392 lines
15 KiB
HTML
392 lines
15 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>Electricity - Electric Current - Physics 299</title>
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<meta content="C. L. Davis" name="author">
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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>
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<h1> <img src="ULPhys1.gif" align="texttop" height="50"
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width="189"></h1>
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</center>
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<center>
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<h1>Electric Current, Resistance and Power<br>
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</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>"When I find myself in the company of
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scientists, I feel like a shabby curate who has strayed by
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mistake into a drawing room full of dukes"</i></font><br>
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W. H. Auden<br>
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</center>
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<img src="netbar.gif" align="middle" height="40" width="100%"> <br>
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<br>
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<blockquote>
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<h2><font color="#3333ff"><u>Electric Current</u></font><br>
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</h2>
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</blockquote>
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<ul>
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<li> Electric current is equal to the rate at which charge passes
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a fixed point in space.</li>
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<br>
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<div align="center"><img alt="eqn9" src="elec_current_eqn9.jpg"
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height="49" width="55"></div>
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<center><br>
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</center>
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Current is measured in <a
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href="http://www-gap.dcs.st-and.ac.uk/%7Ehistory/Mathematicians/Ampere.html">Amperes:</a>
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<img src="Ampere.jpg" align="middle" height="109" width="90"> <br>
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<br>
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<center>1 <a
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href="http://www.npl.co.uk/server.php?show=ConWebDoc.1559">
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Ampere</a> = 1 Coulomb/second</center>
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<br>
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Although from the above definition it looks as though the Ampere
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is defined in terms of the Coulomb in fact it is the Ampere which
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is the basic unit, the Coulomb is the dervived unit. The Ampere is
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defined in terms of the force between two parallel wires carrying
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current as we will see later. <br>
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<br>
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<li>It is important to realize that the value of the current is
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constant, whatever the cross section of the conductor. If
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this were not so then charge would "pile up" at points along a
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conductor.</li>
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<br>
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<li>When you flip a switch a light bulb turn on instantly.
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In fact the current moves at speeds close to the speed of
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light. However, the charge carriers, electrons in a
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metallic wire, travel at a much slower velocity - the <span
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style="font-weight: bold;">drift velocity</span>. <br>
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Consider a wire of length l, cross section A, with n conduction
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electrons per unit volume. The current in the wire can be
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written,</li>
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</ul>
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<div style="text-align: center;"><img style="width: 202px; height:
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60px;" alt="eqn1" src="elec_current_eqn1.jpg"><br>
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<div style="text-align: left; margin-left: 40px;">where e is the
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charge on the electron and v<sub>d</sub> is the drift velocity.<br>
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</div>
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<div style="text-align: left;">
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<ul>
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<li><span style="font-weight: bold; font-style: italic;
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text-decoration: underline;">Current Density, J</span>
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(A/m<sup>2</sup>) is defined by,</li>
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</ul>
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<div style="text-align: center;"><img style="width: 126px;
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height: 54px;" alt="eqn2" src="elec_current_eqn2.jpg"><br>
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<br>
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<div style="text-align: left; margin-left: 40px;">physically,
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J represents charge movement at a particular place within a
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conductor, e.g. when A is large J is small, when A is small
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J is large.<br>
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The general relationship between I and J is<br>
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<div style="text-align: center;"><img style="width: 103px;
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height: 38px;" alt="eqn3" src="elec_current_eqn3.jpg"><br>
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<div style="text-align: left;">The current is the flux of
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J through a surface.<br>
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<br>
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<img style="width: 31px; height: 30px;"
