441 lines
21 KiB
HTML
441 lines
21 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>Magnetism - Faraday's Law of Induction - 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" vlink="#551a8b" alink="#ff0000">
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<center>
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<h1> <img src="ULPhys1.gif" height="50" width="189"
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align="texttop"></h1>
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</center>
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<center>
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<h1>Faraday's Law of Induction<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" width="100%" align="middle">
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<blockquote> </blockquote>
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<ul>
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<li>So far we have treated electricity and magnetism as almost
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separate subjects. We now begin to discuss phenomena which
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show that electricity and magnetism are inextricably connected,
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hence the term <i><b>electromagnetism</b></i>. The first
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of these properties is known as <i><b>Faraday's Law of
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Induction</b></i>.</li>
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</ul>
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<blockquote><img alt="exclamation" src="exclamation-icon.gif"
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height="30" width="31"> Formally, time <i>independent</i>
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electrical and magnetic properties can be described by considering
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electricity and magnetism as largely separate phenomena.
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However, when time dependence becomes part of the "equation"
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we find that electrical and magnetic properties become
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inextricably linked - electromagnetism.<br>
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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>This law is conveniently written in terms of magnetic flux,
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which is defined in the same way as electric flux.</li>
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</ul>
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<div align="center"><img alt="magfaradayeqn1"
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src="mag_faraday_eqn1.jpg" height="47" width="144"><br>
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<blockquote>
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<div align="left">where S is the surface over which the flux is
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evaluated.<br>
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<br>
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For constant <b>B,</b> perpendicular to the surface, Φ<sub>B</sub>
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= BA where A is the surface area of S.<br>
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<br>
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<img alt="exclamation" src="exclamation-icon.gif" height="30"
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width="31"> The magnetic flux, Φ<sub>B</sub>, is so
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important it has its own unit the Weber - 1 Weber
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= 1 T.m<sup>2</sup> . In the early days of
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electromagnetism it was common to measure the magnetic (<b>B</b>)
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field in Weber/m<sup>2</sup> .<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>In term of the magnetic flux Faraday's Law of Induction is
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given by,</li>
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</ul>
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<div align="center"><img alt="magfaradayeqn2"
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src="mag_faraday_eqn2.jpg" height="66" width="108"><br>
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<blockquote>
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<div align="left">The induced electromotive force (<i>emf</i>)
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in a circuit is equal to the rate of change of magnetic
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flux through the circuit.<br>
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<br>
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<img alt="exclamation" src="exclamation-icon.gif"
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height="30" width="31"> An <i>emf</i> is not a force,
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rather it can be considered as the voltage <i>induced</i>
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in a closed circuit.<br>
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<br>
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<img alt="exclamation" src="exclamation-icon.gif"
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height="30" width="31"> Faraday experimentally
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determined his law in the form presented above.<br>
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<br>
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<hr size="2" width="100%"><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>One of the easiest ways to change the magnetic flux
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through a circuit is to move a permanent (bar) magnet
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towards or away from the circuit as shown in the
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diagrams below.</li>
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</ul>
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<div align="center"><img alt="magfaradayfig1"
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src="mag_faraday_fig1.jpg" height="401" width="698"><br>
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<blockquote>
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<div align="left">(a) <img alt="magfaradayfig2"
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src="mag_faraday_fig2.jpg" height="191" width="267"
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align="right">Magnetic flux passes through the
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circuit, but does not change with time, so there is no
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induced <i>emf</i> and so no induced current.<br>
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<br>
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(b) The flux through the circuit increases with
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time causing an induced <i>emf</i> and current.<br>
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<br>
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(c) As the magnet moves faster the rate of
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change of flux with time is increased causing a larger
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<i>emf</i> and current.<br>
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<br>
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(d) When the magnet moves away from the circuit
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the flux decreases with time so the induced <i>emf</i>
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and current are reversed.<br>
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<br>
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<hr size="2" width="100%"><br>
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<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 origin of the changing magnetic flux (field)
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is not limited to permanent magnets. The
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magnetic field due to a second circuit can produce a
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similar effect, as described in the examples
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below. <br>
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</li>
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</ul>
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<div align="center">
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<blockquote>
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<div align="left"><img alt="magfaradayfig3"
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src="mag_faraday_fig3.jpg" height="95"
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width="267" align="right">In the diagram at
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right the current in the left circuit is constant,
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but the flux through the other circuit increases
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as the two circuits get closer.<br>
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<br>
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<br>
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<br>
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<br>
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<img alt="magfaradayfig4"
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src="mag_faraday_fig4.jpg" height="90"
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width="264" align="left">In the situation at
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left both circuits are stationary. The
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current in the left circuit is initially zero, but
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rapidly increases to a constant value when the
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switch is closed. As the current reaches its
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final (constant) value the flux through the right
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circuit is increasing with time, thus by Faraday's
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Law, causing a brief pulse of induced
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current in the second circuit. When the
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switch is opened the flux in the right circuit
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rapidly decreases causing a short induced current
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pulse in the opposite direction.<br>
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<br>
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<hr size="2" width="100%"></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"> Important ! In both the
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above examples a magnetic field (<b>B</b>)
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changing with time results in a changing
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magnetic flux. But it is possible to have
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a changing flux with a constant <b>B</b> if the
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cross sectional area - <b>dA</b> - can be made
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to change with time as in the case of the
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(electric) generator.<br>
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</li>
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</ul>
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<div align="center"><img alt="magfaradayfig5"
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src="mag_faraday_fig5.jpg" height="295"
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width="348"></div>
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<ul>
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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="393"><br>
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<div align="center">
