Question #43262

A thin non-conducting disk of radius $R$ lies in the $xy$-plane and rotates with a constant angular velocity $\vec{\omega} = \omega \hat{k}$ about the $z$-axis. The disk has a non-uniform mass surface density $\sigma_m = \alpha r$ and a non-uniform charge surface density $\sigma_q = \beta r^2$, where $r$ is the radial distance from the centre, and $\alpha, \beta$ are positive dimensional constants. If $Q$ is the total charge and $M$ is the total mass of the disk, determine the effective gyromagnetic ratio $\gamma = \frac{|\vec{\mu}|}{|\vec{L}|}$ of the disk, where $\vec{\mu}$ is the magnetic dipole moment and $\vec{L}$ is the angular momentum about the $z$-axis. Assume all quantities are in standard SI units.