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<meta name="Author" content="C. L. Davis">
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<title>Contents - Physics 299</title>
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<body alink="#ff0000" bgcolor="#ffffff" link="#0000ee" text="#000000"
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vlink="#551a8b">
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<center>
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<h1><img src="ULPhys1.gif" align="texttop" height="50" width="189"></h1>
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</center>
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<center>
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<h1> CONTENTS</h1>
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</center>
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<center><img src="celticbar.gif" height="22" width="576"> </center>
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<br>
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<br>
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<center><font color="#ff0000"><i><font face="Times New Roman, Times,
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serif"> "All science is either physics or stamp collecting"</font></i></font><br>
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Ernest Rutherford<br>
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</center>
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<font><i>Visitors since 20 January, 2011</i><br>
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--> <img src="netbar.gif" align="middle" height="40" width="100%"> <br>
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<ul>
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<li>Classical Electromagnetism
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<ul>
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<li>Electricity
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<ul>
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<li><a href="elec_stat.html">Electric Charge</a></li>
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<li><a href="elec_strmatt.html">Structure of Matter</a><br>
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</li>
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<li><a href="elec_coulomb.html">Coulomb's Law</a></li>
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<li><a href="elec_efield.html">Electric Field</a></li>
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<ul>
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<li><a href="elec_dipole.html">Electric Dipole Field</a></li>
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<li><a href="elec_contdist.html">Electric Field due to
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Continuous Charge Distributions</a></li>
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||||||
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<li><a href="elec_chargemotion.html">Motion of a Point
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Charge in an Electric Field</a><br>
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</li>
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<li><a href="elec_dipexte.html">Electric Dipole in
|
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External Field</a></li>
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</ul>
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<li><a href="elec_gauss.html">Gauss' Law</a></li>
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<ul>
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||||||
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<li><a href="elec_gauss_cond.html">Gauss and Conductors</a></li>
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||||||
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<li><a href="elec_gauss_apps.html">Quantitative use of
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||||||
|
Gauss Law</a></li>
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</ul>
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<li><a href="elec_potential.html">Electric Potential</a></li>
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<ul>
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<li><a href="elec_potential_dipole.html">Dipole Electric
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||||||
|
Potential</a></li>
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<li><a href="elec_potential_efromV.html">Determining "E
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from V" </a></li>
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</ul>
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<li><a href="elec_potenergy.html">Electric Potential
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Energy</a></li>
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<li><a href="elec_capacitors.html">Capacitors</a></li>
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<li><a href="elec_dielectrics.html">Dielectric Materials</a></li>
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||||||
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<li><a href="elec_condins.html">Conductors and Insulators</a></li>
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||||||
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<li><a href="elec_current.html">Electric Current,
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|
Resistance and Power</a></li>
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<li>Electric Circuits</li>
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<ul>
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<li><a href="elec_circuits_sp.html">Series and Parallel</a></li>
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<li><a href="elec_circuits_kirchoff.html">Kirchhoff's
|
||||||
|
Laws</a></li>
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<li><a href="elec_circuits_RC.html">RC Circuits</a><br>
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</li>
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</ul>
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</ul>
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<!-- Start comment
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end comment--> </li>
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<li>Magnetism</li>
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<ul>
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<li><a href="mag_intro.html">Introduction</a><br>
|
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</li>
|
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</ul>
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<ul>
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<li><a href="mag_force_charge.html">Magnetic Force on
|
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Charges</a></li>
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<li><a href="mag_force_current.html">Magnetic Forces on
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Currents</a></li>
|
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<li><a href="mag_dipole.html">Magnetic Dipoles</a></li>
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<li><a href="mag_motionch.html">Motion of Charged Particles
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||||||
|
in Magnetic and Electric Fields</a></li>
|
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<li><a href="mag_biotsavart.html">Biot-Savart Law</a></li>
|
||||||
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<li><a href="mag_ampere.html">Ampere's Law</a></li>
|
||||||
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<li><a href="mag_force_2wires.html">Force Between Two
|
||||||
|
Parallel Wires: Ampere Definition</a><br>
|
||||||
|
</li>
|
||||||
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<li><a href="mag_faraday.html">Faraday's Law of Induction</a></li>
|
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|
<ul>
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||||||
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<li><a href="mag_mutualind.html">Mutual Inductance</a></li>
|
||||||
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<li><a href="mag_selfind.html">Self Inductance</a></li>
|
||||||
|
<li><a href="mag_LR.html">LR Circuits</a></li>
|
||||||
|
</ul>
|
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|
<li><a href="mag_energy.html">Magnetic Energy</a><br>
|
||||||
|
</li>
|
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<ul>
|
||||||
|
</ul>
|
||||||
|
<li><a href="mag_monopoles.html">Magnetic Monopoles and
|
||||||
|
"Gauss's Law for Magnetism"</a></li>
|
||||||
|
<li>Magnetic Properties of Matter<br>
|
||||||
|
</li>
|
||||||
|
<li><a href="mag_displacement.html">Displacement Current</a></li>
|
||||||
|
</ul>
|
||||||
|
<li>Maxwell's Equations</li>
|
||||||
|
<li>Electromagnetic Oscillations</li>
|
||||||
|
<ul>
|
||||||
|
<li>LC Circuits</li>
|
||||||
|
<li>LCR Circuits</li>
|
||||||
|
<li>Alternating Current<br>
|
||||||
|
</li>
|
||||||
|
</ul>
|
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</ul>
|
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</li>
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<li>Light and Optics</li>
|
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|
<ul>
|
||||||
|
<li><a href="lo_emwaves.html">Electromagnetic Waves</a></li>
|
||||||
|
<li><a href="lo_polarisation.html">Polarisation</a></li>
|
||||||
|
<!--
|
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-->
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<li>Geometric Optics</li>
|
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<ul>
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<li><a href="lo_reflection.html">Reflection</a></li>
|
||||||
|
<li><a href="lo_refraction.html">Refraction</a></li>
|
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|
<ul>
|
||||||
|
<li><a href="lo_appdepth.html">Apparent Depth</a></li>
|
||||||
|
<li><a href="lo_tir.html">Total Internal Reflection</a></li>
|
||||||
|
</ul>
|
||||||
|
<li><a href="lo_brewster.html">Brewster's Law</a></li>
|
||||||
|
<li><a href="lo_dispersion.html">Dispersion</a></li>
|
||||||
|
<li><a href="lo_spmirror.html">Spherrical Mirrors</a></li>
|
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|
<li><a href="lo_lenses.html">Thin Lenses</a><br>
|
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|
</li>
|
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|
</ul>
|
||||||
|
<li>Wave (Physical) Optics</li>
|
||||||
|
<ul>
|
||||||
|
<li> <a href="lo_interference.html">Double Slit Interference</a>
|
||||||
|
</li>
|
||||||
|
<li> <a href="lo_intthinfilm.html">Interference from Thin
|
||||||
|
Films</a> </li>
|
||||||
|
<li> <a href="lo_ssdiffraction.html">Single Slit Difraction</a>
|
||||||
|
</li>
|
||||||
|
<li> <a href="lo_dsdiffraction.html">Double Slit
|
||||||
|
Diffraction/Interference</a> </li>
|
||||||
|
<li> <a href="lo_msgratings.html">Multiple Slit
|
||||||
|
Diffraction/Interference - Diffraction Gratings</a> </li>
|
||||||
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</ul>
|
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</ul>
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</ul>
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<!-- Start comment
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end comment-->
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<p><font><br>
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<img src="netbar.gif" height="40" width="100%"> </font></p>
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<center><font><img src="celticbar.gif" height="22" width="576"> <br>
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<br>
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<span style="font-style: italic; color: red;">"Physics is,
|
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|
hopefully, simple. Physicists are not"</span><br>
|
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Edward Teller </font>
|
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<p><font><i>Dr. C. L. Davis</i><br>
|
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|
<i>Physics Department</i><br>
|
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<i>University of Louisville</i><br>
|
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<i>email</i>: <a href="mailto:c.l.davis@louisville.edu">c.l.davis@louisville.edu</a>
|
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<br>
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</font></p>
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<p><font><img src="header-index.gif" height="51" width="92"> </font></p>
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</center>
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<div align="center">
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<p><a href="http://www.nike.com/wearyellow/index_f.html"><font
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color="#ffff00">wearyellow.com</font></a> <br>
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<meta http-equiv="Content-Type" content="text/html;
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<meta name="Author" content="C. L. Davis">
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<title>Electricty - Capacitors - Physics 299</title>
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<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
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<center><img src="ULPhys1.gif" height="50" align="texttop"
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width="189"></center>
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<center>
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<h1>Capacitors<br>
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</h1>
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</center>
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<center><img src="celticbar.gif" height="22" width="576"><br>
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<br>
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<font color="#ff0000"><i>"</i></font><font color="#ff0000"><i>
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<meta http-equiv="content-type" content="text/html;
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charset=windows-1252">
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I used to wonder how it comes about that the electron is
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negative. Negative-positive—these are perfectly symmetric in
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physics. There is no reason whatever to prefer one to the
|
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other. Then why is the electron negative? I thought about this
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for a long time and at last all I could think was 'It won the
|
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fight!' "</i></font><br>
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Albert Einstein<br>
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</center>
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<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
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<blockquote>
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<h2><u>Calculating Capacitance</u></h2>
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</blockquote>
|
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<ul>
|
||||||
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</ul>
|
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<ul>
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||||||
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<li>A capacitor is a system of two insulated conductors. <br>
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</li>
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</ul>
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<ul>
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</ul>
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<ul>
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<li><img alt="elec cap fig1" src="elec_cap_fig1.jpg" height="411"
|
||||||
|
align="right" width="700">The parallel plate capacitor is the
|
||||||
|
simplest example. When the two conductors have equal but
|
||||||
|
opposite charge, the <b>E</b> field between the plates can be
|
||||||
|
found by simple application of Gauss's Law.</li>
|
||||||
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</ul>
|
||||||
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<blockquote>Assuming the plates are large enough so that the <b>E</b>
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||||||
|
field between them is uniform and directed perpendicular, then
|
||||||
|
applying Gauss's Law over surface S<sub>1</sub> we find,<br>
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||||||
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<div align="center"><img alt="elec cap eqn1"
|
||||||
|
src="elec_cap_eqn1.png" height="64" width="191"><br>
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||||||
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<div align="left">where A is the area of S<sub>1</sub>
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||||||
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perpendicular to the <b>E</b> field and σ is the surface
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||||||
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charge density on the plate (assumed uniform).
|
||||||
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Therefore, <br>
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<div align="center"><img alt="elec cap eqn2"
|
||||||
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src="elec_cap_eqn2.png" height="60" width="67"><br>
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||||||
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<br>
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||||||
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<div align="left">everywhere between the plates.<br>
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</div>
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</div>
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||||||
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</div>
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</div>
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</blockquote>
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<div align="center">
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<div align="left">
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<div align="center">
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<div align="left">
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<ul>
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<li>The potential difference between the plates can be
|
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found from</li>
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||||||
|
</ul>
|
||||||
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<div align="center"><img alt="elec cap eqn3"
|
||||||
|
src="elec_cap_eqn3.png" height="64" width="335"><br>
|
||||||
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<blockquote>
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||||||
|
<div align="left">where A and B are points, one on each
|
||||||
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plate, and we integrate along an <b>E</b> field line,
|
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d is the plate separation, A the plate area and q the
|
||||||
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total charge on either plate.<br>
|
||||||
|
</div>
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||||||
|
</blockquote>
|
||||||
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<div align="left">
|
||||||
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<ul>
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||||||
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<li>The capacitance (capacity) of this capacitor is
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||||||
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defined as,</li>
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||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="elec cap eqn4"
|
||||||
|
src="elec_cap_eqn4.png" height="63" width="148"><br>
|
||||||
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<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li>The expression for C for all capacitors is the
|
||||||
|
ratio of the magnitude of the total charge (on
|
||||||
|
either plate) to the magnitude of the potential
|
||||||
|
difference between the plates.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>Units of
|
||||||
|
C:
|
||||||
|
Coulomb/Volt = Farad, 1 C/V =
|
||||||
|
1 F</li>
|
||||||
|
</ul>
|
||||||
|
<blockquote><img alt="exclamation"
|
||||||
|
src="exclamation-icon.gif" height="30"
|
||||||
|
width="31"> Note that since the Coulomb is a
|
||||||
|
very large unit of charge the Farad is also a very
|
||||||
|
large unit of capacitance. Typical
|
||||||
|
capacitors in circuits are measured in μF (10<sup>-6</sup>)
|
||||||
|
or pF (10<sup>-12</sup>).<br>
|
||||||
|
</blockquote>
|
||||||
|
<ul>
|
||||||
|
<li><img alt="exclamation"
|
||||||
|
src="exclamation-icon.gif" height="30"
|
||||||
|
width="31"> Note that the expression for the
|
||||||
|
capacitance of the parallel plate capacitor
|
||||||
|
depends on the geometric properties (A and
|
||||||
|
d). Even though it appears that there is
|
||||||
|
also a dependence on the charge and potential
|
||||||
|
difference (q/ΔV), what happens is that whatever
|
||||||
|
charge you place on the capacitor the pd adjusts
|
||||||
|
itself so that the ratio q/ΔV remains
|
||||||
|
constant. This is a general rule for
|
||||||
|
all capacitors. The capacitance is set by
|
||||||
|
the construction of the capacitor - not the
|
||||||
|
charge or voltage applied.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li><img alt="exclamation"
|
||||||
|
src="exclamation-icon.gif" height="30"
|
||||||
|
width="31"> The above expression for the
|
||||||
|
parallel plate capacitor is strictly only true
|
||||||
|
for an infinite parallel plate capacitor - in
|
||||||
|
which "fringing" (see above) does not
|
||||||
|
occur. However, so long as d is small
|
||||||
|
compared to the "size" of the plates, the simple
|
||||||
|
expression above is a good approximation.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li><img alt="exclamation"
|
||||||
|
src="exclamation-icon.gif" height="30"
|
||||||
|
width="31"> The parallel plate capacitor
|
||||||
|
provides an easy way to "measure" ε<sub>0</sub>
|
||||||
|
<br>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
<blockquote>
|
||||||
|
<div align="center"><img alt="elec cap eqn5"
|
||||||
|
src="elec_cap_eqn5.png" height="54" width="93"><br>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<div align="center">
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li>As indicated above the parallel plate
|
||||||
|
capacitor is the most basic capacitor.
