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[parent] Off axis example of Biot Savart Law (Example)

$r = x \hat{x} + z \hat{z}$
$r' = r \cos \phi' \hat{x} + r \sin \phi' \hat{y}$
$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} $

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

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 ] $

$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] $



"Off axis example of Biot Savart Law" is owned by bloftin.
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This is version 10 of Off axis example of Biot Savart Law, born on 2006-03-11, modified 2006-03-11.
Object id is 120, canonical name is OffAxisExampleOfBiotSavartLaw.
Accessed 1084 times total.

Classification:
Physics Classification: 41.20.Gz (Magnetostatics; magnetic shielding, magnetic induction, boundary-value problems)
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