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alt="exclamation" src="exclamation-icon.gif"> <span
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style="font-weight: bold; text-decoration: underline;">Important:</span>
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The
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current,
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I,
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is
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a scalar quantity, whereas J is a vector. I has a
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"sense" in that we draw arrows to represent its
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"direction", but does not obey the rules of vector
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algebra.<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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<ul>
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<br>
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<li> <img style="width: 15px; height: 22px;" alt="confused"
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src="confused_smiley.gif"> <span style="font-weight: bold;
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text-decoration: underline;">Historical quirk.</span>
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The direction of current flow is defined as the direction in
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which a positive charge will move. But in solid metallic
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conductors the charge carriers are electrons (negative charges)
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which actually move in the opposite direction. Negative
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charges moving right to left are exactly equivalent to positive
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charges moving left to right.</li>
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</ul>
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<div align="center"><img alt="divider" src="divider_ornbarblu.gif"
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height="64" width="393"><br>
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<blockquote>
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<div align="left">
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<h2><font color="#3333ff"><u>Resistance</u></font></h2>
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</div>
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</blockquote>
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</div>
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<ul>
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</ul>
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<ul>
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<li>In metallic conductors the electric field and current density
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are in the same direction and are found to be proportional to
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each other,</li>
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</ul>
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<div style="text-align: center;"><img style="width: 70px; height:
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24px;" alt="eqn4" src="elec_current_eqn4.jpg"><br>
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<br>
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<div style="text-align: left; margin-left: 40px;">where ρ is the
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resistivity of the conductor - characteristic of the
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conductor. The conductivity of a conducting material is
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defined by, σ = 1/ρ.<br>
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For a uniform conductor, length l, cross section A, we have E =
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V/l and J = i/A, so that<br>
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<br>
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<div style="text-align: center;"><img style="width: 367px;
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height: 54px;" alt="eqn5" src="elec_current_eqn5.jpg"><br>
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<br>
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<div style="text-align: left;">The resistance of the conductor
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R, is defined by,<br>
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<div style="text-align: center;"><img style="width: 110px;
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height: 54px;" alt="eqn6" src="elec_current_eqn6.jpg"><br>
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<div style="text-align: left;"><br>
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Resistance is measured in ohms (Ω), then resistivity has
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units ohm.metre and conductivity (ohm.metre)<sup>-1</sup>
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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="text-align: left;">
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<ul>
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<li><img style="width: 31px; height: 30px;"
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alt="exclamation" src="exclamation-icon.gif"> <span
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style="font-weight: bold; text-decoration:
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underline;">Important:</span> The relationship V =
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IR is <span style="font-weight: bold;">NOT</span>
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Ohm's Law !</li>
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</ul>
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<div style="margin-left: 40px;"><a
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href="http://www.juliantrubin.com/bigten/ohmlawexperiments.html"><span
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style="font-weight: bold;"><a
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href="http://www.juliantrubin.com/bigten/ohmlawexperiments.html"><img
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alt="Ohm" src="Ohm.jpg" align="left"
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height="122" border="0" width="95"></a>Ohm's
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Law</span></a>:<br>
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<div style="text-align: center;"><span
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style="font-style: italic;">"If the ratio of
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voltage across a conductor to the current through
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it is constant for all voltages then that
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conductor obeys Ohm's Law"</span><br>
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<div style="text-align: left;"><br>
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Ohm's law holds for metallic conductors, but not
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for devices such as transistors, diodes etc.