|
||||||
|
You should also be able to determine the
|
||||||
|
expressions for the capacitance of spherical
|
||||||
|
and cylindrical capacitors,</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="elec cap fig3"
|
||||||
|
src="elec_cap_fig3.jpg" height="239"
|
||||||
|
width="311">
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
<img
|
||||||
|
alt="elec cap fig2" src="elec_cap_fig2.jpg"
|
||||||
|
height="313" width="419"><br>
|
||||||
|
<br>
|
||||||
|
<img alt="divider" src="divider_ornbarblu.gif"
|
||||||
|
height="64" width="393"><br>
|
||||||
|
<blockquote>
|
||||||
|
<div align="left">
|
||||||
|
<h2><u>Energy and Capacitors</u></h2>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li> One of the most important uses of
|
||||||
|
capacitors is to store electrical
|
||||||
|
energy.</li>
|
||||||
|
</ul>
|
||||||
|
<blockquote>If a capacitor is placed in a
|
||||||
|
circuit with a battery, the potential
|
||||||
|
difference (voltage) of the battery will
|
||||||
|
force electric charge to appear on the
|
||||||
|
plates of the capacitor. The work
|
||||||
|
done by the battery in charging the
|
||||||
|
capacitor is stored as electrical
|
||||||
|
(potential) energy in the capacitor.
|
||||||
|
This energy can be released at a later
|
||||||
|
time to perform work.<br>
|
||||||
|
<br>
|
||||||
|
<div align="center"><img alt="elec cap
|
||||||
|
fig4" src="elec_cap_fig4.jpg"
|
||||||
|
height="204" width="297"></div>
|
||||||
|
</blockquote>
|
||||||
|
<div align="center">
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li>The work necessary to move a
|
||||||
|
charge dq onto one of the plates is
|
||||||
|
given by, dW = Vdq, where V is the
|
||||||
|
pd (voltage) of the battery (=
|
||||||
|
q/C). The total work to place
|
||||||
|
Q on the plate is given by,</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="elec cap
|
||||||
|
eqn6" src="elec_cap_eqn6.png"
|
||||||
|
height="58" width="423"><br>
|
||||||
|
<blockquote>
|
||||||
|
<div align="left">which is equal to
|
||||||
|
the stored electrical potential
|
||||||
|
energy, U.<br>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li>The electrical energy actually
|
||||||
|
resides in the electric field
|
||||||
|
between the plates of the
|
||||||
|
capacitor. For a parallel
|
||||||
|
plate capacitor using C =
|
||||||
|
Aε<sub>0</sub>/d and E =
|
||||||
|
Q/Aε<sub>0</sub> we may write
|
||||||
|
the electrical potential energy,
|
||||||
|
<br>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="elec
|
||||||
|
cap eqn7"
|
||||||
|
src="elec_cap_eqn7.png"
|
||||||
|
height="68" width="339"><br>
|
||||||
|
<blockquote>
|
||||||
|
<div align="left">(Ad) is the
|
||||||
|
volume between the plates,
|
||||||
|
therefore we define the energy
|
||||||
|
density,<br>
|
||||||
|
<br>
|
||||||
|
<div align="center"><img
|
||||||
|
alt="elec cap eqn8"
|
||||||
|
src="elec_cap_eqn8.png"
|
||||||
|
height="54" width="181"><br>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<div align="left">
|
||||||
|
<div align="center">
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li>Although we have
|
||||||
|
evaluated this
|
||||||
|
expression for the
|
||||||
|
energy density for a
|
||||||
|
parallel plate capacitor
|
||||||
|
it is actually a general
|
||||||
|
expression.
|
||||||
|
Wherever there is an
|
||||||
|
electric field the
|
||||||
|
energy density is given
|
||||||
|
by the above.</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img
|
||||||
|
alt="divider"
|
||||||
|
src="divider_ornbarblu.gif"
|
||||||
|
height="64" width="393"><br>
|
||||||
|
<blockquote>
|
||||||
|
<div align="left">
|
||||||
|
<h2><u>Combinations of
|
||||||
|
Capacitors</u></h2>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<div align="left">
|
||||||
|
<blockquote>It is common
|
||||||
|
to find multiple
|
||||||
|
combinations of
|
||||||
|
capacitors in
|
||||||
|
electrical
|
||||||
|
circuits. In the
|
||||||
|
simplest situations
|
||||||
|
capacitors can be
|
||||||
|
considered to be
|
||||||
|
connected in <b><i>series</i></b>
|
||||||
|
or in <i><b>parallel</b></i>.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
<br>
|
||||||
|
</blockquote>
|
||||||
|
<ul>
|
||||||
|
<ul>
|
||||||
|
<li><big><b>Capacitors
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
in Series</b></big></li>
|
||||||
|
</ul>
|
||||||
|
</ul>
|
||||||
|
<blockquote>
|
||||||
|
<blockquote>When
|
||||||
|
different capacitors
|
||||||
|
are connected in
|
||||||
|
series the charge on
|
||||||
|
each capacitor is
|
||||||
|
the same but the
|
||||||
|
voltage (pd) across
|
||||||
|
each capacitor is
|
||||||
|
different<br>
|
||||||
|
<div align="center"><img
|
||||||
|
alt="elec cap
|
||||||
|
fig5"
|
||||||
|
src="elec_cap_fig5.jpg"
|
||||||
|
height="180"
|
||||||
|
width="312"></div>
|
||||||
|
</blockquote>
|
||||||
|
</blockquote>
|
||||||
|
<div align="center">
|
||||||
|
<div align="left"><br>
|
||||||
|
<blockquote>
|
||||||
|
<blockquote>
|
||||||
|
<div
|
||||||
|
align="left">In
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
this
|
||||||
|
situation,
|
||||||
|
using the fact
|
||||||
|
that V = V<sub>1</sub>
|
||||||
|
+ V<sub>2</sub>
|
||||||
|
+V<sub>3</sub>
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
we can show
|
||||||
|
that, as far
|
||||||
|
as the voltage
|
||||||
|
source is
|
||||||
|
concerned, the
|
||||||
|
capacitors can
|
||||||
|
be replaced by
|
||||||
|
a single
|
||||||
|
"equivalent"
|
||||||
|
capacitor C<sub>eq</sub>
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
given by, <br>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
</blockquote>
|
||||||
|
<div align="center"><img
|
||||||
|
alt="elec cap
|
||||||
|
fig9"
|
||||||
|
src="elec_cap_eqn9.png"
|
||||||
|
height="63"
|
||||||
|
width="182"><br>
|
||||||
|
</div>
|
||||||
|
<br>
|
||||||
|
<ul>
|
||||||
|
<ul>
|
||||||
|
<li><big><b>Capacitors
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
in Parallel</b></big></li>
|
||||||
|
</ul>
|
||||||
|
</ul>
|
||||||
|
<blockquote>
|
||||||
|
<blockquote>For
|
||||||
|
capacitors
|
||||||
|
connected in
|
||||||
|
parallel it is
|
||||||
|
the voltage
|
||||||
|
which is same
|
||||||
|
for each
|
||||||
|
capacitor, the
|
||||||
|
charge being
|
||||||
|
different.<br>
|
||||||
|
<br>
|
||||||
|
<div
|
||||||
|
align="center"><img
|
||||||
|
alt="elec cap
|
||||||
|
fig6"
|
||||||
|
src="elec_cap_fig6.jpg"
|
||||||
|
height="176"
|
||||||
|
width="360"><br>
|
||||||
|
<div
|
||||||
|
align="left"><br>
|
||||||
|
Using the fact
|
||||||
|
that Q<sub>Total</sub>=
|
||||||
|
Q<sub>1</sub>
|
||||||
|
+ Q<sub>2</sub>
|
||||||
|
+ Q<sub>3</sub>
|
||||||
|
we can show
|
||||||
|
that the
|
||||||
|
equivalent
|
||||||
|
capacitor, C<sub>eq</sub>
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
is given by,<br>
|
||||||
|
<br>
|
||||||
|
<div
|
||||||
|
align="center"><img
|
||||||
|
alt="elec cap
|
||||||
|
eqn10"
|
||||||
|
src="elec_cap_eqn10.png"
|
||||||
|
height="29"
|
||||||
|
width="172"><br>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
</blockquote>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<blockquote> </blockquote>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<blockquote> </blockquote>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<blockquote>
|
||||||
|
<div align="center"> </div>
|
||||||
|
</blockquote>
|
||||||
|
<p> <img src="netbar.gif" height="40" width="100%"> </p>
|
||||||
|
<center>
|
||||||
|
<p class="MsoNormal"><span style="color: rgb(255, 0, 0);
|
||||||
|
font-style: italic;">
|
||||||
|
<meta http-equiv="content-type" content="text/html;
|
||||||
|
charset=windows-1252">
|
||||||
|
At the electric company: <i>"We would be delighted if you
|
||||||
|
send in your bill. However, if you don't, you will be."</i></span><br>
|
||||||
|
<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>
|
||||||
Binary file not shown.
|
After Width: | Height: | Size: 8.0 KiB |
|
|
@ -0,0 +1,101 @@
|
||||||
|
<!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 - Charged Particle Motion in an Electric Field -
|
||||||
|
Physics 299</title>
|
||||||
|
</head>
|
||||||
|
<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
|
||||||
|
255);" link="#0000ee" alink="#ff0000" vlink="#551a8b">
|
||||||
|
<center>
|
||||||
|
<h1> <img src="ULPhys1.gif" height="50" align="texttop"
|
||||||
|
width="189"></h1>
|
||||||
|
</center>
|
||||||
|
<center>
|
||||||
|
<h1>Charged Particle Motion in an Electric Field <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">
|
||||||
|
The hardest thing in the world to understand is the income tax"</i></font><br>
|
||||||
|
Albert Einstein<br>
|
||||||
|
</center>
|
||||||
|
<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>Having determined the electric field we now want to determine
|
||||||
|
the behaviour of a point charge, q<sub>0</sub>, placed in this
|
||||||
|
field.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>The force on the charge is given by <b>F</b> = q<sub>0</sub><b>E</b>.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
But Newton's second law tells us that <b>F</b> = m<b>a</b>, so
|
||||||
|
that the acceleration of the particle can be written, <b>a</b>
|
||||||
|
= (q<sub>0</sub>/m)<b>E</b>.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>Once we have an expression for the acceleration it is usually
|
||||||
|
possible to determine the trajectory of the particle, although
|
||||||
|
in the general case this will involve solving differential
|
||||||
|
equations. However, when <b>E</b> is constant the
|
||||||
|
acceleration is constant which allows us to use the kinematic
|
||||||
|
equations describing motion under constant acceleration from the
|
||||||
|
beginning of the first semester of this course (Physics
|
||||||
|
298). Important equations in Physics should never be
|
||||||
|
forgotten <img alt="sadface" src="sadface.jpg" height="24"
|
||||||
|
width="24">. <br>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>In two dimensions, with <b>E</b> constant in one direction
|
||||||
|
and zero in the other, charged particle motion can be treated in
|
||||||
|
the same way as projectile motion of a particle under the
|
||||||
|
influence of a (constant) gravitational field.</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="charged particle motion"
|
||||||
|
src="elec_charged_particle_motion.jpg" height="172" width="292"><br>
|
||||||
|
</div>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
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|
<ul>
|
||||||
|
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|
||||||
|
<div align="center"> </div>
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<div style="text-align: left;"><img src="netbar.gif" height="40"
|
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|
width="100%"> </div>
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<center><span style="font-size: 12pt; font-family: "Times New
|
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Roman";"><span style="color: rgb(255, 0, 0); font-style:
|
||||||
|
italic;"></span></span><span style="font-size: 12pt;
|
||||||
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font-family: "Times New Roman";"><span style="color:
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rgb(255, 0, 0); font-style: italic;">
|
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<meta http-equiv="content-type" content="text/html;
|
||||||
|
charset=windows-1252">
|
||||||
|
</span></span><br>
|
||||||
|
<i><font color="#ff0000">What do you get if you have Avogadro's
|
||||||
|
number of donkeys?
