Talk:PlanetPhysics/Quarter Loop Example of Biot Savart Law
Original TeX Content from PlanetPhysics Archive
edit%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: quarter loop example of Biot-Savart law %%% Primary Category Code: 41.20.Gz %%% Filename: QuarterLoopExampleOfBiotSavartLaw.tex %%% Version: 3 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}
\setlength{\topmargin}{0.00in} \setlength{\headsep}{0.00in} \setlength{\headheight}{0.00in} \setlength{\evensidemargin}{0.00in} \setlength{\oddsidemargin}{0.00in} \setlength{\textwidth}{6.5in} \setlength{\textheight}{9.00in} \setlength{\voffset}{0.00in} \setlength{\hoffset}{0.00in} \setlength{\marginparwidth}{0.00in} \setlength{\marginparsep}{0.00in} \setlength{\parindent}{0.00in} \setlength{\parskip}{0.15in}
\usepackage{html}
% 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
\begin{document}
A simple example of the \htmladdnormallink{Biot-Savart law}{http://planetphysics.us/encyclopedia/BiotSavartLaw.html} is given with a current carrying loop for a quater of a circle as shown in Figure 1.
\begin{figure}
\includegraphics[scale=.8]{QuarterCurrentLoop.eps}
\vspace{10 pt}
\caption{Figure 1: Quarter Current Loop}
\end{figure}
The differential line element d{\bf l} is perpendicular to $\hatTemplate:\bf r$, so the Biot-Savart law simplifies from the definition of the \htmladdnormallink{cross product}{http://planetphysics.us/encyclopedia/VectorProduct.html} $$ d{\bf B} = -\frac{\mu_0}{4 \pi}\frac{I dl \, sin (\pi/2)}{r^2} \hatTemplate:\bf k $$
Note that the direction is into the web browser, so we have a $-\hatTemplate:\bf k$. For the differential dl of a circular arc
$$dl = r d\theta$$
plugging this in and setting up the integral gives
$$ {\bf B} = - \hatTemplate:\bf k \frac{\mu_0 I}{4 \pi r} \int_0^{\frac{\pi}{2}} \, d \theta $$
which simply yields
$$ {\bf B} = -\frac{\mu_0 I \hatTemplate:\bf k}{8 r} $$
\end{document}