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<!DOCTYPE html PUBLIC "-//w3c//dtd html 4.0 transitional//en">
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<meta http-equiv="Content-Type" content="text/html;
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<meta name="GENERATOR" content="Mozilla/4.7 [en] (X11; U; OSF1 V4.0
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<meta name="Author" content="C. L. Davis">
<title>Electricity - Electric Current - Physics 299</title>
<meta content="C. L. Davis" name="author">
</head>
<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
255);" alink="#ff0000" link="#0000ee" vlink="#551a8b">
<center>
<h1> <img src="ULPhys1.gif" align="texttop" height="50"
width="189"></h1>
</center>
<center>
<h1>Electric Current, Resistance and Power<br>
</h1>
</center>
<center><img src="celticbar.gif" height="22" width="576"><br>
<br>
<font color="#ff0000"><i>"When I find myself in the company of
scientists, I feel like a shabby curate who has strayed by
mistake into a drawing room full of dukes"</i></font><br>
W. H. Auden<br>
</center>
<img src="netbar.gif" align="middle" height="40" width="100%"> <br>
<br>
<blockquote>
<h2><font color="#3333ff"><u>Electric Current</u></font><br>
</h2>
</blockquote>
<ul>
<li> Electric current is equal to the rate at which charge passes
a fixed point in space.</li>
<br>
<div align="center"><img alt="eqn9" src="elec_current_eqn9.jpg"
height="49" width="55"></div>
<center><br>
</center>
Current is measured in <a
href="http://www-gap.dcs.st-and.ac.uk/%7Ehistory/Mathematicians/Ampere.html">Amperes:</a>
<img src="Ampere.jpg" align="middle" height="109" width="90"> <br>
<br>
<center>1 <a
href="http://www.npl.co.uk/server.php?show=ConWebDoc.1559">
Ampere</a> = 1 Coulomb/second</center>
<br>
Although from the above definition it looks as though the Ampere
is defined in terms of the Coulomb in fact it is the Ampere which
is the basic unit, the Coulomb is the dervived unit. The Ampere is
defined in terms of the force between two parallel wires carrying
current as we will see later. <br>
<br>
<li>It is important to realize that the value of the current is
constant, whatever the cross section of the conductor.&nbsp; If
this were not so then charge would "pile up" at points along a
conductor.</li>
<br>
<li>When you flip a switch a light bulb turn on instantly.&nbsp;
In fact the current moves at speeds close to the speed of
light.&nbsp; However, the charge carriers, electrons in a
metallic wire, travel at a much slower velocity - the <span
style="font-weight: bold;">drift velocity</span>. <br>
Consider a wire of length l, cross section A, with n conduction
electrons per unit volume.&nbsp; The current in the wire can be
written,</li>
</ul>
<div style="text-align: center;"><img style="width: 202px; height:
60px;" alt="eqn1" src="elec_current_eqn1.jpg"><br>
<div style="text-align: left; margin-left: 40px;">where e is the
charge on the electron and v<sub>d</sub> is the drift velocity.<br>
</div>
<div style="text-align: left;">
<ul>
<li><span style="font-weight: bold; font-style: italic;
text-decoration: underline;">Current Density, J</span>
(A/m<sup>2</sup>) is defined by,</li>
</ul>
<div style="text-align: center;"><img style="width: 126px;
height: 54px;" alt="eqn2" src="elec_current_eqn2.jpg"><br>
<br>
<div style="text-align: left; margin-left: 40px;">physically,
J represents charge movement at a particular place within a
conductor, e.g. when A is large J is small, when A is small
J is large.<br>
The general relationship between I and J is<br>
<div style="text-align: center;"><img style="width: 103px;
height: 38px;" alt="eqn3" src="elec_current_eqn3.jpg"><br>
<div style="text-align: left;">The current is the flux of
J through a surface.<br>
<br>
<img style="width: 31px; height: 30px;"
alt="exclamation" src="exclamation-icon.gif"> <span
style="font-weight: bold; text-decoration: underline;">Important:</span>&nbsp;
The
current,
I,
is
a scalar quantity, whereas J is a vector.&nbsp; I has a