|
||||||
|
Answer: molasses (a mole of asses)</font></i><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>
|
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|
<p><br>
|
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<html>
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<head>
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<meta http-equiv="Content-Type" content="text/html;
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charset=windows-1252">
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alpha) [Netscape]">
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<meta name="Author" content="C. L. Davis">
|
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<title>Electricity - RC circuits - Physics 299</title>
|
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|
<meta content="C. L. Davis" name="author">
|
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|
</head>
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<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
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255);" link="#0000ee" alink="#ff0000" vlink="#551a8b">
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<center>
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<h1> <img src="ULPhys1.gif" height="50" align="texttop"
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|
width="189"></h1>
|
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|
</center>
|
||||||
|
<center>
|
||||||
|
<h1>RC Circuits<br>
|
||||||
|
</h1>
|
||||||
|
</center>
|
||||||
|
<center><img src="celticbar.gif" height="22" width="576"><br>
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<br>
|
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|
<font color="#ff0000"><i>
|
||||||
|
<meta http-equiv="content-type" content="text/html;
|
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|
charset=windows-1252">
|
||||||
|
</i></font><font color="#ff0000"><i>
|
||||||
|
<meta http-equiv="content-type" content="text/html;
|
||||||
|
charset=windows-1252">
|
||||||
|
</i></font>
|
||||||
|
<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. It is innocent,
|
||||||
|
unless found guilty. A hypothesis is a novel
|
||||||
|
suggestion that no one wants to believe. It is
|
||||||
|
guilty, until found effective</span></i><span></span>"</font><br>
|
||||||
|
</div>
|
||||||
|
<font color="#ff0000"><i> </i><font color="#000000">Edward Teller</font></font><br>
|
||||||
|
</center>
|
||||||
|
<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
|
||||||
|
<br>
|
||||||
|
<div align="center">
|
||||||
|
<h2><u><font color="#3333ff">CHARGING</font></u></h2>
|
||||||
|
</div>
|
||||||
|
<ul>
|
||||||
|
<li><img alt="fig1" src="elec_RC_fig1.jpg" height="344"
|
||||||
|
align="right" width="532">An example of a series RC circuit is
|
||||||
|
shown at right. With the emf included in the circuit,
|
||||||
|
applying the loop theorem we find</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="eqn1" src="elec_RC_eqn1.jpg"
|
||||||
|
height="53" width="165"><br>
|
||||||
|
<blockquote>
|
||||||
|
<div align="left">where q/C is the voltage drop across the
|
||||||
|
capacitor and i is the current in the circuit.<br>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li>Using the fact that i = dq/dt, we obtain</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="eqn2" src="elec_RC_eqn2.jpg"
|
||||||
|
height="76" width="155"><br>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li>This is a "simple" differential equation the solution
|
||||||
|
of which can be written</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="eqn3" src="elec_RC_eqn3.jpg"
|
||||||
|
height="38" width="197"><br>
|
||||||
|
<blockquote>
|
||||||
|
<div align="left">or<br>
|
||||||
|
<div align="center"><img alt="eqn4"
|
||||||
|
src="elec_RC_eqn4.jpg" height="68" width="169"><br>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li>The voltage across the capacitor, V<sub>C</sub> =
|
||||||
|
q/C and the voltage across the resistor, V<sub>R</sub>
|
||||||
|
= iR. Using the equations above we find that
|
||||||
|
the dependence of these voltages on time is shown
|
||||||
|
below</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="fig2"
|
||||||
|
src="elec_RC_fig2.jpg" height="286" width="399"><br>
|
||||||
|
<br>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li><img alt="exclamation"
|
||||||
|
src="exclamation-icon.gif" height="30"
|
||||||
|
width="31"> Note that the time on the
|
||||||
|
horizontal axis is measured in units of τ = RC,
|
||||||
|
the capacitative time constant.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li><img alt="exclamation"
|
||||||
|
src="exclamation-icon.gif" height="30"
|
||||||
|
width="31"> After one time constant V<sub>C</sub>
|
||||||
|
has reached 63% (1 - e<sup>-1</sup>) of its
|
||||||
|
maximum value and V<sub>R</sub> has 37% (1/e) of
|
||||||
|
its final value.</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="divider"
|
||||||
|
src="divider_ornbarblu.gif" height="64"
|
||||||
|
width="100%"><br>
|
||||||
|
<h2><font color="#3333ff"><u>DISCHARGING</u></font></h2>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li><img alt="fig1" src="elec_RC_fig1.jpg"
|
||||||
|
height="344" align="right" width="532">Now
|
||||||
|
switch the emf out of the circuit and
|
||||||
|
reapply the loop theorem</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="eqn5"
|
||||||
|
src="elec_RC_eqn5.jpg" height="53"
|
||||||
|
width="132"><br>
|
||||||
|
<blockquote>
|
||||||
|
<div align="left">which gives<br>
|
||||||
|
<div align="center"><img alt="eqn6"
|
||||||
|
src="elec_RC_eqn6.jpg" height="65"
|
||||||
|
width="137"><br>
|
||||||
|
<br>
|
||||||
|
<div align="left">which has the solution<br>
|
||||||
|
<div align="center"><img alt="eqn7"
|
||||||
|
src="elec_RC_eqn7.jpg" height="44"
|
||||||
|
width="171"><br>
|
||||||
|
<div align="left">and<br>
|
||||||
|
<div align="center"><img
|
||||||
|
alt="eqn8"
|
||||||
|
src="elec_RC_eqn8.jpg"
|
||||||
|
height="70" width="192"><br>
|
||||||
|
</div>
|
||||||
|
<br>
|
||||||
|
</div>
|
||||||
|
<br>
|
||||||
|
<div align="left">where Cε is the
|
||||||
|
initial charge on the capacitor
|
||||||
|
and ε/R is the initial voltage
|
||||||
|
across the capacitor.<br>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<div align="left">
|
||||||
|
<div align="center">
|
||||||
|
<div align="left">
|
||||||
|
<div align="center">
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li><img alt="fig3"
|
||||||
|
src="elec_RC_fig3.jpg"
|
||||||
|
height="410" align="right"
|
||||||
|
width="316">The time
|
||||||
|
dependence of V<sub>C</sub> and
|
||||||
|
V<sub>R</sub> (where V<sub>R</sub>
|
||||||
|
is proportional to the current
|
||||||
|
in the capacitor) are shown at
|
||||||
|
right.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li><img alt="exclamation"
|
||||||
|
src="exclamation-icon.gif"
|
||||||
|
height="30" width="31"> Once
|
||||||
|
again the time axis is measured
|
||||||
|
in units of RC.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li><img alt="exclamation"
|
||||||
|
src="exclamation-icon.gif"
|
||||||
|
height="30" width="31"> After
|
||||||
|
one time constant V<sub>C</sub>
|
||||||
|
has decreased to 37% (1/e) of
|
||||||
|
its initial value and |V<sub>R</sub>|
|
||||||
|
has decreased to 37% (1/e) of
|
||||||
|
its initial value. </li>
|
||||||
|
</ul>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<blockquote>
|
||||||
|
<div align="left">
|
||||||
|
<div align="center">
|
||||||
|
<div align="left">
|
||||||
|
<div align="center">
|
||||||
|
<div align="left"> </div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<div align="left">
|
||||||
|
<div align="center"><br>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<div align="left"> </div>
|
||||||
|
<img src="netbar.gif" height="40" width="100%">
|
||||||
|
<center>
|
||||||
|
<p style="color: rgb(255, 0, 0); font-style: italic;"
|
||||||
|
class="MsoNormal">
|
||||||
|
<meta http-equiv="content-type" content="text/html;
|
||||||
|
charset=windows-1252">
|
||||||
|
</p>
|
||||||
|
<font color="#ff0000"><i>Q: What did one quantum physicist say
|
||||||
|
when he wanted to fight another quantum physicist?<br>
|
||||||
|
A: Let me <font color="#ff0000">atom</font>. </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>
|
||||||
|
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|
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|
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|
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|
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<html>
|
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|
<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 - Kirchoff's Laws - Physics 299</title>
|
||||||
|
<meta content="C. L. Davis" name="author">
|
||||||
|
</head>
|
||||||
|
<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
|
||||||
|
255);" link="#0000ee" alink="#ff0000" vlink="#551a8b">
|
||||||
|
<center>
|
||||||
|
<h1> <img src="ULPhys1.gif" height="50" align="texttop"
|
||||||
|
width="189"></h1>
|
||||||
|
</center>
|
||||||
|
<center>
|
||||||
|
<h1>Kirchhoff's Laws<br>
|
||||||
|
</h1>
|
||||||
|
</center>
|
||||||
|
<center><img src="celticbar.gif" height="22" width="576"><br>
|
||||||
|
<br>
|
||||||
|
<font color="#ff0000"><i>
|
||||||
|
<meta http-equiv="content-type" content="text/html;
|
||||||
|
charset=windows-1252">
|
||||||
|
</i></font><font color="#ff0000"><i>
|
||||||
|
<meta http-equiv="content-type" content="text/html;
|
||||||
|
charset=windows-1252">
|
||||||
|
</i></font>
|
||||||
|
<div class="copy-paste-block"><font color="#ff0000"><i><span
|
||||||
|
class="bqQuoteLink">"An expert is a man who has made all
|
||||||
|
the mistakes which can be made, in a narrow field.</span></i><span></span>
|
||||||
|
</font>"<br>
|
||||||
|
</div>
|
||||||
|
<font color="#ff0000"><i> </i><font color="#000000">Niels Bohr</font></font><br>
|
||||||
|
</center>
|
||||||
|
<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
|
||||||
|
<ul>
|
||||||
|
<li>
|
||||||
|
<div align="left">The most common general method to analyze
|
||||||
|
electrical circuits is by use of Kirchhoff's Laws.<img
|
||||||
|
alt="kirchoff" src="kirchhoff.jpg" height="133"
|
||||||
|
align="middle" width="84"></div>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
<blockquote>
|
||||||
|
<h3><u><img alt="staricon" src="StarIconGreen.png" height="48"
|
||||||
|
align="middle" width="50"> Junction Theorem</u></h3>
|
||||||
|
</blockquote>
|
||||||
|
<blockquote>
|
||||||
|
<div align="center"><b><big><font color="#3333ff"><i>At any
|
||||||
|
junction in a circuit the current entering the junction
|
||||||
|
must equal the current leaving the junction.</i><i><br>
|
||||||
|
</i></font></big></b></div>
|
||||||
|
<br>
|
||||||
|
<div align="center">(This is nothing more than a statement of
|
||||||
|
conservation of charge)<br>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<blockquote>
|
||||||
|
<h3><u><img alt="staricon" src="StarIconGreen.png" height="52"
|
||||||
|
align="middle" width="53"> Loop Theorem</u></h3>
|
||||||
|
<p align="center"><font color="#3333ff"><b><big><i>The sum of the
|
||||||
|
changes in potential when traversing any complete</i><i>
|
||||||
|
loop is zero.</i></big></b></font><br>
|
||||||
|
</p>
|
||||||
|
<p align="center">(This is equivalent to conservation of energy)<br>
|
||||||
|
</p>
|
||||||
|
</blockquote>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="divider" src="divider_ornbarblu.gif"
|
||||||
|
height="64" width="393"></div>
|
||||||
|
<ul>
|
||||||
|
<li>
|
||||||
|
<h3><u>Conventions</u></h3>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
<blockquote>As usual, in order to ensure consistent results from
|
||||||
|
application of these laws, we must adhere to several conventions
|
||||||
|
concerning the currents and potentials in circuits.<br>
|
||||||
|
<u><b><br>
|
||||||
|
Potentials</b></u>:<br>
|
||||||
|
<ol>
|
||||||
|
<li>When a resistive device is traversed in the direction of
|
||||||
|
current flow the change in potential is -iR. Conversely,
|
||||||
|
if the resistance is traversed opposite to the direction of
|
||||||
|
the current the potential change is +iR.</li>
|
||||||
|
<li>When an emf is traversed in the direction of the emf the
|
||||||
|
change in potential is +ε. Conversely, if the emf is
|
||||||
|
traversed opposite to the emf direction the change in
|
||||||
|
potential is -ε.</li>
|
||||||
|
</ol>
|
||||||
|
<p><br>
|
||||||
|
<u><b>Currents:</b></u><br>
|
||||||
|
</p>
|
||||||
|
<blockquote>
|
||||||
|
<p>In setting up a problem, the current direction in any
|
||||||
|
particular circuit element is assigned arbitrarily.
