davisnotes/mag_dipole.html

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<title>Magnetism - Dipoles - 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 Dipoles <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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<img src="netbar.gif" align="middle" height="40" width="100%">
<blockquote> </blockquote>
<ul>
<li>Shown below is a rectangular current loop sides a and b placed
in a region of space with uniform magnetic field (<b>B</b>).<br>
</li>
</ul>
<div align="center"><img alt="magdipfig1" src="mag_dipole_fig1.jpg"
height="368" width="328">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp;&nbsp; <img
alt="magdipfig2" src="mag_dipole_fig2.jpg" height="346"
width="309"><br>
<br>
<div align="left">
<ul>
<li>Application of the expression for the force on a current
carrying wire to the top and bottom sides of the rectangle
leads to cancellation of forces as shown.</li>
</ul>
<ul>
<li>The forces on the other two sides (length a) are also
equal and opposite, as shown in (b) above, <img
alt="magdipeqn3" src="mag_dipole_eqn3.jpg" align="middle"
height="42" width="117">.&nbsp; Although the net <i><b>force</b></i>
in this case is zero, because the forces are not co-linear,
there is a net <i><b>torque</b></i> on the current loop,
given by</li>
</ul>
<div align="center"><img alt="magdipeqn1"
src="mag_dipole_eqn1.jpg" height="37" width="279"><br>
<blockquote>
<div align="left">The magnitude of this torque is given by<br>
<div align="center"><img alt="magdipeqn2"
src="mag_dipole_eqn2.jpg" height="49" width="423"><br>
<br>
<div align="left">where A = ab is the cross section area
of the coil.<br>
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<div align="center">
<div align="left">
<ul>
<li>The magnitude of the magnetic dipole moment of the
loop with N turns is defined by,</li>
</ul>
<div align="center"><img alt="magdipeqn4"
src="mag_dipole_eqn4.jpg" height="42" width="99"><br>
<blockquote>
<div align="left"><img alt="magdipfig3"
src="mag_dipole_fig3.jpg" align="right"
height="149" width="234">&#956; is a vector quantity
whose direction is perpendicular to the loop with
a sense defined by (another) <b>Right Hand Rule</b>.<br>
</div>
</blockquote>
<div align="left">
<ul>
<li>With this definition of &#956; the torque on the
current loop is given by</li>
</ul>
<div align="center">&nbsp;<img alt="magdipeqn5"
src="mag_dipole_eqn5.jpg" height="33" width="99"><br>
<div align="left">
<ul>
<li>Without proof, the potential energy of a
magnetic dipole in an external magnetic
field is given by</li>
</ul>
<div align="center"><img alt="magdipeqn6"
src="mag_dipole_eqn6.jpg" height="33"
width="117"><br>
<br>
<div align="left"><img alt="hot" src="hot.gif"
height="43" width="79"> Note that the
above expressions for the torque and
potential energy of a magnetic dipole in an
external magnetic field are identical in
form to those obtained for an electric
dipole in an external electric field.<br>
<br>
<div align="center"><img alt="elecdipext"
src="elec_dipexte_eqn1.jpg" height="45"
width="117">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
and&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img
alt="elecdipextf2"
src="elec_dipexte_eqn2.jpg" height="41"
width="117"><br>
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<div align="left"> </div>
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<font color="#ff0000"><i>I stayed up all night to see where the
sun went.&nbsp; Then it dawned on me.</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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