"sense" in that we draw arrows to represent its
"direction", but does not obey the rules of vector
algebra.<br>
</div>
</div>
</div>
</div>
</div>
</div>
<ul>
<br>
<li> <img style="width: 15px; height: 22px;" alt="confused"
src="confused_smiley.gif"> <span style="font-weight: bold;
text-decoration: underline;">Historical quirk.</span>&nbsp;
The direction of current flow is defined as the direction in
which a positive charge will move.&nbsp; But in solid metallic
conductors the charge carriers are electrons (negative charges)
which actually move in the opposite direction.&nbsp; Negative
charges moving right to left are exactly equivalent to positive
charges moving left to right.</li>
</ul>
<div align="center"><img alt="divider" src="divider_ornbarblu.gif"
height="64" width="393"><br>
<blockquote>
<div align="left">
<h2><font color="#3333ff"><u>Resistance</u></font></h2>
</div>
</blockquote>
</div>
<ul>
</ul>
<ul>
<li>In metallic conductors the electric field and current density
are in the same direction and are found to be proportional to
each other,</li>
</ul>
<div style="text-align: center;"><img style="width: 70px; height:
24px;" alt="eqn4" src="elec_current_eqn4.jpg"><br>
<br>
<div style="text-align: left; margin-left: 40px;">where &#961; is the
resistivity of the conductor - characteristic of the
conductor.&nbsp; The conductivity of a conducting material is
defined by, &#963; = 1/&#961;.<br>
For a uniform conductor, length l, cross section A, we have E =
V/l and J = i/A, so that<br>
<br>
<div style="text-align: center;"><img style="width: 367px;
height: 54px;" alt="eqn5" src="elec_current_eqn5.jpg"><br>
<br>
<div style="text-align: left;">The resistance of the conductor
R, is defined by,<br>
<div style="text-align: center;"><img style="width: 110px;
height: 54px;" alt="eqn6" src="elec_current_eqn6.jpg"><br>
<div style="text-align: left;"><br>
Resistance is measured in ohms (&#937;), then resistivity has
units ohm.metre and conductivity (ohm.metre)<sup>-1</sup>
<br>
</div>
</div>
</div>
</div>
</div>
<div style="text-align: left;">
<div style="text-align: center;">
<div style="text-align: left;">
<div style="text-align: center;">
<div style="text-align: left;">
<ul>
<li><img style="width: 31px; height: 30px;"
alt="exclamation" src="exclamation-icon.gif"> <span
style="font-weight: bold; text-decoration:
underline;">Important:</span> The relationship V =
IR is <span style="font-weight: bold;">NOT</span>
Ohm's Law !</li>
</ul>
<div style="margin-left: 40px;"><a
href="http://www.juliantrubin.com/bigten/ohmlawexperiments.html"><span
style="font-weight: bold;"><a
href="http://www.juliantrubin.com/bigten/ohmlawexperiments.html"><img
alt="Ohm" src="Ohm.jpg" align="left"
height="122" border="0" width="95"></a>Ohm's
Law</span></a>:<br>
<div style="text-align: center;"><span
style="font-style: italic;">"If the ratio of
voltage across a conductor to the current through
it is constant for all voltages then that
conductor obeys Ohm's Law"</span><br>
<div style="text-align: left;"><br>
Ohm's law holds for metallic conductors, but not
for devices such as transistors, diodes etc.&nbsp;
The relationship V = IR can always be used to
determine the resistance at some particular I and
V for any device.<br>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div style="text-align: left; margin-left: 40px;">
<div style="text-align: center;">
<div style="text-align: left;">
<div style="text-align: center;"> </div>
</div>
</div>
</div>
</div>
<ul>
<br>
<li> Even in conductors current will only flow between two points
A and B when</li>
<br>
<ol>
<li> There is a potential difference between A and B (producing
the electric field which forces the charges to move) and,</li>
<li> A and B form part of a complete circuit.<br>
</li>
</ol>
<center><img src="elec_circuit.jpg" align="texttop" height="330"
width="300"><br>
<img alt="divider" src="divider_ornbarblu.gif" height="64"
width="393"><br>
</center>
<div align="left">
<h2><font color="#3333ff"><u>Power</u></font></h2>
</div>
</ul>
<ul>
