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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 open="">
<summary class="toc-currentpage">
</li>
<li class="toc-currentpage">
<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>
@@ -1591,33 +1586,197 @@ 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>Steady Currents: the Biot-Savart Law</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ms_lf_c.html">Currents&emsp;<small>[ems.ms.lf.c]</small></a></li><li>Next:&nbsp;<a href="ems_ms_BS_sc.html">The Magnetic Field issuing from a Steady Current&emsp;<small>[ems.ms.BS.sc]</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>Steady Currents: the Biot-Savart Law</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ms_ce.html">Charge Conservation and the Continuity Equation&emsp;<small>[ems.ms.ce]</small></a></li><li>Next:&nbsp;<a href="ems_ms_dcB.html">Divergence and Curl of \({\bf B}\)&emsp;<small>[ems.ms.dcB]</small></a></li><li>Up:&nbsp;<a href="ems_ms.html">Magnetostatics&emsp;<small>[ems.ms]</small></a></li></ul><div id="outline-container-ems_ms_BS" class="outline-4">
<h4 id="ems_ms_BS">Steady Currents: the Biot-Savart Law<a class="headline-permalink" href="./ems_ms_BS.html#ems_ms_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"/>
</svg></a><span class="headline-id">ems.ms.BS</span></h4>
<div class="outline-text-4" id="text-ems_ms_BS">
<p>
Steady currents lead to constant magnetic fields: {\bf magnetostatics}.
The magnetic field issuing from a steady surrent
is given experimentally (around 1820) by the
</p>
<div class="core div" id="orgc94cef6">
<p>
<b>Biot-Savart law</b>
</p>
<div class="eqlabel" id="org8869dd0">
<p>
<a id="BiotSavart"></a><a href="./ems_ms_BS.html#BiotSavart"><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="orge6449ea">
<ul class="org-ul">
<li>FLS II (14.43)</li>
<li>Gr (5.34)</li>
<li>PM (6.49)</li>
</ul>
</div>
</div>
<p>
\[
{\bf B} ({\bf r}) = \frac{\mu_0}{4\pi} \int dl' \frac{{\bf I} \times ({\bf r} - {\bf r}')}{|{\bf r} - {\bf r}'|^3}
\tag{BiotSavart}\label{BiotSavart}
\]
</p>
</div>
<p>
in which \(\mu_0\) is the <b>vacuum permeability</b> (or alternately <i>permeability of free space</i>,
<i>permeability of vacuum</i> or <i>magnetic constant</i>),
\[
\mu_0 = 1.25663706212(19)x10^{-6} H/m
\]
with the <i>henry</i> \(H = kg ~m^2 / s^2 A^2\) being the unit for inductance.
</p>
<p>
Then, since \(\frac{\partial \rho}{\partial t} = 0\), the continuity equation leads to
For surface and volume density currents:
</p>
<div class="main div" id="org97608f1">
<div class="eqlabel" id="org32040cc">
<p>
<a id="BiotSavart_s"></a><a href="./ems_ms_BS.html#BiotSavart_s"><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="org2c84d27">
</div>
</div>
<p>
\[
{\boldsymbol \nabla} \cdot {\bf J} = 0
\label{Gr(5.31)}
{\bf B} ({\bf r}) = \frac{\mu_0}{4\pi} \int da' \frac{{\bf K} ({\bf r}') \times ({\bf r} - {\bf r}')}{|{\bf r} - {\bf r}'|^3},
\tag{BiotSavart_s}\label{BiotSavart_s}
\]
</p>
<div class="eqlabel" id="org56b5fb8">
<p>
<a id="BiotSavart_v"></a><a href="./ems_ms_BS.html#BiotSavart_v"><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="orgc00a3a8">
</div>
</div>
<p>
\[
{\bf B} ({\bf r}) = \frac{\mu_0}{4\pi} \int d\tau' \frac{{\bf J} ({\bf r}') \times ({\bf r} - {\bf r}')}{|{\bf r} - {\bf r}'|^3}
\tag{BiotSavart_v}\label{BiotSavart_v}
\]
</p>
</div>
<p>
N.B.: there is no such thing as a Biot-Savart law for a point charge, since this cannot
represent a steady current.
