## Magnetic dipole moment of a sphere

1. The problem statement, all variables and given/known data
Find the magnetic dipole moment of a spherical shell of radiu R carrying a uniform surface charge sigma, set spinning at angular velocity omega.

2. Relevant equations
$$\vec{m} = \frac{1}{2} \int_{S} \vec{r'} \times \vec{K} (\vec{r'}) da'$$

3. The attempt at a solution
So we gotta figure out the surface charge density (since it is a spherical shell)

$$K = \sigma v$$

and $$v = \omega times R$$
this is where i am doubtful...
the angle between v and R varies from 0 to 2 pi

so this cross product is not unique...
 Recognitions: Science Advisor You are right that the angle changes but v will always point in the correct direction. The changing angle ($$sin(\theta)$$) accounts for the "azimuthal radius", that is, distance from the spin axis z to the shell measured parallel to the xy plane, that changes with polar angle. That distance is also the r you need to use in your integral. BTW, polar angle only varies from 0 to $$\pi$$