<li> Suppose a charge dq moves from point A to point B, where the
potential difference between A and B is V<sub>AB</sub>, then the
energy released in time dt is given by</li>
</ul>
<div align="center"><img alt="elec current eqn7"
src="elec_current_eqn7.png" height="26" width="200"><br>
<br>
<blockquote>
<div align="left">so that the rate at which energy is
transferred (power), P, is given by,<br>
<div align="center"><img alt="elec current eqn8"
src="elec_current_eqn8.png" height="54" width="281"><br>
<br>
<div align="left">In terms of units we can state that&nbsp;
Amps x Volts = Watts.<br>
</div>
</div>
</div>
</blockquote>
<div align="left">
<div align="center">
<div align="left">
<ul>
<li>The form of the energy "released" depends on the
electrical component placed between A and B, for
example,</li>
</ul>
<ul>
<ul>
<li>Motor - mechanical energy (work) released&nbsp;</li>
<li>Battery - chemical energy stored in the battery</li>
<li>Resistance - thermal energy (heat) released<br>
</li>
</ul>
</ul>
</div>
</div>
</div>
</div>
<ul>
</ul>
<div align="center"><img alt="divider" src="divider_ornbarblu.gif"
height="64" width="393"><br>
<blockquote>
<div align="left">
<h2><font color="#3333ff"><u>Electro-motive Force - "emf"</u></font></h2>
</div>
</blockquote>
<div align="left">
<ul>
<li><img alt="fig2" src="elec_current_fig2.jpg" align="right"
height="300" width="370">In discussing electric circuits
you may come across the term "emf" - electro-motive
force.&nbsp; <b>It is important to realize that an "emf" is
NOT a force !</b></li>
</ul>
<ul>
<li>If a device has an "emf" it has the ability to maintain a
potential difference (voltage).&nbsp; Thus, for example, a
battery maintains an emf between its positive and negative
terminals.</li>
</ul>
<ul>
<li>The emf of a device can be defined by &#949; = dW/dq, where dW
is the work done on a positive charge dq in taking it
acrosss the potential difference of the device.&nbsp; In the
case of a simple circuit with a battery (see above) as a
charge traverses the external (to the battery) circuit it
loses energy.&nbsp; In the circuit above the energy appeara
as heat and light in the light bulb.&nbsp; When the
charge&nbsp; returns to the battery the emf of the battery
replenishes its energy.</li>
</ul>
<ul>
<li>At this introductory level we can consider the emf of a
"source" (battery, generator etc) to be exactly equivalent
to the voltage provided by the source.</li>
</ul>
<ul>
<li>The direction of the emf always represents the direction a
positive charge would move in the external circuit.&nbsp;
See circuit at right.&nbsp; The emf direction is an
important factor when we use Kirchoff's laws to analyze
circuits.</li>
<br>
<br>
</ul>
<ul>
</ul>
</div>
</div>
<br>
<div align="center"><img alt="divider" src="divider_ornbarblu.gif"
height="64" width="393"><br>
<blockquote>
<div align="left">
<h2><font color="#3333ff"><u>Internal Resistance</u></font></h2>
</div>
</blockquote>
<div align="left">
<ul>
<li>All emfs - batteries, generators etc - and electrical
measuring devices - ammeters, voltmeters etc - have an
"internal resistance".</li>
</ul>
<ul>
<li><img alt="fig4" src="elec_current_fig4.jpg" align="right"
height="158" width="152">As far as circuit analysis is
concerned these internal resistances can simply be
considered as resistors in series with the "ideal"
emf/meter.</li>
</ul>
<ul>
<li>For ammeters (current measuring devices) the goal is to
have as low an internal resistance as possible so that the
current is not affected.</li>
</ul>
<p align="center"><img alt="fig3" src="elec_current_fig3.jpg"
align="middle" height="96" width="134"></p>
<ul>
<li>For a voltmeter the internal resistance should be as large
as possible.<br>
</li>
</ul>
</div>
<br>
<div align="left"><br>
</div>
</div>
<img src="netbar.gif" height="40" width="100%">
<center>
<p style="color: rgb(255, 0, 0); font-style: italic;"
class="MsoNormal">Q: Does light have mass?<br>
A: Of course not. It's not even Catholic!!!</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>
</html>