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Update 2022-02-09 07:44

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Jean-Sébastien
2022-02-09 07:44:58 +01:00
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@@ -1,7 +1,7 @@
<!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_sc.html">Straight-line Currents&emsp;<small>[ems.ms.dcB.sc]</small></a></li><li>Next:&nbsp;<a href="ems_ms_vp.html">The Vector Potential&emsp;<small>[ems.ms.vp]</small></a></li><li>Up:&nbsp;<a href="ems_ms_dcB.html">Divergence and Curl of \({\bf B}\)&emsp;<small>[ems.ms.dcB]</small></a></li></ul><div id="outline-container-ems_ms_dcB_BS" class="outline-5">
<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><a class="breadcrumb-link"href="ems_ms_dcB.html">Divergence and Curl of \({\bf B}\)</a></li><li>Divergence and Curl of \({\bf B}\) from Biot-Savart</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ms_dcB_sc.html">Straight-line Currents&emsp;<small>[ems.ms.dcB.sc]</small></a></li><li>Next:&nbsp;<a href="ems_ms_vp.html">The Vector Potential&emsp;<small>[ems.ms.vp]</small></a></li><li>Up:&nbsp;<a href="ems_ms_dcB.html">Divergence and Curl of \({\bf B}\)&emsp;<small>[ems.ms.dcB]</small></a></li></ul><div id="outline-container-ems_ms_dcB_BS" class="outline-5">
<h5 id="ems_ms_dcB_BS">Divergence and Curl of \({\bf B}\) from Biot-Savart<a class="headline-permalink" href="./ems_ms_dcB_BS.html#ems_ms_dcB_BS"><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"/>
@@ -1650,7 +1650,7 @@ we get
But \({\bf J}\) depends only on \({\bf r}'\) so \({\boldsymbol \nabla} \times {\bf J} ({\bf r}') = 0\), and since
the curl of a gradient always vanishes, we obtain
</p>
<div class="core div" id="org8f6dda2">
<div class="core div" id="orgd962857">
<p>
\[
{\boldsymbol \nabla} \cdot {\bf B} = 0
@@ -1718,7 +1718,7 @@ at infinity), and in the third step we have used the assumption of steady-state
<p>
We thus obtain in total
</p>
<div class="core div" id="orga0245b5">
<div class="core div" id="orgcfe305c">
<p>
<b>Ampère's law</b>
\[
@@ -1735,7 +1735,7 @@ We thus obtain in total
\]
so
</p>
<div class="core div" id="orgdbc9d47">
<div class="core div" id="org5d8a7de">
<p>
\[
\oint_{\cal P} {\bf B} \cdot d{\bf l} = \mu_0 I_{enc} \hspace{2cm}
@@ -1757,7 +1757,7 @@ Sign ambiguity: resolved by right-hand rule as usual.
Ampère's law in magnetostatics takes a parallel role to Gauss's law in electrostatics.
</p>
<div class="example div" id="orge2cd1ba">
<div class="example div" id="orgdb60bd0">
<p>
\paragraph{Example 5.7:} same as Example 5.5, but now with Ampère.
\paragraph{Solution:} by symmetry, \({\bf B}\) is circumferential and can only depend on \(s\). Then,
@@ -1769,7 +1769,7 @@ choosing an amperian loop at a fixed radius \(s\), we get
</div>
<div class="example div" id="orgf268569">
<div class="example div" id="orgd1801de">
<p>
\paragraph{Example 5.8:} uniform surface current \({\bf K} = K \hat{\bf x}\) flowing in \(xy\) plane.
\paragraph{Solution:} Biot-Savart: \({\bf B}\) must be perpendicular to \({\bf K}\). Intuition:
@@ -1786,7 +1786,7 @@ and along \(\hat{\bf y}\) for \(z &lt; 0\). Amperian loop of width \(l\) punchi
</div>
<div class="example div" id="org5e0a2eb">
<div class="example div" id="org808fe1f">
<p>
\paragraph{Example 5.9:} solenoid along \(\hat{\bf z}\), wire carrying current \(I\) doing \(n\) turns per unit length on cylinder of radius \(R\).
\paragraph{Solution:} by symmetry, \({\bf B}\) must be along axis of solenoid. Outside: infinitely far away, \({\bf B}\) must vanish.
@@ -1807,7 +1807,7 @@ Amperian loop of length \(l\), half-inside and half-outside:
i) infinite straight lines, ii) infinite planes, iii) infinite solenoids, iv) toroids.
</p>
<div class="example div" id="org2a86e5f">
<div class="example div" id="org7db2d3a">
<p>
\paragraph{Example 5.10:} toroidal coil (no matter the shape, as long as it is rotationally symmetric).
\paragraph{Solution:} magnetic field is circumferential everywhere. Outside coil, field again zero.
@@ -1825,10 +1825,21 @@ Amperian loop half inside, half outside:
</div>
<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>