Evaluating Magnetic Dipole Moment Integral in Spherical Coordinates

Question

I need to evaluate the $z$ component of the magnetic dipole moment integral for a current distribution, which is given by the following:

$$\mathbf M = \frac 12 \int_V \; \mathbf r \times \mathbf J \; dv $$

Where $ \mathbf M $ is the magnetic dipole moment; $ \mathbf r $ is the vector from the origin to the point of interest; and $ \mathbf J $ is the current density, defined as $ \mathbf J=J_0 \hat \phi $

The current density is uniform, and this is for a sphere.

I'm confused about what I'm being asked to do, this is asking for the $z$ component of the integral. But since the current density is in spherical co-ordinates, am I supposed to calculate the integral in spherical co-ordinates? If so, is there a $z$ component for spherical co-ordinates?

Answer 1

$${\bf r}\times{\bf J}=r\hat{r}\times J_0\hat{\phi}=J_0 r (-\hat{\theta})$$ The z-component is $$J_0r \sin\theta$$ So you have $${\bf M}=\hat{{\bf z}}\frac{1}{2}\int_VJ_0 r \sin\theta dv$$