228 lines
10 KiB
HTML
228 lines
10 KiB
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<!DOCTYPE html PUBLIC "-//w3c//dtd html 4.0 transitional//en">
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<html>
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<head>
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<meta http-equiv="Content-Type" content="text/html;
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charset=windows-1252">
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<meta name="GENERATOR" content="Mozilla/4.7 [en] (X11; U; OSF1 V4.0
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alpha) [Netscape]">
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<meta name="Author" content="C. L. Davis">
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<title>Electricity - Gauss's Law - Physics 299</title>
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</head>
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<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
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255);" alink="#ff0000" link="#0000ee" vlink="#551a8b">
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<center>
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<h1> <img src="ULPhys1.gif" align="texttop" height="50"
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width="189"></h1>
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</center>
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<center>
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<h1>Gauss's Law<br>
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</h1>
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</center>
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<center><img src="celticbar.gif" height="22" width="576"><br>
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<br>
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<font color="#ff0000"><i>"</i></font><font color="#ff0000"><i>
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<meta http-equiv="content-type" content="text/html;
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charset=windows-1252">
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Equations are just the boring part of mathematics. I attempt
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to see things in terms of geometry."</i></font><br>
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Stephen Hawking<br>
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</center>
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<img src="netbar.gif" align="middle" height="40" width="100%"> <br>
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<ul>
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<li><img alt="gauss" src="gauss.jpg" align="left" height="101"
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width="83">Gauss's Law is the first of Maxwell's equations we
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will consider. At first the whole concept of Gauss's Law
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will seem to be very abstract and confusing, <img
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alt="confused" src="confused_smiley.gif" height="22"
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width="15"> hopefully at least some of the confusion
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will pass as you become more familiar with the idea. </li>
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</ul>
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<p><br>
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<br>
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</p>
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<ul>
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</ul>
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<ul>
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<li>At the outset it is important to realize that <b>Gauss's Law
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and Coulomb's Law are different statements of the same
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physical concept.</b> Which of the two is used in any
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particular situation depends on the particular application and
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what you are asked to determine.</li>
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</ul>
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<ul>
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<li>Before stating Gauss's Law we must first define the concept of
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<b>FLUX</b> - in particular the flux of the electric field.</li>
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</ul>
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<div align="left">
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<ul>
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<li><img alt="Electric flux" src="elec_gauss_figure1.jpg"
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align="right" height="433" width="415">At every point on a
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surface we can calculate an "element" of the electric flux
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given by</li>
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</ul>
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<div align="center"><img alt="defn of E flux"
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src="elec_gauss_eqn1.jpg" height="38" width="117"><br>
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<blockquote>
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<div align="left">so that the total electric flux passing
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through a surface, S, is given by,<br>
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<br>
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<div align="center"><img alt="elec gauss eqn2"
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src="elec_gauss_eqn2.png" height="56" width="133"><br>
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</div>
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</div>
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</blockquote>
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<div align="left">
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<div align="center">
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<div align="left">
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<ul>
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<li>Gauss's Law then states that,</li>
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</ul>
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<div align="center"><img alt="red tick" src="tickred1.gif"
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align="top" height="48" width="48"> <img
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alt="elec gauss 3" src="elec_gauss_eqn3.jpg"
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height="84" width="233"> <img
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alt="red tick" src="tickred1.gif" align="top"
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height="48" width="48"><br>
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<blockquote>
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<div align="left">where the circle on the integral
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means that the surface is <i><b>closed</b></i> and
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q<sub>inside</sub> is the net charge inside this
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closed surface.<br>
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</div>
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</blockquote>
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<div align="left">
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<ul>
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<li><img alt="exclamation"
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src="exclamation-icon.gif" height="30"
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width="31"> A closed surface has a definite
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inside and outside differentiated by the surface,
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e.g. the surface of a sphere.</li>
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<li><img alt="exclamation"
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src="exclamation-icon.gif" height="30"
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width="31"> The <b>dA</b> vector of a closed
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surface is always directed from the inside to the
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outside of the surface.</li>
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<li><img alt="exclamation"
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src="exclamation-icon.gif" height="30"
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width="31"> The exact location of the charges
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inside the closed surface is not important, all
