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