><span class="headline-id">ems.ca</span></h3>
<div class="outline-text-3" id="text-ems_ca">
<details class="prereq" id="orga055812">
<summary id="org2cb689b">
<details class="prereq" id="org0bf319c">
<summary id="orga92f3c7">
Prerequisites
</summary>
<ul class="org-ul">
@ -1607,8 +1607,8 @@ Prerequisites
</ul>
</details>
<details class="objectives" id="orgb006f6c">
<summary id="org037b124">
<details class="objectives" id="org87cd363">
<summary id="org64f736b">
Objectives
</summary>
<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>
</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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-10 Thu 08:32 -->
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<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
from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
</p>
<div class="main div" id="org27034ee">
<div class="main div" id="org98cdfd8">
<p>
</p>
\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}
\end{equation*}
@ -1611,44 +1615,53 @@ from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_v
</div>
<p>
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>
<div class="main div" id="org1620b08">
<div class="main div" id="org12e97ac">
<p>
</p>
<p>
\[
V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|}.
\tag{\ref{eq:V_from_rho}}
\]
\phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|}.
\]
</p>
</div>
<p>
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)
condition (\ref{Gr(2.20)}) can be expressed as the single
'local' (differential) condition (Poisson's equation) (\ref{eq:Poisson})
electrostatic field, Gauss's law <a href="./ems_es_ef_Gl.html#Gl_d">Gl_d</a> and the vanishing curl
condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as the single
<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>
<div class="core div" id="org1361b41">
<p>
\[
{\boldsymbol \nabla}^2 V = -\frac{\rho}{\varepsilon_0}.
\tag{\ref{eq:Poisson}}
\]
{\boldsymbol \nabla}^2 \phi = -\frac{\rho}{\varepsilon_0}.
\]
</p>
</div>
<p>
In the specific case where the charge density vanishes, we fall back onto the simpler
Laplace equation
In the specific case where the charge density vanishes, we fall back onto the simpler Laplace equation <a href="./ems_es_ep_PL.html#Lap">Lap</a>
</p>
<div class="core div" id="org2305746">
<div class="core div" id="orgea7b7e8">
<p>
</p>
<p>
\[
{\boldsymbol \nabla}^2 V = 0
\tag{\ref{eq:Laplace}}
\]
{\boldsymbol \nabla}^2 \phi = 0
\]
</p>
</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>
</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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

