jscaux/pqed_html
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Update 2022-02-09 22:41

Browse Source
This commit is contained in:
Jean-Sébastien
2022-02-09 22:41:42 +01:00
parent 3c40f5bfe8
commit f3c2446d19
208 changed files with 1583 additions and 12916 deletions
Download Patch File Download Diff File Expand all files Collapse all files
build/ems_es_ep_PL.html
+49 -42
View File
@@ -1,7 +1,7 @@
<!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 class="toc-currentpage">
<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>
@@ -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>
@@ -1616,8 +1591,8 @@ 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>The Poisson Equation and the Laplace Equation</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_ep_ex.html">Example calculations for the potential&emsp;<small>[ems.es.ep.ex]</small></a></li><li>Next:&nbsp;<a href="ems_es_ep_bc.html">Electrostatic Boundary Conditions&emsp;<small>[ems.es.ep.bc]</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_PL" class="outline-5">
<h5 id="ems_es_ep_PL">The Poisson Equation and the Laplace Equation<a class="headline-permalink" href="./ems_es_ep_PL.html#ems_es_ep_PL"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<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>Poisson's and Laplace's Equations</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_ep_ex.html">Example calculations for the potential&emsp;<small>[ems.es.ep.ex]</small></a></li><li>Next:&nbsp;<a href="ems_es_ep_bc.html">Electrostatic Boundary Conditions&emsp;<small>[ems.es.ep.bc]</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_PL" class="outline-5">
<h5 id="ems_es_ep_PL">Poisson's and Laplace's Equations<a class="headline-permalink" href="./ems_es_ep_PL.html#ems_es_ep_PL"><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.PL</span></h5>
@@ -1629,35 +1604,67 @@ Our two fundamental equations for the electrostatic field are
\hspace{2cm}
{\boldsymbol \nabla} \times {\bf E} = 0.
\]
For the electrostatic potential, Gauss' law becomes
For the electrostatic potential, by <a href="./ems_es_ep_fp.html#Emgp">Emgp</a>, Gauss's law becomes
</p>
<div class="core div" id="orgf8d5b14">
<div class="eqlabel" id="org57d3945">
<p>
<b>The Poisson equation</b>
<a id="Poi"></a><a href="./ems_es_ep_PL.html#Poi"><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="orgcc8bc83">
<ul class="org-ul">
<li>Gr (2.24)</li>
<li>PM (2.72)</li>
</ul>
</div>
</div>
<div class="core div" id="orgae79cc1">
<p>
<b>Poisson's equation</b>
\[
{\boldsymbol \nabla}^2 V = -\frac{\rho}{\varepsilon_0}
\label{eq:Poisson}
\]
{\boldsymbol \nabla}^2 \phi = -\frac{\rho}{\varepsilon_0}
\tag{🐟}\label{Poi}
\]
</p>
</div>
<p>
When the charge density vanishes, it becomes more simply
</p>
<div class="core div" id="org109885a">
<div class="eqlabel" id="orgc27352d">
<p>
<b>The Laplace Equation</b>
<a id="Lap"></a><a href="./ems_es_ep_PL.html#Lap"><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="org391f57f">
<ul class="org-ul">
<li>Gr (2.25)</li>
<li>PM (2.75)</li>
</ul>
</div>
</div>
<div class="core div" id="org21d0c34">
<p>
<b>Laplace's Equation</b>
\[
{\boldsymbol \nabla}^2 V = 0
\label{eq:Laplace}
\]
{\boldsymbol \nabla}^2 \phi = 0
\tag{Lap}\label{Lap}
\]
</p>
</div>
<p>
Since the curl of a gradient is always zero, we by construction have
\({\boldsymbol \nabla} \times {\bf E} = - {\boldsymbol \nabla} \times ({\boldsymbol \nabla} V) = 0\).
\({\boldsymbol \nabla} \times {\bf E} = - {\boldsymbol \nabla} \times ({\boldsymbol \nabla} \phi) = 0\).
</p>
</div>
</div>
@@ -1678,7 +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: 2022-02-09 Wed 07:31</p>
<p class="date">Created: 2022-02-09 Wed 22:40</p>
<p class="validation"></p>
</div>