483 lines
24 KiB
HTML
483 lines
24 KiB
HTML
<!DOCTYPE html PUBLIC "-//w3c//dtd html 4.0 transitional//en">
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<html>
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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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<meta name="Author" content="C. L. Davis">
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<title>Electricty - Capacitors - 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);" link="#0000ee" alink="#ff0000" vlink="#551a8b">
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<center><img src="ULPhys1.gif" height="50" align="texttop"
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width="189"></center>
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<center>
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<h1>Capacitors<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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I used to wonder how it comes about that the electron is
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negative. Negative-positive—these are perfectly symmetric in
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physics. There is no reason whatever to prefer one to the
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other. Then why is the electron negative? I thought about this
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for a long time and at last all I could think was 'It won the
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fight!' "</i></font><br>
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Albert Einstein<br>
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</center>
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<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
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<blockquote>
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<h2><u>Calculating Capacitance</u></h2>
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</blockquote>
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<ul>
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</ul>
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<ul>
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<li>A capacitor is a system of two insulated conductors. <br>
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</li>
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</ul>
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<ul>
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</ul>
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<ul>
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<li><img alt="elec cap fig1" src="elec_cap_fig1.jpg" height="411"
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align="right" width="700">The parallel plate capacitor is the
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simplest example. When the two conductors have equal but
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opposite charge, the <b>E</b> field between the plates can be
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found by simple application of Gauss's Law.</li>
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</ul>
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<blockquote>Assuming the plates are large enough so that the <b>E</b>
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field between them is uniform and directed perpendicular, then
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applying Gauss's Law over surface S<sub>1</sub> we find,<br>
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<div align="center"><img alt="elec cap eqn1"
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src="elec_cap_eqn1.png" height="64" width="191"><br>
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<div align="left">where A is the area of S<sub>1</sub>
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perpendicular to the <b>E</b> field and σ is the surface
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charge density on the plate (assumed uniform).
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Therefore, <br>
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<div align="center"><img alt="elec cap eqn2"
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src="elec_cap_eqn2.png" height="60" width="67"><br>
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<br>
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<div align="left">everywhere between the plates.<br>
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</div>
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</div>
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</div>
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</div>
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</blockquote>
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<div align="center">
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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>The potential difference between the plates can be
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found from</li>
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</ul>
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<div align="center"><img alt="elec cap eqn3"
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src="elec_cap_eqn3.png" height="64" width="335"><br>
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<blockquote>
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<div align="left">where A and B are points, one on each
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plate, and we integrate along an <b>E</b> field line,
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d is the plate separation, A the plate area and q the
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total charge on either plate.<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>The capacitance (capacity) of this capacitor is
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defined as,</li>
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</ul>
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<div align="center"><img alt="elec cap eqn4"
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src="elec_cap_eqn4.png" height="63" width="148"><br>
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<div align="left">
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<ul>
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<li>The expression for C for all capacitors is the
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ratio of the magnitude of the total charge (on
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either plate) to the magnitude of the potential
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difference between the plates.</li>
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</ul>
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<ul>
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<li>Units of
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C:
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Coulomb/Volt = Farad, 1 C/V =
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1 F</li>
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</ul>
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<blockquote><img alt="exclamation"
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src="exclamation-icon.gif" height="30"
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width="31"> Note that since the Coulomb is a
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very large unit of charge the Farad is also a very
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large unit of capacitance. Typical
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capacitors in circuits are measured in μF (10<sup>-6</sup>)
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or pF (10<sup>-12</sup>).<br>
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</blockquote>
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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"> Note that the expression for the
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capacitance of the parallel plate capacitor
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depends on the geometric properties (A and
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d). Even though it appears that there is
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also a dependence on the charge and potential
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difference (q/ΔV), what happens is that whatever
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charge you place on the capacitor the pd adjusts
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itself so that the ratio q/ΔV remains
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constant. This is a general rule for
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all capacitors. The capacitance is set by
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the construction of the capacitor - not the
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charge or voltage applied.</li>
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</ul>
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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"> The above expression for the
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parallel plate capacitor is strictly only true
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for an infinite parallel plate capacitor - in
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which "fringing" (see above) does not
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occur. However, so long as d is small
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compared to the "size" of the plates, the simple
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expression above is a good approximation.</li>
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</ul>
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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"> The parallel plate capacitor
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provides an easy way to "measure" ε<sub>0</sub>
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<br>
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</li>
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</ul>
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<blockquote>
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<div align="center"><img alt="elec cap eqn5"
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src="elec_cap_eqn5.png" height="54" width="93"><br>
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</div>
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</blockquote>
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<div align="center">
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<div align="left">
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<ul>
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<li>As indicated above the parallel plate
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capacitor is the most basic capacitor.
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You should also be able to determine the
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expressions for the capacitance of spherical
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and cylindrical capacitors,</li>
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</ul>
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<div align="center"><img alt="elec cap fig3"
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src="elec_cap_fig3.jpg" height="239"
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width="311">
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<img
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alt="elec cap fig2" src="elec_cap_fig2.jpg"
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height="313" width="419"><br>
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<br>
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<img alt="divider" src="divider_ornbarblu.gif"
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height="64" width="393"><br>
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<blockquote>
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<div align="left">
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<h2><u>Energy and Capacitors</u></h2>
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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> One of the most important uses of
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capacitors is to store electrical
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energy.</li>
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</ul>
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<blockquote>If a capacitor is placed in a
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circuit with a battery, the potential
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difference (voltage) of the battery will
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force electric charge to appear on the
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plates of the capacitor. The work
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done by the battery in charging the
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capacitor is stored as electrical
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(potential) energy in the capacitor.
