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<title>Magnetism - Force on Charges - Physics 299</title>
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<h1> <img src="ULPhys1.gif" align="texttop" height="50"
width="189"></h1>
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<h1>Magnetic Forces on Charged Particles <br>
</h1>
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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>
</div>
<font color="#ff0000"><i> </i><font color="#000000">Edward Teller</font></font><br>
</center>
<img src="netbar.gif" align="middle" height="40" width="100%"> <br>
<ul>
<li> Proceeding under the assumption that magnetic fields exist -
created by an as yet not described mechanism - the magnetic
force on a <b>positively</b> charged particle is found
experimentally to be given by
<div align="center"><img alt="magforcechargeeqn1"
src="mag_force_charge_eqn1.jpg" align="middle" height="33"
width="135"></div>
</li>
</ul>
<blockquote>
<p>where q is the charge (positive) of the particle, <b>v </b>its
velocity, <b>F<sub>B</sub> </b>the force it experiences and <b>B</b>
the magnetic field causing the force.<br>
</p>
</blockquote>
<ul>
<li>
<div align="center">
<div align="left"> <img alt="exclamation"
src="exclamation-icon.gif" height="30" width="31"> Note
that this equation is the magnetic equivalent of the
electric expression&nbsp; <b>F<sub>E</sub> = </b>q<b>E</b></div>
</div>
</li>
</ul>
<ul>
<div align="left">
<li>The expression for the magnetic force is written in terms of
a vector (cross) product, which means, like it or not, you
have to work in three dimensions.&nbsp; Several important
facts emerge from this equation. <br>
<ol>
<br>
<li>If <b>v</b> = 0 there is no force.&nbsp; Electric
charges at rest in a magnetic field do not feel a magnetic
force. </li>
<br>
<li>The magnitude of <b>F<sub>B</sub> </b>is given by
<div align="center"><img alt="magforcechargeeqn2"
src="mag_force_charge_eqn2.jpg" height="37"
width="162"></div>
<br>
where &#966; is the angle between <b>v</b> and <b>B</b>.
<ul>
</ul>
<ul>
<li> So that, when <b>v</b> is parallel to <b>B</b> (&#966;
= 0) or antiparallel to <b>B (</b>&#966; = 180<sup>o</sup>)
then sin&#966; = 0 and <b>F<sub>B</sub> </b>= 0.&nbsp;
Therefore, charged particles moving along magnetic
field lines experience no magnetic force. </li>
</ul>
<ul>
<li> When <b>v</b> and <b>B</b> are at 90<sup>o</sup>
<b>F<sub>B</sub> </b>has its maximum value, F<sub>B</sub>
= qvB. </li>
</ul>
</li>
<br>
<li> <b>F<sub>B</sub> </b>is at right angles to <b>v</b>
and <b>B</b>.&nbsp; The "sense" is given by the usual
rules for the vector (cross) product, sometimes called
(the) "<b>Right Hand Rule"</b>.
<div align="center"><img alt="magforcechargefig1"
src="mag_force_charge_fig1.jpg" height="243"
width="479"><br>
</div>
<br>
This leads to circular (or spiral) motion around the <b>B</b>
field lines.<br>
<br>
<div align="center"><img alt="magforcechargefig2"
src="mag_force_charge_fig2.jpg" height="378"
width="337"> &nbsp; &nbsp; &nbsp; <img
alt="magforcechargefig3"
src="mag_force_charge_fig3.jpg" height="315"
width="401"><br>
<br>
</div>
<div align="center">
<div><img alt="hot" src="hot.gif" height="43" width="79">Don't
forget that the direction of the magnetic force
obtained above is for a <b>positive</b> charge.&nbsp;
For a negative charge the direction of <b>F<sub>B</sub>&nbsp;</b>
is reversed. <br>
</div>
</div>
</li>
<div align="left"> <br>
<li>The work done by the magnetic force when a charged
particle is displaced by <b>ds</b> is given by, </li>
</div>
<br>
<div align="center"><br>
<img alt="magforcechargeeqn3"
src="mag_force_charge_eqn3.jpg" height="32" width="244">
</div>
<div align="left">where &#952; is the angle between <b>F<sub>B</sub></b><sub>&nbsp;
</sub>and <b>ds</b>.<br>
</div>
<div align="left"> The displacement <b>ds</b> and the
velocity of the particle, <b>v</b>, are in the same
direction, so &#952; is also the angle between <b>F<sub>B</sub></b><sub>&nbsp;
</sub>and <b>v</b>.&nbsp; But from the form of the
magnetic force we know <b>F<sub>B</sub></b><sub>&nbsp; </sub>and
<b>v </b>are perpendicular (&#952; = 90<sup>o</sup>) therefore
dW is always zero.&nbsp; In other words the <i><b>magnetic
force does no work,</b></i> which means, unlike the
electric force, it cannot give a charged particle energy
(increase or decrease its kinteic energy). The effect of a
magnetic force is to change the direction not the kinetic
energy.</div>
</ol>
</li>
</div>
<br>
</ul>
<div align="center"><img alt="divider" src="divider_ornbarblu.gif"
height="64" width="393"><br>
</div>
<ul>
<li> If a charged particle feels both a magnetic and electric
field the resultant force is given by </li>
</ul>
<br>
<div align="center"><img alt="magforcechargeeqn4"
src="mag_force_charge_eqn4.jpg" height="35" width="189"><br>
<blockquote> </blockquote>
<blockquote>
<div align="left">this is known as the Lorentz Force Law.<br>
<br>
<div align="center"><img alt="divider"
src="divider_ornbarblu.gif" height="64" width="393"><br>
<br>
</div>
</div>
</blockquote>
<div align="left">
<div align="left">
<ul>
<li><big><u><b>UNITS</b></u></big></li>
</ul>
<blockquote>
<p>From the form of the magnetic force we see that the units
of <b>B</b> are&nbsp; N/(C.m/s) = N/(A.m).&nbsp; This
combination of basic units is defined as the Tesla.<br>
</p>
<div align="center"> <img alt="magforcechargeeqn5"
src="mag_force_charge_eqn5.jpg" height="56" width="153">
</div>
<p>The Tesla is a very large unit of magnetic field.&nbsp;
For this reason you may occasionally come across the
smaller unit of magnteic field, the Gauss; where&nbsp; 1
Tesla = 10<sup>4</sup> Gauss.<br>
</p>
</blockquote>
</div>
</div>
<blockquote>
<div align="left"> </div>
</blockquote>
<blockquote> </blockquote>
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<font color="#ff0000"><i>Q: What did one quantum physicist say
when he wanted to fight another quantum physicist?<br>
A: Let me <font color="#ff0000">atom</font>. </i></font><br>
<br>
&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>
<br>
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