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<title>Magnetism - Faraday's Law of Induction - Physics 299</title>
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<h1> <img src="ULPhys1.gif" height="50" width="189"
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<h1>Faraday's Law of Induction<br>
</h1>
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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>
</div>
<font color="#ff0000"><i> </i><font color="#000000">Edward Teller</font></font><br>
</center>
<img src="netbar.gif" height="40" width="100%" align="middle">
<blockquote> </blockquote>
<ul>
<li>So far we have treated electricity and magnetism as almost
separate subjects.&nbsp; We now begin to discuss phenomena which
show that electricity and magnetism are inextricably connected,
hence the term <i><b>electromagnetism</b></i>.&nbsp; The first
of these properties is known as <i><b>Faraday's Law of
Induction</b></i>.</li>
</ul>
<blockquote><img alt="exclamation" src="exclamation-icon.gif"
height="30" width="31">&nbsp; Formally, time <i>independent</i>
electrical and magnetic properties can be described by considering
electricity and magnetism as largely separate phenomena.&nbsp;
However, when time dependence becomes part of the "equation"&nbsp;
we find that electrical and magnetic properties become
inextricably linked - electromagnetism.<br>
</blockquote>
<ul>
</ul>
<ul>
<li>This law is conveniently written in terms of magnetic flux,
which is defined in the same way as electric flux.</li>
</ul>
<div align="center"><img alt="magfaradayeqn1"
src="mag_faraday_eqn1.jpg" height="47" width="144"><br>
<blockquote>
<div align="left">where S is the surface over which the flux is
evaluated.<br>
<br>
For constant <b>B,</b> perpendicular to the surface, &#934;<sub>B</sub>
= BA where A is the surface area of S.<br>
<br>
<img alt="exclamation" src="exclamation-icon.gif" height="30"
width="31">&nbsp; The magnetic flux, &#934;<sub>B</sub>, is so
important it has its own unit the Weber&nbsp; -&nbsp; 1 Weber
= 1 T.m<sup>2</sup> .&nbsp; In the early days of
electromagnetism it was common to measure the magnetic (<b>B</b>)
field in Weber/m<sup>2</sup> .<br>
</div>
</blockquote>
<div align="left">
<ul>
<li>In term of the magnetic flux Faraday's Law of Induction is
given by,</li>
</ul>
<div align="center"><img alt="magfaradayeqn2"
src="mag_faraday_eqn2.jpg" height="66" width="108"><br>
<blockquote>
<div align="left">The induced electromotive force (<i>emf</i>)
in a circuit is equal to the rate of change of magnetic
flux through the circuit.<br>
<br>
<img alt="exclamation" src="exclamation-icon.gif"
height="30" width="31"> An <i>emf</i> is not a force,
rather it can be considered as the voltage <i>induced</i>
in a closed circuit.<br>
<br>
<img alt="exclamation" src="exclamation-icon.gif"
height="30" width="31"> Faraday experimentally
determined his law in the form presented above.<br>
<br>
<hr size="2" width="100%"><br>
</div>
</blockquote>
<div align="left">
<ul>
<li>One of the easiest ways to change the magnetic flux
through a circuit is to move a permanent (bar) magnet
towards or away from the circuit as shown in the
diagrams below.</li>
</ul>
<div align="center"><img alt="magfaradayfig1"
src="mag_faraday_fig1.jpg" height="401" width="698"><br>
<blockquote>
<div align="left">(a)&nbsp; <img alt="magfaradayfig2"
src="mag_faraday_fig2.jpg" height="191" width="267"
align="right">Magnetic flux passes through the
circuit, but does not change with time, so there is no
induced <i>emf</i> and so no induced current.<br>
<br>
(b)&nbsp; The flux through the circuit increases with
time causing an induced <i>emf</i> and current.<br>
<br>
(c)&nbsp; As the magnet moves faster the rate of
change of flux with time is increased causing a larger
<i>emf</i> and current.<br>
<br>
(d)&nbsp; When the magnet moves away from the circuit
the flux decreases with time so the induced <i>emf</i>
and current are reversed.<br>
<br>
<hr size="2" width="100%"><br>
<br>
</div>
</blockquote>
<div align="left">
<ul>
<li>The origin of the changing magnetic flux (field)
is not limited to permanent magnets.&nbsp; The
magnetic field due to a second circuit can produce a
similar effect, as described in the examples
below.&nbsp;<br>
</li>
</ul>
<div align="center">
<blockquote>
<div align="left"><img alt="magfaradayfig3"
src="mag_faraday_fig3.jpg" height="95"
width="267" align="right">In the diagram at
right the current in the left circuit is constant,
but the flux through the other circuit increases
as the two circuits get closer.<br>
<br>
<br>
<br>
<br>
<img alt="magfaradayfig4"
src="mag_faraday_fig4.jpg" height="90"
width="264" align="left">In the situation at
left both circuits are stationary.&nbsp; The
current in the left circuit is initially zero, but
rapidly increases to a constant value when the
switch is closed.&nbsp; As the current reaches its
final (constant) value the flux through the right
circuit is increasing with time, thus by Faraday's
Law,&nbsp; causing a brief pulse of induced