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<blockquote>
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<div>
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<h3><font color="#cc33cc"><u><b>LENZ'S LAW</b></u></font></h3>
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</div>
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</blockquote>
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<div>
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<div align="left">
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<ul>
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<li>Mathematically the negative sign in
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Faraday's Law tells us about the
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direction of the induced (<i>emf</i>)
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current. Practically we use Lenz's
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Law to determine the direction in
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specific cases. <br>
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</li>
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</ul>
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</div>
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</div>
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<blockquote>
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<div><img alt="confused"
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src="confused_smiley.gif" height="22"
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width="15"> <big><i><b>"The induced
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current will appear in such a
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direction that it opposes the change
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that produced it"</b></i></big>
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<img alt="confused"
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src="confused_smiley.gif" height="22"
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width="15"><br>
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<br>
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<b>Confusing ? Yes !</b><br>
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<br>
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</div>
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</blockquote>
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<div>
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<div align="left">
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<div align="center">
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<img alt="magfaradayfig7"
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src="mag_faraday_fig7.jpg" height="526"
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width="684"><br>
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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><img alt="exclamation"
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src="exclamation-icon.gif" height="30"
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width="31"> The induced (flux)
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magnetic field (associated with the
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induced current) does not necessarily
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oppose the field which causes the change
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in flux, rather it opposes the <b>CHANGE</b>
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in this field.<br>
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</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"> Lenz's Law always ensures
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that there is a force resisting the
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motion of the magnet. It is the
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work done against this force which
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appears as the energy of the moving
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charges of the induced current. <br>
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</li>
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</ul>
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<p align="center"><img alt="magfaradayfig6"
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src="mag_faraday_fig6.jpg" height="260"
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width="275"></p>
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<ul>
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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="393"><br>
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<br>
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<font color="#cc33cc"><big><u><b>FARADAY'S
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LAW</b></u><u><b> = MAXWELL
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EQUATION</b></u></big></font><br>
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<div align="left">
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<ul>
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<li>Whenever there is an induced
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electric current there must also be
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an induced electric field, <b>E</b>.
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The work dW done by this induced
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field moving a charge q<sub>0</sub>
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a distance <b>ds</b> around a loop
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is given by,</li>
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</ul>
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<div align="center"><img
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alt="magfaradayeqn3"
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src="mag_faraday_eqn3.jpg"
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height="36" width="288"><br>
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<blockquote>
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<div align="left">where dε is the
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potential difference in <b>ds</b>.<br>
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<br>
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Therefore, <br>
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<div align="center"><img
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alt="magfaradayeqn4"
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src="mag_faraday_eqn4.jpg"
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height="31" width="117"><br>
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<div align="left">So that the <i>emf
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</i>around the whole loop is<br>
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<div align="center"><img
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alt="magfaradayeqn5"
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src="mag_faraday_eqn5.jpg"
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height="46" width="189"><br>
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<br>
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<div align="left">Equating
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this <i>emf</i> to that
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given by Faraday's Law we
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obtain the integral form
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of Faraday's Law, the
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third of Maxwell's
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equations we have
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encountered so far,<br>
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<div align="center"><img
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alt="magfaradayeqn6"
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src="mag_faraday_eqn6.jpg"
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height="58"
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width="297"><br>
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<br>
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<div align="left"><img
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alt="exclamation"
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src="exclamation-icon.gif"
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height="30"
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width="31">
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Note that the line
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integral of <b>E</b>
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must be round a closed
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loop (circuit).<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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</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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<div align="center">
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<div align="left">
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<ul>
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<li>Written in the
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above form the
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relationship between
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the <b>E</b> and <b>B</b>
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fields is clear</li>
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</ul>
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|
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<p align="center"><big><i><big><b>"A
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magnetic field
|
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changing with
|
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time induces
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an electric
|
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field"</b></big></i></big><br>
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</p>
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<blockquote>
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<div align="left">Shortly
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we will see that the
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reverse of this
|
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statement is also
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true.<br>
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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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</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"> </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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</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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||
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</div>
|
||
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</div>
|
||
|
|
<blockquote>
|
||
|
|
<div align="left"> </div>
|
||
|
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</blockquote>
|
||
|
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</div>
|
||
|
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</div>
|
||
|
|
<blockquote> </blockquote>
|
||
|
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</div>
|
||
|
|
<div align="center"><img src="netbar.gif" height="40" width="100%"></div>
|
||
|
|
<div align="center">
|
||
|
|
<div align="center"> </div>
|
||
|
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<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>They told me I had type A blood, but it
|
||
|
|
was a Type O.</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>
|
||
|
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<p><br>
|
||
|
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</p>
|
||
|
|
<blockquote> </blockquote>
|
||
|
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</div>
|
||
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</body>
|
||
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</html>
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