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<title>Magnetism - Ampere's Law - Physics 299</title>
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<h1>Ampere's Law <br>
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<div class="copy-paste-block"><font color="#ff0000"><i><span
class="bqQuoteLink">"A</span></i></font><font
color="#ff0000"><i><span class="bqQuoteLink"> fact is a simple
statement that everyone believes.&nbsp; It is innocent,
unless found guilty.&nbsp; A hypothesis is a novel
suggestion that no one wants to believe.&nbsp; It is
guilty, until found effective</span></i><span></span>"</font><br>
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<font color="#ff0000"><i> </i><font color="#000000">Edward Teller</font></font><br>
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<blockquote> </blockquote>
<ul>
<li><img alt="magamperefig3" src="mag_ampere_fig3.gif"
height="206" width="267" align="right">We have just seen that
the Biot-Savart Law is in some sense the magnetic equivalent of
Coulomb's Law.&nbsp; Is there a magnetic equivalent of Gauss's
Law ?&nbsp; The answer is (of course) yes - Ampere's Law. <br>
</li>
</ul>
<div align="center"><img alt="magampereeqn1"
src="mag_ampere_eqn1.jpg" height="60" width="180"><br>
<blockquote>
<div align="left">where <b>ds</b> is an element of length
around an arbitrary closed loop "C", called an Amperian loop
and the summation is over all currents passing <u><i><b>through</b></i></u>
the loop.<br>
<br>
<img alt="exclamation" src="exclamation-icon.gif" height="30"
width="31"> Currents passing "out of" the loop are defined
as positive, currents passing into the loop are negative,
whereas current which do not pass through the loop are not
included in the summation.<br>
<br>
<img alt="exclamation" src="exclamation-icon.gif" height="30"
width="31"> Ampere's Law forms part of the second of
Maxwell's equations.&nbsp;&nbsp; We will shortly adjust it
slightly (following Maxwell), to complete the second of
Maxwell's equations.&nbsp; Remember, Gauss's Law was the first
of Maxwell's equations.<br>
</div>
</blockquote>
<div align="left">
<ul>
<li>As a reminder, Gauss's Law appears below,<br>
</li>
</ul>
</div>
<img alt="elecgausseqn3" src="elec_gauss_eqn3.jpg" height="84"
width="233"><br>
<blockquote>
<div align="left"><img alt="exclamation"
src="exclamation-icon.gif" height="30" width="31"> Note that
Gauss's Law involves a <u><i><b>surface</b></i></u> integral
of <b>E</b> over a closed Gaussian surface "S".&nbsp;
Ampere's Law involves a <u><i><b>line</b></i></u> integral
around a closed Amperian loop "C".<br>
<br>
<img alt="exclamation" src="exclamation-icon.gif" height="30"
width="31"> Gauss's Law is valid for any arbitrary closed
Gaussian surface.&nbsp; Similarly Ampere's Law is valid for
any closed (Amperian) loop.<br>
</div>
</blockquote>
<div align="left">
<ul>
<li>Although Ampere's Law is true for any closed loop "C", it
is only useful to calculate <b>B</b> for some very
symmetric cases, where we already know (from symmetry) some
of the properties of <b>B</b>. </li>
</ul>
<div align="center"><img alt="divider"
src="divider_ornbarblu.gif" height="64" width="393"></div>
<div align="center">
<h3><u><font color="#cc33cc"><b>Simple Applications</b></font></u></h3>
</div>
<ul>
<li><b>B</b><b> due to an infinite straight current carrying
wire.</b> <br>
<ul>
<br>
<li><u><i><img alt="magamperefig4"
src="mag_ampere_fig4.gif" height="268" width="341"
align="right">Symmetry argument:</i></u>
<p>Since the wire is infinite, we know from the
Biot-Savart Law that <b>B</b> is perpendicular to <b>dl</b>
