OffAxisExampleOfBiotSavartLaw 2006-03-11 12:29:40 2006-03-11 16:24:51 Example Biot-Savart law % 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 $r = x \hat{x} + z \hat{z}$ \\ $r' = r \cos \phi' \hat{x} + r \sin \phi' \hat{y}$ \\ $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 formula $(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 ] $\\