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<title>Electricty - Electric Potential and Potential Difference -
Physics 299</title>
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<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%">
&nbsp;
<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>&nbsp; =&nbsp; -V<sub>AB</sub> q </center>
<p style="text-align: left;">where V<sub>AB</sub> = V<sub>B</sub>
- V<sub>A</sub>&nbsp; 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.&nbsp;
The work done by the "external agent"&nbsp; is -W<sub>AB</sub>.</div>
<ul>
<li>Units of potential difference are volts <br>
<center>
<p>1 Volt&nbsp; =&nbsp; 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.&nbsp; [ 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>.&nbsp;</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.&nbsp;
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.&nbsp; Changes in gravitational energy are
independent of the path taken.&nbsp; 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).&nbsp; 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.&nbsp; To obtain absolute values
of V physicists usually define V = 0 at
infinity.&nbsp; 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>= &#8734; 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).&nbsp;</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.&nbsp; 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.&nbsp; Since
all points have the same potential
it requires no work to move a charge
on such a surface.&nbsp; 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">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<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>
&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>
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