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@@ -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>
<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>
@@ -1623,16 +1598,20 @@ Table of contents
</svg></a><span class="headline-id">ems.es.ep.d</span></h5>
<div class="outline-text-5" id="text-ems_es_ep_d">
<p>
Since this work is independent of the path chosen, we can define a function called the <b>electrostatic potential</b>
We have seen that the total work done by moving a test charge (think of a point particle, so
its position is well-defined) in the presence of an electrostatic
field is independent of the path, and that it is proportional to the test charge.
We can thus define a function representing the work per unit charge and
called the <b>electrostatic potential</b>
</p>
<div class="eqlabel" id="orge6c02ac">
<div class="eqlabel" id="orgdfab1f5">
<p>
<a id="es_pot"></a><a href="./ems_es_ep_d.html#es_pot"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<a id="p"></a><a href="./ems_es_ep_d.html#p"><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="orgb12d22e">
<div class="alteqlabels" id="org527e1db">
<ul class="org-ul">
<li>Gr (2.21)</li>
</ul>
@@ -1640,68 +1619,209 @@ Since this work is independent of the path chosen, we can define a function call
</div>
</div>
<p>
\[
V({\bf r}) \equiv -\int_{\cal O}^{\bf r} {\bf E} \cdot d{\bf l}
\tag{es_pot}\label{es_pot}
\phi({\bf r}) \equiv -\int_{\cal O}^{\bf r} {\bf E} \cdot d{\bf l}
\tag{p}\label{p}
\]
where \({\cal O}\) is some chosen reference point. The potential difference between two points is
</p>
<aside id="org4cc55f0">
<p>
\(\phi\) carries as unit: Newton-meters per coulomb or joules per coulomb, which is called a <b>volt</b>.
In the SI system where the ampere is a base unit, a volt is a \(kg m^2/s^3 A\).
</p>
</aside>
<p>
where \({\cal O}\) is some agreed-upon reference point (remember that the actual value of the
energy isn't really important in physics: we can fix the zero where we
want; it's energy <i>differences</i> which matter).
</p>
<p>
The potential difference between two arbitrary points \({\bf a}\) and \({\bf b}\) is
well-defined without the need to specify the reference point,
</p>
<div class="main div" id="orgb1934a8">
<div class="eqlabel" id="orga996849">
<p>
<a id="p_diff"></a><a href="./ems_es_ep_d.html#p_diff"><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="orge766074">
<ul class="org-ul">
<li>Gr (2.21)</li>
</ul>
</div>
</div>
<div class="main div" id="orgb6a531d">
<p>
\[
V({\bf b}) - V({\bf a}) = - \int_{\bf a}^{\bf b} {\bf E} \cdot d{\bf l}
\label{Gr(2.22)}
\phi({\bf b}) - \phi({\bf a}) = - \int_{\bf a}^{\bf b} {\bf E} \cdot d{\bf l}
\tag{p_diff}\label{p_diff}
\]
</p>
</div>
<p>
Thus, the electrostatic potential is interpreted as the potential energy which a unit charge would
have obtained if brought to the specified point from the reference point, in other words the work you need to
do <i>on</i> the unit charge to bring it there.
Often, we put the reference point at infinity. The electrostatic potential coming from a single
point charge \(q\) at the origin then becomes
Thus, the electrostatic potential is interpreted as the potential energy which
a unit charge would have obtained if brought to the specified point from the
reference point, in other words the work you need to do <i>on</i> the unit charge to bring it there.
