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Update 2022-02-21 20:42

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Jean-Sébastien
2022-02-21 20:42:13 +01:00
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-21 Mon 10:33 -->
<!-- 2022-02-21 Mon 20:41 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -706,28 +706,41 @@ Table of contents
</summary>
<ul>
<li>
<a href="./emsm_esm_p.html#emsm_esm_p">Polarization</a><span class="headline-id">emsm.esm.p</span>
</li>
<li>
<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
</li>
<li>
<details>
<summary>
<a href="./emsm_esm_fpo.html#emsm_esm_fpo">The Field of a Polarized Object</a><span class="headline-id">emsm.esm.fpo</span>
<a href="./emsm_esm_mE.html#emsm_esm_mE">Matter Bathed in E Fields; Polarization</a><span class="headline-id">emsm.esm.mE</span>
</summary>
<ul>
<li>
<a href="./emsm_esm_fpo_pibc.html#emsm_esm_fpo_pibc">Physical Interpretation of Bound Charges</a><span class="headline-id">emsm.esm.fpo.pibc</span>
<a href="./emsm_esm_mE_o.html#emsm_esm_mE_o">Overview</a><span class="headline-id">emsm.esm.mE.o</span>
</li>
<li>
<a href="./emsm_esm_fpo_fid.html#emsm_esm_fpo_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.fpo.fid</span>
<a href="./emsm_esm_mE_P.html#emsm_esm_mE_P">Polarization</a><span class="headline-id">emsm.esm.mE.P</span>
</li>
</ul>
</details>
</li>
<li>
<details>
<summary>
<a href="./emsm_esm_po.html#emsm_esm_po">Polarized Objects; Bound Charges</a><span class="headline-id">emsm.esm.po</span>
</summary>
<ul>
<li>
<a href="./emsm_esm_po_pibc.html#emsm_esm_po_pibc">Physical Interpretation of Bound Charges</a><span class="headline-id">emsm.esm.po.pibc</span>
</li>
<li>
<a href="./emsm_esm_po_fid.html#emsm_esm_po_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.po.fid</span>
</li>
@@ -750,18 +763,34 @@ Table of contents
</ul>
</details>
</li>
<li>
<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
</li>
<li>
<details>
<summary>
<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
<a href="./emsm_esm_ld.html#emsm_esm_ld">Linear Dielectrics</a><span class="headline-id">emsm.esm.ld</span>
</summary>
<ul>
<li>
<a href="./emsm_esm_di_ld.html#emsm_esm_di_ld">Linear Dielectrics</a><span class="headline-id">emsm.esm.di.ld</span>
<a href="./emsm_esm_ld_sp.html#emsm_esm_ld_sp">Susceptibility, Permittivity, Dielectric Constant</a><span class="headline-id">emsm.esm.ld.sp</span>
</li>
<li>
<a href="./emsm_esm_ld_bvp.html#emsm_esm_ld_bvp">Boundary Value Problems with Linear Dielectrics</a><span class="headline-id">emsm.esm.ld.bvp</span>
</li>
<li>
<a href="./emsm_esm_ld_e.html#emsm_esm_ld_e">Energy in Dielectric Systems</a><span class="headline-id">emsm.esm.ld.e</span>
</li>
<li>
<a href="./emsm_esm_ld_f.html#emsm_esm_ld_f">Forces on Dielectrics</a><span class="headline-id">emsm.esm.ld.f</span>
</li>
@@ -1598,14 +1627,14 @@ calculated from Coulomb's law using the superposition principle. Since each inf
volume element \(d\tau' = dx' dy' dz'\) contains a charge \(dq' = \rho({\bf r}') d\tau'\), we have
</p>
<div class="eqlabel" id="org5dd289a">
<div class="eqlabel" id="orgf8f5b46">
<p>
<a id="E_vcd"></a><a href="./ems_es_ef_ccd.html#E_vcd"><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="org95f40c5">
<div class="alteqlabels" id="org9e69130">
<ul class="org-ul">
<li>Gr4 (2.8)</li>
</ul>
@@ -1613,7 +1642,7 @@ volume element \(d\tau' = dx' dy' dz'\) contains a charge \(dq' = \rho({\bf r}')
</div>
</div>
<div class="main div" id="org9e06dd7">
<div class="main div" id="org339b083">
<p>
</p>
@@ -1633,14 +1662,14 @@ Similarly, if the charge is spread out over a two-dimensional surface \({\cal S}
\(\sigma({\bf r})\), we have over an infinitesimal area \(da'\) a charge \(dq' = \sigma({\bf r}') da'\), so
</p>
<div class="eqlabel" id="orge696187">
<div class="eqlabel" id="orgafdc565">
<p>
<a id="E_scd"></a><a href="./ems_es_ef_ccd.html#E_scd"><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="org1eaf297">
<div class="alteqlabels" id="org93c3527">
<ul class="org-ul">
<li>Gr4(2.7)</li>
</ul>
@@ -1648,7 +1677,7 @@ Similarly, if the charge is spread out over a two-dimensional surface \({\cal S}
</div>
</div>
<div class="main div" id="org10d1c89">
<div class="main div" id="org3a4c97a">
<p>
</p>
@@ -1665,14 +1694,14 @@ Similarly, if the charge is spread out over a two-dimensional surface \({\cal S}
Finally, for a line path \({\cal P}\) with linear charge density \(\lambda({\bf r}')\),
</p>
<div class="eqlabel" id="org62aff9d">
<div class="eqlabel" id="orgaa31dee">
<p>
<a id="E_lcd"></a><a href="./ems_es_ef_ccd.html#E_lcd"><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="org472d25c">
<div class="alteqlabels" id="org0653e7e">
<ul class="org-ul">
<li>Gr (2.6)</li>
</ul>
@@ -1680,7 +1709,7 @@ Finally, for a line path \({\cal P}\) with linear charge density \(\lambda({\bf
</div>
</div>
<div class="main div" id="orgbef4f59">
<div class="main div" id="org77d757c">
<p>
</p>
@@ -1694,7 +1723,7 @@ Finally, for a line path \({\cal P}\) with linear charge density \(\lambda({\bf
</div>
<div class="example div" id="orgdccabd6">
<div class="example div" id="org8d8dd55">
<p>
<b>Example</b>
</p>
@@ -1728,7 +1757,7 @@ most easily by observing that \(\frac{d}{dx} \left( \frac{x}{\sqrt{z^2 + x^2}} \
= \frac{1}{\sqrt{z^2 + x^2}} - \frac{x^2}{(z^2 + x^2)^{3/2}} = \frac{z^2}{(z^2 + x^2)^{3/2}}\),
leading to
</p>
<aside id="org1062a5f">
<aside id="org8c89679">
<p>
You could alternately proceed by using changes of variables \(y = zx\) followed by \(y = \tanh \alpha\):
\(\int_{-L}^L \frac{dx}{(z^2 + x^2)^{3/2}} = \frac{1}{z^2}
@@ -1785,7 +1814,7 @@ 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: 2022-02-21 Mon 10:33</p>
<p class="date">Created: 2022-02-21 Mon 20:41</p>
<p class="validation"></p>
</div>