</p>
<p>
The <b>superposition principle</b> applies here as well: a collection of currents generates a \({\bf B}\) field which is
the vector sum of the fields generated by the individual currents.
</p>
<div class="example div" id="orgb52405d">
<p>
<b>Example: \({\bf B}\) from long straight wire</b>
</p>
<p>
<b>Task</b>: find \({\bf B}\) a distance \(s\) from a long straight wire carrying steady current \(I\).
</p>
<p>
<b>Solution</b>:
\(dl {\bf I} \times ({\bf r} - {\bf r}')\) points out of the page, and has
magnitude \(dl' \sin \alpha = dl' \cos \theta\). But \(l' = s \tan \theta\) so \(dl' = \frac{s}{\cos^2 \theta} d\theta\),
and \(s = |{\bf r} - {\bf r}'| \cos \theta\). Then,
</p>
<p>
\[
B = \frac{\mu_0}{4\pi} I \int_{\theta_1}^{\theta_2} d\theta \cos \theta \frac{\cos^2 \theta}{s^2} \frac{s}{\cos^2 \theta}
= \frac{\mu_0 I}{4\pi s} (\sin \theta_2 - \sin \theta_1)
\]
For infinite wire: \(\theta_1 = -\pi/2\), \(\theta_2 = \pi/2\), so
</p>
<div class="eqlabel" id="orga23cca2">
<p>
<a id="Bwire1"></a><a href="./ems_ms_BS.html#Bwire1"><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="org5708c99">
<ul class="org-ul">
<li>Gr (5.38)</li>
</ul>
</div>
</div>
<p>
\[
B = \frac{\mu_0 I}{2\pi s}
\tag{Bwire1}\label{Bwire1}
\]
</p>
</div>
<p>
As an immediate consequence, we see that the force per unit length between two wires with
currents \(I_1\) and \(I_2\) separated by distance \(d\) is
</p>
<p>
\[
f = \frac{\mu_0}{2\pi} \frac{I_1 I_2}{d}
\]
(like currents attract).
</p>
<div class="example div" id="org585c93b">
<p>
<b>Example: \({\bf B}\) above a circular loop</b>
</p>
<p>
<b>Task</b>: find {\bf B} a distance \(z\) above the center of a circular loop of radius \(R\),
carrying a steady counterclockwise current \(I\).
</p>
<p>
<b>Solution</b>: By symmetry, only the vertical component doesn't cancel.
</p>
<p>
\[
B(z) = \frac{\mu_0 I}{4\pi} \int dl' \frac{\cos \theta}{|{\bf r} - {\bf r}'|}
= \frac{\mu_0 I}{4\pi} \frac{\cos \theta}{R^2 + z^2} \int dl' = \frac{\mu_0 I}{2} \frac{R^2}{(R^2 + z^2)^{3/2}}
\]
(since \(\cos \theta = R/\sqrt{R^2 + z^2}\)).
</p>
</div>
</div>
</div>
<h5>In this section:</h5>
<ul class="child-links-list">
<li><a href="ems_ms_BS_sc.html">The Magnetic Field issuing from a Steady Current</a><span class="headline-id">ems.ms.BS.sc</span></li>
</ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ms_lf_c.html">Currents&emsp;<small>[ems.ms.lf.c]</small></a></li><li>Next:&nbsp;<a href="ems_ms_BS_sc.html">The Magnetic Field issuing from a Steady Current&emsp;<small>[ems.ms.BS.sc]</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_ce.html">Charge Conservation and the Continuity Equation&emsp;<small>[ems.ms.ce]</small></a></li><li>Next:&nbsp;<a href="ems_ms_dcB.html">Divergence and Curl of \({\bf B}\)&emsp;<small>[ems.ms.dcB]</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">
@@ -1632,7 +1791,7 @@ 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>