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<title>Electricity - Kirchoff's Laws - Physics 299</title>
<meta content="C. L. Davis" name="author">
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<center>
<h1> <img src="ULPhys1.gif" height="50" align="texttop"
width="189"></h1>
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<center>
<h1>Kirchhoff's Laws<br>
</h1>
</center>
<center><img src="celticbar.gif" height="22" width="576"><br>
<br>
<font color="#ff0000"><i>
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<div class="copy-paste-block"><font color="#ff0000"><i><span
class="bqQuoteLink">"An expert is a man who has made all
the mistakes which can be made, in a narrow field.</span></i><span></span>
</font>"<br>
</div>
<font color="#ff0000"><i> </i><font color="#000000">Niels Bohr</font></font><br>
</center>
<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
<ul>
<li>
<div align="left">The most common general method to analyze
electrical circuits is by use of Kirchhoff's Laws.<img
alt="kirchoff" src="kirchhoff.jpg" height="133"
align="middle" width="84"></div>
</li>
</ul>
<blockquote>
<h3><u><img alt="staricon" src="StarIconGreen.png" height="48"
align="middle" width="50"> Junction Theorem</u></h3>
</blockquote>
<blockquote>
<div align="center"><b><big><font color="#3333ff"><i>At any
junction in a circuit the current entering the junction
must equal the current leaving the junction.</i><i><br>
</i></font></big></b></div>
<br>
<div align="center">(This is nothing more than a statement of
conservation of charge)<br>
</div>
</blockquote>
<blockquote>
<h3><u><img alt="staricon" src="StarIconGreen.png" height="52"
align="middle" width="53"> Loop Theorem</u></h3>
<p align="center"><font color="#3333ff"><b><big><i>The sum of the
changes in potential when traversing any complete</i><i>
loop is zero.</i></big></b></font><br>
</p>
<p align="center">(This is equivalent to conservation of energy)<br>
</p>
</blockquote>
<ul>
</ul>
<div align="center"><img alt="divider" src="divider_ornbarblu.gif"
height="64" width="393"></div>
<ul>
<li>
<h3><u>Conventions</u></h3>
</li>
</ul>
<blockquote>As usual, in order to ensure consistent results from
application of these laws, we must adhere to several conventions
concerning the currents and potentials in circuits.<br>
<u><b><br>
Potentials</b></u>:<br>
<ol>
<li>When a resistive device is traversed in the direction of
current flow the change in potential is -iR.&nbsp; Conversely,
if the resistance is traversed opposite to the direction of
the current the potential change is +iR.</li>
<li>When an emf is traversed in the direction of the emf the
change in potential is +&#949;.&nbsp; Conversely, if the emf is
traversed opposite to the emf direction the change in
potential is -&#949;.</li>
</ol>
<p><br>
<u><b>Currents:</b></u><br>
</p>
<blockquote>
<p>In setting up a problem, the current direction in any
particular circuit element is assigned arbitrarily.&nbsp;
Kitchoff's laws are then applied to the circuit using these
current directions.&nbsp; After solving the resulting
equations if a current is negative that means the "actual"
current direction is opposite the arbitrarily chosen
direction.<br>
</p>
</blockquote>
<ol>
</ol>
<div align="center"><img alt="divider" src="divider_ornbarblu.gif"
height="64" width="393"><br>
</div>
</blockquote>
<ul>
<li>
<h3><u>Application of Kirchhoff's Laws</u></h3>
</li>
</ul>
<blockquote>
<p>Kirchhoff's laws can be applied to <b>any circuit</b> to
obtain a set of equations relating the currents, resistances and
emfs in the circuit.&nbsp; These equations can then be solved
for the unknown quantities in the circuit.&nbsp; For any circuit
follow the steps below.<br>
</p>
</blockquote>
<blockquote>
<ol>
<li>Label the current flowing in each part of the circuit,
bearing in mind that current will "split" on reaching a
junction.&nbsp; The direction of the defined direction of the
current does not matter - see current convention above.</li>
<li>At each junction in the circuit use the junction theorem to
write down the equations relating the currents entering and
leaving.&nbsp;</li>
<li>Define all possible loops in the circuit and label.</li>
<li>For each loop choose a starting location then use the loop
theorem to write down the equation relating changes in
potential which must be zero after traversing the complete
loop.</li>
<li>Solve the set of equations from 2. and 4. to obtain the
unknown parameters of the circuit.</li>
</ol>
<p><br>
As an example, consider the circuit below.&nbsp; With the 3 emfs
we cannot use the series/parallel analysis.<br>
</p>
<div align="center"><img alt="fig1" src="elec_kirch_fig1.gif"
height="185" width="435"></div>
</blockquote>
<blockquote><u>Junctions:</u><br>
<blockquote>a:&nbsp;&nbsp; I<sub>1</sub> = I<sub>2</sub> + I<sub>3</sub>
<br>
b:&nbsp;&nbsp; I<sub>3</sub> + I<sub>2</sub> = I<sub>3</sub> <br>
<br>
</blockquote>
<u>Loops:</u><br>
<blockquote>1 (including &#949;<sub>1</sub> starting at a traversing
clockwise):&nbsp; - I<sub>3</sub>R<sub>4</sub> - &#949;<sub>3</sub> -
I<sub>1</sub>R<sub>2</sub> + &#949;<sub>1</sub> - I<sub>1</sub>R<sub>1</sub>
= 0<br>
2 (including &#949;<sub>2</sub> starting at a traversing clockwise):
&nbsp; - I<sub>2</sub>R<sub>3</sub> - &#949;<sub>2</sub> + &#949;<sub>3</sub>
+ I<sub>3</sub>R<sub>4</sub> = 0<br>
3 (including &#949;1 and &#949;<sub>2</sub> starting at a traversing
clockwise):&nbsp; - I<sub>2</sub>R<sub>3</sub> - &#949;<sub>2</sub> -
I<sub>1</sub>R<sub>2</sub> + &#949;<sub>1</sub> - I<sub>1</sub>R<sub>1</sub>
= 0<br>
</blockquote>
Looking at these equations it is clear that the two junction
equations are equivalent, and that loop equation 3 is simply the
sum of loop equations 1 and 2.&nbsp; Therefore there are only 3
independent equations (a, 1 and 2), which we can solve for, say,
the currents I<sub>1</sub>, I<sub>2</sub> and I<sub>3</sub>.<br>
<br>
<img alt="exlamation" src="exclamation-icon.gif" height="30"
width="31"> Note that in more complicated circuits there will be
many more junctions and a large number of possible loops.&nbsp;
You only need apply the loop theorem to as many loops to obtain
the number of independent equations necessary to determine the
unknown parameters.&nbsp; That is if you have 3 unknown
quantities, you'll need a total of 3 independent equations.<br>
</blockquote>
<ul>
</ul>
<div align="left"> </div>
<img src="netbar.gif" height="40" width="100%">
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</p>
<font color="#ff0000"><i>What do you get if you have Avogadro's
number of donkeys?<br>
&nbsp;Answer: molasses (a mole of asses)</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>
&nbsp; </p>
<p><img src="header-index.gif" height="51" width="92"> </p>
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<p><br>
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