init commit
This commit is contained in:
517
elec_gauss_apps.html
Normal file
517
elec_gauss_apps.html
Normal file
@@ -0,0 +1,517 @@
|
||||
<!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>Electricity - Quantitative use of Gauss's Law - Physics 299</title>
|
||||
</head>
|
||||
<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
|
||||
255);" link="#0000ee" alink="#ff0000" bgcolor="#3333ff"
|
||||
text="#000000" vlink="#551a8b">
|
||||
<center>
|
||||
<h1> <img src="ULPhys1.gif" height="50" align="texttop"
|
||||
width="189"></h1>
|
||||
</center>
|
||||
<center>
|
||||
<h1>Quantitative Use of Gauss's Law <br>
|
||||
</h1>
|
||||
</center>
|
||||
<center><img src="celticbar.gif" height="22" width="576"><br>
|
||||
<br>
|
||||
<font color="#ff0000"><i>"</i></font><font color="#ff0000"><i>
|
||||
<meta http-equiv="content-type" content="text/html;
|
||||
charset=windows-1252">
|
||||
It has been said that democracy is the worst form of
|
||||
government except all the others that have been tried."</i></font><br>
|
||||
Winston Churchill<br>
|
||||
</center>
|
||||
<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
|
||||
<div align="center"> <br>
|
||||
<img alt="elec gauss 3" src="elec_gauss_eqn3.jpg"
|
||||
height="84" width="233"> </div>
|
||||
<ul>
|
||||
<li>Gauss's Law is valid for any closed surface (a Gaussian
|
||||
surface) and any distribution of charges. If the electric
|
||||
field is known at every point on the surface S the integral can
|
||||
in principle be evaluated and will be seen to be equal to the
|
||||
sum of the enclosed charges divided by ε<sub>0</sub>.
|
||||
However, only in certain very symmetric situations,
|
||||
where we can infer a great deal of information about the
|
||||
electric field, can it be used to actually calculate <b>E</b>.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
In such cases Gauss's Law provides a short cut to determining <b>E</b>.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
The key is to be able to "extract" the <b>E</b> from the flux
|
||||
integral.</li>
|
||||
</ul>
|
||||
<ul>
|
||||
<li>We will consider three possible geometric situations in which
|
||||
we can obtain <b>E</b> from Gauss's Law:</li>
|
||||
<ul>
|
||||
<li>Spherical symmetry - three dimensions</li>
|
||||
<li>Rectangular symmetry - two dimensions</li>
|
||||
<li>Cylindrical symmetry - one dimension</li>
|
||||
</ul>
|
||||
</ul>
|
||||
<div align="center"><img alt="divider bar"
|
||||
src="divider_ornbarblu.gif" height="64" width="393"><br>
|
||||
</div>
|
||||
<div align="center"><big><font color="#3333ff"><u><big><b>SPHERICAL
|
||||
SYMMETRY</b></big></u></font></big></div>
|
||||
<ul>
|
||||
</ul>
|
||||
<ul>
|
||||
<li><big><b>Single Point Charge</b></big></li>
|
||||
</ul>
|
||||
<blockquote><img alt="elec gauss figure 5"
|
||||
src="elec_gauss_figure5.jpg" height="313" align="right"
|
||||
width="237">Consider a single point charge +Q and a spherical
|
||||
surface, S, of radius r and center at the location of
|
||||
+Q. From the symmetry of this situation we can conclude
|
||||
that, everywhere on the surface S, <b>E</b> has the same value
|
||||
and is directed radially outwards (normal to the surface).
