The question deals with finding the magnetic field of a ball with radius R at its "northern pole" (at point (0,0,R) in cylindrical) My usage of cylindrical is $(r,\theta, \phi)$
So I've calculated the magnetic field of a disc on it's axis of symmetry:
$$\vec B=\frac{\mu_0 \omega}{2}\left( \frac{R^2+2z^2}{\sqrt{R^2+z^2}}-2|z|\right)\hat z$$
Now of course I need to sum up all the rings in order to find the total contribution of each ring.
My questions are as follows:
1.) I need to convert dz to spherical coordinates. Do I just use the connection $z=rcos\phi$ and $dz=rsin\phi$ d$\phi$?
2.) Limit of integration from $0\le \phi \le \pi$?
3.) Can I get rid of the absolute value on $-2|z|$ after switching to spherical since it will always be positive in our limits of integration? $(-2|z|=2|rcos\phi|)$
EDIT Just kidding about #3, since cosx is non-positive on a portion of the range.
I appreciate any and all tips or hints.