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