|
||||||
|
Kitchoff's laws are then applied to the circuit using these
|
||||||
|
current directions. After solving the resulting
|
||||||
|
equations if a current is negative that means the "actual"
|
||||||
|
current direction is opposite the arbitrarily chosen
|
||||||
|
direction.<br>
|
||||||
|
</p>
|
||||||
|
</blockquote>
|
||||||
|
<ol>
|
||||||
|
</ol>
|
||||||
|
<div align="center"><img alt="divider" src="divider_ornbarblu.gif"
|
||||||
|
height="64" width="393"><br>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<ul>
|
||||||
|
<li>
|
||||||
|
<h3><u>Application of Kirchhoff's Laws</u></h3>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
<blockquote>
|
||||||
|
<p>Kirchhoff's laws can be applied to <b>any circuit</b> to
|
||||||
|
obtain a set of equations relating the currents, resistances and
|
||||||
|
emfs in the circuit. These equations can then be solved
|
||||||
|
for the unknown quantities in the circuit. For any circuit
|
||||||
|
follow the steps below.<br>
|
||||||
|
</p>
|
||||||
|
</blockquote>
|
||||||
|
<blockquote>
|
||||||
|
<ol>
|
||||||
|
<li>Label the current flowing in each part of the circuit,
|
||||||
|
bearing in mind that current will "split" on reaching a
|
||||||
|
junction. The direction of the defined direction of the
|
||||||
|
current does not matter - see current convention above.</li>
|
||||||
|
<li>At each junction in the circuit use the junction theorem to
|
||||||
|
write down the equations relating the currents entering and
|
||||||
|
leaving. </li>
|
||||||
|
<li>Define all possible loops in the circuit and label.</li>
|
||||||
|
<li>For each loop choose a starting location then use the loop
|
||||||
|
theorem to write down the equation relating changes in
|
||||||
|
potential which must be zero after traversing the complete
|
||||||
|
loop.</li>
|
||||||
|
<li>Solve the set of equations from 2. and 4. to obtain the
|
||||||
|
unknown parameters of the circuit.</li>
|
||||||
|
</ol>
|
||||||
|
<p><br>
|
||||||
|
As an example, consider the circuit below. With the 3 emfs
|
||||||
|
we cannot use the series/parallel analysis.<br>
|
||||||
|
</p>
|
||||||
|
<div align="center"><img alt="fig1" src="elec_kirch_fig1.gif"
|
||||||
|
height="185" width="435"></div>
|
||||||
|
</blockquote>
|
||||||
|
<blockquote><u>Junctions:</u><br>
|
||||||
|
<blockquote>a: I<sub>1</sub> = I<sub>2</sub> + I<sub>3</sub>
|
||||||
|
<br>
|
||||||
|
b: I<sub>3</sub> + I<sub>2</sub> = I<sub>3</sub> <br>
|
||||||
|
<br>
|
||||||
|
</blockquote>
|
||||||
|
<u>Loops:</u><br>
|
||||||
|
<blockquote>1 (including ε<sub>1</sub> starting at a traversing
|
||||||
|
clockwise): - I<sub>3</sub>R<sub>4</sub> - ε<sub>3</sub> -
|
||||||
|
I<sub>1</sub>R<sub>2</sub> + ε<sub>1</sub> - I<sub>1</sub>R<sub>1</sub>
|
||||||
|
= 0<br>
|
||||||
|
2 (including ε<sub>2</sub> starting at a traversing clockwise):
|
||||||
|
- I<sub>2</sub>R<sub>3</sub> - ε<sub>2</sub> + ε<sub>3</sub>
|
||||||
|
+ I<sub>3</sub>R<sub>4</sub> = 0<br>
|
||||||
|
3 (including ε1 and ε<sub>2</sub> starting at a traversing
|
||||||
|
clockwise): - I<sub>2</sub>R<sub>3</sub> - ε<sub>2</sub> -
|
||||||
|
I<sub>1</sub>R<sub>2</sub> + ε<sub>1</sub> - I<sub>1</sub>R<sub>1</sub>
|
||||||
|
= 0<br>
|
||||||
|
</blockquote>
|
||||||
|
Looking at these equations it is clear that the two junction
|
||||||
|
equations are equivalent, and that loop equation 3 is simply the
|
||||||
|
sum of loop equations 1 and 2. Therefore there are only 3
|
||||||
|
independent equations (a, 1 and 2), which we can solve for, say,
|
||||||
|
the currents I<sub>1</sub>, I<sub>2</sub> and I<sub>3</sub>.<br>
|
||||||
|
<br>
|
||||||
|
<img alt="exlamation" src="exclamation-icon.gif" height="30"
|
||||||
|
width="31"> Note that in more complicated circuits there will be
|
||||||
|
many more junctions and a large number of possible loops.
|
||||||
|
You only need apply the loop theorem to as many loops to obtain
|
||||||
|
the number of independent equations necessary to determine the
|
||||||
|
unknown parameters. That is if you have 3 unknown
|
||||||
|
quantities, you'll need a total of 3 independent equations.<br>
|
||||||
|
</blockquote>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<div align="left"> </div>
|
||||||
|
<img src="netbar.gif" height="40" width="100%">
|
||||||
|
<center>
|
||||||
|
<p style="color: rgb(255, 0, 0); font-style: italic;"
|
||||||
|
class="MsoNormal">
|
||||||
|
<meta http-equiv="content-type" content="text/html;
|
||||||
|
charset=windows-1252">
|
||||||
|
</p>
|
||||||
|
<font color="#ff0000"><i>What do you get if you have Avogadro's
|
||||||
|
number of donkeys?<br>
|
||||||
|
Answer: molasses (a mole of asses)</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>
|
||||||
|
|
@ -0,0 +1,140 @@
|
||||||
|
<!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 - Series and Parallel Circuits - Physics 299</title>
|
||||||
|
<meta content="C. L. Davis" name="author">
|
||||||
|
</head>
|
||||||
|
<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
|
||||||
|
255);" link="#0000ee" alink="#ff0000" vlink="#551a8b">
|
||||||
|
<center>
|
||||||
|
<h1> <img src="ULPhys1.gif" height="50" align="texttop"
|
||||||
|
width="189"></h1>
|
||||||
|
</center>
|
||||||
|
<center>
|
||||||
|
<h1>Series and Parallel Circuits<br>
|
||||||
|
</h1>
|
||||||
|
</center>
|
||||||
|
<center><img src="celticbar.gif" height="22" width="576"><br>
|
||||||
|
<br>
|
||||||
|
<font color="#ff0000"><i>"</i></font><font color="#ff0000"><i> </i></font><font
|
||||||
|
color="#ff0000"><i>
|
||||||
|
<meta http-equiv="content-type" content="text/html;
|
||||||
|
charset=windows-1252">
|
||||||
|
It would be better for the true physics if there were no
|
||||||
|
mathematicians on earth." <br>
|
||||||
|
</i><font color="#000000">Daniel Bernoulli</font></font><br>
|
||||||
|
</center>
|
||||||
|
<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
|
||||||
|
<ul>
|
||||||
|
<li>You may already be familiar with the idea of circuits with
|
||||||
|
resistors in series or parallel or some combination of the two.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<ul>
|
||||||
|
<li>
|
||||||
|
<h3><font color="#3333ff">Series</font></h3>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
</ul>
|
||||||
|
<blockquote>
|
||||||
|
<blockquote><img alt="fig1" src="elec_circuits_fig1.jpg"
|
||||||
|
height="168" align="right" width="273">The same current flows
|
||||||
|
in each resistor, the voltages across them are typically
|
||||||
|
different, where V = V<sub>1</sub> + V<sub>2</sub> + V<sub>3</sub>
|
||||||
|
which leads to the equivalent resistance formula<br>
|
||||||
|
<br>
|
||||||
|
<div align="center">R<sub>eq</sub> = R<sub>1</sub> + R<sub>2</sub>
|
||||||
|
+ R<sub>3</sub><br>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
</blockquote>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<ul>
|
||||||
|
<li>
|
||||||
|
<h3><font color="#3333ff">Parallel</font></h3>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
</ul>
|
||||||
|
<blockquote>
|
||||||
|
<blockquote>The potential difference across each resistor is the
|
||||||
|
same, but the currents through them are typically different,
|
||||||
|
where I = I<sub>1</sub> + I<sub>2</sub> + I<sub>3</sub>.
|
||||||
|
This lead to the equivalent resistance formula,<br>
|
||||||
|
<div align="center"><img alt="eqn1"
|
||||||
|
src="elec_circuits_eqn1.jpg" height="58" width="152"><br>
|
||||||
|
<br>
|
||||||
|
<div align="left">Note that both of the diagrams below
|
||||||
|
represent resistors in parallel.<br>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<img alt="fig2" src="elec_circuits_fig2.jpg" height="173"
|
||||||
|
width="312">
|
||||||
|
|
||||||
|
<img alt="fig3" src="elec_circuits_fig3.jpg" height="179"
|
||||||
|
width="301"><br>
|
||||||
|
<br>
|
||||||
|
</blockquote>
|
||||||
|
</blockquote>
|
||||||
|
<ul>
|
||||||
|
<ul>
|
||||||
|
<li>
|
||||||
|
<h3><font color="#3333ff">Combinations</font></h3>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
</ul>
|
||||||
|
<blockquote>
|
||||||
|
<blockquote>Some circuits can be analysed as combinations of
|
||||||
|
series and parallel circuits. In the circuit below R<sub>2</sub>
|
||||||
|
and R<sub>3</sub> are in parallel, their equivalent resistance
|
||||||
|
is then in series with R<sub>1</sub>.<br>
|
||||||
|
<div align="center"><img alt="fig4"
|
||||||
|
src="elec_circuits_fig4.jpg" height="181" width="390"><br>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<img alt="exclamation" src="exclamation-icon.gif" height="30"
|
||||||
|
width="31"> Note that it is not possible to represent all
|
||||||
|
circuits as combinations of series and parallel elements, this
|
||||||
|
is most obvious in many cases where there is more than one
|
||||||
|
battery in the circuit, see example below. To analyse this
|
||||||
|
type of circuit we must use Kirchhoff's Laws.<br>
|
||||||
|
<br>
|
||||||
|
<div align="center"><img alt="fig5" src="elec_circuits_fig5.jpg"
|
||||||
|
height="185" width="435"><br>
|
||||||
|
</div>
|
||||||
|
<div align="center"><br>
|
||||||
|
</div>
|
||||||
|
<blockquote>
|
||||||
|
<div align="center"> </div>
|
||||||
|
</blockquote>
|
||||||
|
</blockquote>
|
||||||
|
</div>
|
||||||
|
<img src="netbar.gif" height="40" width="100%">
|
||||||
|
<center>
|
||||||
|
<p style="color: rgb(255, 0, 0); font-style: italic;"
|
||||||
|
class="MsoNormal">
|
||||||
|
<meta http-equiv="content-type" content="text/html;
|
||||||
|
charset=windows-1252">
|
||||||
|
</p>
|
||||||
|
<font color="#ff0000"><i>Got mole problems? Call Avogadro at
|
||||||
|
602-1023.</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>
|
||||||
|
|
@ -0,0 +1,147 @@
|
||||||
|
<!DOCTYPE HTML PUBLIC "-//w3c//dtd html 4.0 transitional//en">
|
||||||
|
<html>
|
||||||
|
<head>
|
||||||
|
<meta http-equiv="Content-Type"
|
||||||
|
content="text/html; charset=ISO-8859-1">
|
||||||
|
<meta name="GENERATOR"
|
||||||
|
content="Mozilla/4.7 [en] (X11; U; OSF1 V4.0 alpha) [Netscape]">
|
||||||
|
<meta name="Author" content="C. L. Davis">
|
||||||
|
<title>Electricity - Conductors and Insulators - 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>Conductors and Insulators</h1>
|
||||||
|
</center>
|
||||||
|
<center><img src="celticbar.gif" height="22" width="576"><br>
|
||||||
|
<br>
|
||||||
|
<font color="#ff0000"><i>"What is the use of a new-born child ?"</i></font><br>
|
||||||
|
Benjamin Franklin<br>
|
||||||
|
<small><small>(when asked what was the use of a new invention)</small></small><br>
|
||||||
|
</center>
|
||||||
|
<img src="netbar.gif" align="middle" height="40" width="100%"> <br>
|
||||||
|
<center><img src="misc1.gif" align="middle" height="32" width="288">
|
||||||
|
</center>
|
||||||
|
<ul>
|
||||||
|
<li> Moving electric charges constitute what is know as an electric
|
||||||
|
current. It is the electric currents in semi-conductor devices
|
||||||
|
which are responsible for the electronic technology in today's society.<br>
|
||||||
|
<br>
|
||||||
|
</li>
|
||||||
|
<ul>
|
||||||
|
<li> <b>Conductors</b> are materials which allow the free movement
|
||||||
|
of electric charge. Examples include,</li>
|
||||||
|
<ul>
|
||||||
|
<li> Metals</li>
|
||||||
|
<li> Some liquids</li>
|
||||||
|
<li> Gas plasmas<br>
|
||||||
|
<br>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
<li> <b>Insulators</b> (or non conductors) are materials which
|
||||||
|
provide
|
||||||
|
significant resistance to the flow of electric charge. Examples
|
||||||
|
include,</li>
|
||||||
|
<ul>
|
||||||
|
<li> Non metals - plastic, wood, glass, rubber etc.</li>
|
||||||
|
<li> Gases<br>
|
||||||
|
<br>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
<li> <b>Semi-conductors</b> are materials whose resistance to
|
||||||
|
current flow
|
||||||
|
falls between conductors and insulators. There are very few such
|
||||||
|
materials, but their importance in electronic technology cannot be
|
||||||
|
emphasized enough. Examples,</li>
|
||||||
|
<ul>
|
||||||
|
<li> Silicon</li>
|
||||||
|
<li> Germanium<br>
|
||||||
|
<br>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
</ul>
|
||||||
|
<li> <b>Mechanisms of conduction:</b><img src="science.gif"
|
||||||
|
align="middle" height="32" width="288"> </li>
|
||||||
|
<ul>
|
||||||
|
<li> Metals (solid)</li>
|
||||||
|
<ul>
|
||||||
|
<li> Each atom in the solid is "fixed", forming a lattice.</li>
|
||||||
|
<li> Outer electrons in a metal are weakly bound to the atomic
|
||||||
|
nucleus.</li>
|
||||||
|
<li> When an external electric field is applied these outer
|
||||||
|
electrons move
|
||||||
|
through the material creating an electric current.</li>
|
||||||
|
</ul>
|
||||||
|
<li> Liquid conductors and gas plasmas</li>
|
||||||
|
<ul>
|
||||||
|
<li> Conducting liquids and gases are comprised of positive and
|
||||||
|
negative
|
||||||
|
ions (charged particles).</li>
|
||||||
|
<li> Both positive and negative ions move when an external
|
||||||
|
electric field
|
||||||
|
is applied, thus creating the current.</li>
|
||||||
|
<li> A positive charge moving to the right creates the same
|
||||||
|
current as
|
||||||
|
an equal negative charge moving to the left.</li>
|
||||||
|
</ul>
|
||||||
|
<li> Insulators</li>
|
||||||
|
<ul>
|
||||||
|
<li> All electrons in these materials are tightly bound to the
|
||||||
|
atomic nuclei. External electric fields are typically not large
|
||||||
|
enough to cause any flow of charge.</li>
|
||||||
|
</ul>
|
||||||
|
<li> Semi-conductors</li>
|
||||||
|
<ul>
|
||||||
|
<li> These materials have a small number of weakly bound
|
||||||
|
electrons, the
|
||||||
|
number of which is very dependent on the temperature and
|
||||||
|
potential
|
||||||
|
difference applied across the material.</li>
|
||||||
|
<br>
|
||||||
|
|
||||||
|
</ul>
|
||||||
|
<li> It is important to realise that because sustained electric
|
||||||
|
currents
|
||||||
|
only occur when a potential difference is maintained in a closed
|
||||||
|
circuit,
|
||||||
|
as many charge carriers enter as leave any part of the circuit.