|
||||
This is the same as the direction of <b>dA</b>. Therefore,<br>
|
||||
<div align="center"><img alt="elec gauss eqn5"
|
||||
src="elec_gauss_eqn5.png" height="62" width="515"><br>
|
||||
<div align="left">so that,<br>
|
||||
<div align="center"><img alt="elec gauss eqn6"
|
||||
src="elec_gauss_eqn6.png" height="64" width="117"><br>
|
||||
<div align="left">which is exactly Coulomb's Law !!<br>
|
||||
<br>
|
||||
As has already been stated - <font color="#ff0000"><big><b>Gauss's
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
Law and Coulomb's Law are different statements of
|
||||
the same physical principle.<br>
|
||||
<font color="#330033"><br>
|
||||
<br>
|
||||
</font></b></big></font></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</blockquote>
|
||||
<div align="center">
|
||||
<div align="left">
|
||||
<div align="center">
|
||||
<div align="left">
|
||||
<ul>
|
||||
<li><big><b>Spherical Charge Distribution with Uniform
|
||||
Charge Density</b></big></li>
|
||||
</ul>
|
||||
<blockquote>Charge<b> </b>is distributed uniformly
|
||||
throughout the volume of the sphere (this means that the
|
||||
sphere must be a non-conductor since as we have seen the
|
||||
charge on a conductor must reside on the surface) such
|
||||
that the total charge Q is given by,<br>
|
||||
<br>
|
||||
<div align="center"><img alt="elec gauss eqn7"
|
||||
src="elec_gauss_eqn7.png" height="41" width="117"><br>
|
||||
<br>
|
||||
<div align="left">where ρ is the (volume) charge
|
||||
density, in units of Coulombs/m<sup>3</sup>.<br>
|
||||
<br>
|
||||
What is the electric field at any point either outside
|
||||
or inside the sphere ?<br>
|
||||
Due to the symmetry of this configuration we can
|
||||
conclude that <b>E</b> is directed radially outwards
|
||||
everywhere and can (at most) depend only on the
|
||||
(radial) distance from the center of the sphere.
|
||||
There are two distinct regions to consider:<br>
|
||||
<br>
|
||||
<b><img alt="elec gauss figure 5"
|
||||
src="elec_gauss_figure6.png" height="215"
|
||||
align="right" width="155"><u>Outside the
|
||||
sphere, r > R</u></b><br>
|
||||
<br>
|
||||
Applying Gauss's Law over a Gaussian surface (sphere)
|
||||
of radius r, then,<br>
|
||||
<div align="center"><img alt="elec gauss eqn 5"
|
||||
src="elec_gauss_eqn5.png" height="62" width="515"><br>
|
||||
<br>
|
||||
<div align="left">so that,<br>
|
||||
<div align="center"><img alt="elec gauss eqn 6"
|
||||
src="elec_gauss_eqn6.png" height="64"
|
||||
width="117"><br>
|
||||
<br>
|
||||
<div align="left">In other words, for points
|
||||
outside the sphere, the sphere behaves as a
|
||||
point charge located the sphere's center.<br>
|
||||
<img alt="hot" src="hot.gif" height="43"
|
||||
align="middle" width="79"> We saw
|
||||
exactly the same type of behavior when
|
||||
considering the gravitational effect of a
|
||||
spherical mass.<br>
|
||||
<br>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<b><img alt="elec gauss figure 7"
|
||||
src="elec_gauss_figure7.png" height="230"
|
||||
align="right" width="167"><u>Inside the sphere, r
|
||||
< R</u></b><br>
|
||||
<br>
|
||||
Applying Gauss's Law over a Gaussian surface (sphere)
|
||||
of radius r, then,<br>
|
||||
<br>
|
||||
<div align="center"><img alt="elec gauss eqn8"
|
||||
src="elec_gauss_eqn8.jpg" height="63" width="563"><br>
|
||||
<div align="left">Or in terms of Q and R,<br>
|
||||
<br>
|
||||
<div align="center"><img alt="elec gauss eqn9"
|
||||
src="elec_gauss_eqn9.jpg" height="64"
|
||||
width="123"><br>
|
||||
<br>
|
||||
<div align="left">Note that for r < R only
|
||||
the charge inside a sphere of radius r
|
||||
contributes to <b>E</b>. The charge
|
||||
between r and R has no effect.<br>
|
||||
<br>
|
||||
<img alt="exclamation"
|
||||
src="exclamation-icon.gif" height="30"
|
||||
width="31"> It is important to realize that
|
||||
without using Gauss's Law, these results could
|
||||
be obtained via Coulomb's Law, but would
|
||||
involve considerably more work - setting
|
||||
up a non-trivial multiple integral to
|
||||
consider every point charge in the sphere....<br>
|
||||
<br>
|
||||
<div align="center"><img alt="divider bar"
|
||||
src="divider_ornbarblu.gif" height="64"
|
||||
width="393"><br>
|
||||
<big><font color="#3333ff"><u><big><b>CYLINDRICAL
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
SYMMETRY<br>
|
||||
<br>
|
||||
</b></big></u></font></big></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</blockquote>
|
||||
<div align="center">
