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<!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 class="toc-currentpage">
<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 class="toc-currentpage">
<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
</li>
<li>
<details open="">
<summary class="toc-currentpage">
<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
<details>
<summary>
<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>
@@ -1586,22 +1615,62 @@ Table of contents
</ul>
</details>
</nav>
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emsm.html">Electromagnetostatics in matter</a></li><li><a class="breadcrumb-link"href="emsm_esm.html">Electrostatics in matter</a></li><li>Dielectrics</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emsm_esm_p.html">Polarization&emsp;<small>[emsm.esm.p]</small></a></li><li>Next:&nbsp;<a href="emsm_esm_fpo.html">The Field of a Polarized Object&emsp;<small>[emsm.esm.fpo]</small></a></li><li>Up:&nbsp;<a href="emsm_esm.html">Electrostatics in matter&emsp;<small>[emsm.esm]</small></a></li></ul>
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emsm.html">Electromagnetostatics in matter</a></li><li><a class="breadcrumb-link"href="emsm_esm.html">Electrostatics in matter</a></li><li>Dielectrics</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emsm_esm_D_bc.html">Boundary Conditions&emsp;<small>[emsm.esm.D.bc]</small></a></li><li>Next:&nbsp;<a href="emsm_esm_ld.html">Linear Dielectrics&emsp;<small>[emsm.esm.ld]</small></a></li><li>Up:&nbsp;<a href="emsm_esm.html">Electrostatics in matter&emsp;<small>[emsm.esm]</small></a></li></ul><div id="outline-container-emsm_esm_di" class="outline-4">
<h4 id="emsm_esm_di">Dielectrics<a class="headline-permalink" href="./emsm_esm_di.html#emsm_esm_di"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a><span class="headline-id">emsm.esm.di</span></h4>
<div class="outline-text-4" id="text-emsm_esm_di">
<p>
Let's quickly pause to give a classification of the types of responses that might be expected from
materials.
</p>
<p>
<b>Permanent polarization</b>: here, \({\bf P} \neq 0\) even if \({\bf E} = 0\).
These are the <b>electrets</b>.
</p>
<p>
<b>Nonlinear dielectrics</b>: for these, there are non-negligible corrections to the purely linear
behaviour. One can express these nonlinearities through an expansion of the form
\[
P_i = \sum_j \alpha_{ij} E_j + \sum_{jk} \beta_{ijkl} E_j E_k E_l + ...
\]
(even-order terms usually absent, but could be there in principle).
</p>
<p>
<b>Linear dielectrics</b>: for these, the only significant term is the linear term. The most general
form is then
\[
P_i = \varepsilon_0 \chi_{ij} E_j
\]
where \({\bf \chi}\) is the <b>electric susceptibility tensor</b> (it's a tensor of rank 2),
which cannot depend on \({\bf E}\) (otherwise we go back to the nonlinear case).
This is common in crystals, leading to phenomena such as double refraction.
</p>
<p>
<b>Linear isotropic dielectrics</b>: at a given point, the electrical properties of
the dielectric are independent of the direction of \({\bf E}\) (isotropy). Liquids fall into
this category. We here have
</p>
<div class="main div" id="org6bed2ea">
<p>
\[
{\bf P} = \varepsilon_0 \chi_e {\bf E}
\]
</p>
</div>
</div>
</div>
<h5>In this section:</h5>
<ul class="child-links-list">
<li><a href="emsm_esm_di_ld.html">Linear Dielectrics</a><span class="headline-id">emsm.esm.di.ld</span></li>
</ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emsm_esm_p.html">Polarization&emsp;<small>[emsm.esm.p]</small></a></li><li>Next:&nbsp;<a href="emsm_esm_fpo.html">The Field of a Polarized Object&emsp;<small>[emsm.esm.fpo]</small></a></li><li>Up:&nbsp;<a href="emsm_esm.html">Electrostatics in matter&emsp;<small>[emsm.esm]</small></a></li></ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emsm_esm_D_bc.html">Boundary Conditions&emsp;<small>[emsm.esm.D.bc]</small></a></li><li>Next:&nbsp;<a href="emsm_esm_ld.html">Linear Dielectrics&emsp;<small>[emsm.esm.ld]</small></a></li><li>Up:&nbsp;<a href="emsm_esm.html">Electrostatics in matter&emsp;<small>[emsm.esm]</small></a></li></ul>
<br>
<hr>
<div class="license">
@@ -1616,7 +1685,7 @@ 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>