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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-09 Wed 07:31 -->
<!-- 2022-02-09 Wed 22:40 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -408,17 +408,13 @@ Table of contents
<li>
<a href="./ems_es_ep_fp.html#ems_es_ep_fp">Field in terms of the potential</a><span class="headline-id">ems.es.ep.fp</span>
</li>
<li>
<a href="./ems_es_ep_c.html#ems_es_ep_c">Comments on the Electrostatic Potential</a><span class="headline-id">ems.es.ep.c</span>
</li>
<li>
<a href="./ems_es_ep_ex.html#ems_es_ep_ex">Example calculations for the potential</a><span class="headline-id">ems.es.ep.ex</span>
</li>
<li>
<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
<a href="./ems_es_ep_PL.html#ems_es_ep_PL">Poisson's and Laplace's Equations</a><span class="headline-id">ems.es.ep.PL</span>
</li>
<li class="toc-currentpage">
@@ -430,29 +426,8 @@ Table of contents
</details>
</li>
<li>
<details>
<summary>
<a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>
</summary>
<ul>
<li>
<a href="./ems_es_e_pcd.html#ems_es_e_pcd">The Energy of a Point Charge Distribution</a><span class="headline-id">ems.es.e.pcd</span>
</li>
<li>
<a href="./ems_es_e_ccd.html#ems_es_e_ccd">The Energy of a Continuous Charge Distribution</a><span class="headline-id">ems.es.e.ccd</span>
</li>
<li>
<a href="./ems_es_e_c.html#ems_es_e_c">Comments on Electrostatic Energy</a><span class="headline-id">ems.es.e.c</span>
</li>
</ul>
</details>
</li>
<li>
@@ -1616,29 +1591,40 @@ Table of contents
</ul>
</details>
</nav>
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="ems.html">Electromagnetostatics</a></li><li><a class="breadcrumb-link"href="ems_es.html">Electrostatics</a></li><li><a class="breadcrumb-link"href="ems_es_ep.html">The Electrostatic Potential</a></li><li>Electrostatic Boundary Conditions</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_ep_PL.html">The Poisson Equation and the Laplace Equation&emsp;<small>[ems.es.ep.PL]</small></a></li><li>Next:&nbsp;<a href="ems_es_e.html">Electrostatic Energy from the Potential&emsp;<small>[ems.es.e]</small></a></li><li>Up:&nbsp;<a href="ems_es_ep.html">The Electrostatic Potential&emsp;<small>[ems.es.ep]</small></a></li></ul><div id="outline-container-ems_es_ep_bc" class="outline-5">
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="ems.html">Electromagnetostatics</a></li><li><a class="breadcrumb-link"href="ems_es.html">Electrostatics</a></li><li><a class="breadcrumb-link"href="ems_es_ep.html">The Electrostatic Potential</a></li><li>Electrostatic Boundary Conditions</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_ep_PL.html">Poisson's and Laplace's Equations&emsp;<small>[ems.es.ep.PL]</small></a></li><li>Next:&nbsp;<a href="ems_es_e.html">Electrostatic Energy from the Potential&emsp;<small>[ems.es.e]</small></a></li><li>Up:&nbsp;<a href="ems_es_ep.html">The Electrostatic Potential&emsp;<small>[ems.es.ep]</small></a></li></ul><div id="outline-container-ems_es_ep_bc" class="outline-5">
<h5 id="ems_es_ep_bc">Electrostatic Boundary Conditions<a class="headline-permalink" href="./ems_es_ep_bc.html#ems_es_ep_bc"><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><span class="headline-id">ems.es.ep.bc</span></h5>
<div class="outline-text-5" id="text-ems_es_ep_bc">
<p>
For a surface, Gauss's law states
For a surface, Gauss's law gives us that
\[
\oint_{\cal S} {\bf E} \cdot d{\bf a} = \frac{Q_{\mbox{enc}}}{\varepsilon_0} = \frac{1}{\varepsilon_0} \sigma A
\]
where \(A\) is the area of the Gaussian pillbox and \(\sigma\) the surface charge density.
The sides contribute nothing if the pillbox is thin. Taking its area very small, we get
where \(A\) is the area of the Gaussian pillbox and \(\sigma\) the surface charge density,
which is here assumed to be constant.
The sides contribute nothing if the pillbox is made infinitesimally thin, in which case
we get
</p>
<p>
\[
{\bf E}^{\perp}_{\mbox{above}} - {\bf E}^{\perp}_{\mbox{below}} = \frac{\sigma}{\varepsilon_0},
\label{Gr(2.31)}
\]
</p>
<p>
so the normal component of \({\bf E}\) is discontinuous at the boundary by an amount \(\sigma/\varepsilon_0\).
