328 lines
16 KiB
HTML
328 lines
16 KiB
HTML
<!DOCTYPE html PUBLIC "-//w3c//dtd html 4.0 transitional//en">
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<html>
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<head>
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<meta http-equiv="Content-Type" content="text/html;
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charset=windows-1252">
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<meta name="GENERATOR" content="Mozilla/4.7 [en] (X11; U; OSF1 V4.0
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alpha) [Netscape]">
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<meta name="Author" content="C. L. Davis">
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<title>Electricty - Electric Potential and Potential Difference -
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Physics 299</title>
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</head>
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<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
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255);" alink="#ff0000" link="#0000ee" vlink="#551a8b">
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<center><img src="ULPhys1.gif" align="texttop" height="50"
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width="189"></center>
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<center>
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<h1>Electric Potential and Potential Difference</h1>
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</center>
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<center><img src="celticbar.gif" height="22" width="576"><br>
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<br>
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<font color="#ff0000"><i>"The outcome of any serious research can
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only be to make two questions grow where one question grew
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before"</i></font><br>
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Thorstein Veblen<br>
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</center>
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<img src="netbar.gif" align="middle" height="40" width="100%">
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<ul>
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<li> <b><i>The field</i></b><i> near </i>a system of charges can
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also be described by a scalar quantity known as the "Electric
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Potential". <br>
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<br>
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<b><i>" A potential difference of one volt exists between two
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points when one Joule of work is required to move one
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Coulomb of charge from one point to the other"</i></b> <br>
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<br>
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Between two points A and B we may write
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<center><br>
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W<sub>AB</sub> = -V<sub>AB</sub> q </center>
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<p style="text-align: left;">where V<sub>AB</sub> = V<sub>B</sub>
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- V<sub>A</sub> is the potential difference between A
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and B. </p>
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</li>
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</ul>
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<div style="margin-left: 40px;"><img style="width: 31px; height:
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30px;" alt="exclamation" src="exclamation-icon.gif"> Note that W<sub>AB</sub>
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is the work done by the electric field in moving the charge.
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The work done by the "external agent" is -W<sub>AB</sub>.</div>
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<ul>
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<li>Units of potential difference are volts <br>
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<center>
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<p>1 Volt = 1 Joule/Coulomb (J/C)</p>
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</center>
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</li>
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<li>In a region of space where there is an electric field the work
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done by the electric field, dW, when a positive point charge, q,
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is displaced by <span style="font-weight: bold;">ds</span> is
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given by,</li>
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</ul>
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<div style="text-align: center;"><img style="width: 128px; height:
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29px;" alt="eqn1" src="elec_potential_eqn1.jpg"><img
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style="width: 251px; height: 227px;" alt="fig1"
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src="elec_potential_fig1.jpg" align="right">
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<div style="text-align: left; margin-left: 40px;">Therefore,<br>
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<div style="text-align: center;"><img style="width: 519px;
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height: 75px;" alt="eqn2" src="elec_potential_eqn2.jpg"
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height="75" width="519"><br>
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<br>
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<div style="text-align: left;">For a uniform electric field we
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obtain,<br>
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<div style="text-align: center;"><img style="width: 446px;
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height: 43px;" alt="eqn3" src="elec_potential_eqn3b.jpg"
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height="44" width="357"><br>
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<br>
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<div align="left">where an arbitrary path can always be
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split into sections along <b>E</b> and sections
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perpendicular to <b>E</b>.<br>
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<br>
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</div>
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</div>
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</div>
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</div>
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</div>
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<div style="text-align: left;">
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<div style="text-align: center;">
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<div style="text-align: left;">
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<div style="text-align: center;">
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<div style="margin-left: 40px; text-align: left;"><img
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style="width: 31px; height: 30px;" alt="exclamation"
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src="exclamation-icon.gif"> Note that this means that
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the electric field can be expressed in the units
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V/m. [ 1 N/C = 1V/m ]<br>
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<div style="text-align: center;"><img style="width:
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319px; height: 266px;" alt="fig2"
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src="elec_potential_fig2.jpg" align="right"><img
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style="width: 393px; height: 64px;" alt="divider"
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src="divider_ornbarblu.gif"> </div>
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</div>
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<div style="text-align: left;">
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<div style="margin-left: 40px;"><img style="width: 79px;
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height: 43px;" alt="hot" src="hot.gif"><big><big>The
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electric field is a <span style="font-weight:
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bold;">conservative field</span></big>. </big>
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<img style="width: 79px; height: 43px;" alt="hot"
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src="hot.gif"></div>
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<ul>
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</ul>
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<ul>
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<li>This means that the potential difference between
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two points is independent of the path taken.
