davisnotes/mag_motionch.html

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<title>Magnetism - Motion of Charged Particles in B and E - Physics
299</title>
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<h1>Motion of Charged Particles in Magnetic and Electric Fields<br>
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<div class="copy-paste-block"><font color="#ff0000"><i><span
class="bqQuoteLink">"A</span></i></font><font
color="#ff0000"><i><span class="bqQuoteLink"> fact is a simple
statement that everyone believes.&nbsp; It is innocent,
unless found guilty.&nbsp; A hypothesis is a novel
suggestion that no one wants to believe.&nbsp; It is
guilty, until found effective</span></i><span></span>"</font><br>
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<font color="#ff0000"><i> </i><font color="#000000">Edward Teller</font></font><br>
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<div align="center"><font color="#6600cc"><big><u><b>MAGNETIC
FIELDS</b></u></big></font></div>
<ul>
<li>Return now to the case of a "point" charge
moving with velocity <b>v</b> in a region
of constant magnetic field (<b>B</b>).&nbsp;
We previously stated (without proof) that
such a particle would move in a circle or
helix.&nbsp; Now let's prove it.</li>
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<li>&nbsp;<img alt="magforcechargefig2"
src="mag_force_charge_fig2.jpg"
height="324" align="right" width="292">Consider
the positive charge (e) at the top of the
figure at right.&nbsp; With the directions
of <b>v</b> and <b>B</b> indicated the
right-hand-rule leads to a downwards force
as shown.&nbsp; This force changes the
direction of <b>v</b> but does not change
the magnitude of <b>v</b>, as shown.</li>
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<ul>
<li>Quantitatively we find</li>
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<div align="center"><img alt="magmotionch"
src="mag_motionch_eqn1.jpg" height="37"
width="180"><br>
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<div align="left">Assuming that this force
causes circular motion, <b>F<sub>B</sub>
</b>is the centripetal force so that<br>
<br>
<div align="center"><img
alt="magmotioneqn2"
src="mag_motionch_eqn2.jpg"
height="64" width="108"><br>
<div align="left">which leads to a
radius<br>
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<div align="center"><img alt="magmotioncheqn3"
src="mag_motionch_eqn3.jpg" height="63"
width="72"><br>
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<div align="left">The period (time taken to
perform one revolution) is given by<br>
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<div align="center"><img
alt="magmotioncheqn4"
src="mag_motionch_eqn4.jpg"
height="56" width="225"><br>
<div align="left">The frequency, f, is<br>
<div align="center"><img
alt="magmotioncheqn5"
src="mag_motionch_eqn5.jpg"
height="62" width="135"><br>
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<ul>
<li><img alt="exclamation"
src="exclamation-icon.gif" height="30"
width="31"> Notice that this frequency
is independent of the velocity.&nbsp;
That is fast particles will move in
large&nbsp; radius circles, slow
particles will have smaller radii.&nbsp;
This phenomenon is the basis for the
cyclotron one of the earliest charged
particle accelerators.</li>
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<ul>
<li><img alt="exclamation"
src="exclamation-icon.gif" height="30"
width="31">&nbsp; If the velocity of
the particle has a component along <b>B</b>
this will be unchanged and the resultant
trajectory will be helical.</li>
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src="divider_ornbarblu.gif" height="64"
width="393"><br>
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<font color="#6600cc"><big><u><b>MAGNETIC
and ELECTRIC FIELDS</b></u></big></font><br>
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<ul>
<li>There are many useful applications in
which <b>B</b> and <b>E</b> fields are
applied simultaneously.&nbsp; Perhaps
the simplest application is the
"crossed" <b>B</b> and <b>E</b>
fields.&nbsp;&nbsp;</li>
</ul>
<div align="center"><img
alt="magmotionchfig1"
src="mag_motionch_fig1.gif" height="174"
width="311"></div>
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<p>For this configuration the electric and
magnetic forces are in opposite
directions.&nbsp; If a charged particle
moves through this region of space with
no deflection then</p>
<p align="center"><img
alt="magmotioncheqn6"
src="mag_motionch_eqn6.jpg"
height="32" width="189"><br>
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<div align="left">so that<br>
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<div align="center"><img
alt="magmotioncheqn7"
src="mag_motionch_eqn7.jpg"
height="54" width="54"><br>
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<li>Other applications include
mass spectrometers and particle
accelerators.&nbsp; (<b>EXPAND
?)</b><br>
</li>
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<font color="#ff0000"><i>I'm reading a book about
anti-gravity.&nbsp;&nbsp; I can't put it down. </i></font><br>
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&nbsp;<img src="celticbar.gif" height="22" width="576"> <br>
&nbsp;
<p><i>Dr. C. L. Davis</i> <br>
<i>Physics Department</i> <br>
<i>University of Louisville</i> <br>
<i>email</i>: <a href="mailto:c.l.davis@louisville.edu">c.l.davis@louisville.edu</a>
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