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<!DOCTYPE html PUBLIC "-//w3c//dtd html 4.0 transitional//en">
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<meta name="Author" content="C. L. Davis">
<title>Electricty - Capacitors - Physics 299</title>
</head>
<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
255);" link="#0000ee" alink="#ff0000" vlink="#551a8b">
<center><img src="ULPhys1.gif" height="50" align="texttop"
width="189"></center>
<center>
<h1>Capacitors<br>
</h1>
</center>
<center><img src="celticbar.gif" height="22" width="576"><br>
<br>
<font color="#ff0000"><i>"</i></font><font color="#ff0000"><i>
<meta http-equiv="content-type" content="text/html;
charset=windows-1252">
I used to wonder how it comes about that the electron is
negative. Negative-positive<76>these are perfectly symmetric in
physics. There is no reason whatever to prefer one to the
other. Then why is the electron negative? I thought about this
for a long time and at last all I could think was 'It won the
fight!' "</i></font><br>
Albert Einstein<br>
</center>
<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
<blockquote>
<h2><u>Calculating Capacitance</u></h2>
</blockquote>
<ul>
</ul>
<ul>
<li>A capacitor is a system of two insulated conductors.&nbsp; <br>
</li>
</ul>
<ul>
</ul>
<ul>
<li><img alt="elec cap fig1" src="elec_cap_fig1.jpg" height="411"
align="right" width="700">The parallel plate capacitor is the
simplest example.&nbsp; When the two conductors have equal but
opposite charge, the <b>E</b> field between the plates can be
found by simple application of Gauss's Law.</li>
</ul>
<blockquote>Assuming the plates are large enough so that the <b>E</b>
field between them is uniform and directed perpendicular, then
applying Gauss's Law over surface S<sub>1</sub> we find,<br>
<div align="center"><img alt="elec cap eqn1"
src="elec_cap_eqn1.png" height="64" width="191"><br>
<div align="left">where A is the area of S<sub>1</sub>
perpendicular to the <b>E</b> field and &#963; is the surface
charge density on the plate (assumed uniform).&nbsp;
Therefore, <br>
<div align="center"><img alt="elec cap eqn2"
src="elec_cap_eqn2.png" height="60" width="67"><br>
<br>
<div align="left">everywhere between the plates.<br>
</div>
</div>
</div>
</div>
</blockquote>
<div align="center">
<div align="left">
<div align="center">
<div align="left">
<ul>
<li>The potential difference between the plates can be
found from</li>
</ul>
<div align="center"><img alt="elec cap eqn3"
src="elec_cap_eqn3.png" height="64" width="335"><br>
<blockquote>
<div align="left">where A and B are points, one on each
plate, and we integrate along an <b>E</b> field line,
d is the plate separation, A the plate area and q the
total charge on either plate.<br>
</div>
</blockquote>
<div align="left">
<ul>
<li>The capacitance (capacity) of this capacitor is
defined as,</li>
</ul>
<div align="center"><img alt="elec cap eqn4"
src="elec_cap_eqn4.png" height="63" width="148"><br>
<div align="left">
<ul>
<li>The expression for C for all capacitors is the
ratio of the magnitude of the total charge (on
either plate) to the magnitude of the potential
difference between the plates.</li>
</ul>
<ul>
<li>Units of
C:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
Coulomb/Volt = Farad,&nbsp;&nbsp;&nbsp; 1 C/V =
1 F</li>
</ul>
<blockquote><img alt="exclamation"
src="exclamation-icon.gif" height="30"
width="31"> Note that since the Coulomb is a
very large unit of charge the Farad is also a very
large unit of capacitance.&nbsp; Typical
capacitors in circuits are measured in &#956;F (10<sup>-6</sup>)
or pF (10<sup>-12</sup>).<br>
</blockquote>
<ul>
<li><img alt="exclamation"
src="exclamation-icon.gif" height="30"
width="31"> Note that the expression for the
capacitance of the parallel plate capacitor
depends on the geometric properties (A and
d).&nbsp; Even though it appears that there is
also a dependence on the charge and potential
difference (q/&#916;V), what happens is that whatever
charge you place on the capacitor the pd adjusts
itself so that the ratio&nbsp; q/&#916;V remains
constant.&nbsp;&nbsp; This is a general rule for
all capacitors.&nbsp; The capacitance is set by
the construction of the capacitor - not the
charge or voltage applied.</li>
</ul>
<ul>
<li><img alt="exclamation"
src="exclamation-icon.gif" height="30"
width="31"> The above expression for the
parallel plate capacitor is strictly only true
for an infinite parallel plate capacitor - in
which "fringing" (see above) does not
occur.&nbsp; However, so long as d is small
compared to the "size" of the plates, the simple
expression above is a good approximation.</li>
</ul>
<ul>
<li><img alt="exclamation"
src="exclamation-icon.gif" height="30"
width="31"> The parallel plate capacitor
provides an easy way to "measure" &#949;<sub>0</sub>
<br>
</li>
</ul>
<blockquote>
<div align="center"><img alt="elec cap eqn5"
src="elec_cap_eqn5.png" height="54" width="93"><br>
</div>
</blockquote>
<div align="center">
<div align="left">
<ul>
<li>As indicated above the parallel plate
capacitor is the most basic capacitor.&nbsp;
You should also be able to determine the
expressions for the capacitance of spherical
and cylindrical capacitors,</li>
</ul>
<div align="center"><img alt="elec cap fig3"
