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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>
<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>
@@ -1586,7 +1615,7 @@ 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>The Electric Displacement</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emsm_esm_fpo_fid.html">The Field Inside a Dielectric&emsp;<small>[emsm.esm.fpo.fid]</small></a></li><li>Next:&nbsp;<a href="emsm_esm_D_bc.html">Boundary Conditions&emsp;<small>[emsm.esm.D.bc]</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>The Electric Displacement</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emsm_esm_po_fid.html">The Field Inside a Dielectric&emsp;<small>[emsm.esm.po.fid]</small></a></li><li>Next:&nbsp;<a href="emsm_esm_D_bc.html">Boundary Conditions&emsp;<small>[emsm.esm.D.bc]</small></a></li><li>Up:&nbsp;<a href="emsm_esm.html">Electrostatics in matter&emsp;<small>[emsm.esm]</small></a></li></ul>
<h4 id="emsm_esm_D">The Electric Displacement<a class="headline-permalink" href="./emsm_esm_D.html#emsm_esm_D"><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"/>
@@ -1596,7 +1625,7 @@ Table of contents
<p>
Field due to polarization: effectively comes from bound surface and volume charges,
</p>
<div class="main div" id="org48e839f">
<div class="main div" id="orgd45c81c">
<p>
\[
\rho_b = -{\boldsymbol \nabla} \cdot {\bf P}, \hspace{1cm} \sigma_b = {\bf P} \cdot \hat{\bf n}.
@@ -1605,8 +1634,8 @@ Field due to polarization: effectively comes from bound surface and volume char
</div>
<p>
Rest of charges: {\bf free charge} (electrons in conductors, ions embedded in dielectric,
{\it i.e.} any charge which doesn't come from polarization).
Rest of charges: <b>free charge</b> (electrons in conductors, ions embedded in dielectric,
<i>i.e.</i> any charge which doesn't come from polarization).
</p>
<p>
@@ -1625,59 +1654,119 @@ Convenient way of writing:
\]
Defining the
</p>
<div class="core div" id="orge3cf467">
<div class="core div" id="orgf83cc52">
<p>
<b>Electric displacement</b> \({\bf D}\)
\[
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
\label{Gr(4.21)}
\]
</p>
<div class="eqlabel" id="org3a6db49">
<p>
<a id="D"></a><a href="./emsm_esm_D.html#D"><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="org6817622">
<ul class="org-ul">
<li>Gr (4.21)</li>
</ul>
</div>
</div>
<p>
\[
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
\tag{D}\label{D}
\]
</p>
</div>
<p>
Gauss's law becomes
</p>
<div class="core div" id="orgb7bc0a4">
<div class="core div" id="org19849d8">
<div class="eqlabel" id="org144633a">
<p>
<a id="GlD"></a><a href="./emsm_esm_D.html#GlD"><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="orgcee8169">
<ul class="org-ul">
<li>Gr (4.22)</li>
</ul>
</div>
</div>
<p>
\[
{\boldsymbol \nabla} \cdot {\bf D} = \rho_f
\label{Gr(4.22)}
\]
{\boldsymbol \nabla} \cdot {\bf D} = \rho_f
\tag{GlD}\label{GlD}
\]
</p>
</div>
<p>
or in integral form
</p>
<div class="core div" id="org84b108a">
<div class="core div" id="org08e5fb5">
<div class="eqlabel" id="org6a7a09f">
<p>
<a id="GlD_int"></a><a href="./emsm_esm_D.html#GlD_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgca18aaf">
<ul class="org-ul">
<li>Gr (4.23)</li>
</ul>
</div>
</div>
<p>
\[
\oint d{\bf a} \cdot {\bf D} = Q_{f_{enc}}
\label{Gr(4.23)}
\]
\oint d{\bf a} \cdot {\bf D} = Q_{f_{enc}}
\tag{GlD_int}\label{GlD_int}
\]
</p>
</div>
<div class="example div" id="org3fd517d">
<div class="example div" id="orgd8ebcbd">
<p>
\paragraph{Example 4.4:} long straight wire, uniform line charge \(\lambda\), surrounded by rubber insulation
to radius \(a\). Find \({\bf D}\).