and <b>r</b> and thus lines of <b>B</b> must form
concentric circles around the current.&nbsp; Also, <b>B</b>
can, at most, depend only on the distance from the
wire, r.<br>
</p>
</li>
<li><u><i> Choice of Amperian Loop:</i></u>
<p> The Amperian loop is chosen so that <b>B</b> is
constant on the loop and in the same direction as <b>ds</b>
- that is a circle whose plane is perpendicular to the
wire and centered on the wire.&nbsp; This allows us to
take <b>B</b> "out of the integral". </p>
</li>
<li><u><i> Evaluation of <b>B</b>:</i></u>
<p> With the Amperian loop above we have<br>
</p>
<div align="center"><img alt="magampereeqn2"
src="mag_ampere_eqn2.jpg" height="60" width="493"><br>
<div align="left">so that<br>
<div align="center"><img alt="magampereeqn3"
src="mag_ampere_eqn3.jpg" height="71" width="99"><br>
<br>
<div align="left">directed "circumferentially"
around the loop, with the sense given by the
right-hand-rule described under the Biot-Savart
law.</div>
</div>
</div>
</div>
</li>
</ul>
<div align="center"><img alt="magamperefig5"
src="mag_ampere_fig5.jpg" height="246" width="313"><br>
</div>
<ul>
</ul>
</li>
<br>
</ul>
<hr size="2" width="100%">
<ul>
<br>
<li><b><img alt="magamperefig1" src="mag_ampere_fig1.jpg"
height="127" width="175" align="right">B</b><b> due to
an infinite solenoid</b>
<ul>
<br>
<li><u><i>Symmetry argument:</i></u>
<p>Since the solenoid is infinite, we conclude that <b>B</b>
is directed along the axis of the solenoid.&nbsp;
Also, <b>B</b> can, at most, depend only on the
distance from the axis of the solenoid.&nbsp; For an
infinite solenoid <b>B</b> = 0 outside the solenoid.<br>
</p>
</li>
<li><u><i> <img alt="magamperefig2"
src="mag_ampere_fig2.jpg" height="342" width="304"
align="right">Choice of Amperian Loop:</i></u>
<p> The rectangular Amperian loop (at right) is chosen
so that <b>B</b> is constant on the two sides
parallel to the solenoid axis and perpendicular to the
<b>ds</b> on the other two sides. This allows us to
take <b>B</b> "out of the integral". </p>
</li>
<li><u><i> Evaluation of <b>B</b>:</i></u>
<p> With the Amperian loop above we have<br>
</p>
<div align="center"><img alt="magampereeqn4"
src="mag_ampere_eqn4.jpg" height="49" width="542"><br>
<div align="left">so that<br>
<div align="center"><img alt="magampereeqn5"
src="mag_ampere_eqn5.jpg" height="46"
width="117"><br>
<br>
<div align="left">where n is the number of turns
per unit length and the field is directed along
the solenoid axis as shown. (Use
right-hand-rule)&nbsp;</div>
</div>
</div>
</div>
</li>
</ul>
<img alt="exclamation" src="exclamation-icon.gif"
height="30" width="31"> Note that, the above analysis is
true only for an <i><b>infinite</b></i> solenoid.&nbsp; For
a real solenoid it is a good approximation inside, away from
the ends.&nbsp; The <b>B</b> field outside is not zero, but
much smaller than the field inside.</li>
</ul>
<blockquote><img alt="exclamation" src="exclamation-icon.gif"
height="30" width="31"> The shape of the <b>B</b> field due
to a solenoid is the same as that of a bar magnet and a
magnetic dipole.<br>
<div align="center"><img alt="magamperefig6"
src="mag_ampere_fig6.gif" height="257" width="437"><br>
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</blockquote>
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<font color="#ff0000"><i>What does a clock do when it's hungry ?
It goes back four seconds. </i></font><br>
<br>
&nbsp;<img src="celticbar.gif" height="22" width="576"> <br>
&nbsp;
<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>
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