davisnotes/mag_biotsavart.html

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<title>Magnetism - Biot-Savart Law - Physics 299</title>
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<h1>The Law of Biot-Savart<br>
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<div class="copy-paste-block"><font color="#ff0000"><i><span
class="bqQuoteLink">"A</span></i></font><font
color="#ff0000"><i><span class="bqQuoteLink"> fact is a simple
statement that everyone believes.&nbsp; It is innocent,
unless found guilty.&nbsp; A hypothesis is a novel
suggestion that no one wants to believe.&nbsp; It is
guilty, until found effective</span></i><span></span>"</font><br>
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<font color="#ff0000"><i> </i><font color="#000000">Edward Teller</font></font><br>
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<li>We have seen that a magnetic field exerts
a force on a moving charge or current, just
as an electric field exerts a force on a
charge.</li>
</ul>
<div align="center"><img
alt="magforcechargeeqn1"
src="mag_force_charge_eqn1.jpg"
align="middle" height="33" width="135">
&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; and &nbsp;
&nbsp; &nbsp; &nbsp;<img
alt="magforcecurrenteqn3"
src="mag_force_current_eqn3.jpg"
align="middle" height="31" width="126"><br>
<br>
<img alt="magbiotsavarteqn1"
src="mag_biotsavart_eqn1.jpg" height="31"
width="81"><br>
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<ul>
<li>In the electric case Coulomb's Law
gives us a method of determining <b>E</b>
from an arbitrary distribution of
charges.&nbsp; For a point charge we
have<br>
</li>
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<div align="center"><img
alt="magbiotsavarteqn3"
src="mag_biotsavart_eqn2.jpg"
height="61" width="180"><br>
<blockquote>
<div align="left">where <img
alt="magbiotsavarteqn3"
src="mag_biotsavart_eqn3.jpg"
height="27" width="19">is a unit
vector from the point charge to
location at which the <b>E</b> field
is to be determined.<br>
<br>
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<ul>
<li><img alt="magbiotsavartfig1"
src="mag_biotsavart_fig1.jpg"
align="right" height="354"
width="341">The Biot-Savart Law
provides a general method of
determining the <b>B</b> field from
an arbitrary current distribution.</li>
</ul>
<div align="center"><img
alt="magbiotsavarteqn4"
src="mag_biotsavart_eqn4.jpg"
height="57" width="144"><br>
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<div align="left">where &#956;<sub>0</sub>
is the permeability of the vacuum
(free space) = 4&#960; x 10<sup>-7</sup>
T.m/A, dl is a "<i>current element</i>"
directed along the current in the
wire and <img
alt="magbiotsavarteqn3"
src="mag_biotsavart_eqn3.jpg"
height="27" width="19">is a unit
vector from <b>dl</b> to where
the <b>B</b> field is to be
calculated as in the diagram at
right.<br>
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<li>There are a number of
important points to be aware of:</li>
</ul>
<blockquote><img alt="exclamation"
src="exclamation-icon.gif"
height="30" width="31"> The law
must be written in differential
form since there is no such thing
as a "point current".&nbsp; Note
that a moving point charge is
equivalent to a current and a
stationary point charge does not
create a magnetic field. <br>
<br>
<img alt="exclamation"
src="exclamation-icon.gif"
height="30" width="31">&nbsp;
Due to the cross product <b>dB</b>
is at right angles to both <b>dl</b>
and <img alt="magbiotsavarteqn3"
src="mag_biotsavart_eqn3.jpg"
height="27" width="19">.&nbsp;
You can't avoid considering 3
dimensions.<br>
<br>
<img alt="exclamation"
src="exclamation-icon.gif"
height="30" width="31">&nbsp; <img
alt="magbiotsavartfig2"
src="mag_biotsavart_fig2.jpg"
align="right" height="274"
width="294">The "sense" of <b>dB</b>
(into or out of the plane in the
above diagram) is determined by
the basic definition of a vector
cross product or equivalently
another "<i>right-hand-rule</i>"
shown at right.<br>
<br>
<img alt="exclamation"
src="exclamation-icon.gif"
height="30" width="31">&nbsp; (&#956;<sub>0</sub>/4&#960;)
plays a similar role to the
Coulomb constant, k (= 1/4&#960;&#949;<sub>0</sub>).&nbsp;
But compare the values of &#956;<sub>0</sub>
and k; this is an indication of
the relative strengths of the
electric and magnetic interactions
of charges.<br>
<br>
<img alt="exclamation"
src="exclamation-icon.gif"
height="30" width="31">&nbsp;
Similar to our initial statement
of Coulomb's Law, the above
expression of the Biot-Savart Law
gives <b>dB</b> due to a current
element in a vacuum.&nbsp; For
real world applications wheres the
current is not in a vacuum a
slight adjustment must be made to
take into account the magnetic
properties of the medium.&nbsp;
This can be done in a similar
manner to the adjustment of
Coulomb's Law in a <a
href="elec_dielectrics.html">dielectric
medium</a>, but is beyond the
scope of this course.<br>
<br>
</blockquote>
<ul>
<li>Needless to say determining
the magnetic field is typically
significantly more complicated
than the electric field. <img
alt="sadface"
src="sadface.jpg"
align="middle" height="52"
width="52"><br>
</li>
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<font color="#ff0000"><i>Energizer bunny arrested.&nbsp; Charged
with battery.&nbsp; </i></font><br>
<br>
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&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>
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