Finding the gyromagnetic ratio of an axially symmetric body

[\Tau_{\mu\ni}]

Homework Statement

So, we can presumably write that m = g L, where L is the angular momentum, g the ratio wanted, and m magnetic dipole moment of an axially symmetric body. Total mass is M, total charge Q, mass density $\rho_m(r)=\frac{M}{Q}\rho_e(r)$, where $\rho_e(r)$ is charge density.

Homework Equations

Moment of inertia can also be written $I_\omega=\int \rho_m d^2 d\tau$ where d is the distance from the axis of symmetry.

The Attempt at a Solution

I guess the dipole moment is in the direction of the axis of the symmetry, as is the angular velocity and it can be written:
m = g I$_\omega$ ω

RHS:
= g ω $\hat{z}$ I$_\omega$

LHS:
m = I $\int d\vec{a}$

Here is where I guess I'm having conceptual problems. I is the total current, part of which should be a current flowing through a differential circular loop inside the body, at distance d, that is to say:

I = $\int \left| v \right| \rho_e d\tau$ , where $\left|v\right|$ is the module of radial speed of a differential volume element (at distance d) that can be written: $\left|v\right|=\left|\omega\right|\left|r\right|$sin$\theta$= ωd. $\int d\vec{a}$ of a differential loop in question is simply $d^2 \pi$. So, I am getting:

g ω $\int \rho_m d^2 d\tau$ =ω $\int d^3 \pi \frac{M}{Q} \rho_m d\tau$

Where did I go wrong, how to find g?

 

[\Tau_{\mu\ni}]

anyone? :( the integrals are supposed to annihilate each other leaving me with a factor ~M/Q