|
||||
<div align="left">
|
||||
<div align="center">
|
||||
<div align="left">
|
||||
<div align="center">
|
||||
<div align="left">
|
||||
<div align="center">
|
||||
<div align="left">
|
||||
<ul>
|
||||
<li><big><b>Infinite </b><b>Line Charge</b></big>
|
||||
<big><b>with Linear Charge Density λ</b></big></li>
|
||||
</ul>
|
||||
<blockquote><img alt="elec gauss figure8"
|
||||
src="elec_gauss_figure8.png"
|
||||
height="212" align="right" width="331">Determine
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
the <b>E</b> field a distance r from the
|
||||
line charge. (Note that the units of
|
||||
λ are Coulombs/meter)<br>
|
||||
<br>
|
||||
Symmetry tells us that <b>E</b> can only
|
||||
have a component perpendicular to the line
|
||||
charge, that is perpendicular to the
|
||||
cylindrical surface shown.<br>
|
||||
<br>
|
||||
Applying Gauss's Law over the cylindrical
|
||||
Gaussian surface, radius r and length l,
|
||||
as shown, there will in principle be three
|
||||
contributions - one from the curved
|
||||
surface and one from each of the two
|
||||
ends. However, on the ends <b>E</b>
|
||||
and <b>dA</b> are perpendicular, so that
|
||||
<b>E<EFBFBD>dA</b> = 0, therefore there is no
|
||||
contribution to the flux through S.
|
||||
On the curved surface <b>E</b> and <b>dA</b>
|
||||
are parallel, thus,<br>
|
||||
<br>
|
||||
<div align="center"><img alt="elec gauss
|
||||
eqn10" src="elec_gauss_eqn10,jpg.jpg"
|
||||
height="51" width="577"><br>
|
||||
<div align="left">so that,<br>
|
||||
<div align="center"><img alt="elec
|
||||
gauss eqn11"
|
||||
src="elec_gauss_eqn11.jpg"
|
||||
height="75" width="117"><br>
|
||||
<div align="left"><br>
|
||||
We can extend this analysis to the
|
||||
case of a uniformly charged
|
||||
infinite cylinder in a similar
|
||||
manner to the extension of the
|
||||
point charge to the spherical
|
||||
charge distribution above.<br>
|
||||
<br>
|
||||
<div align="center"><img
|
||||
alt="divider bar"
|
||||
src="divider_ornbarblu.gif"
|
||||
height="64" width="393"><br>
|
||||
<big><font color="#3333ff"><u><big><b>RECTANGULAR
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
SYMMETRY</b></big></u></font></big><br>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</blockquote>
|
||||
<div align="center">
|
||||
<div align="left">
|
||||
<div align="center">
|
||||
<div align="left">
|
||||
<div align="center">
|
||||
<div align="left">
|
||||
<ul>
|
||||
<li><big><b>Infinite plane of
|
||||
charge</b></big></li>
|
||||
</ul>
|
||||
<blockquote><img alt="elec gauss
|
||||
figure 9"
|
||||
src="elec_gauss_figure9.jpg"
|
||||
height="200" align="right"
|
||||
width="246">Determine the <b>E</b>
|
||||
field at any distance above or
|
||||
below an infinite plane with
|
||||
charge density σ (Coulombs/m<sup>2</sup>).<br>
|
||||
<br>
|
||||
Symmetry dictates the <b>E</b>
|
||||
must be perpendicular to the
|
||||
surface everywhere.<br>
|
||||
<br>
|
||||
Applying Gauss's Law over the
|
||||
cylindrical surface shown,
|
||||
then the curved surface of the
|
||||
cylinder contributes
|
||||
nothing to the flux since <b>E</b>
|
||||
and <b>dA</b> are
|
||||
perpendicular. But on
|
||||
the ends <b>E</b> and <b>dA</b>
|
||||
are parallel. Therefore,<br>
|
||||
<br>
|
||||
<div align="center"><img
|
||||
alt="elec gauss eqn12"
|
||||
src="elec_gauss_eqn12.jpg"
|
||||
height="56" width="536"><br>
|
||||
<br>
|
||||
<div align="left">so that,<br>
|
||||
<div align="center"><img
|
||||
alt="elec gauss eqn13"
|
||||
src="elec_gauss_eqn13.jpg" height="96" width="117"><br>
|
||||
<br>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</blockquote>
|
||||
<blockquote>That is the electric
|
||||
field is constant - it does
|
||||
not depend on how far the
|
||||
field point is from the plane
|
||||
!! <br>
|
||||
<br>
|
||||
<img alt="exclamation"
|
||||
src="exclamation-icon.gif"
|
||||
height="30" width="31"> Note
|
||||
that this is only true for an
|
||||
infinite plane of
|
||||
charge. If the distance
|
||||
of the field point from the
|
||||
plane is small compared to the
|
||||
"size" of the plane, the above
|
||||
expression is a good
|
||||
approximation.<br>
|
||||
<br>
|
||||
<div align="center"><img
|
||||
alt="divider"
|
||||
src="divider_ornbarblu.gif"
|
||||
height="64" width="393"><br>
|
||||
</div>
|
||||
</blockquote>
|
||||
<div align="center">
|
||||
<div align="left">
|
||||
<ul>
|
||||
<li>In all the above
|
||||
situations the key to
|
||||
using Gauss's Law is <b>SYMMETRY</b>.
