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2022-02-21 10:35:02 +01:00
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@@ -1,7 +1,7 @@
<!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 class="toc-currentpage">
<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>
@@ -1591,8 +1586,8 @@ 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><a class="breadcrumb-link"href="ems_ms_lf.html">Charges in Motion: the Lorentz Force Law</a></li><li>Point Charge</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ms_lf.html">Charges in Motion: the Lorentz Force Law&emsp;<small>[ems.ms.lf]</small></a></li><li>Next:&nbsp;<a href="ems_ms_lf_c.html">Currents&emsp;<small>[ems.ms.lf.c]</small></a></li><li>Up:&nbsp;<a href="ems_ms_lf.html">Charges in Motion: the Lorentz Force Law&emsp;<small>[ems.ms.lf]</small></a></li></ul><div id="outline-container-ems_ms_lf_pc" class="outline-5">
<h5 id="ems_ms_lf_pc">Point Charge<a class="headline-permalink" href="./ems_ms_lf_pc.html#ems_ms_lf_pc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<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_lf.html">Charges in Motion: the Lorentz Force Law</a></li><li>Point Charges</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ms_lf.html">Charges in Motion: the Lorentz Force Law&emsp;<small>[ems.ms.lf]</small></a></li><li>Next:&nbsp;<a href="ems_ms_lf_sc.html">Steady Currents&emsp;<small>[ems.ms.lf.sc]</small></a></li><li>Up:&nbsp;<a href="ems_ms_lf.html">Charges in Motion: the Lorentz Force Law&emsp;<small>[ems.ms.lf]</small></a></li></ul><div id="outline-container-ems_ms_lf_pc" class="outline-5">
<h5 id="ems_ms_lf_pc">Point Charges<a class="headline-permalink" href="./ems_ms_lf_pc.html#ems_ms_lf_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><span class="headline-id">ems.ms.lf.pc</span></h5>
@@ -1600,7 +1595,7 @@ Table of contents
<p>
Force on a point charge \(q\) moving at velocity \({\bf v}\) in magnetic field \({\bf B}\):
</p>
<div class="core div" id="org734fc6a">
<div class="core div" id="org6f47374">
<p>
\[
{\bf F}_{mag} = q {\bf v} \times {\bf B}
@@ -1610,49 +1605,72 @@ Force on a point charge \(q\) moving at velocity \({\bf v}\) in magnetic field \
</div>
<p>
Units of \({\bf B}\): \(N/(A~m)\) is called a {\bf tesla} (symbol: \(T\)). Total electromagnetic force:
Units of \({\bf B}\): \(N/(A~m)\) is called a <b>tesla</b> (symbol: \(T\)).
The total electromagnetic force on a point charge \(q\) moving at velocity
\({\bf v}\) is given by the <b>Lorentz force</b>:
</p>
<div class="core div" id="org85a6786">
<div class="eqlabel" id="org8abe978">
<p>
<a id="LorFo"></a><a href="./ems_ms_lf_pc.html#LorFo"><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="orgca3a8b5">
<ul class="org-ul">
<li>FLS II (13.1)</li>
<li>Gr (5.2)</li>
<li>PM (6.1)</li>
</ul>
</div>
</div>
<div class="core div" id="org7f05f5e">
<p>
\[
{\bf F}_{mag} = q ({\bf E} + {\bf v} \times {\bf B})
\label{eq:EMForce}
\]
{\bf F} = q ({\bf E} + {\bf v} \times {\bf B})
\tag{LorFo}\label{LorFo}
\]
</p>
</div>
<div class="example div" id="org3866a61">
<div class="example div" id="orgfcfa3dd">
<p>
<b>Cyclotron motion</b>
</p>
<p>
Consider a magnetic field \({\bf B}\) pointing into the page.
A charge \(q &gt; 0\) moves counterclockwise in the plane of the pagewith speed \(v\) on a circle of radius \(R\).
The magnetic force points inwards.
Equating the centrifugal and centripetal accelerations, we obtain the <b>cyclotron formula</b>
</p>
<p>
\paragraph{Example 5.1:} cyclotron motion. Field \({\bf B}\) pointing into page. Charge \(q &gt; 0\) moves
counterclockwise with speed \(v\) on a circle of radius \(R\). Magnetic force points inwards.
Equating, obtain the {\bf cyclotron formula}
\[
q v B = m \frac{v^2}{R} ~~\rightarrow~~ p = mv = q B R.
\label{Gr(5.3)}
\]
The {\bf cyclotron frequency} is
q v B = m \frac{v^2}{R} ~~\rightarrow~~ p = mv = q B R
\]
in which the <b>cyclotron frequency</b> is
</p>
<p>
\[
\omega = 2\pi \frac{v}{2\pi R} = \frac{q B}{m}
\label{Gr(5.4)}
\]
\omega = 2\pi \frac{v}{2\pi R} = \frac{q B}{m}
\]
</p>
</div>
<div class="example div" id="orgbb3483d">
<p>
\paragraph{Example 5.2:} cycloid motion. Recommendation: {\it look at it!!}
Due to the perpendicularity between the velocity and the magnetic force,
<b>magnetic forces do no work</b>, as we can see by computing a differential work element
</p>
</div>
<p>
Important point: {\bf magnetic forces do no work}. Work:
\[
dW_{mag} = {\bf F}_{mag} \cdot d{\bf l} = q ({\bf v} \times {\bf B}) \times {\bf v} dt = 0
\label{Gr(5.11)}
\]
</p>
</div>
@@ -1661,7 +1679,7 @@ dW_{mag} = {\bf F}_{mag} \cdot d{\bf l} = q ({\bf v} \times {\bf B}) \times {\bf
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ms_lf.html">Charges in Motion: the Lorentz Force Law&emsp;<small>[ems.ms.lf]</small></a></li><li>Next:&nbsp;<a href="ems_ms_lf_c.html">Currents&emsp;<small>[ems.ms.lf.c]</small></a></li><li>Up:&nbsp;<a href="ems_ms_lf.html">Charges in Motion: the Lorentz Force Law&emsp;<small>[ems.ms.lf]</small></a></li></ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ms_lf.html">Charges in Motion: the Lorentz Force Law&emsp;<small>[ems.ms.lf]</small></a></li><li>Next:&nbsp;<a href="ems_ms_lf_sc.html">Steady Currents&emsp;<small>[ems.ms.lf.sc]</small></a></li><li>Up:&nbsp;<a href="ems_ms_lf.html">Charges in Motion: the Lorentz Force Law&emsp;<small>[ems.ms.lf]</small></a></li></ul>
<br>
<hr>
<div class="license">
@@ -1676,7 +1694,7 @@ 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>