|
||||||
|
In
|
||||||
|
other words electric current is not "used up"; it has the same value
|
||||||
|
everywhere in the circuit.<br>
|
||||||
|
<br>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
</ul>
|
||||||
|
<p><br>
|
||||||
|
<img src="netbar.gif" height="40" width="100%"> </p>
|
||||||
|
<center>
|
||||||
|
<p class="MsoNormal"><span
|
||||||
|
style="color: rgb(255, 0, 0); font-style: italic;">Marilyn Monroe
|
||||||
|
suggests to Einstein: What do you say,
|
||||||
|
professor, shouldn't we marry and have a little baby together: what a
|
||||||
|
baby it
|
||||||
|
would be - my looks and your intelligence!</span><br
|
||||||
|
style="color: rgb(255, 0, 0); font-style: italic;">
|
||||||
|
<span style="color: rgb(255, 0, 0); font-style: italic;">Einstein: I'm
|
||||||
|
afraid, dear lady, it might be the other way around...</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>
|
||||||
|
|
@ -0,0 +1,96 @@
|
||||||
|
<!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 - Electric Field Due to Continuous Charge
|
||||||
|
Distributions - Physics 299</title>
|
||||||
|
</head>
|
||||||
|
<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
|
||||||
|
255);" link="#0000ee" alink="#ff0000" bgcolor="#ff0000"
|
||||||
|
text="#000000" vlink="#551a8b">
|
||||||
|
<center>
|
||||||
|
<h1> <img src="ULPhys1.gif" height="50" align="texttop"
|
||||||
|
width="189"></h1>
|
||||||
|
</center>
|
||||||
|
<center>
|
||||||
|
<h1>Electric Field Due Continuous Charge Distribtuions<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">
|
||||||
|
Common sense is the collection of prejudices acquired by age
|
||||||
|
eighteen"</i></font><br>
|
||||||
|
Albert Einstein<br>
|
||||||
|
</center>
|
||||||
|
<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>Electric charge is a property of individual particles -
|
||||||
|
protons, electrons etc. But since these particles are
|
||||||
|
extremely small it is often convenient to consider charge to be
|
||||||
|
continuously distributed. <br>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>These distributions can be over a line (one dimension), an
|
||||||
|
area (two dimensions) or a volume (three dimensions).</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>In order to determine the electric field due to a continuous
|
||||||
|
charge distribution we "sum" the fields due to the individual
|
||||||
|
"elements" that comprise the distribution, by integrating over
|
||||||
|
the line, area or volume in question. For example in the
|
||||||
|
example below charge is distributed uniformly over the rod on
|
||||||
|
the x axis. To determine the electric field at point P, we
|
||||||
|
write down the expression for the field at P due to the "point"
|
||||||
|
charge dq located at "x" as shown, then integrate over x from x
|
||||||
|
= 0 to x = x. <br>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="contin charge dist"
|
||||||
|
src="elec_continchgdist.jpg" height="244" width="272"><br>
|
||||||
|
</div>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<br>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<div align="center"> </div>
|
||||||
|
<div style="text-align: left;"><img src="netbar.gif" height="40"
|
||||||
|
width="100%"> </div>
|
||||||
|
<center><span style="font-size: 12pt; font-family: "Times New
|
||||||
|
Roman";"><span style="color: rgb(255, 0, 0); font-style:
|
||||||
|
italic;"></span></span><span style="font-size: 12pt;
|
||||||
|
font-family: "Times New Roman";"><span style="color:
|
||||||
|
rgb(255, 0, 0); font-style: italic;">
|
||||||
|
<meta http-equiv="content-type" content="text/html;
|
||||||
|
charset=windows-1252">
|
||||||
|
</span></span><br>
|
||||||
|
<i><font color="#ff0000">Overheard after a student failed a
|
||||||
|
physics test miserably:
|
||||||
|
Nuclear, Hydrogen, Atomic, My test- They can all be bombs.</font></i><br>
|
||||||
|
<span style="font-size: 12pt; font-family: "Times New
|
||||||
|
Roman";"><span style="color: rgb(255, 0, 0); font-style:
|
||||||
|
italic;"></span></span> <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>
|
||||||
Binary file not shown.
|
After Width: | Height: | Size: 4.3 KiB |
|
|
@ -0,0 +1,154 @@
|
||||||
|
<!DOCTYPE HTML PUBLIC "-//w3c//dtd html 4.0 transitional//en">
|
||||||
|
<html>
|
||||||
|
<head>
|
||||||
|
<meta http-equiv="Content-Type"
|
||||||
|
content="text/html; charset=ISO-8859-1">
|
||||||
|
<meta name="GENERATOR"
|
||||||
|
content="Mozilla/4.7 [en] (X11; U; OSF1 V4.0 alpha) [Netscape]">
|
||||||
|
<meta name="Author" content="C. L. Davis">
|
||||||
|
<title>Electricity - Coulomb's Law - Physics 299</title>
|
||||||
|
</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>Coulomb's Law</h1>
|
||||||
|
</center>
|
||||||
|
<center><img src="celticbar.gif" height="22" width="576"><br>
|
||||||
|
<br>
|
||||||
|
<font color="#ff0000"><i>"When man wanted to make a machine that would
|
||||||
|
walk
|
||||||
|
he created the wheel, which does not resemble a leg"</i></font><br>
|
||||||
|
Guillaume Apollinaire<br>
|
||||||
|
</center>
|
||||||
|
<img src="netbar.gif" align="middle" height="40" width="100%"> <br>
|
||||||
|
|
||||||
|
<ul>
|
||||||
|
<li> The magnitude of the force of attraction (or repulsion), F<sub>12</sub>
|
||||||
|
between two point charges q<sub>1</sub> and q<sub> 2 </sub> is
|
||||||
|
given by Coulomb's Law.</li>
|
||||||
|
<center>
|
||||||
|
<p><br>
|
||||||
|
<img alt="" src="elec_coulomb_eqn1.gif"
|
||||||
|
style="width: 80px; height: 47px;"> <br>
|
||||||
|
</p>
|
||||||
|
</center>
|
||||||
|
<p>where R<sub>12</sub> is the distance between the
|
||||||
|
charges. k is a constant of proportionality known as the Coulomb
|
||||||
|
constant, having the value 9 x
|
||||||
|
10<sup>9</sup> N.m<sup>2</sup> / C<sup>2</sup> in a
|
||||||
|
vacuum. </p>
|
||||||
|
<p><img style="width: 31px; height: 30px;" alt="exclamation"
|
||||||
|
src="exclamation-icon.gif"> Note that the Coulomb constant, k, is
|
||||||
|
often replaced with (1/4π ε<sub>0</sub>), where
|
||||||
|
ε<sub>0</sub>is the permittivity of the vacuum (more later).<br>
|
||||||
|
</p>
|
||||||
|
<li> The direction of this force is along the line joining the two
|
||||||
|
charges
|
||||||
|
with the sense determined by the relative signs of the charges</li>
|
||||||
|
<p><br>
|
||||||
|
</p>
|
||||||
|
<center>
|
||||||
|
<p><img src="coulaw1.gif" height="112" width="182"> </p>
|
||||||
|
</center>
|
||||||
|
<li> Note that the force on each charge has the same magnitude (as
|
||||||
|
required by Newton's third law of motion).</li>
|
||||||
|
<br>
|
||||||
|
|
||||||
|
<li> For two 1 Coulomb charges separated by 1 metre the
|
||||||
|
magnitude
|
||||||
|
of the force is given by,
|
||||||
|
<center> <br>
|
||||||
|
F = (9 x 10<sup>9</sup> x 1 x 1 )/ 1 = 9 x 10<sup>9</sup>
|
||||||
|
Newtons</center>
|
||||||
|
<p>This is an <b><i>extremely large</i></b> force (sufficient to
|
||||||
|
move Mt. Everest with an acceleration of 1cm/s<sup>2</sup>). The
|
||||||
|
Coulomb
|
||||||
|
is a <b><i>very large</i></b> unit. Typical macroscopic charges
|
||||||
|
are measured in micro-coulombs (10<sup>-6</sup> C). </p>
|
||||||
|
</li>
|
||||||
|
<li>To handle situations with more than one charge, the charges must
|
||||||
|
be treated in pairs, so that the overall force on one charge will be
|
||||||
|
the <span style="font-weight: bold;">vector</span> sum of the force
|
||||||
|
due to each of the other charges. For example the force on q<sub>1</sub>
|
||||||
|
due to all other charges q<sub>2</sub>, q<sub>3</sub> , q<sub>4</sub>...
|
||||||
|
would
|
||||||
|
be
|
||||||
|
given
|
||||||
|
by,</li>
|
||||||
|
</ul>
|
||||||
|
<div style="text-align: center;"><span style="font-weight: bold;">F</span><sub
|
||||||
|
style="font-weight: bold;">1</sub><span style="font-weight: bold;"> = F</span><sub
|
||||||
|
style="font-weight: bold;">21</sub><span style="font-weight: bold;"> +
|
||||||
|
F</span><sub style="font-weight: bold;">31</sub><span
|
||||||
|
style="font-weight: bold;"> + F</span><sub style="font-weight: bold;">41</sub><span
|
||||||
|
style="font-weight: bold;"> + ...</span><br>
|
||||||
|
<div style="text-align: left;">
|
||||||
|
<ul>
|
||||||
|
<li><img style="width: 79px; height: 43px;" alt="hot" src="hot.gif">Notice
|
||||||
|
the
|
||||||
|
similarity
|
||||||
|
of
|
||||||
|
Coulomb's Law to Newton's Law of Gravitation</li>
|
||||||
|
</ul>
|
||||||
|
<div style="text-align: center;"><img
|
||||||
|
style="border: 0px solid ; width: 114px; height: 62px;" alt="eqn1"
|
||||||
|
src="grav_eqn1.jpg"><br>
|
||||||
|
<div style="text-align: left; margin-left: 40px;"><br>
|
||||||
|
both are "inverse square" laws. Substitute charge for mass and
|
||||||
|
"k" for "G" and you have Coulomb's law.<br>
|
||||||
|
<img style="width: 31px; height: 30px;" alt="exclamation"
|
||||||
|
src="exclamation-icon.gif"> The relative magnitudes of the Coulomb
|
||||||
|
constant, k = 9 x 10<sup>9</sup> and the gravitational constant, G =
|
||||||
|
6.67 x 10<sup>-11</sup>, is an indication of the relative strengths of
|
||||||
|
the two forces. The electrical force of attraction is much, much
|
||||||
|
stronger than the gravitational force of attraction.<br>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<p><br>
|
||||||
|
<img src="netbar.gif" height="40" width="100%"> </p>
|
||||||
|
<center><span
|
||||||
|
style="font-size: 12pt; font-family: "Times New Roman"; color: rgb(255, 0, 0); font-style: italic;">"The
|
||||||
|
wireless
|
||||||
|
telegraph
|
||||||
|
is
|
||||||
|
not
|
||||||
|
difficult
|
||||||
|
to
|
||||||
|
understand. The ordinary telegraph is like a very long cat. You pull
|
||||||
|
the tail
|
||||||
|
in </span><st1:state style="color: rgb(255, 0, 0); font-style: italic;"><st1:place><span
|
||||||
|
style="font-size: 12pt; font-family: "Times New Roman";">New York</span></st1:place></st1:state><span
|
||||||
|
style="font-size: 12pt; font-family: "Times New Roman"; color: rgb(255, 0, 0); font-style: italic;">,
|
||||||
|
and
|
||||||
|
it
|
||||||
|
meows
|
||||||
|
in
|
||||||
|
</span><st1:city style="color: rgb(255, 0, 0); font-style: italic;"><st1:place><span
|
||||||
|
style="font-size: 12pt; font-family: "Times New Roman";">Los Angeles</span></st1:place></st1:city><span
|
||||||
|
style="font-size: 12pt; font-family: "Times New Roman";"><span
|
||||||
|
style="color: rgb(255, 0, 0); font-style: italic;">. The wireless is
|
||||||
|
the same, only without the cat."</span><br>
|
||||||
|
Albert Einstein<br>
|
||||||
|
</span><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>
|
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<p><img src="header-index.gif" height="51" width="92">
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charset=windows-1252">
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alpha) [Netscape]">
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<meta name="Author" content="C. L. Davis">
|
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<title>Electricity - Electric Current - Physics 299</title>
|
||||||
|
<meta content="C. L. Davis" name="author">
|
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</head>
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<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
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255);" alink="#ff0000" link="#0000ee" vlink="#551a8b">
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<center>
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<h1> <img src="ULPhys1.gif" align="texttop" height="50"
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width="189"></h1>
|
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</center>
|
||||||
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<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. 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.