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Every point in space has a single value of V and E.</li>
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</ul>
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<div style="margin-left: 40px;"><br>
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<img style="width: 64px; height: 41px;"
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alt="hamburger" src="hamburger.gif"> <span
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style="font-style: italic; font-weight: bold;">Food
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for thought....</span><br>
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<br>
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The gravitational field behaves in exactly the same
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way. Changes in gravitational energy are
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independent of the path taken. Climbing stairs
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from one floor to another involve the same amount of
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work against gravity as riding an elevator.<br>
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Note that in simple gravitational applications we
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don't usually define a gravitational potential only
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gravitational potential energy (mgh). In this
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case the gravitational potential is defined as gh.<br>
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<br>
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</div>
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<ul>
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<li><img style="width: 17px; height: 23px;" alt="idea"
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src="idea2.gif">Exactly equivalent to gravity, it
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is <span style="font-weight: bold;">CHANGES</span>
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in potential difference, <img style="width: 34px;
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height: 18px;" alt="delta V" src="deltaV.jpg">,
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which are defined. To obtain absolute values
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of V physicists usually define V = 0 at
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infinity. But this is an arbitary definition;
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in engineering applications it is often convenient
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to define the earth as V = 0.</li>
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</ul>
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<div style="text-align: center;"><img style="width:
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393px; height: 64px;" alt="divider"
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src="divider_ornbarblu.gif"><br>
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</div>
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<ul>
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<li><big><span style="text-decoration: underline;
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font-weight: bold;">Potential due to a point
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charge </span></big><br>
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<br>
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For a single point charge Q the potential difference
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between A and B is given by,<img style="width:
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186px; height: 184px;" alt="fig3"
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src="elec_potential_fig3.gif" align="right"></li>
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</ul>
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<div style="text-align: center;"><img style="width:
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519px; height: 43px;" alt="eqn4"
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src="elec_potential_eqn4.jpg"><br>
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<br>
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<div style="text-align: left; margin-left: 40px;">where
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E is the field due to a point charge and ds = dr ,
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so that,<br>
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<br>
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<div style="text-align: center;"><img style="width:
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484px; height: 68px;" alt="eqn5"
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src="elec_potential_eqn5.jpg"><br>
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<div style="text-align: left;"><br>
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If we assume r<sub>B</sub>= ∞ then V<sub>B</sub>
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= 0 and,<br>
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<br>
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<div style="text-align: center;"><img
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style="width: 177px; height: 60px;"
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alt="eqn6" src="elec_potential_eqn6.jpg"><br>
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<br>
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<div style="text-align: left;">Note that the
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potential is inversely proportional to r,
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rather than r<sup>2</sup> as in the case of
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the electric field.<br>
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</div>
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</div>
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</div>
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</div>
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</div>
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<div style="text-align: left;">
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<div style="text-align: center;">
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<div style="text-align: left;">
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<div style="text-align: center;">
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<div style="text-align: left;">
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<ul>
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<li>Since V is a scalar the potential due
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to multiple point charges is found by
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adding the potential due to the
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individual charges (taking into account
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the sign of the charge). </li>
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</ul>
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<ul>
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<li><u><big><b>Continuous Charge
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Distributions</b></big></u></li>
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</ul>
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<ul>
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</ul>
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<blockquote>For continuous distributions of
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charge we may write,<br>
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</blockquote>
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<div style="margin-left: 40px;">
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<div style="text-align: center;"><img
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style="width: 141px; height: 60px;"
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alt="eqn7"
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src="elec_potential_eqn7.jpg"><br>
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<div style="text-align: left;">
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<p>The electric potential due to a
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continuous charge distribution can
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be calculated in a similar manner to
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the electric field due to such a
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distribution. For example the
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potential at point P due to a
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uniform ring of charge (below).<br>
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</p>
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<div align="center"><img alt="elec pot
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fig4"
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src="elec_potential_fig4.jpg"
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height="253" width="374"></div>
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<div style="text-align: center;"><img
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style="width: 393px; height:
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64px;" alt="divider"
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src="divider_ornbarblu.gif"><br>
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</div>
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</div>
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</div>
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</div>
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<div style="text-align: center;">
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<div style="text-align: left;">
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<ul>
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<li><big><span style="font-weight:
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bold; text-decoration:
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underline;">Equipotential
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Surfaces </span></big><br>
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<br>
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An Equipotential Surface is a
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surface in space on which all points
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have the same potential. Since
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all points have the same potential
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it requires no work to move a charge
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on such a surface. This means
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that there is no component of E in
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the plane of the surface, in other
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words E must be at right angles to
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the surface.</li>
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</ul>
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<div style="text-align: center;"><span
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style="font-weight: bold;">Electric
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Field Lines and Equipotential
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Surfaces are at right angles</span><span
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style="font-weight: bold;
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font-style: italic;"><br>
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</span><span style="font-style:
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italic;">(Red dotted lines below)</span><span
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style="font-weight: bold;
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font-style: italic;"><br>
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</span><img style="width: 345px;
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height: 337px;" alt="equip2"
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src="elec_potential_equip2.gif">
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<img style="width: 319px; height:
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312px;" alt="equip1"
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src="elec_potential_equip1.gif"><br>
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</div>
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</div>
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</div>
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</div>
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</div>
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</div>
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</div>
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</div>
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<div style="text-align: left; margin-left: 40px;"
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align="center"> <br>
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</div>
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</div>
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</div>
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</div>
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</div>
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</div>
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</div>
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</div>
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<ul>
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</ul>
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<div align="center"><br>
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</div>
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<p> <img src="netbar.gif" height="40" width="100%"> </p>
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<center>
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<p class="MsoNormal"><span style="color: rgb(255, 0, 0);
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font-style: italic;">In the period that Einstein was active as
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a professor, one of his students came to him and said: "The
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questions of this year's exam are the same as last years!"
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"True," Einstein said, "but this year all answers are
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different."</span><br>
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Albert Einstein<br>
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</p>
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<img src="celticbar.gif" height="22" width="576"> <br>
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<p><i>Dr. C. L. Davis</i> <br>
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<i>Physics Department</i> <br>
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<i>University of Louisville</i> <br>
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<i>email</i>: <a href="mailto:c.l.davis@louisville.edu">c.l.davis@louisville.edu</a>
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<br>
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</p>
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<p><img src="header-index.gif" height="51" width="92"> </p>
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</center>
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<p><br>
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</p>
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</body>
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</html>
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