260 lines
13 KiB
HTML
260 lines
13 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 - Biot-Savart 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);" alink="#ff0000" link="#0000ee" vlink="#551a8b">
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<center>
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<h1> <img src="ULPhys1.gif" align="texttop" height="50"
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width="189"></h1>
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</center>
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<center>
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<h1>The Law of Biot-Savart<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" align="middle" height="40" width="100%">
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<blockquote> </blockquote>
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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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<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>We have seen that a magnetic field exerts
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a force on a moving charge or current, just
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as an electric field exerts a force on a
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charge.</li>
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</ul>
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<div align="center"><img
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alt="magforcechargeeqn1"
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src="mag_force_charge_eqn1.jpg"
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align="middle" height="33" width="135">
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and
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<img
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alt="magforcecurrenteqn3"
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src="mag_force_current_eqn3.jpg"
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align="middle" height="31" width="126"><br>
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<br>
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<img alt="magbiotsavarteqn1"
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src="mag_biotsavart_eqn1.jpg" height="31"
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width="81"><br>
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<div align="left">
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<ul>
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<li>In the electric case Coulomb's Law
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gives us a method of determining <b>E</b>
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from an arbitrary distribution of
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charges. For a point charge we
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have<br>
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</li>
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</ul>
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<div align="center"><img
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alt="magbiotsavarteqn3"
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src="mag_biotsavart_eqn2.jpg"
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height="61" width="180"><br>
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<blockquote>
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<div align="left">where <img
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alt="magbiotsavarteqn3"
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src="mag_biotsavart_eqn3.jpg"
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height="27" width="19">is a unit
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vector from the point charge to
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location at which the <b>E</b> field
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is to be determined.<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><img alt="magbiotsavartfig1"
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src="mag_biotsavart_fig1.jpg"
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align="right" height="354"
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width="341">The Biot-Savart Law
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provides a general method of
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determining the <b>B</b> field from
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an arbitrary current distribution.</li>
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</ul>
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<div align="center"><img
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alt="magbiotsavarteqn4"
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src="mag_biotsavart_eqn4.jpg"
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height="57" width="144"><br>
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<br>
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<blockquote>
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<div align="left">where μ<sub>0</sub>
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is the permeability of the vacuum
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(free space) = 4π x 10<sup>-7</sup>
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T.m/A, dl is a "<i>current element</i>"
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directed along the current in the
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wire and <img
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alt="magbiotsavarteqn3"
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src="mag_biotsavart_eqn3.jpg"
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height="27" width="19">is a unit
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vector from <b>dl</b> to where
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the <b>B</b> field is to be
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calculated as in the diagram at
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right.<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>There are a number of
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important points to be aware of:</li>
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</ul>
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<blockquote><img alt="exclamation"
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src="exclamation-icon.gif"
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height="30" width="31"> The law
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must be written in differential
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form since there is no such thing
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as a "point current". Note
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that a moving point charge is
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equivalent to a current and a
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stationary point charge does not
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create a magnetic field. <br>
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<br>
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<img alt="exclamation"
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src="exclamation-icon.gif"
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height="30" width="31">
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Due to the cross product <b>dB</b>
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is at right angles to both <b>dl</b>
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and <img alt="magbiotsavarteqn3"
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src="mag_biotsavart_eqn3.jpg"
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height="27" width="19">.
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You can't avoid considering 3
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dimensions.<br>
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<br>
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<img alt="exclamation"
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src="exclamation-icon.gif"
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height="30" width="31"> <img
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alt="magbiotsavartfig2"
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src="mag_biotsavart_fig2.jpg"
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align="right" height="274"
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width="294">The "sense" of <b>dB</b>
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(into or out of the plane in the
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above diagram) is determined by
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the basic definition of a vector
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cross product or equivalently
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another "<i>right-hand-rule</i>"
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shown at right.<br>
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<br>
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<img alt="exclamation"
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src="exclamation-icon.gif"
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height="30" width="31"> (μ<sub>0</sub>/4π)
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plays a similar role to the
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Coulomb constant, k (= 1/4πε<sub>0</sub>).
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But compare the values of μ<sub>0</sub>
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and k; this is an indication of
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the relative strengths of the
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electric and magnetic interactions
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of charges.<br>
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<br>
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<img alt="exclamation"
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src="exclamation-icon.gif"
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height="30" width="31">
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Similar to our initial statement
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of Coulomb's Law, the above
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expression of the Biot-Savart Law
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gives <b>dB</b> due to a current
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element in a vacuum. For
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real world applications wheres the
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current is not in a vacuum a
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slight adjustment must be made to
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take into account the magnetic
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properties of the medium.
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This can be done in a similar
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manner to the adjustment of
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Coulomb's Law in a <a
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href="elec_dielectrics.html">dielectric
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medium</a>, but is beyond the
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scope of this course.<br>
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<br>
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</blockquote>
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<ul>
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<li>Needless to say determining
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the magnetic field is typically
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significantly more complicated
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than the electric field. <img
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alt="sadface"
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src="sadface.jpg"
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align="middle" height="52"
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width="52"><br>
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</li>
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</ul>
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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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<img src="netbar.gif" height="40" width="100%"></div>
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<div align="center"> </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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</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>Energizer bunny arrested. Charged
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with battery. </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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