PlanetPhysics/Off Axis Example of Biot Savart Law

$r=x\hat{x}+z\hat{z}$ \\ $r^{\prime }=r\cos {\varphi }^{\prime }\hat{x}+r\sin {\varphi }^{\prime }\hat{y}$ \\ $dl=-rd{\varphi }^{\prime }\sin {\varphi }^{\prime }\hat{x}+rd{\varphi }^{\prime }\cos {\varphi }^{\prime }\hat{y}$\\ $r-r^{\prime }=(x-r^{\prime }\cos {\varphi }^{\prime })\hat{x}+r\sin {\varphi }^{\prime }\hat{y}+z\hat{z}$ \\ $dl\times (r-r^{\prime })=zrd\varphi \cos {\varphi }^{\prime }\hat{x}+zrd\varphi \sin {\varphi }^{\prime }\hat{y}+[-xrd\varphi cos{\varphi }^{\prime }+r^{2}d\varphi ]\hat{z}$ \\

$|r-r^{\prime }|=\sqrt{x^2-2rx\cos {\varphi }^{\prime }+r^2+z^2}$ \\

with

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

expand

$(\alpha -\beta \cos {\varphi }^{\prime }{)}^{-\frac{3}{2}}$\\

rewrite as

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

use expansion [[../Formula/|formula]] $(1-x{)}^{-n}=1+\frac{nx}{1!}+\frac{n(n+1)x^2}{2!}$\\

$1+\frac{3}{2}\frac{\beta }{\alpha }\cos {\varphi }^{\prime }+\frac{15}{8}\frac{{\beta }^2}{{\alpha }^2}{\cos }^2{\varphi }^{\prime }$

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

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