</p>
<p>
The tangential component is continuous: from the curlless condition \ref{Gr(2.19)} applied to
a small loop straddling the surface,
The tangential component is continuous: from the curlless condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a>
applied to a small loop straddling the surface:
</p>
<p>
\[
{\bf E}^{\parallel}_{\mbox{above}} = {\bf E}^{\parallel}_{\mbox{below}}
\label{Gr(2.32)}
@@ -1646,38 +1632,73 @@ a small loop straddling the surface,
</p>
<p>
Put together:
We can unify both boundary conditions into a single equation:
</p>
<div class="eqlabel" id="org17fbc45">
<p>
<a id="Edisc"></a><a href="./ems_es_ep_bc.html#Edisc"><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="org7a21d0c">
<ul class="org-ul">
<li>Gr (2.33)</li>
</ul>
</div>
</div>
<p>
\[
{\bf E}_{\mbox{above}} - {\bf E}_{\mbox{below}} = \frac{\sigma}{\varepsilon_0} \hat{\bf n}
\label{Gr(2.33)}
\tag{Edisc}\label{Edisc}
\]
with \(\hat{\bf n}\) a unit vector normal to the surface, pointing 'out'.
</p>
<p>
The potential is continuous across any boundary: since
The potential is however continuous across any boundary: since
\[
V_{\mbox{above}} - V_{\mbox{below}} = -\int_{\bf a}^{\bf b} {\bf E} \cdot d{\bf l}
\phi_{\mbox{above}} - \phi_{\mbox{below}} = -\int_{\bf a}^{\bf b} {\bf E} \cdot d{\bf l}
\]
where the path shrinks to zero,
where the path shrinks to zero, we conclude that
\[
V_{\mbox{above}} = V_{\mbox{below}}
\phi_{\mbox{above}} = \phi_{\mbox{below}}
\label{Gr(2.34)}
\]
The gradient however inherits the discontinuity of the electrostatic field, since
\({\bf E} = -{\boldsymbol \nabla} V\):
\({\bf E} = -{\boldsymbol \nabla} \phi\):
\[
{\boldsymbol \nabla} V_{\mbox{above}} - {\boldsymbol \nabla} V_{\mbox{below}} = -\frac{\sigma}{\varepsilon_0} \hat{\bf n}
{\boldsymbol \nabla} \phi_{\mbox{above}} - {\boldsymbol \nabla} \phi_{\mbox{below}} = -\frac{\sigma}{\varepsilon_0} \hat{\bf n}
\label{Gr(2.35)}
\]
or
</p>
<div class="eqlabel" id="orgc1e3983">
<p>
<a id="dpdisc"></a><a href="./ems_es_ep_bc.html#dpdisc"><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="org84bec06">
<ul class="org-ul">
<li>Gr (2.36)</li>
</ul>
</div>
</div>
<p>
\[
\frac{\partial V_{\mbox{above}}}{\partial n} - \frac{\partial V_{\mbox{below}}}{\partial n} = -\frac{\sigma}{\varepsilon_0}
\label{Gr(2.36)}
\frac{\partial \phi_{\mbox{above}}}{\partial n} - \frac{\partial \phi_{\mbox{below}}}{\partial n} = -\frac{\sigma}{\varepsilon_0}
\tag{dpdisc}\label{dpdisc}
\]
where
\[
\frac{\partial V}{\partial n} = {\boldsymbol \nabla} V \cdot \hat{\bf n}
\frac{\partial \phi}{\partial n} = {\boldsymbol \nabla} \phi \cdot \hat{\bf n}
\label{Gr(2.37)}
\]
is the <b>normal derivative</b> of the potential.
@@ -1685,9 +1706,9 @@ is the <b>normal derivative</b> of the potential.
<p>
This is the kind of boundary condition that we need to fix a unique solution to Poisson's equation:
our only problem is that \ref{Gr(2.36)} gives the change of the normal derivative of \(V\), not
our only problem is that <a href="./ems_es_ep_bc.html#dpdisc">dpdisc</a> gives the change of the normal derivative of \(V\), not
its value. However, if we assume (as in our first case corollary) that there are no charges living outside
of our volume \({\cal V}\), we find that \ref{Gr(2.36)} fully specifies the potential's normal derivative
of our volume \({\cal V}\), we find that <a href="./ems_es_ep_bc.html#dpdisc">dpdisc</a> fully specifies the potential's normal derivative
if the surface charge is known.
</p>
</div>
@@ -1707,7 +1728,7 @@ 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: 2022-02-09 Wed 07:31</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
<p class="validation"></p>
</div>