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<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-17 Thu 08:42 -->
<!-- 2022-02-21 Mon 10:33 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -602,11 +602,11 @@ Table of contents
</summary>
<ul>
<li>
<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charge</a><span class="headline-id">ems.ms.lf.pc</span>
<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charges</a><span class="headline-id">ems.ms.lf.pc</span>
</li>
<li>
<a href="./ems_ms_lf_c.html#ems_ms_lf_c">Currents</a><span class="headline-id">ems.ms.lf.c</span>
<a href="./ems_ms_lf_sc.html#ems_ms_lf_sc">Steady Currents</a><span class="headline-id">ems.ms.lf.sc</span>
</li>
@@ -614,21 +614,12 @@ Table of contents
</details>
</li>
<li>
<a href="./ems_ms_ce.html#ems_ms_ce">Charge Conservation and the Continuity Equation</a><span class="headline-id">ems.ms.ce</span>
<details>
<summary>
</li>
<li>
<a href="./ems_ms_BS.html#ems_ms_BS">Steady Currents: the Biot-Savart Law</a><span class="headline-id">ems.ms.BS</span>
</summary>
<ul>
<li>
<a href="./ems_ms_BS_sc.html#ems_ms_BS_sc">The Magnetic Field issuing from a Steady Current</a><span class="headline-id">ems.ms.BS.sc</span>
</li>
</ul>
</details>
</li>
<li>
@@ -640,11 +631,15 @@ Table of contents
</summary>
<ul>
<li>
<a href="./ems_ms_dcB_sc.html#ems_ms_dcB_sc">Straight-line Currents</a><span class="headline-id">ems.ms.dcB.sc</span>
<a href="./ems_ms_dcB_iw.html#ems_ms_dcB_iw">Simplistic case: infinite wire</a><span class="headline-id">ems.ms.dcB.iw</span>
</li>
<li>
<a href="./ems_ms_dcB_BS.html#ems_ms_dcB_BS">Divergence and Curl of \({\bf B}\) from Biot-Savart</a><span class="headline-id">ems.ms.dcB.BS</span>
<a href="./ems_ms_dcB_d.html#ems_ms_dcB_d">Divergence of \({\bf B}\) from Biot-Savart</a><span class="headline-id">ems.ms.dcB.d</span>
</li>
<li>
<a href="./ems_ms_dcB_c.html#ems_ms_dcB_c">Curl of \({\bf B}\) from Biot-Savart; Ampère's Law</a><span class="headline-id">ems.ms.dcB.c</span>
</li>
@@ -661,6 +656,10 @@ Table of contents
</summary>
<ul>
<li>
<a href="./ems_ms_vp_A.html#ems_ms_vp_A">Definition; Gauge Choices</a><span class="headline-id">ems.ms.vp.A</span>
</li>
<li>
<a href="./ems_ms_vp_mbc.html#ems_ms_vp_mbc">Magnetic Boundary Conditions</a><span class="headline-id">ems.ms.vp.mbc</span>
</li>
@@ -698,10 +697,6 @@ Table of contents
</summary>
<ul>
<li>
<a href="./emsm_esm_s.html#emsm_esm_s">A proper definition of "statics"</a><span class="headline-id">emsm.esm.s</span>
</li>
<li>
<details>
<summary>
@@ -1435,7 +1430,7 @@ Table of contents
</li>
<li>
<a href="./c_m_dc_pr.html#c_m_dc_pr">Product Rules</a><span class="headline-id">c.m.dc.pr</span>
<a href="./c_m_dc_pr.html#c_m_dc_pr">Product arguments</a><span class="headline-id">c.m.dc.pr</span>
</li>
<li>
@@ -1591,181 +1586,26 @@ 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_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>
<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_c.html">Curl of \({\bf B}\) from Biot-Savart; Ampère's Law&emsp;<small>[ems.ms.dcB.c]</small></a></li><li>Next:&nbsp;<a href="ems_ms_vp_A.html">Definition; Gauge Choices&emsp;<small>[ems.ms.vp.A]</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"/>
</svg></a><span class="headline-id">ems.ms.vp</span></h4>
<div class="outline-text-4" id="text-ems_ms_vp">
<p>
Since \({\boldsymbol \nabla} \cdot {\bf B} = 0\) in magnetostatics, following Helmholtz's theorem we can write
</p>
<div class="core div" id="org5373f0d">
<p>
\[
{\bf B} = {\boldsymbol \nabla} \times {\bf A}
\label{Gr(5.59)}
\]
</p>
</div>
<p>
Connection with Ampère's law:
\[
{\boldsymbol \nabla} \times {\bf B} = {\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf A}
) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf A}) - {\boldsymbol \nabla}^2 {\bf A}
= \mu_0 {\bf J}
\label{Gr(5.60)}
\]
Electrostatics: you could add any constant to electrostatic potential. Here: you can
add any curlless function (so gradient of a scalar field) to the vector potential,
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="org4bf3f0a">
<p>
{\bf Example gauge choice:}
\[
{\boldsymbol \nabla} \cdot {\bf A} = 0.
