What is the Magnetic Dipole Moment of a Spinning Spherical Shell?

[stunner5000pt]

Homework Statement

Find the magnetic dipole moment of a spherical shell of radiu R carrying a uniform surface charge sigma, set spinning at angular velocity omega.

Homework Equations

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

The Attempt at a Solution

So we got to 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...
or am i thinking about this the wrong way??

please help!

 

[marcusl]

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 $$\pi$$