src="elec_cap_fig3.jpg" height="239"
width="311"> &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; &nbsp; &nbsp;&nbsp; <img
alt="elec cap fig2" src="elec_cap_fig2.jpg"
height="313" width="419"><br>
<br>
<img alt="divider" src="divider_ornbarblu.gif"
height="64" width="393"><br>
<blockquote>
<div align="left">
<h2><u>Energy and Capacitors</u></h2>
</div>
</blockquote>
<div align="left">
<ul>
<li> One of the most important uses of
capacitors is to store electrical
energy.</li>
</ul>
<blockquote>If a capacitor is placed in a
circuit with a battery, the potential
difference (voltage) of the battery will
force electric charge to appear on the
plates of the capacitor.&nbsp; The work
done by the battery in charging the
capacitor is stored as electrical
(potential) energy in the capacitor.&nbsp;
This energy can be released at a later
time to perform work.<br>
<br>
<div align="center"><img alt="elec cap
fig4" src="elec_cap_fig4.jpg"
height="204" width="297"></div>
</blockquote>
<div align="center">
<div align="left">
<ul>
<li>The work necessary to move a
charge dq onto one of the plates is
given by, dW = Vdq, where V is the
pd (voltage) of the battery (=
q/C).&nbsp; The total work to place
Q on the plate is given by,</li>
</ul>
<div align="center"><img alt="elec cap
eqn6" src="elec_cap_eqn6.png"
height="58" width="423"><br>
<blockquote>
<div align="left">which is equal to
the stored electrical potential
energy, U.<br>
</div>
</blockquote>
<div align="left">
<ul>
<li>The electrical energy actually
resides in the electric field
between the plates of the
capacitor.&nbsp; For a parallel
plate capacitor using&nbsp; C =
A&#949;<sub>0</sub>/d and&nbsp; E =
Q/A&#949;<sub>0</sub> we may write
the electrical potential energy,
<br>
</li>
</ul>
<div align="center"><img alt="elec
cap eqn7"
src="elec_cap_eqn7.png"
height="68" width="339"><br>
<blockquote>
<div align="left">(Ad) is the
volume between the plates,
therefore we define the energy
density,<br>
<br>
<div align="center"><img
alt="elec cap eqn8"
src="elec_cap_eqn8.png"
height="54" width="181"><br>
</div>
</div>
</blockquote>
<div align="left">
<div align="center">
<div align="left">
<ul>
<li>Although we have
evaluated this
expression for the
energy density for a
parallel plate capacitor
it is actually a general
expression.&nbsp;
Wherever there is an
electric field the
energy density is given
by the above.</li>
</ul>
<div align="center"><img
alt="divider"
src="divider_ornbarblu.gif"
height="64" width="393"><br>
<blockquote>
<div align="left">
<h2><u>Combinations of
Capacitors</u></h2>
</div>
</blockquote>
<div align="left">
<blockquote>It is common
to find multiple
combinations of
capacitors in
electrical
circuits.&nbsp; In the
simplest situations
capacitors can be
considered to be
connected in <b><i>series</i></b>
or in <i><b>parallel</b></i>.&nbsp;
<br>
</blockquote>
<ul>
<ul>
<li><big><b>Capacitors
in Series</b></big></li>
</ul>
</ul>
<blockquote>
<blockquote>When
different capacitors
are connected in
series the charge on
each capacitor is
the same but the
voltage (pd) across
each capacitor is
different<br>
<div align="center"><img
alt="elec cap
fig5"
src="elec_cap_fig5.jpg"
height="180"
width="312"></div>
</blockquote>
</blockquote>
<div align="center">
<div align="left"><br>
<blockquote>
<blockquote>
<div
align="left">In
this
situation,
using the fact
that V = V<sub>1</sub>
+ V<sub>2</sub>
+V<sub>3</sub>&nbsp;
we can show
that, as far
as the voltage
source is
concerned, the
capacitors can
be replaced by
a single
"equivalent"
capacitor C<sub>eq</sub>&nbsp;
given by, <br>
</div>
</blockquote>
</blockquote>
<div align="center"><img
alt="elec cap
fig9"
src="elec_cap_eqn9.png"
height="63"
width="182"><br>
</div>
<br>
<ul>
<ul>
<li><big><b>Capacitors
in Parallel</b></big></li>
</ul>
</ul>
<blockquote>
<blockquote>For
capacitors
connected in
parallel it is
the voltage
which is same
for each
capacitor, the
charge being
different.<br>
<br>
<div
align="center"><img
alt="elec cap
fig6"
src="elec_cap_fig6.jpg"
height="176"
width="360"><br>
<div
align="left"><br>
Using the fact
that Q<sub>Total</sub>=
Q<sub>1</sub>
+ Q<sub>2</sub>
+ Q<sub>3</sub>
we can show
that the
equivalent
capacitor, C<sub>eq</sub>&nbsp;
is given by,<br>
<br>
<div
align="center"><img
alt="elec cap
eqn10"
src="elec_cap_eqn10.png"
height="29"
width="172"><br>
</div>
</div>
</div>
</blockquote>
</blockquote>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<ul>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<blockquote> </blockquote>
<ul>
</ul>
<blockquote> </blockquote>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<blockquote>
<div align="center"> </div>
</blockquote>
<p> <img src="netbar.gif" height="40" width="100%"> </p>
<center>
<p class="MsoNormal"><span style="color: rgb(255, 0, 0);
font-style: italic;">
<meta http-equiv="content-type" content="text/html;
charset=windows-1252">
At the electric company: <i>"We would be delighted if you
send in your bill. However, if you don't, you will be."</i></span><br>
<br>
</p>
<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>
&nbsp; </p>
<p><img src="header-index.gif" height="51" width="92"> </p>
</center>
<p><br>
</p>
</body>
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