current in the second circuit.&nbsp; When the
switch is opened the flux in the right circuit
rapidly decreases causing a short induced current
pulse in the opposite direction.<br>
<br>
<hr size="2" width="100%"></div>
</blockquote>
<div align="left">
<ul>
<li><img alt="exclamation"
src="exclamation-icon.gif" height="30"
width="31"> Important !&nbsp; In both the
above examples a magnetic field (<b>B</b>)
changing with time results in a changing
magnetic flux.&nbsp; But it is possible to have
a changing flux with a constant <b>B</b> if the
cross sectional area - <b>dA</b> - can be made
to change with time as in the case of the
(electric) generator.<br>
</li>
</ul>
<div align="center"><img alt="magfaradayfig5"
src="mag_faraday_fig5.jpg" height="295"
width="348"></div>
<ul>
</ul>
<div align="center"><img alt="divider"
src="divider_ornbarblu.gif" height="64"
width="393"><br>
<div align="center">
<blockquote>
<div>
<h3><font color="#cc33cc"><u><b>LENZ'S LAW</b></u></font></h3>
</div>
</blockquote>
<div>
<div align="left">
<ul>
<li>Mathematically the negative sign in
Faraday's Law tells us about the
direction of the induced (<i>emf</i>)
current.&nbsp; Practically we use Lenz's
Law to determine the direction in
specific cases. <br>
</li>
</ul>
</div>
</div>
<blockquote>
<div><img alt="confused"
src="confused_smiley.gif" height="22"
width="15"> <big><i><b>"The induced
current will appear in such a
direction that it opposes the change
that produced it"</b></i></big>&nbsp;
<img alt="confused"
src="confused_smiley.gif" height="22"
width="15"><br>
<br>
<b>Confusing ?&nbsp; Yes !</b><br>
<br>
</div>
</blockquote>
<div>
<div align="left">
<div align="center">
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img alt="magfaradayfig7"
src="mag_faraday_fig7.jpg" height="526"
width="684"><br>
</div>
<ul>
</ul>
<ul>
<li><img alt="exclamation"
src="exclamation-icon.gif" height="30"
width="31"> The induced (flux)
magnetic field (associated with the
induced current) does not necessarily
oppose the field which causes the change
in flux, rather it opposes the <b>CHANGE</b>
in this field.<br>
</li>
</ul>
<ul>
<li><img alt="exclamation"
src="exclamation-icon.gif" height="30"
width="31"> Lenz's Law always ensures
that there is a force resisting the
motion of the magnet.&nbsp; It is the
work done against this force which
appears as the energy of the moving
charges of the induced current. <br>
</li>
</ul>
<p align="center"><img alt="magfaradayfig6"
src="mag_faraday_fig6.jpg" height="260"
width="275"></p>
<ul>
</ul>
<div align="center"><img alt="divider"
src="divider_ornbarblu.gif" height="64"
width="393"><br>
<br>
<font color="#cc33cc"><big><u><b>FARADAY'S
LAW</b></u><u><b> = MAXWELL
EQUATION</b></u></big></font><br>
<div align="left">
<ul>
<li>Whenever there is an induced
electric current there must also be
an induced electric field, <b>E</b>.&nbsp;
The work dW done by this induced
field moving a charge q<sub>0</sub>
a distance <b>ds</b> around a loop
is given by,</li>
</ul>
<div align="center"><img
alt="magfaradayeqn3"
src="mag_faraday_eqn3.jpg"
height="36" width="288"><br>
<blockquote>
<div align="left">where d&#949; is the
potential difference in <b>ds</b>.<br>
<br>
Therefore, <br>
<div align="center"><img
alt="magfaradayeqn4"
src="mag_faraday_eqn4.jpg"
height="31" width="117"><br>
<div align="left">So that the <i>emf
</i>around the whole loop is<br>
<div align="center"><img
alt="magfaradayeqn5"
src="mag_faraday_eqn5.jpg"
height="46" width="189"><br>
<br>
<div align="left">Equating
this <i>emf</i> to that
given by Faraday's Law we
obtain the integral form
of Faraday's Law, the
third of Maxwell's
equations we have
encountered so far,<br>
<div align="center"><img
alt="magfaradayeqn6"
src="mag_faraday_eqn6.jpg"
height="58"
width="297"><br>
<br>
<div align="left"><img
alt="exclamation"
src="exclamation-icon.gif"
height="30"
width="31">&nbsp;
Note that the line
integral of <b>E</b>
must be round a closed
loop (circuit).<br>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</blockquote>
<div align="left">
<div align="center">
<div align="left">
<div align="center">
<div align="left">
<div align="center">
<div align="left">
<ul>
<li>Written in the
above form the
relationship between
the <b>E</b> and <b>B</b>
fields is clear</li>
</ul>
<p align="center"><big><i><big><b>"A
magnetic field
changing with
time induces
an electric
field"</b></big></i></big><br>
</p>
<blockquote>
<div align="left">Shortly
we will see that the
reverse of this
statement is also
true.<br>
</div>
</blockquote>
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<blockquote>
<div align="left"> </div>
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<blockquote>
<div align="left"> </div>
</blockquote>
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<blockquote> </blockquote>
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<font color="#ff0000"><i>They told me I had type A blood, but it
was a Type O.</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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