243 lines
11 KiB
HTML
243 lines
11 KiB
HTML
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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 - RC circuits - Physics 299</title>
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<meta content="C. L. Davis" name="author">
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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>
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<h1> <img src="ULPhys1.gif" height="50" align="texttop"
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width="189"></h1>
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</center>
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<center>
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<h1>RC Circuits<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>
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<meta http-equiv="content-type" content="text/html;
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charset=windows-1252">
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</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></font>
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<div class="copy-paste-block"><font color="#ff0000"><i><span
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class="bqQuoteLink">"A</span></i></font><font
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color="#ff0000"><i><span class="bqQuoteLink"> fact is a simple
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statement that everyone believes. It is innocent,
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unless found guilty. A hypothesis is a novel
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suggestion that no one wants to believe. It is
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guilty, until found effective</span></i><span></span>"</font><br>
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</div>
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<font color="#ff0000"><i> </i><font color="#000000">Edward Teller</font></font><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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<br>
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<div align="center">
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<h2><u><font color="#3333ff">CHARGING</font></u></h2>
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</div>
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<ul>
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<li><img alt="fig1" src="elec_RC_fig1.jpg" height="344"
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align="right" width="532">An example of a series RC circuit is
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shown at right. With the emf included in the circuit,
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applying the loop theorem we find</li>
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</ul>
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<div align="center"><img alt="eqn1" src="elec_RC_eqn1.jpg"
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height="53" width="165"><br>
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<blockquote>
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<div align="left">where q/C is the voltage drop across the
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capacitor and i is the current in the circuit.<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>Using the fact that i = dq/dt, we obtain</li>
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</ul>
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<div align="center"><img alt="eqn2" src="elec_RC_eqn2.jpg"
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height="76" width="155"><br>
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<div align="left">
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<ul>
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<li>This is a "simple" differential equation the solution
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of which can be written</li>
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</ul>
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<div align="center"><img alt="eqn3" src="elec_RC_eqn3.jpg"
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height="38" width="197"><br>
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<blockquote>
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<div align="left">or<br>
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<div align="center"><img alt="eqn4"
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src="elec_RC_eqn4.jpg" height="68" width="169"><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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<ul>
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<li>The voltage across the capacitor, V<sub>C</sub> =
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q/C and the voltage across the resistor, V<sub>R</sub>
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= iR. Using the equations above we find that
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the dependence of these voltages on time is shown
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below</li>
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</ul>
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<div align="center"><img alt="fig2"
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src="elec_RC_fig2.jpg" height="286" width="399"><br>
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<br>
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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"> Note that the time on the
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horizontal axis is measured in units of τ = RC,
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the capacitative time constant.</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"> After one time constant V<sub>C</sub>
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has reached 63% (1 - e<sup>-1</sup>) of its
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maximum value and V<sub>R</sub> has 37% (1/e) of
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its final value.</li>
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</ul>
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<div align="center"><img alt="divider"
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src="divider_ornbarblu.gif" height="64"
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width="100%"><br>
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<h2><font color="#3333ff"><u>DISCHARGING</u></font></h2>
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<div align="left">
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<ul>
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<li><img alt="fig1" src="elec_RC_fig1.jpg"
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height="344" align="right" width="532">Now
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switch the emf out of the circuit and
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reapply the loop theorem</li>
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</ul>
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<div align="center"><img alt="eqn5"
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src="elec_RC_eqn5.jpg" height="53"
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width="132"><br>
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<blockquote>
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<div align="left">which gives<br>
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<div align="center"><img alt="eqn6"
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src="elec_RC_eqn6.jpg" height="65"
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width="137"><br>
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<br>
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<div align="left">which has the solution<br>
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<div align="center"><img alt="eqn7"
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src="elec_RC_eqn7.jpg" height="44"
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width="171"><br>
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<div align="left">and<br>
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<div align="center"><img
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alt="eqn8"
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src="elec_RC_eqn8.jpg"
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height="70" width="192"><br>
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</div>
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<br>
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</div>
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<br>
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<div align="left">where Cε is the
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initial charge on the capacitor
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and ε/R is the initial voltage
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across the capacitor.<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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</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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<div align="center">
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<div align="left">
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<ul>
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<li><img alt="fig3"
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src="elec_RC_fig3.jpg"
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height="410" align="right"
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width="316">The time
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dependence of V<sub>C</sub> and
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V<sub>R</sub> (where V<sub>R</sub>
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is proportional to the current
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in the capacitor) are shown at
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right.</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"
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height="30" width="31"> Once
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again the time axis is measured
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in units of RC.</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"
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height="30" width="31"> After
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one time constant V<sub>C</sub>
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has decreased to 37% (1/e) of
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its initial value and |V<sub>R</sub>|
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has decreased to 37% (1/e) of
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its initial value. </li>
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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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<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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<div align="center">
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<div align="left"> </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>
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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 align="left">
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<div align="center"><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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</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 align="left"> </div>
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<img src="netbar.gif" height="40" width="100%">
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<center>
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<p style="color: rgb(255, 0, 0); font-style: italic;"
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class="MsoNormal">
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<meta http-equiv="content-type" content="text/html;
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charset=windows-1252">
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</p>
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<font color="#ff0000"><i>Q: What did one quantum physicist say
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when he wanted to fight another quantum physicist?<br>
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A: Let me <font color="#ff0000">atom</font>. </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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