davisnotes/elec_circuits_RC.html

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<title>Electricity - RC circuits - Physics 299</title>
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<h1> <img src="ULPhys1.gif" height="50" align="texttop"
width="189"></h1>
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<h1>RC Circuits<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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<h2><u><font color="#3333ff">CHARGING</font></u></h2>
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<ul>
<li><img alt="fig1" src="elec_RC_fig1.jpg" height="344"
align="right" width="532">An example of a series RC circuit is
shown at right.&nbsp; With the emf included in the circuit,
applying the loop theorem we find</li>
</ul>
<div align="center"><img alt="eqn1" src="elec_RC_eqn1.jpg"
height="53" width="165"><br>
<blockquote>
<div align="left">where q/C is the voltage drop across the
capacitor and i is the current in the circuit.<br>
</div>
</blockquote>
<div align="left">
<ul>
<li>Using the fact that i = dq/dt, we obtain</li>
</ul>
<div align="center"><img alt="eqn2" src="elec_RC_eqn2.jpg"
height="76" width="155"><br>
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<ul>
<li>This is a "simple" differential equation the solution
of which can be written</li>
</ul>
<div align="center"><img alt="eqn3" src="elec_RC_eqn3.jpg"
height="38" width="197"><br>
<blockquote>
<div align="left">or<br>
<div align="center"><img alt="eqn4"
src="elec_RC_eqn4.jpg" height="68" width="169"><br>
</div>
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</blockquote>
<div align="left">
<ul>
<li>The voltage across the capacitor, V<sub>C</sub> =
q/C and the voltage across the resistor, V<sub>R</sub>
= iR.&nbsp; Using the equations above we find that
the dependence of these voltages on time is shown
below</li>
</ul>
<div align="center"><img alt="fig2"
src="elec_RC_fig2.jpg" height="286" width="399"><br>
<br>
<div align="left">
<ul>
<li><img alt="exclamation"
src="exclamation-icon.gif" height="30"
width="31"> Note that the time on the
horizontal axis is measured in units of &#964; = RC,
the capacitative time constant.</li>
</ul>
<ul>
<li><img alt="exclamation"
src="exclamation-icon.gif" height="30"
width="31"> After one time constant V<sub>C</sub>
has reached 63% (1 - e<sup>-1</sup>) of its
maximum value and V<sub>R</sub> has 37% (1/e) of
its final value.</li>
</ul>
<div align="center"><img alt="divider"
src="divider_ornbarblu.gif" height="64"
width="100%"><br>
<h2><font color="#3333ff"><u>DISCHARGING</u></font></h2>
<div align="left">
<ul>
<li><img alt="fig1" src="elec_RC_fig1.jpg"
height="344" align="right" width="532">Now
switch the emf out of the circuit and
reapply the loop theorem</li>
</ul>
<div align="center"><img alt="eqn5"
src="elec_RC_eqn5.jpg" height="53"
width="132"><br>
<blockquote>
<div align="left">which gives<br>
<div align="center"><img alt="eqn6"
src="elec_RC_eqn6.jpg" height="65"
width="137"><br>
<br>
<div align="left">which has the solution<br>
<div align="center"><img alt="eqn7"
src="elec_RC_eqn7.jpg" height="44"
width="171"><br>
<div align="left">and<br>
<div align="center"><img
alt="eqn8"
src="elec_RC_eqn8.jpg"
height="70" width="192"><br>
</div>
<br>
</div>
<br>
<div align="left">where C&#949; is the
initial charge on the capacitor
and &#949;/R is the initial voltage
across the capacitor.<br>
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</div>
</div>
</div>
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</blockquote>
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<div align="center">
<div align="left">
<div align="center">
<div align="left">
<ul>
<li><img alt="fig3"
src="elec_RC_fig3.jpg"
height="410" align="right"
width="316">The time
dependence of V<sub>C</sub> and
V<sub>R</sub> (where V<sub>R</sub>
is proportional to the current
in the capacitor) are shown at
right.</li>
</ul>
<ul>
<li><img alt="exclamation"
src="exclamation-icon.gif"
height="30" width="31"> Once
again the time axis is measured
in units of RC.</li>
</ul>
<ul>
<li><img alt="exclamation"
src="exclamation-icon.gif"
height="30" width="31"> After
one time constant V<sub>C</sub>
has decreased to 37% (1/e) of
its initial value and |V<sub>R</sub>|
has decreased to 37% (1/e) of
its initial value. </li>
</ul>
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<div align="center">
<div align="left">
<div align="center">
<div align="left"> </div>
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<font color="#ff0000"><i>Q: What did one quantum physicist say
when he wanted to fight another quantum physicist?<br>
A: Let me <font color="#ff0000">atom</font>. </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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