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The relationship V = IR can always be used to
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determine the resistance at some particular I and
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V for any device.<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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<div style="text-align: center;">
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<div style="text-align: left;">
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<div style="text-align: center;"> </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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<br>
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<li> Even in conductors current will only flow between two points
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A and B when</li>
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<br>
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<ol>
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<li> There is a potential difference between A and B (producing
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the electric field which forces the charges to move) and,</li>
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<li> A and B form part of a complete circuit.<br>
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</li>
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</ol>
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<center><img src="elec_circuit.jpg" align="texttop" height="330"
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width="300"><br>
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<img alt="divider" src="divider_ornbarblu.gif" height="64"
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width="393"><br>
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</center>
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<div align="left">
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<h2><font color="#3333ff"><u>Power</u></font></h2>
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</div>
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</ul>
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<ul>
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<li> Suppose a charge dq moves from point A to point B, where the
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potential difference between A and B is V<sub>AB</sub>, then the
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energy released in time dt is given by</li>
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</ul>
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<div align="center"><img alt="elec current eqn7"
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src="elec_current_eqn7.png" height="26" width="200"><br>
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<br>
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<blockquote>
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<div align="left">so that the rate at which energy is
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transferred (power), P, is given by,<br>
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<div align="center"><img alt="elec current eqn8"
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src="elec_current_eqn8.png" height="54" width="281"><br>
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<br>
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<div align="left">In terms of units we can state that
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Amps x Volts = Watts.<br>
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</div>
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</div>
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</div>
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</blockquote>
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<div align="left">
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<div align="center">
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<div align="left">
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<ul>
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<li>The form of the energy "released" depends on the
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electrical component placed between A and B, for
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example,</li>
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</ul>
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<ul>
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<ul>
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<li>Motor - mechanical energy (work) released </li>
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<li>Battery - chemical energy stored in the battery</li>
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<li>Resistance - thermal energy (heat) released<br>
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</li>
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</ul>
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</ul>
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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"><img alt="divider" src="divider_ornbarblu.gif"
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height="64" width="393"><br>
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<blockquote>
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<div align="left">
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<h2><font color="#3333ff"><u>Electro-motive Force - "emf"</u></font></h2>
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</div>
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</blockquote>
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<div align="left">
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<ul>
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<li><img alt="fig2" src="elec_current_fig2.jpg" align="right"
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height="300" width="370">In discussing electric circuits
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you may come across the term "emf" - electro-motive
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force. <b>It is important to realize that an "emf" is
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NOT a force !</b></li>
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</ul>
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<ul>
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<li>If a device has an "emf" it has the ability to maintain a
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potential difference (voltage). Thus, for example, a
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battery maintains an emf between its positive and negative
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terminals.</li>
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</ul>
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<ul>
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<li>The emf of a device can be defined by ε = dW/dq, where dW
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is the work done on a positive charge dq in taking it
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acrosss the potential difference of the device. In the
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case of a simple circuit with a battery (see above) as a
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charge traverses the external (to the battery) circuit it
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loses energy. In the circuit above the energy appeara
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as heat and light in the light bulb. When the
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charge returns to the battery the emf of the battery
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replenishes its energy.</li>
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</ul>
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<ul>
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<li>At this introductory level we can consider the emf of a
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"source" (battery, generator etc) to be exactly equivalent
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to the voltage provided by the source.</li>
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</ul>
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<ul>
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<li>The direction of the emf always represents the direction a
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positive charge would move in the external circuit.
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See circuit at right. The emf direction is an
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important factor when we use Kirchoff's laws to analyze
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circuits.</li>
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<br>
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<br>
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</ul>
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<ul>
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</ul>
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</div>
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</div>
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<br>
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<div align="center"><img alt="divider" src="divider_ornbarblu.gif"
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height="64" width="393"><br>
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<blockquote>
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<div align="left">
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<h2><font color="#3333ff"><u>Internal Resistance</u></font></h2>
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</div>
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</blockquote>
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<div align="left">
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<ul>
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<li>All emfs - batteries, generators etc - and electrical
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measuring devices - ammeters, voltmeters etc - have an
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"internal resistance".</li>
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</ul>
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<ul>
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<li><img alt="fig4" src="elec_current_fig4.jpg" align="right"
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height="158" width="152">As far as circuit analysis is
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concerned these internal resistances can simply be
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considered as resistors in series with the "ideal"
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emf/meter.</li>
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</ul>
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<ul>
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<li>For ammeters (current measuring devices) the goal is to
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have as low an internal resistance as possible so that the
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current is not affected.</li>
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</ul>
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<p align="center"><img alt="fig3" src="elec_current_fig3.jpg"
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align="middle" height="96" width="134"></p>
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<ul>
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<li>For a voltmeter the internal resistance should be as large
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as possible.<br>
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</li>
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</ul>
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</div>
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<br>
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<div align="left"><br>
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</div>
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</div>
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<img src="netbar.gif" height="40" width="100%">
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<center>
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<p style="color: rgb(255, 0, 0); font-style: italic;"
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class="MsoNormal">Q: Does light have mass?<br>
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A: Of course not. It's not even Catholic!!!</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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