243 lines
11 KiB
HTML
243 lines
11 KiB
HTML
<!DOCTYPE html PUBLIC "-//w3c//dtd html 4.0 transitional//en">
|
|
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html;
|
|
charset=windows-1252">
|
|
<meta name="GENERATOR" content="Mozilla/4.7 [en] (X11; U; OSF1 V4.0
|
|
alpha) [Netscape]">
|
|
<meta name="Author" content="C. L. Davis">
|
|
<title>Electricity - RC circuits - Physics 299</title>
|
|
<meta content="C. L. Davis" name="author">
|
|
</head>
|
|
<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
|
|
255);" link="#0000ee" alink="#ff0000" vlink="#551a8b">
|
|
<center>
|
|
<h1> <img src="ULPhys1.gif" height="50" align="texttop"
|
|
width="189"></h1>
|
|
</center>
|
|
<center>
|
|
<h1>RC Circuits<br>
|
|
</h1>
|
|
</center>
|
|
<center><img src="celticbar.gif" height="22" width="576"><br>
|
|
<br>
|
|
<font color="#ff0000"><i>
|
|
<meta http-equiv="content-type" content="text/html;
|
|
charset=windows-1252">
|
|
</i></font><font color="#ff0000"><i>
|
|
<meta http-equiv="content-type" content="text/html;
|
|
charset=windows-1252">
|
|
</i></font>
|
|
<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. It is innocent,
|
|
unless found guilty. A hypothesis is a novel
|
|
suggestion that no one wants to believe. 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" height="40" align="middle" width="100%"> <br>
|
|
<br>
|
|
<div align="center">
|
|
<h2><u><font color="#3333ff">CHARGING</font></u></h2>
|
|
</div>
|
|
<ul>
|
|
<li><img alt="fig1" src="elec_RC_fig1.jpg" height="344"
|
|
align="right" width="532">An example of a series RC circuit is
|
|
shown at right. With the emf included in the circuit,
|
|
applying the loop theorem we find</li>
|
|
</ul>
|
|
<div align="center"><img alt="eqn1" src="elec_RC_eqn1.jpg"
|
|
height="53" width="165"><br>
|
|
<blockquote>
|
|
<div align="left">where q/C is the voltage drop across the
|
|
capacitor and i is the current in the circuit.<br>
|
|
</div>
|
|
</blockquote>
|
|
<div align="left">
|
|
<ul>
|
|
<li>Using the fact that i = dq/dt, we obtain</li>
|
|
</ul>
|
|
<div align="center"><img alt="eqn2" src="elec_RC_eqn2.jpg"
|
|
height="76" width="155"><br>
|
|
<div align="left">
|
|
<ul>
|
|
<li>This is a "simple" differential equation the solution
|
|
of which can be written</li>
|
|
</ul>
|
|
<div align="center"><img alt="eqn3" src="elec_RC_eqn3.jpg"
|
|
height="38" width="197"><br>
|
|
<blockquote>
|
|
<div align="left">or<br>
|
|
<div align="center"><img alt="eqn4"
|
|
src="elec_RC_eqn4.jpg" height="68" width="169"><br>
|
|
</div>
|
|
</div>
|
|
</blockquote>
|
|
<div align="left">
|
|
<ul>
|
|
<li>The voltage across the capacitor, V<sub>C</sub> =
|
|
q/C and the voltage across the resistor, V<sub>R</sub>
|
|
= iR. Using the equations above we find that
|
|
the dependence of these voltages on time is shown
|
|
below</li>
|
|
</ul>
|
|
<div align="center"><img alt="fig2"
|
|
src="elec_RC_fig2.jpg" height="286" width="399"><br>
|
|
<br>
|
|
<div align="left">
|
|
<ul>
|
|
<li><img alt="exclamation"
|
|
src="exclamation-icon.gif" height="30"
|
|
width="31"> Note that the time on the
|
|
horizontal axis is measured in units of τ = RC,
|
|
the capacitative time constant.</li>
|
|
</ul>
|
|
<ul>
|
|
<li><img alt="exclamation"
|
|
src="exclamation-icon.gif" height="30"
|
|
