# Magnetic dipole moment of a sphere

#### stunner5000pt

1. 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.

2. Homework 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...

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#### 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 $$\pi$$