\label{Gr(5.61)}
\]
</p>
</div>
<p>
Proof: suppose our starting \({\bf A}_0\) is not divergenceless. We add \({\boldsymbol \nabla} \lambda\)
to the vector potential, so \({\bf A} = {\bf A}_0 + {\boldsymbol \nabla} \lambda\). Then,
\[
{\boldsymbol \nabla} \cdot {\bf A} = {\boldsymbol \nabla} \cdot {\bf A}_0 + {\boldsymbol \nabla}^2 \lambda.
\]
The scalar field then obeys a Poisson-like equation,
\[
{\boldsymbol \nabla}^2 \lambda = -{\boldsymbol \nabla} \cdot {\bf A}_0,
\]
whose solution we know how to find. Provided \({\boldsymbol \nabla} \cdot {\bf A}_0\) goes to
zero at infinity,
\[
\lambda ({\bf r}) = \frac{1}{4\pi} \int d\tau' \frac{{\boldsymbol \nabla}' \cdot{\bf A}_0 ({\bf r}')}{|{\bf r} - {\bf r}'|}.
\]
{\it Provide proof in one line using Laplacian leading to delta function.}
</p>
<p>
Under this gauge choice, Ampère's law becomes
</p>
<div class="main div" id="orgba724ea">
<p>
\[
{\boldsymbol \nabla}^2 {\bf A} = -\mu_0 {\bf J}
\label{Gr(5.62)}
\]
</p>
</div>
<p>
Note: this is a Poisson equation for each component.
For currents falling off sufficiently rapidly at infinity,
</p>
<div class="core div" id="org8c1c094">
<p>
\[
{\bf A} ({\bf r}) = \frac{\mu_0}{4\pi} \int d\tau' \frac{J({\bf r}')}{|{\bf r} - {\bf r}'|}
\label{Gr(5.63)}
\]
</p>
</div>
<p>
For line and surface currents, <i>(beware Griffiths' <b>horrendous</b> notation)</i>
</p>
<div class="main div" id="orgd9e9a57">
<p>
\[
{\bf A}({\bf r}) = \frac{\mu_0}{4\pi} \int dl' \frac{{\bf I ({\bf r}')}}{|{\bf r} - {\bf r}'|},
\hspace{2cm}
{\bf A}({\bf r}) = \frac{\mu_0}{4\pi} \int da' \frac{{\bf K ({\bf r}')}}{|{\bf r} - {\bf r}'|}.
\label{Gr(5.64)}
\]
</p>
</div>
<div class="example div" id="org8d24f49">
<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 \({\bf r}\).
\paragraph{Solution:} do it by yourselves. Fun conclusion: the field inside
the sphere is uniform !
\[
{\bf B} = \frac{2}{3} \mu_0 \sigma R {\boldsymbol \omega}.
