209 lines
7.7 KiB
HTML
209 lines
7.7 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>Magnetism - LR 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" vlink="#551a8b" alink="#ff0000">
|
|
<center>
|
|
<h1> <img src="ULPhys1.gif" height="50" width="189"
|
|
align="texttop"></h1>
|
|
</center>
|
|
<center>
|
|
<h1>LR 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" width="100%" align="middle">
|
|
<blockquote> </blockquote>
|
|
<ul>
|
|
<li>As we have seen, due to Faraday's Law, an induced emf will
|
|
appear in any loop/coil/circuit in which the current (and
|
|
therefore the magnetic field) changes with time. Certain
|
|
circuit elements have a larger (self) inductance than others -
|
|
the solenoid is the most obvious example of a circuit element
|
|
with a large inductance.</li>
|
|
</ul>
|
|
<ul>
|
|
<li>An ideal inductance (L) has no resistance and is represented
|
|
by the following symbol <img alt="magLRfig1"
|
|
src="mag_LR_fig1.jpg" height="30" width="143" align="middle">
|
|
.</li>
|
|
</ul>
|
|
<div align="left">
|
|
<ul>
|
|
<li><img alt="magLRfig2" src="mag_LR_fig2.jpg" height="207"
|
|
width="300" align="right">Placing an inductance (L) and a
|
|
resistance (R) in a simple circuit gives us the "LR" circuit,
|
|
at right. When the switch is closed to include the power
|
|
supply in the circuit we can apply Kirchoff's loop theorem to
|
|
obtain the following equation,</li>
|
|
</ul>
|
|
<div align="center"><img alt="magLReqn1" src="mag_LR_eqn1.jpg"
|
|
height="84" width="158"><br>
|
|
<blockquote>
|
|
<div align="left"><img alt="exclamation"
|
|
src="exclamation-icon.gif" height="30" width="31"> Note
|
|
that in applying the loop theorem we make use of the fact
|
|
that the induced emf across the inductance opposes the
|
|
changing emf that causes it. <br>
|
|
</div>
|
|
</blockquote>
|
|
<div align="left">
|
|
<ul>
|
|
<li><img alt="magLRfig3" src="mag_LR_fig3.jpg" height="240"
|
|
width="419" align="right">The solution of this
|
|
differential equation is given by,</li>
|
|
</ul>
|
|
<div align="center"><img alt="magLReqn2" src="mag_LR_eqn2.jpg"
|
|
height="57" width="236"><br>
|
|
<blockquote>
|
|
<div align="left">shown at right, where I<sub>max</sub> is
|
|
ε/R . Also shown is the time dependence of
|
|
the voltage across the inductance.<br>
|
|
</div>
|
|
</blockquote>
|
|
<div align="left">
|
|
<ul>
|
|
<li>The time constant of the inductance, given by
|
|
<img alt="magLReqn3" src="mag_LR_eqn3.jpg" height="38"
|
|
width="112" align="middle"> is a measure of the time
|
|
needed for the current in the circuit to reach 63% of
|
|
its maximum value.</li>
|
|
</ul>
|
|
<p><br>
|
|
</p>
|
|
<ul>
|
|
</ul>
|
|
<ul>
|
|
<li><img alt="magLRfig4" src="mag_LR_fig4.jpg"
|
|
height="247" width="270" align="right">Taking the
|
|
power supply out of the circuit leads to an even
|
|
simpler differential equation<br>
|
|
</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<div align="center"><img alt="magLReqn4" src="mag_LR_eqn4.jpg"
|
|
height="84" width="161"><br>
|
|
<blockquote>
|
|
<div align="left">the solution of which is<br>
|
|
<div align="center"><img alt="magLReqn5" src="mag_LR_eqn5.jpg"
|
|
height="57" width="180"><br>
|
|
<div align="left">shown at right. In this figure I<sub>0</sub>
|
|
= ε/R. </div>
|
|
</div>
|
|
</div>
|
|
</blockquote>
|
|
</div>
|
|
<ol>
|
|
</ol>
|
|
<div align="center">
|
|
<div align="left">
|
|
<div align="center">
|
|
<div align="left">
|
|
<div align="center">
|
|
<div align="left">
|
|
<div align="center">
|
|
<div align="left">
|
|
<div align="center">
|
|
<div align="left"> </div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<blockquote>
|
|
<div align="left"> </div>
|
|
</blockquote>
|
|
<div align="center">
|
|
<div align="left">
|
|
<div align="center">
|
|
<div align="left">
|
|
<div align="center">
|
|
<div align="left">
|
|
<div align="center">
|
|
<div align="left">
|
|
<div align="center">
|
|
<div align="center">
|
|
<div>
|
|
<div align="left">
|
|
<div align="center">
|
|
<div align="left"> </div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<blockquote>
|
|
<div align="left"> </div>
|
|
</blockquote>
|
|
</div>
|
|
</div>
|
|
<blockquote> </blockquote>
|
|
</div>
|
|
<div align="center"><img src="netbar.gif" height="40" width="100%"></div>
|
|
<div align="center">
|
|
<div align="center"> </div>
|
|
<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>This girl said she recognized me from
|
|
the vegetarian club, but I'd never met herbivore. </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>
|
|
<blockquote> </blockquote>
|
|
</div>
|
|
</body>
|
|
</html>
|