244 lines
11 KiB
HTML
244 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>Magnetism - Ampere's Law - 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>Ampere's Law <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><img alt="magamperefig3" src="mag_ampere_fig3.gif"
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height="206" width="267" align="right">We have just seen that
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the Biot-Savart Law is in some sense the magnetic equivalent of
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Coulomb's Law. Is there a magnetic equivalent of Gauss's
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Law ? The answer is (of course) yes - Ampere's Law. <br>
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</li>
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</ul>
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<div align="center"><img alt="magampereeqn1"
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src="mag_ampere_eqn1.jpg" height="60" width="180"><br>
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<blockquote>
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<div align="left">where <b>ds</b> is an element of length
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around an arbitrary closed loop "C", called an Amperian loop
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and the summation is over all currents passing <u><i><b>through</b></i></u>
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the loop.<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"> Currents passing "out of" the loop are defined
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as positive, currents passing into the loop are negative,
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whereas current which do not pass through the loop are not
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included in the summation.<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"> Ampere's Law forms part of the second of
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Maxwell's equations. We will shortly adjust it
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slightly (following Maxwell), to complete the second of
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Maxwell's equations. Remember, Gauss's Law was the first
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of Maxwell's equations.<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>As a reminder, Gauss's Law appears below,<br>
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</li>
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</ul>
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</div>
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<img alt="elecgausseqn3" src="elec_gauss_eqn3.jpg" height="84"
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width="233"><br>
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<blockquote>
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<div align="left"><img alt="exclamation"
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src="exclamation-icon.gif" height="30" width="31"> Note that
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Gauss's Law involves a <u><i><b>surface</b></i></u> integral
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of <b>E</b> over a closed Gaussian surface "S".
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Ampere's Law involves a <u><i><b>line</b></i></u> integral
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around a closed Amperian loop "C".<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"> Gauss's Law is valid for any arbitrary closed
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Gaussian surface. Similarly Ampere's Law is valid for
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any closed (Amperian) loop.<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>Although Ampere's Law is true for any closed loop "C", it
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is only useful to calculate <b>B</b> for some very
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symmetric cases, where we already know (from symmetry) some
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of the properties of <b>B</b>. </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" width="393"></div>
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<div align="center">
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<h3><u><font color="#cc33cc"><b>Simple Applications</b></font></u></h3>
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</div>
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<ul>
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<li><b>B</b><b> due to an infinite straight current carrying
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wire.</b> <br>
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<ul>
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<br>
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<li><u><i><img alt="magamperefig4"
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src="mag_ampere_fig4.gif" height="268" width="341"
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align="right">Symmetry argument:</i></u>
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<p>Since the wire is infinite, we know from the
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Biot-Savart Law that <b>B</b> is perpendicular to <b>dl</b>
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and <b>r</b> and thus lines of <b>B</b> must form
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concentric circles around the current. Also, <b>B</b>
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can, at most, depend only on the distance from the
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wire, r.<br>
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</p>
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</li>
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<li><u><i> Choice of Amperian Loop:</i></u>
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<p> The Amperian loop is chosen so that <b>B</b> is
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constant on the loop and in the same direction as <b>ds</b>
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- that is a circle whose plane is perpendicular to the
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wire and centered on the wire. This allows us to
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take <b>B</b> "out of the integral". </p>
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</li>
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<li><u><i> Evaluation of <b>B</b>:</i></u>
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<p> With the Amperian loop above we have<br>
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</p>
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<div align="center"><img alt="magampereeqn2"
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src="mag_ampere_eqn2.jpg" height="60" width="493"><br>
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<div align="left">so that<br>
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<div align="center"><img alt="magampereeqn3"
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src="mag_ampere_eqn3.jpg" height="71" width="99"><br>
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<br>
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<div align="left">directed "circumferentially"
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around the loop, with the sense given by the
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right-hand-rule described under the Biot-Savart
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law.</div>
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</div>
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</div>
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</div>
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</li>
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</ul>
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<div align="center"><img alt="magamperefig5"
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src="mag_ampere_fig5.jpg" height="246" width="313"><br>
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</div>
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<ul>
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</ul>
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</li>
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<br>
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</ul>
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<hr size="2" width="100%">
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<ul>
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<br>
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<li><b><img alt="magamperefig1" src="mag_ampere_fig1.jpg"
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height="127" width="175" align="right">B</b><b> due to
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an infinite solenoid</b>
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<ul>
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<br>
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<li><u><i>Symmetry argument:</i></u>
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<p>Since the solenoid is infinite, we conclude that <b>B</b>
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is directed along the axis of the solenoid.
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Also, <b>B</b> can, at most, depend only on the
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distance from the axis of the solenoid. For an
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infinite solenoid <b>B</b> = 0 outside the solenoid.<br>
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</p>
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</li>
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<li><u><i> <img alt="magamperefig2"
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src="mag_ampere_fig2.jpg" height="342" width="304"
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align="right">Choice of Amperian Loop:</i></u>
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<p> The rectangular Amperian loop (at right) is chosen
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so that <b>B</b> is constant on the two sides
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parallel to the solenoid axis and perpendicular to the
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<b>ds</b> on the other two sides. This allows us to
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take <b>B</b> "out of the integral". </p>
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</li>
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<li><u><i> Evaluation of <b>B</b>:</i></u>
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<p> With the Amperian loop above we have<br>
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</p>
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<div align="center"><img alt="magampereeqn4"
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src="mag_ampere_eqn4.jpg" height="49" width="542"><br>
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<div align="left">so that<br>
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<div align="center"><img alt="magampereeqn5"
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src="mag_ampere_eqn5.jpg" height="46"
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width="117"><br>
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<br>
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<div align="left">where n is the number of turns
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per unit length and the field is directed along
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the solenoid axis as shown. (Use
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right-hand-rule) </div>
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</div>
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</div>
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</div>
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</li>
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</ul>
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<img alt="exclamation" src="exclamation-icon.gif"
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height="30" width="31"> Note that, the above analysis is
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true only for an <i><b>infinite</b></i> solenoid. For
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a real solenoid it is a good approximation inside, away from
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the ends. The <b>B</b> field outside is not zero, but
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much smaller than the field inside.</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"> The shape of the <b>B</b> field due
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to a solenoid is the same as that of a bar magnet and a
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magnetic dipole.<br>
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<div align="center"><img alt="magamperefig6"
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src="mag_ampere_fig6.gif" height="257" width="437"><br>
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</div>
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</blockquote>
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<ul>
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</ul>
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<img src="netbar.gif" height="40" width="100%"></div>
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<div align="center"> </div>
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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>What does a clock do when it's hungry ?
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It goes back four seconds. </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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<blockquote> </blockquote>
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</div>
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</body>
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</html>
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