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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>
@@ -1604,14 +1604,14 @@ 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="orgc533ab3">
<div class="eqlabel" id="org89afdc8">
<p>
<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="org15bf0d6">
<div class="alteqlabels" id="org75f1411">
<ul class="org-ul">
<li>Gr (2.21)</li>
</ul>
@@ -1627,7 +1627,7 @@ called the <b>electrostatic potential</b>
\]
</p>
<aside id="orgcf2675e">
<aside id="org0dab7b7">
<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 \(kg m^2/s^3 A\).
@@ -1644,14 +1644,14 @@ want; it's energy <i>differences</i> which matter).
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="eqlabel" id="org9122c30">
<div class="eqlabel" id="org6384e5c">
<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="orga78af3a">
<div class="alteqlabels" id="org5dcaf18">
<ul class="org-ul">
<li>Gr (2.21)</li>
</ul>
@@ -1659,7 +1659,7 @@ well-defined without the need to specify the reference point,
</div>
</div>
<div class="main div" id="orgb652ba9">
<div class="main div" id="orgeb834e4">
<p>
\[
\phi({\bf b}) - \phi({\bf a}) = - \int_{\bf a}^{\bf b} {\bf E} \cdot d{\bf l}
@@ -1676,14 +1676,14 @@ The reference point is typically put at infinity. The electrostatic potential c
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="org6e188d2">
<div class="eqlabel" id="orgeaff4ce">
<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="orgc641f61">
<div class="alteqlabels" id="org359b71c">
<ul class="org-ul">
<li>FLS II (4.20,4.23)</li>
<li>Gr (2.26)</li>
@@ -1706,14 +1706,14 @@ point charge \(q\) at the origin can then be calculated (taking for convenience
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="eqlabel" id="org186810e">
<div class="eqlabel" id="org6e5fe47">
<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="orgc1b749e">
<div class="alteqlabels" id="org22c3c5d">
<ul class="org-ul">
<li>FLS II (4.24)</li>
<li>Gr (2.27)</li>
@@ -1723,7 +1723,7 @@ so for a distribution of point charges \(q_i\) at positions \({\bf r}_i\), we ha
</div>
</div>
<div class="core div" id="orga63a2be">
<div class="core div" id="orgfed3d36">
<p>
</p>
@@ -1739,14 +1739,14 @@ so for a distribution of point charges \(q_i\) at positions \({\bf r}_i\), we ha
<p>
For a continuous charge density in a volume \({\cal V}\), we have
</p>
<div class="eqlabel" id="org1c0d869">
<div class="eqlabel" id="org4cb18bb">
<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="orgcecd7a4">
<div class="alteqlabels" id="orgc8797db">
<ul class="org-ul">
<li>Gr (2.29)</li>
<li>PM (2.18)</li>
@@ -1755,7 +1755,7 @@ For a continuous charge density in a volume \({\cal V}\), we have
</div>
</div>
<div class="main div" id="orge8de361">
<div class="main div" id="org5f30b88">
<p>
</p>
@@ -1772,14 +1772,14 @@ For a continuous charge density in a volume \({\cal V}\), we have
<p>
whereas for a surface or line charge distribution, respectively,
</p>
<div class="eqlabel" id="org3523e8c">
<div class="eqlabel" id="org370294d">
<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="orgb7c9068">
<div class="alteqlabels" id="org4bbddb7">
<ul class="org-ul">
<li>Gr (2.30)b</li>
</ul>
@@ -1787,7 +1787,7 @@ whereas for a surface or line charge distribution, respectively,
</div>
</div>
<div class="main div" id="org6eb9f4a">
<div class="main div" id="orgac4aef3">
<p>
\[
\phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\cal S} da' \frac{\sigma({\bf r}')}{|{\bf r} - {\bf r}'|}
@@ -1797,14 +1797,14 @@ whereas for a surface or line charge distribution, respectively,
</div>
<div class="eqlabel" id="orgb12e0c6">
<div class="eqlabel" id="org0cba1c0">
<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="org47eb94d">
<div class="alteqlabels" id="org6248c1e">
<ul class="org-ul">
<li>Gr (2.30)a</li>
</ul>
@@ -1812,7 +1812,7 @@ whereas for a surface or line charge distribution, respectively,
</div>
</div>
<div class="main div" id="org7efd986">
<div class="main div" id="orgb1662ed">
<p>
</p>
@@ -1830,6 +1830,8 @@ whereas for a surface or line charge distribution, respectively,
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_ep.html">The Electrostatic Potential&emsp;<small>[ems.es.ep]</small></a></li><li>Next:&nbsp;<a href="ems_es_ep_fp.html">Field in terms of the potential&emsp;<small>[ems.es.ep.fp]</small></a></li><li>Up:&nbsp;<a href="ems_es_ep.html">The Electrostatic Potential&emsp;<small>[ems.es.ep]</small></a></li></ul>
<br>
<hr>
<div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@@ -1843,7 +1845,7 @@ 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>