|
||||
There must be enough
|
||||
symmetry in the problem
|
||||
to know the direction of
|
||||
<b>E</b> everywhere in
|
||||
the vicinity of the
|
||||
charge
|
||||
distribution.
|
||||
Knowing the direction of
|
||||
<b>E</b> the trick is
|
||||
then to choose a
|
||||
Gaussian surface over
|
||||
which to apply Gauss's
|
||||
Law such that <b>E</b>
|
||||
can be "taken out" of
|
||||
the flux integral.
|
||||
So when using Gauss's
|
||||
Law to determine <b>E</b>
|
||||
there are three key
|
||||
steps:</li>
|
||||
</ul>
|
||||
<ul>
|
||||
<ol>
|
||||
<li>
|
||||
<h3>State what you are
|
||||
assuming about <b>E</b>
|
||||
based on the
|
||||
symmetry of the
|
||||
problem.</h3>
|
||||
</li>
|
||||
<li>
|
||||
<h3>State clearly the
|
||||
Gaussian surface(s)
|
||||
you will use - often
|
||||
most easily done by
|
||||
sketching the
|
||||
surface(s) on a
|
||||
diagram.</h3>
|
||||
</li>
|
||||
<li>
|
||||
<h3>Evaluate the
|
||||
surface (flux)
|
||||
integral to
|
||||
determine E.
|
||||
The symmetry of E
|
||||
and choice of
|
||||
Gaussian surface
|
||||
should allow "E" to
|
||||
be "taken out" of
|
||||
the integral and
|
||||
thus be determined.<br>
|
||||
</h3>
|
||||
</li>
|
||||
</ol>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
<blockquote> </blockquote>
|
||||
<blockquote> </blockquote>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<blockquote>
|
||||
<div align="center">
|
||||
<div align="left">
|
||||
<div align="center"> </div>
|
||||
</div>
|
||||
</div>
|
||||
</blockquote>
|
||||
<ul>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<blockquote>
|
||||
<div align="center">
|
||||
<div align="left"> </div>
|
||||
</div>
|
||||
</blockquote>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<ul>
|
||||
<font color="#ff0000"><big> </big></font>
|
||||
</ul>
|
||||
<font color="#ff0000"><big><b> </b></big></font><img
|
||||
src="netbar.gif" height="40" width="100%"><br>
|
||||
<br>
|
||||
<center><span style="font-size: 12pt; font-family: "Times New
|
||||
Roman";"><span style="color: rgb(255, 0, 0); font-style:
|
||||
italic;"></span></span><font color="#ff0000"><i>An engineer
|
||||
friend of mine told me of a group of scientists that were
|
||||
nominated for a Nobel prize. Using dental tools, they were
|
||||
able to sort out the smallest particles that mankind has yet
|
||||
discovered. The group became known as " the Graders of the
|
||||
Flossed Quark."</i><i><span style="font-size: 12pt;
|
||||
font-family: "Times New Roman";"></span></i><i><span
|
||||
style="font-size: 12pt; font-family: "Times New
|
||||
Roman";">
|
||||
<meta http-equiv="content-type" content="text/html;
|
||||
charset=windows-1252">
|
||||
</span></i><i> </i></font><br>
|
||||
<br>
|
||||
<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>
|
||||
Reference in New Issue
Block a user