\label{Gr(5.68)}
\]
</p>
</div>
<div class="example div" id="org450360c">
<p>
\paragraph{Example 5.12:} find the vector potential of an infinite solenoid with \(n\) turns
pet unit length, radius \(R\) and current \(I\).
\paragraph{Solution:} cannot use \ref{Gr(5.64)} since the current extends to infinity
({\bf Comment:} check that the integral converges anyway, by combining the integrals
for \(z &gt; 0\) and \(z &lt; 0\) into one).
</p>
<p>
{\bf Nice trick:} notice that
\[
\oint {\bf A} \cdot d{\bf l} = \int ({\boldsymbol \nabla} \times {\bf A}) \cdot d{\bf a}
= \int {\bf B} \cdot d{\bf a} = \Phi.
\label{Gr(5.69)}
\]
This is reminiscent of Ampère's law in integral form, \ref{Gr(5.55)},
\[
\oint {\bf B} \cdot d{\bf l} = \mu_0 I_{enc}.
\]
It's the same equation ! Replacement: \({\bf B} \rightarrow {\bf A}\) and \(\mu_0 I_{enc} \rightarrow \Phi\).
And to paraphrase Feynman's lectures: {\it the same equations have the same solutions.}
</p>
<p>
Use symmetry: vector potential can only be cicumferential. Using an 'amperian' loop at a radius
\(s\) {\it inside} the solenoid, and the fact that the field inside a solenoid is \(\mu_0 n I\)
(\ref{Gr(5.57)}), we get
\[
\oint {\bf A} \cdot d{\bf l} = A (2\pi s) = \int {\bf B} \cdot d{\bf a} = \mu_0 n I (\pi s^2),
\]
so
\[
{\bf A} = \frac{\mu_0 n I}{2} s \hat{\boldsymbol \varphi}, \hspace{1cm} s &lt; R.
\label{Gr(5.70)}
\]
For an 'amperian' loop outside, the flux is always \(\mu_0 n I (\pi R^2)\), so
\[
{\bf A} = \frac{\mu_0 n I}{2} \frac{R^2}{s} \hat{\boldsymbol \varphi}, \hspace{1cm} s &gt; R.
\label{Gr(5.71)}
\]
\paragraph{Exercise:} check that \({\boldsymbol \nabla} \times {\bf A} = {\bf B}\) and that
\({\boldsymbol \nabla} \cdot {\bf A} = 0\).
</p>
</div>
</div>
<h5>In this section:</h5>
<ul class="child-links-list">
<li><a href="ems_ms_vp_A.html">Definition; Gauge Choices</a><span class="headline-id">ems.ms.vp.A</span></li>
<li><a href="ems_ms_vp_mbc.html">Magnetic Boundary Conditions</a><span class="headline-id">ems.ms.vp.mbc</span></li>
<li><a href="ems_ms_vp_me.html">Multipole Expansion of the Vector Potential</a><span class="headline-id">ems.ms.vp.me</span></li>
<li><a href="ems_ms_vp_comp.html">Comparison of Electrostatics and Magnetostatics</a><span class="headline-id">ems.ms.vp.comp</span></li>
<li><a href="ems_ms_vp_LC.html">The Levi-Civita Symbol</a><span class="headline-id">ems.ms.vp.LC</span></li>
</ul>
<br><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>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ms_dcB_c.html">Curl of \({\bf B}\) from Biot-Savart; Ampère's Law&emsp;<small>[ems.ms.dcB.c]</small></a></li><li>Next:&nbsp;<a href="ems_ms_vp_A.html">Definition; Gauge Choices&emsp;<small>[ems.ms.vp.A]</small></a></li><li>Up:&nbsp;<a href="ems_ms.html">Magnetostatics&emsp;<small>[ems.ms]</small></a></li></ul>
<br>
<hr>
<div class="license">
@@ -1780,7 +1620,7 @@ 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: 2022-02-17 Thu 08:42</p>
<p class="date">Created: 2022-02-21 Mon 10:33</p>
<p class="validation"></p>
</div>