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Update 2022-02-14 06:33

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Jean-Sébastien
2022-02-14 06:33:37 +01:00
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-10 Thu 08:32 -->
<!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1603,7 +1603,11 @@ A generic configuration of static charges coupled via the Coulomb interaction
defines an electrostatic problem, whose solution is in principle obtained
from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
</p>
<div class="main div" id="org27034ee">
<div class="main div" id="org98cdfd8">
<p>
</p>
\begin{equation*}
{\bf E} ({\bf r}) = \frac{1}{4\pi\varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') \frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3}
\end{equation*}
@@ -1611,44 +1615,53 @@ from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_v
</div>
<p>
or (often simpler) by calculating the electrostatic potential, using either the
explicit construction (\ref{eq:V_from_rho})
explicit construction <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
</p>
<div class="main div" id="org1620b08">
<div class="main div" id="org12e97ac">
<p>
</p>
<p>
\[
V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|}.
\tag{\ref{eq:V_from_rho}}
\]
\phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|}.
\]
</p>
</div>
<p>
Alternately, we have also seen that the two fundamental equations for the
electrostatic field, Gauss's law (\ref{Gr(2.14)}) and the no-perpetual-machine (vanishing curl)
condition (\ref{Gr(2.20)}) can be expressed as the single
'local' (differential) condition (Poisson's equation) (\ref{eq:Poisson})
electrostatic field, Gauss's law <a href="./ems_es_ef_Gl.html#Gl_d">Gl_d</a> and the vanishing curl
condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as the single
<i>local</i> differential condition (Poisson's equation)
<a href="./ems_es_ep_PL.html#Poi">🐟</a>
</p>
<div class="core div" id="orgcacf446">
<p>
</p>
<div class="core div" id="org1361b41">
<p>
\[
{\boldsymbol \nabla}^2 V = -\frac{\rho}{\varepsilon_0}.
\tag{\ref{eq:Poisson}}
\]
{\boldsymbol \nabla}^2 \phi = -\frac{\rho}{\varepsilon_0}.
\]
</p>
</div>
<p>
In the specific case where the charge density vanishes, we fall back onto the simpler
Laplace equation
In the specific case where the charge density vanishes, we fall back onto the simpler Laplace equation <a href="./ems_es_ep_PL.html#Lap">Lap</a>
</p>
<div class="core div" id="org2305746">
<div class="core div" id="orgea7b7e8">
<p>
</p>
<p>
\[
{\boldsymbol \nabla}^2 V = 0
\tag{\ref{eq:Laplace}}
\]
{\boldsymbol \nabla}^2 \phi = 0
\]
</p>
</div>
@@ -1663,6 +1676,8 @@ Laplace equation
<li><a href="ems_ca_fe_uP.html">Uniqueness of Solution to Poisson's Equation</a><span class="headline-id">ems.ca.fe.uP</span></li>
</ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca.html">Calculating or Approximating the Electrostatic Potential&emsp;<small>[ems.ca]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe_L.html">The Laplace Equation&emsp;<small>[ems.ca.fe.L]</small></a></li><li>Up:&nbsp;<a href="ems_ca.html">Calculating or Approximating the Electrostatic Potential&emsp;<small>[ems.ca]</small></a></li></ul>
<br>
<hr>
<div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@@ -1676,7 +1691,7 @@ 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: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p>
</div>