|
||||||
|
In fact the current moves at speeds close to the speed of
|
||||||
|
light. 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. 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>
|
||||||
|
The
|
||||||
|
current,
|
||||||
|
I,
|
||||||
|
is
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
a scalar quantity, whereas J is a vector. 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>
|
||||||
|
The direction of current flow is defined as the direction in
|
||||||
|
which a positive charge will move. But in solid metallic
|
||||||
|
conductors the charge carriers are electrons (negative charges)
|
||||||
|
which actually move in the opposite direction. 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 ρ is the
|
||||||
|
resistivity of the conductor - characteristic of the
|
||||||
|
conductor. The conductivity of a conducting material is
|
||||||
|
defined by, σ = 1/ρ.<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 (Ω), 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.
|
||||||
|
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
|
||||||
|
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 </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. <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). 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 ε = dW/dq, where dW
|
||||||
|
is the work done on a positive charge dq in taking it
|
||||||
|
acrosss the potential difference of the device. 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. In the circuit above the energy appeara
|
||||||
|
as heat and light in the light bulb. When the
|
||||||
|
charge 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.
|
||||||
|
See circuit at right. 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>
|
||||||
|
|
||||||
|
<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>
|
||||||
|
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|
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|
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<html>
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<head>
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<meta http-equiv="Content-Type" content="text/html;
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charset=windows-1252">
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<meta name="GENERATOR" content="Mozilla/4.7 [en] (X11; U; OSF1 V4.0
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|
alpha) [Netscape]">
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<meta name="Author" content="C. L. Davis">
|
||||||
|
<title>Electricty - Dielectric Materials - Physics 299</title>
|
||||||
|
</head>
|
||||||
|
<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
|
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|
255);" link="#0000ee" alink="#ff0000" bgcolor="#ff0000"
|
||||||
|
text="#000000" vlink="#551a8b">
|
||||||
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<center><img src="ULPhys1.gif" height="50" align="texttop"
|
||||||
|
width="189"></center>
|
||||||
|
<center>
|
||||||
|
<h1>Dielectric Materials<br>
|
||||||
|
</h1>
|
||||||
|
</center>
|
||||||
|
<center><img src="celticbar.gif" height="22" width="576"><br>
|
||||||
|
<br>
|
||||||
|
<font color="#ff0000"><i>
|
||||||
|
<meta http-equiv="content-type" content="text/html;
|
||||||
|
charset=windows-1252">
|
||||||
|
</i></font>
|
||||||
|
<div class="quotation"> <font color="#ff0000"><i>"Basic research
|
||||||
|
is like shooting an arrow into the air and, where it lands,
|
||||||
|
painting a target."</i></font><br>
|
||||||
|
</div>
|
||||||
|
<div class="quotename"> </div>
|
||||||
|
Homer Burton Adkins<br>
|
||||||
|
</center>
|
||||||
|
<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
|
||||||
|
<ul>
|
||||||
|
<li>In all our discussions to date we have implicitly assumed that
|
||||||
|
our charges have been located in a vacuum or on the surface of
|
||||||
|
conductors. We now need to consider how to take into
|
||||||
|
account the presence of non-conducting material in the real
|
||||||
|
world. Dielectric material is simply another way of saying
|
||||||
|
non-conducting material.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li><img alt="elec dielec fig2" src="elec_dielec_fig2.jpg"
|
||||||
|
height="211" align="right" width="436">Imagine a parallel
|
||||||
|
plate capacitor in which a dielectric material is placed between
|
||||||
|
the plates (at right below). The dielectric is composed of
|
||||||
|
atoms/molecules which contain positive and negative
|
||||||
|
charges. The applied electric field between the plates, E<sub>0</sub>,
|
||||||
|
will cause the positive and negative charges of the constituent
|
||||||
|
atoms/molecules to move slightly in opposite directions
|
||||||
|
(right). Electric dipole moments will be "induced" in the
|
||||||
|
material, as shown. The net effect will be for charge to
|
||||||
|
appear on the surface of the dielectric material as shown.
|
||||||
|
The dielectric is said to have been polarized, leading to a
|
||||||
|
polarization electric field, E<sub>P</sub>. <br>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
<blockquote>
|
||||||
|
<div align="left"><img alt="exclamation"
|
||||||
|
src="exclamation-icon.gif" height="30" width="31"> In
|
||||||
|
conductors (metals) there are (almost) free electrons which will
|
||||||
|
move through the material when an electric field is applied,
|
||||||
|
generating an electric current.<br>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li><img alt="elec dielec fig1" src="elec_dielec_fig1.jpg"
|
||||||
|
height="311" align="right" width="495">The net <b>E</b> field
|
||||||
|
between the plates has been reduced, <br>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="elec dielec eqn1"
|
||||||
|
src="elec_dielec_eqn1.png" height="30" width="386"><br>
|
||||||
|
<br>
|
||||||
|
<blockquote>
|
||||||
|
<div align="left">where κ is called the dielectric constant or
|
||||||
|
relative permittivity of the medium.<br>
|
||||||
|
<br>
|
||||||
|
Note that for a vacuum, since E<sub>P</sub> = 0, κ = 1
|
||||||
|
and since E<sub>P</sub> < E<sub>0</sub> for all other
|
||||||
|
materials κ > 1.<br>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li>It is easy to show that for the parallel plate capacitor
|
||||||
|
the voltage (p.d) between the plates and the energy stored
|
||||||
|
are reduced by a factor κ, whereas the capacitance is
|
||||||
|
increased by a factor of κ.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>By application of Gauss's Law to a parallel plate
|
||||||
|
capacitor with a dielectric between the plates it can be
|
||||||
|
shown that to account for the presence of the dielectric
|
||||||
|
Gauss's Law becomes,</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="elec dielec eqn2"
|
||||||
|
src="elec_dielec_eqn2.png" height="60" width="139"><br>
|
||||||
|
<blockquote>
|
||||||
|
<div align="left">As a general rule when dielectric media is
|
||||||
|
present wherever ε<sub>0</sub> appears it must be replaced
|
||||||
|
by ε<sub>0</sub>κ.</div>
|
||||||
|
</blockquote>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<blockquote> </blockquote>
|
||||||
|
</div>
|
||||||
|
<img src="netbar.gif" height="40" width="100%"> <br>
|
||||||
|
<center>
|
||||||
|
<p class="MsoNormal"><span style="color: rgb(255, 0, 0);
|
||||||
|
font-style: italic;">
|
||||||
|
<meta http-equiv="content-type" content="text/html;
|
||||||
|
charset=windows-1252">
|
||||||
|
</span></p>
|
||||||
|
<p><font color="#ff0000"><i>A chemist, a biologist and an
|
||||||
|
electrical engineer were on death row waiting to go in the
|
||||||
|
electric chair.</i></font></p>
|
||||||
|
<font color="#ff0000"><i> </i></font>
|
||||||
|
<p><font color="#ff0000"><i>The chemist was brought forward first.
|
||||||
|
"Do you have anything you want to say?" asked the
|
||||||
|
executioner, strapping him in. "No," replied the chemist.
|
||||||
|
The executioner flicked the switch and nothing happened.
|
||||||
|
Under this particular State's law, if an execution attempt
|
||||||
|
fails, the prisoner is to be released, so the chemist was
|
||||||
|
released.</i></font></p>
|
||||||
|
<font color="#ff0000"><i> </i></font>
|
||||||
|
<p><font color="#ff0000"><i>Then the biologist was brought
|
||||||
|
forward. "Do you have anything you want to say?" "No, just
|
||||||
|
get on with it." The executioner flicked the switch, and
|
||||||
|
again nothing happened, so the biologist was released.</i></font></p>
|
||||||
|
<font color="#ff0000"><i> </i></font>
|
||||||
|
<p><font color="#ff0000"><i> Then the electrical engineer was
|
||||||
|
brought forward. "Do you have anything you want to say?"
|
||||||
|
asked the executioner. "Yes," replied the engineer. "If you
|
||||||
|
swap the red and the blue wires over, you might make this
|
||||||
|
thing work."</i></font></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>
|
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|
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|
||||||
|
<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 - Electric Dipole in an External Field - Physics
|
||||||
|
299</title>
|
||||||
|
</head>
|
||||||
|
<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
|
||||||
|
255);" link="#0000ee" alink="#ff0000" vlink="#551a8b">
|
||||||
|
<center>
|
||||||
|
<h1> <img src="ULPhys1.gif" height="50" align="texttop"
|
||||||
|
width="189"></h1>
|
||||||
|
</center>
|
||||||
|
<center>
|
||||||
|
<h1>Electric Dipole in an External Electric Field <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">
|
||||||
|
When you look at yourself from a universal standpoint,
|
||||||
|
something inside always reminds or informs you that there are
|
||||||
|
bigger and better things to worry about."</i></font><br>
|
||||||
|
Albert Einstein<br>
|
||||||
|
</center>
|
||||||
|
<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>We have already considered the electric field <i><b>created</b></i>
|
||||||
|
by an electric dipole. Now we consider the behavior of an
|
||||||
|
electric dipole placed in a uniform (constant) electric field.</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="elec dip in external field"
|
||||||
|
src="elec_dip_exte.gif" height="157" width="323"><br>
|
||||||
|
<br>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li>Note that since the force on each of the charges are equal
|
||||||
|
in magnitude but opposite in direction there is <b>no net
|
||||||
|
force on the dipole</b>.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>However, since the two forces are not concurrent, there is
|
||||||
|
a non-zero torque about the center of the dipole given by,</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="elec dip ext e eqn1"
|
||||||
|
src="elec_dipexte_eqn1.jpg" height="45" width="117"><br>
|
||||||
|
<br>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li>Using the definition of the work done by a torque
|
||||||
|
(rotational force), it can be shown that the
|
||||||
|
electrical potential energy stored by a dipole in an
|
||||||
|
external field is given by,</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="elec dip in ext e eqn2"
|
||||||
|
src="elec_dipexte_eqn2.jpg" height="41" width="117"><br>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li><img alt="hot" src="hot.gif" height="43"
|
||||||
|
align="middle" width="79">A dipole placed in a
|
||||||
|
uniform electric field will rotate until it is
|
||||||
|
aligned "-" to "+" along the field - this is the
|
||||||
|
lowest energy configuration.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li><img alt="exclamation" src="exclamation-icon.gif"
|
||||||
|
height="30" width="31"> If the external
|
||||||
|
field is not uniform, the net force will not be
|
||||||
|
zero.<br>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<div align="center"> </div>
|
||||||
|
<div style="text-align: left;"><img src="netbar.gif" height="40"
|
||||||
|
width="100%"> </div>
|
||||||
|
<center><span style="font-size: 12pt; font-family: "Times New
|
||||||
|
Roman";"><span style="color: rgb(255, 0, 0); font-style:
|
||||||
|
italic;"></span></span><span style="font-size: 12pt;
|
||||||
|
font-family: "Times New Roman";"><span style="color:
|
||||||
|
rgb(255, 0, 0); font-style: italic;">
|
||||||
|
<meta http-equiv="content-type" content="text/html;
|
||||||
|
charset=windows-1252">
|
||||||
|
</span></span><br>
|
||||||
|
<font color="#ff0000"><i>What is a quantum particle?
