ACatsMeowApplicationOfGaussLaw 2006-12-18 23:14:26 2006-12-18 23:14:26 Example Gauss's Law Gauss % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) \usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here If you know the amount of charge contained within a Gaussian surface, then the total flux of the electric field generated by the enclosed charge is calculated from Gauss' law. As a demonstration, imagine a pair of cats that have charges placed on them by their loyal masters. Although the contours of the cats' elegant frames represent a complicated geometry, calculating the flux is a simple task if the charge on the cats is known. The flux through the Gaussian surface in Figure 1 is given by Gauss' law \begin{equation} \Phi = \frac{q_1 + q_2}{\epsilon_0} \end{equation} \vspace{10 pt} \begin{center} \includegraphics[scale=.6]{Cats.eps} {\bf Figure 1:} Gaussian Surface Encompassing Two Cats \end{center} Note that we add the charges in equation (1) because it is the net enclosed charge. For exampke if the charge on cat 1 is $10.5 \,\, [\mu C]$ and the charge on the cat 2 is $12.2 [\mu C]$, then the total flux through $G$ is $$\Phi = \frac{10.5\times 10^{-8} \, [C] + 12.2\times 10^{-8} \, [C] }{8.85 \times 10^{-12} \, [C^2/N m^2]}$$ $$ \Phi = 3073.4 \,\, [N m^2/C]$$ The reverse of this problem is another important result. If we measure the flux through a given Gaussian surface, then we can calculate the amount of enclosed charge. \begin{thebibliography}{9} \bibitem{Figure1} Figure 1, The Cat Clip art is public domain and was downloaded from \PMlinkexternal{WP Clipart}{ http://www.wpclipart.com/} \end{thebibliography}