# Calculating the magnetic moment of an electron

## Homework Statement

Assume that an electron is a sphere of uniform mass density
$\rho_m=\frac{m_e}{\frac{4}{3} \pi r_e^3}$, uniform charge
density $\rho_e=\frac{-e}{\frac{4}{3} \pi r_e^3}$, and
radius $r_e$ rotating at a frequency $\omega$
about the z-axis. $$m_e=9.109*10^{-31}$$ kg and
$$e=1.602*10^{-19}$$ C

Using the formula $$\vec{m}=\frac{1}{2} \int \vec{r} \times \vec{J(\vec{r})} d\tau$$, compute the magnetic moment of this
electron. Your answer should depend on e, $$\omega$$ and
$$r_e$$

Given above.

## The Attempt at a Solution

Ok, so I know that in general, $$\vec {J}=\rho_e \vec {v}$$. I'm not sure how to proceed from here since writing J in terms of omega yields $$\vec {J}=\frac {\rho_e \omega}{\vec {r}}$$. I've always heard that dividing by a vector is not strictly defined in a math sense. Either I'm not approaching this in the right way, or putting that funkiness into the cross product above yields some magnificence that I am, as of now, incapable of seeing.

Help will be greatly appreciated.