Magnetic dipole moment of a sphere

stunner5000pt

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
[tex] \vec{m} = \frac{1}{2} \int_{S} \vec{r'} \times \vec{K} (\vec{r'}) da' [/tex]

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

[tex] K = \sigma v [/tex]

and [tex] v = \omega times R [/tex]
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 ([tex]sin(\theta)[/tex]) 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 [tex]\pi[/tex]