davisnotes/mag_displacement.html

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<title>Magnetism - Magnetic Energy - Physics 299</title>
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<h1> <img src="ULPhys1.gif" height="50" align="texttop"
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<h1>Displacement Current<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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<ul>
<li> Having discussed Gauss' Law for Magnetism, we now have four
equations describing electromagnetism,</li>
</ul>
<div align="center">Gauss' Law: &nbsp; <img alt="elecgausseqn3"
src="elec_gauss_eqn3.jpg" height="84" align="middle" width="233"><br>
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<div align="center">Ampere's Law:&nbsp; <img alt="magampereeqn1"
src="mag_ampere_eqn1.jpg" height="60" align="middle" width="180"><br>
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<div align="center"> Faraday's Law:&nbsp; <img alt="magfaradayeqn6"
src="mag_faraday_eqn6.jpg" height="58" align="middle"
width="297"><br>
<br>
Nobody's Law:&nbsp; <img alt="magmonopolefig5"
src="mag_monopole_fig5.jpg" height="48" align="middle"
width="171"><br>
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<ul>
<li>Looking carefully at these equations, the two flux
equations are now consistent with our physical understanding
of electric and magnetic charges.&nbsp; <br>
</li>
</ul>
<ul>
<li>The line integrals are a different matter.&nbsp; The right
hand side of Ampere's Law has a summation over electric
currents.&nbsp; Naively we would expect a similar sum over
"magnetic currents" on the right hand side of Faraday's
Law.&nbsp; But since there are no magnetic charges,
"magnetic currents" do not exist and the necessity for such
an additional term disappears.</li>
</ul>
<ul>
<li>However, in words, Faraday's Law states that</li>
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<blockquote>
<p><b>"A changing magnetic field ( </b><b><img
alt="magdisplacementeqn1"
src="mag_displacement_eqn1.jpg" height="31"
align="middle" width="45"></b><b> ) gives rise to an
electric field ( </b><b><img alt="magdisplacementeqn2"
src="mag_displacement_eqn2.jpg" height="29"
align="middle" width="54"></b><b> </b>)"<br>
</p>
<p align="left">For a more complete correspondence between <b>E</b>
and <b>B</b> we would expect a term added to the right
hand side of Ampere's Law which indicates,<br>
</p>
<p align="center">&nbsp;<b>"A changing electric field ( </b><img
alt="magdisplacementeqn3"
src="mag_displacement_eqn3.jpg" height="31"
align="middle" width="45">)<b>&nbsp; gives rise to a
magnetic field ( </b><img alt="magdisplacementeqn4"
src="mag_displacement_eqn4.jpg" height="29"
align="middle" width="54"><b> )</b>"<br>
</p>
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<ul>
<li>Maxwell proposed this addition, the existence of which
is now physically verified.&nbsp; Furthermore, Maxwell
showed that this additional term, known as the
displacement current, is essential for the description
of electromagnetic waves.</li>
</ul>
<ul>
<li>The form of the displacement current term can be
obtained by the argument described below.</li>
</ul>
<ul>
<li>We have seen that current and current density are
related by the following equation</li>
</ul>
<div align="center"><img alt="eleccurrenteqn3"
src="elec_current_eqn3.jpg" height="38" width="103"><br>
<blockquote>
<div align="left">Suppose we consider a closed surface,
then charge conservation tells us that this flux
integral must be zero.&nbsp; Charge cannot be created
or destroyed inside the closed surface, therefore,<br>
<div align="center"><img alt="magdisplacementeqn5"
src="mag_displacement_eqn5.jpg" height="46"
width="117"><br>
<div align="left"><img alt="exclamation"
src="exclamation-icon.gif" height="30"
width="31"> This equation can be considered as a
formal statement of conservation of charge.<br>
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<ul>
<li><img alt="magdisplacementfig1"
src="mag_displacement_fig1.jpg" height="305"
align="right" width="284">Consider a parallel
plate capacitor (at right).&nbsp; The closed surface
over which we will apply the above integral is S<sub>1</sub>
and S<sub>2</sub>.&nbsp; Current I passes in though
S<sub>1</sub>, but since S<sub>2</sub> is between
the capacitor plates, no current passes out of the
closed surface.&nbsp; As it stands, we have violated
conservation of charge... What to do&nbsp; ?&nbsp;
Propose that there is an equal current passing out
of the closed surface through S<sub>2</sub> - the
"displacement current".</li>
</ul>
<ul>
<li>Now</li>
</ul>
<div align="center"><img alt="magdisplacementeqn6"
src="mag_displacement_eqn6.jpg" height="65"
width="441"><br>
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<blockquote>
<div align="left">Therefore, if we define<br>
<div align="center"><img alt="magdisplacementeqn7"
src="mag_displacement_eqn7.jpg" height="67"
width="126"><br>
<div align="left">as the displacement current,
by writing<br>
<div align="center"><img
alt="magdisplacementeqn8"
src="mag_displacement_eqn8.jpg"
height="39" width="135"><br>
<div align="left">we can ensure charge is
conserved.<br>
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<ul>
<li>Therefore, including the displacement
current, Ampere's Law becomes,</li>
</ul>
<div align="center"><img alt=""
src="mag_displacement_eqn9.jpg"
height="64" width="252"></div>
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<font color="#ff0000"><i>This girl said she recognized me from
the vegetarian club, but I'd never met herbivore. </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>
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