|
||||||
|
The dreams that stuff is made of!</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>
|
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|
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|
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|
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|
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|
<head>
|
||||||
|
<meta http-equiv="Content-Type" content="text/html;
|
||||||
|
charset=windows-1252">
|
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|
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alpha) [Netscape]">
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|
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<title>Electricity - Electric Dipole - Physics 299</title>
|
||||||
|
</head>
|
||||||
|
<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
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255);" link="#0000ee" alink="#ff0000" vlink="#551a8b">
|
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<center>
|
||||||
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<h1> <img src="ULPhys1.gif" height="50" align="texttop"
|
||||||
|
width="189"></h1>
|
||||||
|
</center>
|
||||||
|
<center>
|
||||||
|
<h1>Electric Field due to a Dipole<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;
|
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|
charset=windows-1252">
|
||||||
|
To punish me for my contempt for authority, fate made me an
|
||||||
|
authority myself"</i></font><br>
|
||||||
|
Albert Einstein<br>
|
||||||
|
</center>
|
||||||
|
<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>An electric dipole consists of two point charges of equal
|
||||||
|
magnitude, but opposite sign, separated by a short distance.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>The dipole is electrically neutral, but due to the
|
||||||
|
separation of its charges gives rise to an electric field in its
|
||||||
|
vicinity.</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="electric dipole" src="elec_dipole.jpg"
|
||||||
|
height="347" width="461"><br>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li>The electric field at the "field point" is given by
|
||||||
|
<b>E</b> = <b>E</b><sub>+q</sub> + <b>E</b><sub>-q</sub>.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
Note that in adding the two electric fields the y-component
|
||||||
|
cancels leaving only an x-component given by,</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="elec dipole eqn1"
|
||||||
|
src="elec_dipole_eqn1.jpg" height="55" width="261"><br>
|
||||||
|
<blockquote>
|
||||||
|
<div align="left">where R is the distance from the centre of
|
||||||
|
the dipole to the field point and the approximation is
|
||||||
|
valid when r and R are almost equal. In this case
|
||||||
|
the dimension of the dipole (a) is small compared to the
|
||||||
|
field point distance.<br>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li>p (=2aq) is called the electric dipole moment.
|
||||||
|
It's actually a vector pointing from the negative to the
|
||||||
|
positive charge in the dipole so that,</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="elec dipole eqn 2"
|
||||||
|
src="elec_dipole_eqn2.jpg" height="59" width="126"><br>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li><img alt="hot" src="hot.gif" height="43"
|
||||||
|
align="middle" width="79">Many molecules have
|
||||||
|
charge distributions which can be approximated as an
|
||||||
|
electric dipole, water being one of the most common.</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="water molecule"
|
||||||
|
src="elec_water-molecule-and-dipole-moment.jpg"
|
||||||
|
height="214" width="603"><br>
|
||||||
|
</div>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<div style="text-align: left;"><img src="netbar.gif" height="40"
|
||||||
|
width="100%"> </div>
|
||||||
|
<center><br>
|
||||||
|
<span style="font-size: 12pt; font-family: "Times New
|
||||||
|
Roman";"><span style="color: rgb(255, 0, 0); font-style:
|
||||||
|
italic;">"How many Astronomers does it take to change a light
|
||||||
|
bulb ?<br>
|
||||||
|
None, astronomers prfeer the dark"</span></span><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>
|
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|
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|
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|
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<html>
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<head>
|
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<meta http-equiv="Content-Type"
|
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|
content="text/html; charset=ISO-8859-1">
|
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<meta name="GENERATOR"
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content="Mozilla/4.7 [en] (X11; U; OSF1 V4.0 alpha) [Netscape]">
|
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<meta name="Author" content="C. L. Davis">
|
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|
<title>Electricity - Electric Field - Physics 299</title>
|
||||||
|
</head>
|
||||||
|
<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255, 255);"
|
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|
alink="#ff0000" link="#0000ee" vlink="#551a8b">
|
||||||
|
<center>
|
||||||
|
<h1> <img src="ULPhys1.gif" align="texttop" height="50" width="189"></h1>
|
||||||
|
</center>
|
||||||
|
<center>
|
||||||
|
<h1>Electric Field</h1>
|
||||||
|
</center>
|
||||||
|
<center><img src="celticbar.gif" height="22" width="576"><br>
|
||||||
|
<br>
|
||||||
|
<font color="#ff0000"><i>"Discovery consists of seeing what everybody
|
||||||
|
has
|
||||||
|
seen and thinking what nobody has thought"</i></font><br>
|
||||||
|
Albert von Szent-Gyorgyi<br>
|
||||||
|
</center>
|
||||||
|
<img src="netbar.gif" align="middle" height="40" width="100%"> <br>
|
||||||
|
|
||||||
|
<ul>
|
||||||
|
<li>When two charges exert a force on each other, through what means
|
||||||
|
is the force transmitted ?</li>
|
||||||
|
</ul>
|
||||||
|
<div style="margin-left: 40px;">The simplest assumption is "<span
|
||||||
|
style="font-style: italic; font-weight: bold;">action-at-a-distance</span>".
|
||||||
|
They just "<span style="font-weight: bold; font-style: italic;">know</span>"
|
||||||
|
about each other's presence. If one of the charges moves the
|
||||||
|
other charge is aware of this immediately. This sounds like
|
||||||
|
"magic" with today's scientific understanding. <br>
|
||||||
|
The mechanism to remove the magic was proposed by <a
|
||||||
|
href="http://www.bbc.co.uk/history/historic_figures/faraday_michael.shtml">
|
||||||
|
Michael Faraday </a> <a
|
||||||
|
href="http://www.rigb.org/contentControl?action=detail&section=1391&pg=4&filter=pd"><img
|
||||||
|
alt="faraday" src="faraday.jpg"
|
||||||
|
style="border: 0px solid ; width: 80px; height: 121px;" align="middle"></a>
|
||||||
|
- the <span style="font-weight: bold;">electric field</span>.
|
||||||
|
Every charge creates its own electric field in the space around it
|
||||||
|
(actually the space around it means all space); other charges then
|
||||||
|
interact with this field. When a charge moves it creates a
|
||||||
|
disturbance in its electric field which is propagated away from the
|
||||||
|
charge at the speed of light.<br>
|
||||||
|
<br>
|
||||||
|
<img style="width: 31px; height: 30px;" alt="exclamation"
|
||||||
|
src="exclamation-icon.gif"> In developing his theory of gravitation
|
||||||
|
Newton was aware of the same "<span
|
||||||
|
style="font-style: italic; font-weight: bold;">action-at-a-distance</span>"
|
||||||
|
problem. To solve the problem, in a similar manner, we introduce
|
||||||
|
the concept of the gravitational field.<br>
|
||||||
|
<br>
|
||||||
|
<img style="width: 31px; height: 30px;" alt="exclamation"
|
||||||
|
src="exclamation-icon.gif"> Note that the concept of the electric
|
||||||
|
field is a convenient construct to describe electromagnetic
|
||||||
|
phenomena,
|
||||||
|
but its true existence is neither proven nor essential.</div>
|
||||||
|
<ul>
|
||||||
|
<br>
|
||||||
|
|
||||||
|
<li> The electric field vector, <span style="font-weight: bold;">E</span>,
|
||||||
|
is defined in the following way. If a charge, q, feels a force <b><u>F</u></b>,
|
||||||
|
then
|
||||||
|
the
|
||||||
|
electric field, <b><u>E</u></b>, at the location of the charge
|
||||||
|
is
|
||||||
|
given by</li>
|
||||||
|
<center>
|
||||||
|
<p><br>
|
||||||
|
<b><u><img alt="" src="elec_field_eqn1.gif"
|
||||||
|
style="width: 43px; height: 44px;"></u></b><br>
|
||||||
|
</p>
|
||||||
|
</center>
|
||||||
|
<br>
|
||||||
|
<li> Units of electric field are Newtons/Coulomb (N/C) or (as
|
||||||
|
we shall see later) Volts/metre (V/m).</li>
|
||||||
|
<br>
|
||||||
|
|
||||||
|
<li> If the electric field felt by the charge q is due to
|
||||||
|
a point
|
||||||
|
charge Q, located a distance R from q, then its magnitude is given by</li>
|
||||||
|
<br>
|
||||||
|
<center>
|
||||||
|
<p><img alt="" src="elec_field_eqn2.gif"
|
||||||
|
style="width: 55px; height: 41px;"><br>
|
||||||
|
</p>
|
||||||
|
</center>
|
||||||
|
<br>
|
||||||
|
<li> The electric field due to many point charges is given by the
|
||||||
|
vector
|
||||||
|
sum of the fields due to the individual charges</li>
|
||||||
|
<br>
|
||||||
|
<center>
|
||||||
|
<p><b><u>E</u></b> = <b><u>E</u></b><sub>1</sub> +
|
||||||
|
<b><u>
|
||||||
|
E</u></b><sub> 2</sub> + <b><u>E</u></b> <sub>3</sub> +
|
||||||
|
.....</p>
|
||||||
|
</center>
|
||||||
|
<br>
|
||||||
|
<p>In performing this sum the direction of each <b><u>E</u></b>
|
||||||
|
is
|
||||||
|
the
|
||||||
|
same as the force felt by a <b><i>positive</i></b> charge. <br>
|
||||||
|
</p>
|
||||||
|
<li> The electric field in a region of space can be represented by
|
||||||
|
electric field lines otherwise known as "<b><i>lines of force</i></b>".</li>
|
||||||
|
<br>
|
||||||
|
<center>
|
||||||
|
<p><img src="efield_lines1.gif" align="texttop" height="221"
|
||||||
|
width="370"> </p>
|
||||||
|
</center>
|
||||||
|
<br>
|
||||||
|
<p>The direction of the electric field line is the same as that of
|
||||||
|
the force
|
||||||
|
felt by a positive charge. The density of the field lines
|
||||||
|
provides a
|
||||||
|
measure of the magnitude of the field. FIeld lines alway begin on
|
||||||
|
positive
|
||||||
|
charges and end on negative charges; they cannot be left 'hanging' in
|
||||||
|
empty
|
||||||
|
space. <br>
|
||||||
|
</p>
|
||||||
|
</ul>
|
||||||
|
<div style="text-align: center;"><img
|
||||||
|
style="width: 745px; height: 254px;" alt="efield_lines3.jpg"
|
||||||
|
src="efield_lines3.jpg"><br>
|
||||||
|
</div>
|
||||||
|
<ul>
|
||||||
|
<p><br>
|
||||||
|
</p>
|
||||||
|
<center>
|
||||||
|
<p>The diagrams above display the electric field lines in the
|
||||||
|
vicinity of
|
||||||
|
two equal point charges.</p>
|
||||||
|
</center>
|
||||||
|
</ul>
|
||||||
|
<div style="text-align: left;"><img src="netbar.gif" height="40"
|
||||||
|
width="100%">
|
||||||
|
</div>
|
||||||
|
<center><br>
|
||||||
|
<span style="font-size: 12pt; font-family: "Times New Roman";"><span
|
||||||
|
style="color: rgb(255, 0, 0); font-style: italic;">"Two things are
|
||||||
|
infinite: the universe and human
|
||||||
|
stupidity; and I'm not sure about the universe."</span><br style="">
|
||||||
|
<!--[if !supportLineBreakNewLine]-->Albert Einstein</span><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>
|
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|
||||||
|
<!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 - Gauss's Law - Physics 299</title>
|
||||||
|
</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>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">
|
||||||
|
Equations are just the boring part of mathematics. I attempt
|
||||||
|
to see things in terms of geometry."</i></font><br>
|
||||||
|
Stephen Hawking<br>
|
||||||
|
</center>
|
||||||
|
<img src="netbar.gif" align="middle" height="40" width="100%"> <br>
|
||||||
|
<ul>
|
||||||
|
<li><img alt="gauss" src="gauss.jpg" align="left" height="101"
|
||||||
|
width="83">Gauss's Law is the first of Maxwell's equations we
|
||||||
|
will consider. At first the whole concept of Gauss's Law
|
||||||
|
will seem to be very abstract and confusing, <img
|
||||||
|
alt="confused" src="confused_smiley.gif" height="22"
|
||||||
|
width="15"> hopefully at least some of the confusion
|
||||||
|
will pass as you become more familiar with the idea. </li>
|
||||||
|
</ul>
|
||||||
|
<p><br>
|
||||||
|
<br>
|
||||||
|
</p>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>At the outset it is important to realize that <b>Gauss's Law
|
||||||
|
and Coulomb's Law are different statements of the same
|
||||||
|
physical concept.</b> Which of the two is used in any
|
