An insulating rod of length I carries a charge q distrubuted uniformly on it. The rod is pivoted at its mid-point and is rotated at a frequency f about a fixed axis perpendicular to the the rod and passing through the pivot . The magnetic moment of the rod system is

Options

A. `(piQvL^2)/(2)`
B. `(piQvL^2)/(3)`
C. `(piQvL^2)/(6)`
D. `(piQvL^2)/(12)`

Explanation

Consider a small element of rod of length `dx` at distance x from the axis of rotation.

Charge on small element dx, `dQ=Q/Ldx`
Current in the element due to rotation of rod
`dI=vdQ=vQ/Ldx`
Magnetic moment of the small element
`dM=(dI)xxpix^2=((vQ)/(L)dx)pix^2=(pivQ)/(L)x^2dx`
Total magnetic moment of rod system due to rotation of the whole rod is
`M=int_(-1//2)^(+1//2)(pivQ)/(L)x^2dx=(pivQL^2)/(12)`