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@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-10 Thu 08:32 -->
<!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@ -1598,8 +1598,10 @@ Table of contents
</svg></a><span class="headline-id">ems.ca.fe.L</span></h5>
<div class="outline-text-5" id="text-ems_ca_fe_L">
<p>
Of course, the simplest situation is to start by looking at the region of space
where there is no charge density. The potential then solves Laplace's equation. How can it possibly look ?
In regions of space where there is no charge density,
the potential must solve Laplace's equation.
Let us discuss how solutions to this equation look,
in increasingly complicated situations.
</p>
</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>
<div class="outline-text-6" id="text-ems_ca_fe_L_1d">
<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>
\[
\frac{d^2 V(x)}{dx^2} = 0 \Longrightarrow V(x) = mx + b
\label{Gr(3.6)}
\frac{d^2 \phi(x)}{dx^2} = 0.
\tag{Lap_1d}\label{Lap_1d}
\]
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>
Boundary conditions: always work: two end values, one end value + same end derivative value.
Not always: one end value + derivative value at other end, two end derivative values.
The solution to this
</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>
</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>
<div class="outline-text-6" id="text-ems_ca_fe_L_2d">
<p>
\[
\frac{d^2 V}{dx^2} + \frac{d^2 V}{dy^2} = 0.
\]
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.
In two dimensions, the potential becomes a function
of two variables (here: \(x\) and \(y\)), so Laplace's
equation now reads
</p>
<div class="eqlabel" id="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>
@ -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>
<div class="outline-text-6" id="text-ems_ca_fe_L_3d">
<p>
\[
{\boldsymbol \nabla}^2 V = 0
\]
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.
In three dimensions, we will write the potential
as a function of a 3-dimensional vector, \(\phi({\bf r})\).
The Laplace equation is (we repeat)
</p>
<p>
\paragraph{Another way of seeing this} is to write the second derivatives as
\[
\frac{\partial^2 V({\bf r})}{\partial x^2} = f_x ({\bf r}), \hspace{5mm}
\frac{\partial^2 V({\bf r})}{\partial y^2} = f_y ({\bf r}), \hspace{5mm}
\frac{\partial^2 V({\bf r})}{\partial z^2} = f_z ({\bf r}), \hspace{5mm}
{\boldsymbol \nabla}^2 \phi ({\bf r}) = 0
\]
</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>
\[
\frac{\partial^2 \phi({\bf r})}{\partial x^2} = f_x ({\bf r}), \hspace{5mm}
\frac{\partial^2 \phi({\bf r})}{\partial y^2} = f_y ({\bf r}), \hspace{5mm}
\frac{\partial^2 \phi({\bf r})}{\partial z^2} = f_z ({\bf r}), \hspace{5mm}
f_x + f_y + f_z = 0.
\]
The \(f_a ({\bf r})\) represent the three components of the curvature of \(V({\bf r})\).
An extremum of \(V\) at \({\bf r}_e\) would be characterized by \({\boldsymbol \nabla} V |_{{\bf r}_e} \cdot \delta{\bf r} = 0\)
</p>
<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
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,
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
of the \(f_x + f_y + f_z = 0\) condition above.
</p>
<div class="info div" id="orge499be0">
<div class="info div" id="org950ce9d">
<p>
<b>Earnshaw's theorem</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
with a stable equilibrium position for a test charge.
<b>Earnshaw's theorem (physical version)</b> <br>
It is impossible to find a static distribution of charges which generates an electrostatic field
displaying a stable equilibrium position in empty space.
</p>
</div>
@ -1692,47 +1945,56 @@ Going back to Poisson's equation, we can make a few comments:
</p>
<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>
<p>
We therefore want to ask the question: <b>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.
We therefore want to ask the question: <i>under what conditions can an electrostatic problem be fully
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>
<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:
\[
{\boldsymbol \nabla}^2 V ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') {\boldsymbol \nabla}^2 \frac{1}{|{\bf r} - {\bf r}'|}
= \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' (-4\pi) \delta ({\bf r} - {\bf r}') = -\frac{\rho ({\bf r})}{\varepsilon_0}.
\]
where we have used (\ref{Gr(1.102)}), and the fact that the delta function is always resolved since we
integrate over all space. Note: it is implicitly assumed that the integral in (\ref{eq:V_from_rho})
</p>
<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
become singular).
</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
\(\rho ({\bf x})\) 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 \({\cal S}\).
Of course, \({\cal S}\) need not be a connected surface.
</p>
<p>
This is obvious: we know where all the charges are, so this is really the same as the first case.
</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
\(\rho ({\bf x})\) 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.
@ -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
\(V_2 ({\bf r})\). Look at the difference, \(U \equiv V_1 - V_2\). In the bulk, \(U\) obeys the Laplace
equation
</p>
<p>
\[
{\boldsymbol \nabla}^2 U = {\boldsymbol \nabla}^2 V_1 - {\boldsymbol \nabla}^2 V_2 = -\frac{\rho}{\varepsilon_0} + \frac{\rho}{\varepsilon_0} = 0.
\]
</p>
<p>
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 \(U = 0\) \(\forall {\bf r} \in {\cal V}\)
(Griffiths' proof).
</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 ?
</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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

54
build/ems_ca_fe_g.html
View File

@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-10 Thu 08:32 -->
<!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<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"/>
</svg></a><span class="headline-id">ems.ca.fe.g</span></h5>
<div class="outline-text-5" id="text-ems_ca_fe_g">
<div class="info div" id="org5bc5392">
<div class="info div" id="orgc7ef55c">
<p>
<b>George Green</b>
</p>
<aside id="orgb80e217">
<aside id="org5a19053">
<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>
</aside>
<p>
@ -1632,19 +1632,53 @@ and
\[
\phi {\boldsymbol \nabla} \psi \cdot {\bf n} = \phi \frac{\partial \psi}{\partial n}.
\]
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>
\[
\int_{\cal V} d\tau ~(\phi {\boldsymbol \nabla}^2 \psi + {\boldsymbol \nabla} \phi \cdot {\boldsymbol \nabla} \psi) = \oint_{\cal S} da ~\phi \frac{\partial \psi}{\partial n}.
\label{eq:GreensFirstIdentity}
\tag{Green1}\label{Green1}
\]
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 \(\psi\)
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>
\[
\int_{\cal V} d\tau (\phi {\boldsymbol \nabla}^2 \psi - \psi {\boldsymbol \nabla}^2 \phi)
= \oint_{\cal S} da \left(\phi \frac{\partial \psi}{\partial n} - \psi \frac{\partial \phi}{\partial n} \right).
\label{eq:GreensTheorem}
\tag{Green2}\label{Green2}
\]
</p>
</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>
<div class="license">
<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 id="postamble" class="status">
<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>
</div>

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