||||||
|
particular situation depends on the particular application and
|
||||||
|
what you are asked to determine.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>Before stating Gauss's Law we must first define the concept of
|
||||||
|
<b>FLUX</b> - in particular the flux of the electric field.</li>
|
||||||
|
</ul>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li><img alt="Electric flux" src="elec_gauss_figure1.jpg"
|
||||||
|
align="right" height="433" width="415">At every point on a
|
||||||
|
surface we can calculate an "element" of the electric flux
|
||||||
|
given by</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="defn of E flux"
|
||||||
|
src="elec_gauss_eqn1.jpg" height="38" width="117"><br>
|
||||||
|
<blockquote>
|
||||||
|
<div align="left">so that the total electric flux passing
|
||||||
|
through a surface, S, is given by,<br>
|
||||||
|
<br>
|
||||||
|
<div align="center"><img alt="elec gauss eqn2"
|
||||||
|
src="elec_gauss_eqn2.png" height="56" width="133"><br>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<div align="left">
|
||||||
|
<div align="center">
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li>Gauss's Law then states that,</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="red tick" src="tickred1.gif"
|
||||||
|
align="top" height="48" width="48"> <img
|
||||||
|
alt="elec gauss 3" src="elec_gauss_eqn3.jpg"
|
||||||
|
height="84" width="233"> <img
|
||||||
|
alt="red tick" src="tickred1.gif" align="top"
|
||||||
|
height="48" width="48"><br>
|
||||||
|
<blockquote>
|
||||||
|
<div align="left">where the circle on the integral
|
||||||
|
means that the surface is <i><b>closed</b></i> and
|
||||||
|
q<sub>inside</sub> is the net charge inside this
|
||||||
|
closed surface.<br>
|
||||||
|
</div>
|
||||||
|
</blockquote>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li><img alt="exclamation"
|
||||||
|
src="exclamation-icon.gif" height="30"
|
||||||
|
width="31"> A closed surface has a definite
|
||||||
|
inside and outside differentiated by the surface,
|
||||||
|
e.g. the surface of a sphere.</li>
|
||||||
|
<li><img alt="exclamation"
|
||||||
|
src="exclamation-icon.gif" height="30"
|
||||||
|
width="31"> The <b>dA</b> vector of a closed
|
||||||
|
surface is always directed from the inside to the
|
||||||
|
outside of the surface.</li>
|
||||||
|
<li><img alt="exclamation"
|
||||||
|
src="exclamation-icon.gif" height="30"
|
||||||
|
width="31"> The exact location of the charges
|
||||||
|
inside the closed surface is not important, all
|
||||||
|
that matters is the net charge.</li>
|
||||||
|
<li><img alt="exclamation"
|
||||||
|
src="exclamation-icon.gif" height="30"
|
||||||
|
width="31"> ε<sub>0</sub> is the "Permittivity
|
||||||
|
of the Vacuum" a constant whose value is 8.85 x 10<sup>-12</sup>
|
||||||
|
C<sup>2</sup>/(N.m<sup>2</sup>) where the Coulomb
|
||||||
|
constant, k = 1/(4πε<sub>0</sub>). Note that
|
||||||
|
if the charges are not located in vacuum ε<sub>0</sub>
|
||||||
|
must be replaced by the permittivity of the medium
|
||||||
|
in question. </li>
|
||||||
|
<li><img alt="exclamation"
|
||||||
|
src="exclamation-icon.gif" height="30"
|
||||||
|
width="31"> The proof of Gauss's Law is beyond
|
||||||
|
the scope of this course. Suffice to say the
|
||||||
|
inverse square dependence on distance of Coulomb's
|
||||||
|
Law is critical.</li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>Before using Gauss's Law to evaluate electric
|
||||||
|
fields a brief qualitative discussion is
|
||||||
|
worthwhile. Consider the situation of two
|
||||||
|
point charges below. Application of Gauss's
|
||||||
|
Law over each of the closed surfaces:</li>
|
||||||
|
</ul>
|
||||||
|
<blockquote>
|
||||||
|
<ul>
|
||||||
|
<li><img alt="elec gauss figure 2"
|
||||||
|
src="elec_gauss_figure2.png" align="right"
|
||||||
|
height="523" width="359"> S<sub>1</sub>:
|
||||||
|
At every point on this surface both <b>E </b>and
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
<b>dA</b> are directed "outwards", such that the
|
||||||
|
scalar product <b>E·dA</b> = EdAcosθ is always
|
||||||
|
positive. Thus the integral over the
|
||||||
|
surface S<sub>1</sub> will be positive, as it
|
||||||
|
must be if Gauss's Law is to be satisfied, since
|
||||||
|
the net charge enclosed is positive.</li>
|
||||||
|
<li>S<sub>2</sub> : <b>E </b>is directed
|
||||||
|
"inwards", <b>dA</b> "outwards", leading to a
|
||||||
|
negative value for the flux through S<sub>2</sub>,
|
||||||
|
consistent with the fact that the net charge
|
||||||
|
enclosed is negative.</li>
|
||||||
|
<li>S<sub>3</sub>: Some of this surface has
|
||||||
|
E directed "inwards" the remainder has <b>E</b>
|
||||||
|
directed "outwards". <b>dA</b> is
|
||||||
|
"outwards" everywhere on the surface.
|
||||||
|
Therefore the flux integral has both positive
|
||||||
|
and negative contributions. Since there is
|
||||||
|
no net charge enclosed by S<sub>3</sub> by
|
||||||
|
Gauss's Law the net flux will be zero.</li>
|
||||||
|
<li>S<sub>4</sub>: Once again there are negative
|
||||||
|
and positive contributions to the flux integral,
|
||||||
|
so that we can write Gauss's Law,</li>
|
||||||
|
</ul>
|
||||||
|
<div align="center"><img alt="elec_gauss_eqn4"
|
||||||
|
src="elec_gauss_eqn4.png" height="86"
|
||||||
|
width="381"><br>
|
||||||
|
<div align="left"><br>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
</blockquote>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
</div>
|
||||||
|
<ul>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li> Electric flux through a closed box <a
|
||||||
|
href="http://www.youtube.com/watch?v=5ENl4vn82bc">animation.</a></li>
|
||||||
|
</ul>
|
||||||
|
<ul>
|
||||||
|
<li>Coulomb's Law, Electric Field, Electric Flux and Gauss's Law <a
|
||||||
|
href="http://www.veoh.com/collection/APPhysics/watch/v15544578N9Hg8YBK">video</a>.<br>
|
||||||
|
</li>
|
||||||
|
</ul>
|
||||||
|
<div style="text-align: left;"><img src="netbar.gif" height="40"
|
||||||
|
width="100%"></div>
|
||||||
|
<center><span style="font-size: 12pt; font-family: "Times New
|
||||||
|
Roman";"><span style="color: rgb(255, 0, 0); font-style:
|
||||||
|
italic;"></span></span><span style="font-size: 12pt;
|
||||||
|
font-family: "Times New Roman";"><span style="color:
|
||||||
|
rgb(255, 0, 0); font-style: italic;">
|
||||||
|
<meta http-equiv="content-type" content="text/html;
|
||||||
|
charset=windows-1252">
|
||||||
|
</span></span><br>
|
||||||
|
<font color="#ff0000"><i>Did you hear about the French post-doc
|
||||||
|
who went to work at the Fermi Lab, but never went in because
|
||||||
|
the sign over the door always said it was closed.</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>
|
||||||
|
|
@ -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·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>
|
||||||
|
|
@ -0,0 +1,157 @@
|
||||||
|
<!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 - Gauss's Law and Conductors - Physics 299</title>
|
||||||
|
</head>
|
||||||
|
<body style="color: rgb(0, 0, 0); background-color: rgb(255, 255,
|
||||||
|
255);" link="#0000ee" alink="#ff0000" vlink="#551a8b">
|
||||||
|
<center>
|
||||||
|
<h1> <img src="ULPhys1.gif" height="50" align="texttop"
|
||||||
|
width="189"></h1>
|
||||||
|
</center>
|
||||||
|
<center>
|
||||||
|
<h1>Gauss's Law and Conductors<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">
|
||||||
|
We shouldn't be surprised that conditions in the universe are
|
||||||
|
suitable for life, but this is not evidence that the universe
|
||||||
|
was designed to allow for life."</i></font><br>
|
||||||
|
Stephen Hawking<br>
|
||||||
|
</center>
|
||||||
|
<img src="netbar.gif" height="40" align="middle" width="100%"> <br>
|
||||||
|
<ul>
|
||||||
|
<li>In electrostatic conditions - no electric current flow -
|
||||||
|
Gauss's Law applied to conductors (typically metallic objects)
|
||||||
|
leads to some important conclusions.</li>
|
||||||
|
</ul>
|
||||||
|
<blockquote><img alt="exclamation" src="exclamation-icon.gif"
|
||||||
|
height="30" width="31"> Note that since a conductor contains
|
||||||
|
"free" charges if an electric field exists anywhere in the
|
||||||
|
conductor a current will flow. Thus, in electrostatic
|
||||||
|
conditions ("static" means all charges are at rest) there can be
|
||||||
|
no electric field anywhere in the conductor.<br>
|
||||||
|
</blockquote>
|
||||||
|
<ul>
|
||||||
|
<li><b><img alt="elec gauss figure 4" src="elec_gauss_figure3.jpg"
|
||||||
|
height="166" align="right" width="193">Insulated solid
|
||||||
|
conductor having a net charge</b></li>
|
||||||
|
</ul>
|
||||||
|
<blockquote>Under electrostatic conditions, <b>E</b> = 0 throughout
|
||||||
|
the object. Applying Gauss's Law to the closed surface A, we
|
||||||
|
conclude that there can be no charge inside A. But the
|
||||||
|
conductor has a net charge. The only possibility is that the
|
||||||
|
charge resides outside the surface A. If we gradually
|
||||||
|
increase the size of A, so that eventually it lies just below the
|
||||||
|
surface of the conductor, the charge must still reside outside
|
||||||
|
A. Therefore, as a consequence of Gauss's Law, any <b>charge
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
placed in a conductor must reside on its surface.</b><br>
|
||||||
|
<br>
|
||||||
|
</blockquote>
|
||||||
|
<ul>
|
||||||
|
<li><b>Insulated hollow charged conductor (conducting shell)</b></li>
|
||||||
|
</ul>
|
||||||
|
<blockquote>
|
||||||
|
<p>We now hollow out the conductor, changing nothing else.
|
||||||
|
Thus, there is no charge inside the hollowed out conductor, so
|
||||||
|
that <b>E</b> = 0 inside. This fact leads to the
|
||||||
|
necessity for an antenna to pick up radio signals inside a
|
||||||
|
car. Radio waves are comprised of electric and magnetic
|
||||||
|
fields (electromagnetic waves - much more later), which must be
|
||||||
|
received by the radio. But the car is approximately a
|
||||||
|
hollow metallic conductor, which means <b>E</b> = 0
|
||||||
|
inside. Without an antenna the radio waves cannot be
|
||||||
|
received by the radio. The antenna provides a "shielded
|
||||||
|
channel" to direct the radio signal into the car.<br>
|
||||||
|
<br>
|
||||||
|
</p>
|
||||||
|
<center><img alt="elec gauss figure 4"
|
||||||
|
src="elec_gauss_figure4.jpg" height="226" width="700"><br>
|
||||||
|
</center>
|
||||||
|
</blockquote>
|
||||||
|
<center>
|
||||||
|
<div align="left">
|
||||||
|
<ul>
|
||||||
|
<li><b>Faraday Cage</b></li>
|
||||||
|
</ul>
|
||||||
|
<blockquote>
|
||||||
|
<p>A <a
|
||||||
|
href="http://www.princeton.edu/%7Eachaney/tmve/wiki100k/docs/Faraday_cage.html">Faraday
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
cage</a> is a metal container, which is used to shield
|
||||||
|
sensitive electronics from stray electric fields.
|
||||||
|
Fields outside the container cannot penetrate due to the
|
||||||
|
above explanation.<br>
|
||||||
|
<br>
|
||||||
|
</p>
|
||||||
|
</blockquote>
|
||||||
|
<ul>
|
||||||
|
<li><b>Proof of inverse square nature of Coulomb's Law</b></li>
|
||||||
|
</ul>
|
||||||
|
<blockquote>
|
||||||
|
<p>It can be shown mathematically that if Coulomb's Law is not
|
||||||
|
exactly of the inverse square form - 1/r<sup>2</sup> then
|
||||||
|
the electric field inside a closed conductor would not be
|
||||||
|
exactly zero. All experiments to date have failed to
|
||||||
|
measure such an electric field, with an accuracy such that
|
||||||
|
we know that the inverse component of r in Coulomb's Law is
|
||||||
|
2 with an accuracy of 16 decimal places.<br>
|
||||||
|
</p>
|
||||||
|
</blockquote>
|
||||||
|
</div>
|
||||||
|
</center>
|
||||||
|
<br>
|
||||||
|
<img src="netbar.gif" height="40" width="100%"><br>
|
||||||
|
<center><span style="font-size: 12pt; font-family: "Times New
|
||||||
|
Roman";"><span style="color: rgb(255, 0, 0); font-style:
|
||||||
|
italic;"></span></span><span style="font-size: 12pt;
|
||||||
|
font-family: "Times New Roman";"><span style="color:
|
||||||
|
rgb(255, 0, 0); font-style: italic;">
|
||||||
|
<meta http-equiv="content-type" content="text/html;
|
||||||
|
charset=windows-1252">
|
||||||
|
</span></span><br>
|
||||||
|
<font color="#ff0000"><i>A Simpleton's Guide to Science (stolen
|
||||||
|
from UK magazine)
|
||||||
|
<br>
|
||||||
|
Relativity : Family get-togethers at Christmas
|
||||||
|
<br>
|
||||||
|
Gravity : Strength of a glass of beer
|
||||||
|
<br>
|
||||||
|
Time travel : Throwing the alarm clock at the wall
|
||||||
|
<br>
|
||||||
|
Black holes : What you get in black socks
|
||||||
|
<br>
|
||||||
|
Critical mass: A gaggle of film reviewers
|
||||||
|
<br>
|
||||||
|
Hyperspace : Where you park at the superstore</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>
|
||||||
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