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This energy can be released at a later
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time to perform work.<br>
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<br>
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<div align="center"><img alt="elec cap
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fig4" src="elec_cap_fig4.jpg"
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height="204" width="297"></div>
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</blockquote>
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<div align="center">
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<div align="left">
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<ul>
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<li>The work necessary to move a
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charge dq onto one of the plates is
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given by, dW = Vdq, where V is the
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pd (voltage) of the battery (=
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q/C). The total work to place
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Q on the plate is given by,</li>
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</ul>
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<div align="center"><img alt="elec cap
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eqn6" src="elec_cap_eqn6.png"
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height="58" width="423"><br>
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<blockquote>
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<div align="left">which is equal to
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the stored electrical potential
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energy, U.<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>The electrical energy actually
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resides in the electric field
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between the plates of the
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capacitor. For a parallel
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plate capacitor using C =
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Aε<sub>0</sub>/d and E =
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Q/Aε<sub>0</sub> we may write
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the electrical potential energy,
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<br>
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</li>
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</ul>
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<div align="center"><img alt="elec
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cap eqn7"
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src="elec_cap_eqn7.png"
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height="68" width="339"><br>
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<blockquote>
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<div align="left">(Ad) is the
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volume between the plates,
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therefore we define the energy
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density,<br>
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<br>
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<div align="center"><img
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alt="elec cap eqn8"
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src="elec_cap_eqn8.png"
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height="54" width="181"><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>Although we have
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evaluated this
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expression for the
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energy density for a
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parallel plate capacitor
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it is actually a general
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expression.
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Wherever there is an
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electric field the
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energy density is given
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by the above.</li>
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</ul>
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<div align="center"><img
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alt="divider"
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src="divider_ornbarblu.gif"
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height="64" width="393"><br>
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<blockquote>
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<div align="left">
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<h2><u>Combinations of
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Capacitors</u></h2>
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</div>
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</blockquote>
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<div align="left">
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<blockquote>It is common
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to find multiple
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combinations of
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capacitors in
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electrical
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circuits. In the
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simplest situations
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capacitors can be
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considered to be
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connected in <b><i>series</i></b>
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or in <i><b>parallel</b></i>.
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<br>
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</blockquote>
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<ul>
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<ul>
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<li><big><b>Capacitors
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in Series</b></big></li>
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</ul>
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</ul>
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<blockquote>
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<blockquote>When
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different capacitors
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are connected in
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series the charge on
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each capacitor is
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the same but the
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voltage (pd) across
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each capacitor is
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different<br>
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<div align="center"><img
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alt="elec cap
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fig5"
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src="elec_cap_fig5.jpg"
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height="180"
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width="312"></div>
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</blockquote>
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</blockquote>
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<div align="center">
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<div align="left"><br>
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<blockquote>
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<blockquote>
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<div
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align="left">In
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this
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situation,
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using the fact
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that V = V<sub>1</sub>
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+ V<sub>2</sub>
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+V<sub>3</sub>
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we can show
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that, as far
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as the voltage
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source is
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concerned, the
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capacitors can
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be replaced by
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a single
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"equivalent"
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capacitor C<sub>eq</sub>
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given by, <br>
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</div>
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</blockquote>
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</blockquote>
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<div align="center"><img
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alt="elec cap
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fig9"
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src="elec_cap_eqn9.png"
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height="63"
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width="182"><br>
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</div>
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<br>
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<ul>
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<ul>
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<li><big><b>Capacitors
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in Parallel</b></big></li>
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</ul>
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</ul>
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<blockquote>
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<blockquote>For
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capacitors
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connected in
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parallel it is
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the voltage
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which is same
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for each
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capacitor, the
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charge being
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different.<br>
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<br>
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<div
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align="center"><img
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alt="elec cap
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fig6"
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src="elec_cap_fig6.jpg"
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height="176"
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width="360"><br>
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<div
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align="left"><br>
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Using the fact
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that Q<sub>Total</sub>=
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Q<sub>1</sub>
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+ Q<sub>2</sub>
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+ Q<sub>3</sub>
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we can show
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that the
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equivalent
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capacitor, C<sub>eq</sub>
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is given by,<br>
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<br>
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<div
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align="center"><img
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alt="elec cap
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eqn10"
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src="elec_cap_eqn10.png"
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height="29"
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width="172"><br>
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</div>
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</div>
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</div>
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</blockquote>
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</blockquote>
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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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</div>
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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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</div>
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<blockquote> </blockquote>
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<ul>
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</ul>
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<blockquote> </blockquote>
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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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</div>
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<blockquote>
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<div align="center"> </div>
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</blockquote>
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<p> <img src="netbar.gif" height="40" width="100%"> </p>
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<center>
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<p class="MsoNormal"><span style="color: rgb(255, 0, 0);
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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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At the electric company: <i>"We would be delighted if you
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send in your bill. However, if you don't, you will be."</i></span><br>
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<br>
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</p>
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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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