width="31"> After one time constant V<sub>C</sub>
|
|
has reached 63% (1 - e<sup>-1</sup>) of its
|
|
maximum value and V<sub>R</sub> has 37% (1/e) of
|
|
its final value.</li>
|
|
</ul>
|
|
<div align="center"><img alt="divider"
|
|
src="divider_ornbarblu.gif" height="64"
|
|
width="100%"><br>
|
|
<h2><font color="#3333ff"><u>DISCHARGING</u></font></h2>
|
|
<div align="left">
|
|
<ul>
|
|
<li><img alt="fig1" src="elec_RC_fig1.jpg"
|
|
height="344" align="right" width="532">Now
|
|
switch the emf out of the circuit and
|
|
reapply the loop theorem</li>
|
|
</ul>
|
|
<div align="center"><img alt="eqn5"
|
|
src="elec_RC_eqn5.jpg" height="53"
|
|
width="132"><br>
|
|
<blockquote>
|
|
<div align="left">which gives<br>
|
|
<div align="center"><img alt="eqn6"
|
|
src="elec_RC_eqn6.jpg" height="65"
|
|
width="137"><br>
|
|
<br>
|
|
<div align="left">which has the solution<br>
|
|
<div align="center"><img alt="eqn7"
|
|
src="elec_RC_eqn7.jpg" height="44"
|
|
width="171"><br>
|
|
<div align="left">and<br>
|
|
<div align="center"><img
|
|
alt="eqn8"
|
|
src="elec_RC_eqn8.jpg"
|
|
height="70" width="192"><br>
|
|
</div>
|
|
<br>
|
|
</div>
|
|
<br>
|
|
<div align="left">where Cε is the
|
|
initial charge on the capacitor
|
|
and ε/R is the initial voltage
|
|
across the capacitor.<br>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</blockquote>
|
|
<div align="left">
|
|
<div align="center">
|
|
<div align="left">
|
|
<div align="center">
|
|
<div align="left">
|
|
<ul>
|
|
<li><img alt="fig3"
|
|
src="elec_RC_fig3.jpg"
|
|
height="410" align="right"
|
|
width="316">The time
|
|
dependence of V<sub>C</sub> and
|
|
V<sub>R</sub> (where V<sub>R</sub>
|
|
is proportional to the current
|
|
in the capacitor) are shown at
|
|
right.</li>
|
|
</ul>
|
|
<ul>
|
|
<li><img alt="exclamation"
|
|
src="exclamation-icon.gif"
|
|
height="30" width="31"> Once
|
|
again the time axis is measured
|
|
in units of RC.</li>
|
|
</ul>
|
|
<ul>
|
|
<li><img alt="exclamation"
|
|
src="exclamation-icon.gif"
|
|
height="30" width="31"> After
|
|
one time constant V<sub>C</sub>
|
|
has decreased to 37% (1/e) of
|
|
its initial value and |V<sub>R</sub>|
|
|
has decreased to 37% (1/e) of
|
|
its initial value. </li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<blockquote>
|
|
<div align="left">
|
|
<div align="center">
|
|
<div align="left">
|
|
<div align="center">
|
|
<div align="left"> </div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</blockquote>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<ul>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div align="left">
|
|
<div align="center"><br>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<ul>
|
|
</ul>
|
|
<div align="left"> </div>
|
|
<img src="netbar.gif" height="40" width="100%">
|
|
<center>
|
|
<p style="color: rgb(255, 0, 0); font-style: italic;"
|
|
class="MsoNormal">
|
|
<meta http-equiv="content-type" content="text/html;
|
|
charset=windows-1252">
|
|
</p>
|
|
<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>
|
|
<img src="celticbar.gif" height="22" width="576"> <br>
|
|
|
|
<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>
|
|
</p>
|
|
<p><img src="header-index.gif" height="51" width="92"> </p>
|
|
</center>
|
|
<p><br>
|
|
</p>
|
|
</body>
|
|
</html>
|