\paragraph{Solution:} cylindrical Gaussian surface, radius \(s\) and length \(L\). Applying \ref{Gr(4.23)},
<b>Example: long insulated wire</b>
</p>
<p>
Consider a long straight wire, with uniform line charge \(\lambda\), surrounded by rubber insulation
to radius \(a\).
</p>
<p>
<b>Task</b>: find \({\bf D}\).
</p>
<p>
<b>Solution</b>: cylindrical Gaussian surface, radius \(s\) and length \(L\). Applying <a href="./emsm_esm_D.html#GlD_int">GlD_int</a>,
\[
D 2\pi s L = \lambda L,
\]
D 2\pi r L = \lambda L,
\]
so
</p>
<p>
\[
{\bf D} = \frac{\lambda}{2\pi s} \hat{\bf s}
\label{Gr(4.24)}
\]
{\bf D} = \frac{\lambda}{2\pi r} \hat{\bf r}
\]
This holds within insulation and outside of it. Outside, \({\bf P} = 0\) so
\[
{\bf E} = \frac{1}{\varepsilon_0} {\bf D} = \frac{\lambda}{2\pi \varepsilon_0 s} \hat{\bf s},
\hspace{1cm} s &gt; a.
\]
{\bf E} = \frac{1}{\varepsilon_0} {\bf D} = \frac{\lambda}{2\pi \varepsilon_0 r} \hat{\bf r},
\hspace{1cm} r &gt; a.
\]
Inside: can't know the electric field, since \({\bf P}\) isn't known.
</p>
@@ -1685,17 +1774,35 @@ Inside: can't know the electric field, since \({\bf P}\) isn't known.
<p>
\paragraph{Note of caution:} while \ref{Gr(4.23)} might lead you to think that
<b>Warning</b>: while <a href="./emsm_esm_D.html#GlD_int">GlD_int</a> might lead you to think that
\[
{\bf D} ({\bf r}) = \frac{1}{4\pi} \int_{\cal V} d\tau' \rho_f ({\bf r}') \frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3}
\]
but this is {\bf not} correct. In particular, the curl of the displacement isn't always zero,
but this is <i>not</i> correct, because the curl of the displacement isn't necessarily zero,
</p>
<div class="eqlabel" id="org18c21e8">
<p>
<a id="curlDcurlP"></a><a href="./emsm_esm_D.html#curlDcurlP"><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="org68e92f6">
<ul class="org-ul">
<li>Gr (4.25)</li>
</ul>
</div>
</div>
<p>
\[
{\boldsymbol \nabla} \times {\bf D} = \varepsilon_0 ({\boldsymbol \nabla} \times {\bf E} + {\boldsymbol \nabla} \times {\bf P})
= {\boldsymbol \nabla} \times {\bf P}
\label{Gr(4.25)}
\tag{curlDcurlP}\label{curlDcurlP}
\]
There is not 'potential' for \({\bf D}\).
There is therefore no 'potential' for \({\bf D}\).
</p>
</div>
@@ -1706,7 +1813,7 @@ There is not 'potential' for \({\bf D}\).
<li><a href="emsm_esm_D_bc.html">Boundary Conditions</a><span class="headline-id">emsm.esm.D.bc</span></li>
</ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emsm_esm_fpo_fid.html">The Field Inside a Dielectric&emsp;<small>[emsm.esm.fpo.fid]</small></a></li><li>Next:&nbsp;<a href="emsm_esm_D_bc.html">Boundary Conditions&emsp;<small>[emsm.esm.D.bc]</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_po_fid.html">The Field Inside a Dielectric&emsp;<small>[emsm.esm.po.fid]</small></a></li><li>Next:&nbsp;<a href="emsm_esm_D_bc.html">Boundary Conditions&emsp;<small>[emsm.esm.D.bc]</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">
@@ -1721,7 +1828,7 @@ 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>