A charge `q` is unifromly distributed over the volume of a unifrom ball of mass `m` and radius `R` which rotates with an angular velocity `omega` about the axis passing through its centre. Find the respective magnetic moment and its ratio to the mechanical moment.

Explanation

The magentic moment must clearly be along the axis of rotation. Consider a volume element `dV`. It contains a charge `(q)/(4pi//3R^(3)) dV`. The rotation of the sphere causes this charge to resolve around the axis and constiude a current.
`(3q)/(4pi R) dV xx (omega)/(2pi)`
Its magentic moment will be
`(3q)/(4pi R^(3)) dV xx (omega)/(2pi) xx pi r^(2) sin^(2) theta`
So that total magnetic moment is
`p_(m) = int_(0)^(R) int_(0)^(pi) (3q)/(2R^(3)) r^(2) sin theta d theta xx (omega r^(2) sin^(2) theta)/(2) dr = (3q)/(2 R^(3)) xx (omega)/(2) xx (R^(5))/(5) xx (4)/(3) = (1)/(5) qR^(2) omega`
The mechanical moment is
`M = (2)/(5) mR^(2) omega`, So, `(p_(m))/(M) = (q)/(2m)`