Talk:PlanetPhysics/Off Axis Example of Biot Savart Law

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Off axis example of Biot Savart Law %%% Primary Category Code: 41.20.Gz %%% Filename: OffAxisExampleOfBiotSavartLaw.tex %%% Version: 10 %%% 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}

$r = x \hat{x} + z \hat{z}$ \\ $r' = r \cos \phi' \hat{x} + r \sin \phi' \hat{y}$ \\ $dl = -r d\phi' \sin \phi' \hat{x} + r d\phi' \cos \phi' \hat{y} $\\ $ r - r' = (x - r' \cos \phi') \hat{x} + r \sin \phi' \hat{y} + z \hat{z}$ \\ $ dl \times (r - r') = zr d\phi \cos \phi' \hat{x} + zrd\phi \sin \phi' \hat{y} + [- x r d\phi cos\phi' + r^2 d\phi] \hat{z} $ \\

$ \left | r - r' \right | = \sqrt{ x^2 - 2rx \cos \phi'+ r^2 + z^2}$ \\

with

$\alpha = x^2 + r^2 + z^2$ \\ $\beta = 2rx$ \\

expand

$( \alpha - \beta \cos \phi')^{-\frac{3}{2}} $\\

rewrite as

$\alpha^{-\frac{3}{2}}( 1 - \frac{\alpha}{\beta}x)^{-\frac{3}{2}} $\\

use expansion \htmladdnormallink{formula}{http://planetphysics.us/encyclopedia/Formula.html} $(1 - x)^{-n} = 1 + \frac{nx}{1!} + \frac{n(n+1)x^2}{2!} $\\

$1 + \frac{3}{2}\frac{\beta}{\alpha} \cos \phi' + \frac{15}{8}\frac{\beta^2}{\alpha^2} \cos^2 \phi' $

$B_r = \frac{2 \pi I a^2 \cos \theta}{c(a^2 + r^2)^{3/2}} \left [ 1 + \frac{15 a^2 r^2 sin^2\theta}{4(a^2 + r^2)^2} + ... \right ] $\\

$B_{\theta} = -\frac{\pi I a^2 \sin \theta}{c(a^2 + r^2)^5/2} \left [ 2a^2 - r^2 + \frac{15 a^2 r^2 \sin^2 \theta(4a^2 - 3r^2)}{8(a^2 + r^2)^2} + ... \right] $\\

\end{document}