Consider a solid sphere of radius r and mass m having a charge q...

For a rigid body having uniform charge and mass distribution, the ratio of charge to mass for any elemental volume is constant.

Let the uniform charge density be $\rho_q = \dfrac{q}{V}$ and uniform mass density be $\rho_m = \dfrac{m}{V}$.

Consider an elemental volume $dV$ rotating at a perpendicular distance $x$ from the axis of rotation with angular speed $\omega$.

The charge of the element is $dq = \rho_q dV$ and its mass is $dm = \rho_m dV$.

The magnetic moment of this rotating element is $d\mu = \dfrac{1}{2} dq x^2 \omega$.

The angular momentum of the element is $dl = dm x^2 \omega$.

Taking the ratio of the two quantities:

$\dfrac{d\mu}{dl} = \dfrac{\dfrac{1}{2} dq x^2 \omega}{dm x^2 \omega} = \dfrac{1}{2} \dfrac{dq}{dm} = \dfrac{1}{2} \dfrac{\rho_q dV}{\rho_m dV} = \dfrac{q}{2m}$

Since this ratio is constant for all elements, integrating over the entire volume of the sphere yields:

$\mu = \dfrac{q}{2m} l$