The reference point is typically put at infinity. The electrostatic potential coming from a single
point charge \(q\) at the origin can then be calculated (taking for convenience \(d{\bf l} = \hat{\bf r} dr\),
<i>i.e.</i> moving in purely radially) as
</p>
<div class="eqlabel" id="org098df6a">
<p>
<a id="p_pc"></a><a href="./ems_es_ep_d.html#p_pc"><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="org6a5e3e8">
<ul class="org-ul">
<li>FLS II (4.20,4.23)</li>
<li>Gr (2.26)</li>
<li>PM (2.1,2.2)</li>
</ul>
</div>
</div>
<p>
\[
V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{q}{r}
\label{Gr(2.26)}
\phi({\bf r}) = -\int_{\infty}^r dr \frac{1}{4\pi \varepsilon_0} \frac{q}{r^2} =
\frac{1}{4\pi \varepsilon_0} \frac{q}{r}
\tag{p_pc}\label{p_pc}
\]
</p>
<p>
The electrostatic potential moreover inherits the superposition principle from the electric field,
so for a distribution of point charges \(q_i\) at positions \({\bf r}_i\), we have
</p>
<div class="core div" id="orge1ad14a">
<div class="eqlabel" id="orgc89730c">
<p>
<a id="p_pcd"></a><a href="./ems_es_ep_d.html#p_pcd"><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="org55b211e">
<ul class="org-ul">
<li>FLS II (4.24)</li>
<li>Gr (2.27)</li>
<li>PM (2.19)</li>
</ul>
</div>
</div>
<div class="core div" id="org817630e">
<p>
</p>
<p>
\[
V({\bf r}) = \frac{1}{4\pi\varepsilon_0} \sum_{i} \frac{q_i}{|{\bf r} - {\bf r}_i|}
\label{Gr(2.27)}
\]
\phi({\bf r}) = \frac{1}{4\pi\varepsilon_0} \sum_{i} \frac{q_i}{|{\bf r} - {\bf r}_i|}
\tag{p_pcd}\label{p_pcd}
\]
</p>
</div>
<p>
For a continuous charge density in a volume \({\cal V}\), we have
</p>
<div class="main div" id="org977a6c5">
<div class="eqlabel" id="org97f7105">
<p>
<a id="p_vcd"></a><a href="./ems_es_ep_d.html#p_vcd"><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="org43eb984">
<ul class="org-ul">
<li>Gr (2.29)</li>
<li>PM (2.18)</li>
</ul>
</div>
</div>
<div class="main div" id="orga933822">
<p>
</p>
<p>
\[
V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\cal V} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|},
\label{eq:V_from_rho}
\]
\phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\cal V} d\tau'
\frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|},
\tag{p_vcd}\label{p_vcd}
\]
</p>
</div>
<p>
whereas for a surface or line charge distribution, respectively,
</p>
<div class="main div" id="org867be71">
<div class="eqlabel" id="org818d44f">
<p>
<a id="p_scd"></a><a href="./ems_es_ep_d.html#p_scd"><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="orgca813bb">
<ul class="org-ul">
<li>Gr (2.30)b</li>
</ul>
</div>
</div>
<div class="main div" id="orgf1e9775">
<p>
\[
V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\cal S} da' \frac{\sigma({\bf r}')}{|{\bf r} - {\bf r}'|},
\hspace{2cm}
V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\cal P} dl' \frac{\lambda({\bf r}')}{|{\bf r} - {\bf r}'|}.
\label{Gr(2.30)}
\]
\phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\cal P} dl' \frac{\lambda({\bf r}')}{|{\bf r} - {\bf r}'|}
\tag{p_scd}\label{p_scd}
\]
</p>
</div>
<div class="eqlabel" id="orgf55b92e">
<p>
<a id="p_lcd"></a><a href="./ems_es_ep_d.html#p_lcd"><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="org2cbfbf2">
<ul class="org-ul">
<li>Gr (2.30)a</li>
</ul>
</div>
</div>
<div class="main div" id="orga7ba18b">
<p>
</p>
<p>
\[
\phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\cal S} dl' \frac{\sigma({\bf r}')}{|{\bf r} - {\bf r}'|}
\tag{p_lcd}\label{p_lcd}
\]
</p>
</div>
@@ -1723,7 +1843,7 @@ 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>