288 lines
12 KiB
HTML
288 lines
12 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 - Magnetic Energy - 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>Displacement Current<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%">
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<blockquote> </blockquote>
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<ul>
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<li> Having discussed Gauss' Law for Magnetism, we now have four
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equations describing electromagnetism,</li>
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</ul>
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<div align="center">Gauss' Law: <img alt="elecgausseqn3"
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src="elec_gauss_eqn3.jpg" height="84" align="middle" width="233"><br>
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<br>
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</div>
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<div align="center">Ampere's Law: <img alt="magampereeqn1"
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src="mag_ampere_eqn1.jpg" height="60" align="middle" width="180"><br>
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<br>
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</div>
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<div align="center"> Faraday's Law: <img alt="magfaradayeqn6"
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src="mag_faraday_eqn6.jpg" height="58" align="middle"
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width="297"><br>
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<br>
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Nobody's Law: <img alt="magmonopolefig5"
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src="mag_monopole_fig5.jpg" height="48" align="middle"
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width="171"><br>
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<br>
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<div align="left">
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<ul>
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<li>Looking carefully at these equations, the two flux
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equations are now consistent with our physical understanding
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of electric and magnetic charges. <br>
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</li>
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</ul>
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<ul>
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<li>The line integrals are a different matter. The right
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hand side of Ampere's Law has a summation over electric
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currents. Naively we would expect a similar sum over
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"magnetic currents" on the right hand side of Faraday's
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Law. But since there are no magnetic charges,
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"magnetic currents" do not exist and the necessity for such
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an additional term disappears.</li>
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</ul>
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<ul>
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<li>However, in words, Faraday's Law states that</li>
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</ul>
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<div align="center">
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<blockquote>
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<p><b>"A changing magnetic field ( </b><b><img
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alt="magdisplacementeqn1"
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src="mag_displacement_eqn1.jpg" height="31"
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align="middle" width="45"></b><b> ) gives rise to an
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electric field ( </b><b><img alt="magdisplacementeqn2"
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src="mag_displacement_eqn2.jpg" height="29"
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align="middle" width="54"></b><b> </b>)"<br>
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</p>
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<p align="left">For a more complete correspondence between <b>E</b>
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and <b>B</b> we would expect a term added to the right
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hand side of Ampere's Law which indicates,<br>
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</p>
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<p align="center"> <b>"A changing electric field ( </b><img
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alt="magdisplacementeqn3"
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src="mag_displacement_eqn3.jpg" height="31"
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align="middle" width="45">)<b> gives rise to a
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magnetic field ( </b><img alt="magdisplacementeqn4"
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src="mag_displacement_eqn4.jpg" height="29"
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align="middle" width="54"><b> )</b>"<br>
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</p>
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</blockquote>
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<div align="left">
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<ul>
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<li>Maxwell proposed this addition, the existence of which
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is now physically verified. Furthermore, Maxwell
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showed that this additional term, known as the
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displacement current, is essential for the description
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of electromagnetic waves.</li>
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</ul>
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<ul>
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<li>The form of the displacement current term can be
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obtained by the argument described below.</li>
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</ul>
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<ul>
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<li>We have seen that current and current density are
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related by the following equation</li>
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</ul>
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<div align="center"><img alt="eleccurrenteqn3"
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src="elec_current_eqn3.jpg" height="38" width="103"><br>
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<blockquote>
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<div align="left">Suppose we consider a closed surface,
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then charge conservation tells us that this flux
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integral must be zero. Charge cannot be created
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or destroyed inside the closed surface, therefore,<br>
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<div align="center"><img alt="magdisplacementeqn5"
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src="mag_displacement_eqn5.jpg" height="46"
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width="117"><br>
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<div align="left"><img alt="exclamation"
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src="exclamation-icon.gif" height="30"
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width="31"> This equation can be considered as a
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formal statement of conservation of charge.<br>
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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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<ul>
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<li><img alt="magdisplacementfig1"
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src="mag_displacement_fig1.jpg" height="305"
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align="right" width="284">Consider a parallel
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plate capacitor (at right). The closed surface
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over which we will apply the above integral is S<sub>1</sub>
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and S<sub>2</sub>. Current I passes in though
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S<sub>1</sub>, but since S<sub>2</sub> is between
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the capacitor plates, no current passes out of the
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closed surface. As it stands, we have violated
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conservation of charge... What to do ?
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Propose that there is an equal current passing out
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of the closed surface through S<sub>2</sub> - the
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"displacement current".</li>
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</ul>
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<ul>
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<li>Now</li>
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</ul>
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<div align="center"><img alt="magdisplacementeqn6"
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src="mag_displacement_eqn6.jpg" height="65"
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width="441"><br>
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<br>
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<blockquote>
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<div align="left">Therefore, if we define<br>
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<div align="center"><img alt="magdisplacementeqn7"
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src="mag_displacement_eqn7.jpg" height="67"
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width="126"><br>
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<div align="left">as the displacement current,
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by writing<br>
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<div align="center"><img
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alt="magdisplacementeqn8"
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src="mag_displacement_eqn8.jpg"
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height="39" width="135"><br>
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<div align="left">we can ensure charge is
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conserved.<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>Therefore, including the displacement
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current, Ampere's Law becomes,</li>
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</ul>
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<div align="center"><img alt=""
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src="mag_displacement_eqn9.jpg"
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height="64" width="252"></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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<ol>
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</ol>
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<div align="left"> </div>
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<blockquote>
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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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<blockquote> </blockquote>
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</div>
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<div align="center"><img src="netbar.gif" height="40" width="100%"></div>
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<div align="center">
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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>This girl said she recognized me from
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the vegetarian club, but I'd never met herbivore. </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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</body>
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</html>
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