Consider a non conducting plate of radius `a` and mass `m` which has a charge `q` distributed uniformly over it, The plate is rotated about its own axis with a angular speed `omega`. Show that the magnetic moment `M` and the angular momentum `L` of the plate are related as `(M)/(L)=(q)/(2m)`.

Explanation

If `sigma` is the surface charge density, then `q = sigma pi a^2`
Current `i = sigma omega r dr` . The magnetic moment of the element ring `dM = (idA) = sigma omega r dr (pi r^2) = pi sigma omega r^3 dr`
`M = pi sigma omega int_0^a r^3 dr = (pi sigma omega r^4)/(4)`
` = (pi a^2 sigma) (omega a^2)/(4) = (q omega a^2)/(4)`
The angular momentum of the disc about its axis
`L = (ma^2)/(2) omega . ` the ratio `M/L = (4)/( (omega ma^2)/(2) ) = (q)/(2m) `