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Jean-Sébastien
2022-02-09 07:44:58 +01:00
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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-08 Tue 17:21 -->
<!-- 2022-02-09 Wed 07:31 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1616,7 +1616,7 @@ Table of contents
</ul>
</details>
</nav>
<ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ms_dcB_BS.html">Divergence and Curl of \({\bf B}\) from Biot-Savart&emsp;<small>[ems.ms.dcB.BS]</small></a></li><li>Next:&nbsp;<a href="ems_ms_vp_mbc.html">Magnetic Boundary Conditions&emsp;<small>[ems.ms.vp.mbc]</small></a></li><li>Up:&nbsp;<a href="ems_ms.html">Magnetostatics&emsp;<small>[ems.ms]</small></a></li></ul>
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="ems.html">Electromagnetostatics</a></li><li><a class="breadcrumb-link"href="ems_ms.html">Magnetostatics</a></li><li>The Vector Potential</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ms_dcB_BS.html">Divergence and Curl of \({\bf B}\) from Biot-Savart&emsp;<small>[ems.ms.dcB.BS]</small></a></li><li>Next:&nbsp;<a href="ems_ms_vp_mbc.html">Magnetic Boundary Conditions&emsp;<small>[ems.ms.vp.mbc]</small></a></li><li>Up:&nbsp;<a href="ems_ms.html">Magnetostatics&emsp;<small>[ems.ms]</small></a></li></ul>
<h4 id="ems_ms_vp">The Vector Potential<a class="headline-permalink" href="./ems_ms_vp.html#ems_ms_vp"><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"/>
@@ -1626,7 +1626,7 @@ Table of contents
<p>
Since \({\boldsymbol \nabla} \cdot {\bf B} = 0\) in magnetostatics, following Helmholtz's theorem we can write
</p>
<div class="core div" id="org8d029d6">
<div class="core div" id="orgb824ded">
<p>
\[
{\bf B} = {\boldsymbol \nabla} \times {\bf A}
@@ -1648,7 +1648,7 @@ add any curlless function (so gradient of a scalar field) to the vector potentia
without changing the magnetic field. This is called a {\bf gauge choice} in electrodynamics.
For example, we can {\bf always} eliminate the divergence of \({\bf A}\),
</p>
<div class="main div" id="org2f5ec71">
<div class="main div" id="org40f92f3">
<p>
{\bf Example gauge choice:}
\[
@@ -1679,7 +1679,7 @@ zero at infinity,
<p>
Under this gauge choice, Ampère's law becomes
</p>
<div class="main div" id="orga1e23e0">
<div class="main div" id="orgba366e5">
<p>
\[
{\boldsymbol \nabla}^2 {\bf A} = -\mu_0 {\bf J}
@@ -1692,7 +1692,7 @@ Under this gauge choice, Ampère's law becomes
Note: this is a Poisson equation for each component.
For currents falling off sufficiently rapidly at infinity,
</p>
<div class="core div" id="org4174f16">
<div class="core div" id="org576a84c">
<p>
\[
{\bf A} ({\bf r}) = \frac{\mu_0}{4\pi} \int d\tau' \frac{J({\bf r}')}{|{\bf r} - {\bf r}'|}
@@ -1704,7 +1704,7 @@ For currents falling off sufficiently rapidly at infinity,
<p>
For line and surface currents, <i>(beware Griffiths' <b>horrendous</b> notation)</i>
</p>
<div class="main div" id="org712334e">
<div class="main div" id="orgd64eded">
<p>
\[
{\bf A}({\bf r}) = \frac{\mu_0}{4\pi} \int dl' \frac{{\bf I ({\bf r}')}}{|{\bf r} - {\bf r}'|},
@@ -1718,7 +1718,7 @@ For line and surface currents, <i>(beware Griffiths' <b>horrendous</b> notation)
<div class="example div" id="org0baf51b">
<div class="example div" id="orgfd862d3">
<p>
\paragraph{Example 5.11:} a spherical shell of radius \(R\), carrying a uniform surface charge
\(\sigma\), is set spinning at angular velocity \(\omega\). Find the vector potential at \({\bf r}\).
@@ -1732,7 +1732,7 @@ the sphere is uniform !
</div>
<div class="example div" id="org1f6ff4e">
<div class="example div" id="org78723ad">
<p>
\paragraph{Example 5.12:} find the vector potential of an infinite solenoid with \(n\) turns
pet unit length, radius \(R\) and current \(I\).
@@ -1790,10 +1790,21 @@ For an 'amperian' loop outside, the flux is always \(\mu_0 n I (\pi R^2)\), so
<li><a href="ems_ms_vp_LC.html">The Levi-Civita Symbol</a><span class="headline-id">ems.ms.vp.LC</span></li>
</ul>
<hr><div id="postamble" class="status">
<hr>
<div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
target="_blank" class="m-2">
<img alt="Creative Commons License" style="border-width:0"
src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
</a>
Except where otherwise noted, all content is licensed under a
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
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: 2022-02-08 Tue 17:21</p>
<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
<p class="date">Created: 2022-02-09 Wed 07:31</p>
<p class="validation"></p>
</div>