Calculating the magnetic moment of an electron

[PsychoDash]

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
$$r_e$$

Homework Equations

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.

 

[Liquidxlax]

yeah integrating vectors gets really really hairy, when i get home i will help you out. There is a way to define some new quantity "C" per-say and it makes it easier

Unless someone helps you first

 

[PsychoDash]

The assignment has already been turned in. I ended up just not using the given formula. Got the correct answer, even if not really in the correct way. Thanks though.