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that matters is the net charge.</li>
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<li><img alt="exclamation"
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src="exclamation-icon.gif" height="30"
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width="31"> ε<sub>0</sub> is the "Permittivity
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of the Vacuum" a constant whose value is 8.85 x 10<sup>-12</sup>
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C<sup>2</sup>/(N.m<sup>2</sup>) where the Coulomb
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constant, k = 1/(4πε<sub>0</sub>). Note that
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if the charges are not located in vacuum ε<sub>0</sub>
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must be replaced by the permittivity of the medium
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in question. </li>
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<li><img alt="exclamation"
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src="exclamation-icon.gif" height="30"
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width="31"> The proof of Gauss's Law is beyond
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the scope of this course. Suffice to say the
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inverse square dependence on distance of Coulomb's
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Law is critical.</li>
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</ul>
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<ul>
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<li>Before using Gauss's Law to evaluate electric
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fields a brief qualitative discussion is
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worthwhile. Consider the situation of two
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point charges below. Application of Gauss's
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Law over each of the closed surfaces:</li>
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</ul>
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<blockquote>
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<ul>
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<li><img alt="elec gauss figure 2"
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src="elec_gauss_figure2.png" align="right"
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height="523" width="359"> S<sub>1</sub>:
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At every point on this surface both <b>E </b>and
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<b>dA</b> are directed "outwards", such that the
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scalar product <b>E<EFBFBD>dA</b> = EdAcosθ is always
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positive. Thus the integral over the
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surface S<sub>1</sub> will be positive, as it
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must be if Gauss's Law is to be satisfied, since
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the net charge enclosed is positive.</li>
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<li>S<sub>2</sub> : <b>E </b>is directed
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"inwards", <b>dA</b> "outwards", leading to a
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negative value for the flux through S<sub>2</sub>,
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consistent with the fact that the net charge
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enclosed is negative.</li>
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<li>S<sub>3</sub>: Some of this surface has
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E directed "inwards" the remainder has <b>E</b>
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directed "outwards". <b>dA</b> is
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"outwards" everywhere on the surface.
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Therefore the flux integral has both positive
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and negative contributions. Since there is
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no net charge enclosed by S<sub>3</sub> by
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Gauss's Law the net flux will be zero.</li>
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<li>S<sub>4</sub>: Once again there are negative
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and positive contributions to the flux integral,
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so that we can write Gauss's Law,</li>
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</ul>
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<div align="center"><img alt="elec_gauss_eqn4"
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src="elec_gauss_eqn4.png" height="86"
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width="381"><br>
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<div align="left"><br>
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</div>
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</div>
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<ul>
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</ul>
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</blockquote>
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<ul>
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</ul>
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</div>
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</div>
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</div>
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</div>
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</div>
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</div>
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</div>
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<ul>
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</ul>
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<ul>
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<li> Electric flux through a closed box <a
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href="http://www.youtube.com/watch?v=5ENl4vn82bc">animation.</a></li>
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</ul>
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<ul>
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<li>Coulomb's Law, Electric Field, Electric Flux and Gauss's Law <a
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href="http://www.veoh.com/collection/APPhysics/watch/v15544578N9Hg8YBK">video</a>.<br>
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</li>
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</ul>
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<div style="text-align: left;"><img src="netbar.gif" height="40"
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width="100%"></div>
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<center><span style="font-size: 12pt; font-family: "Times New
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Roman";"><span style="color: rgb(255, 0, 0); font-style:
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italic;"></span></span><span style="font-size: 12pt;
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font-family: "Times New Roman";"><span style="color:
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rgb(255, 0, 0); font-style: italic;">
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<meta http-equiv="content-type" content="text/html;
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charset=windows-1252">
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</span></span><br>
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<font color="#ff0000"><i>Did you hear about the French post-doc
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who went to work at the Fermi Lab, but never went in because
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the sign over the door always said it was closed.</i></font><br>
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<br>
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<img src="celticbar.gif" height="22" width="576"> <br>
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<p><i>Dr. C. L. Davis</i> <br>
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<i>Physics Department</i> <br>
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<i>University of Louisville</i> <br>
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<i>email</i>: <a href="mailto:c.l.davis@louisville.edu">c.l.davis@louisville.edu</a>
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<br>
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</p>
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<p><img src="header-index.gif" height="51" width="92"> </p>
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</center>
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<p><br>
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</p>
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</body>
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</html>
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