A thin circular disk of radius `R` is uniformly charged with density `sigma gt 0` per unit area.The disk rotates about its axis with a uniform angular speed `omega`.The magnetic moment of the disk is :

Options

A. `2piR^(4)sigma omega`
B. `piR^(4)sigma omega`
C. `(piR^(4))/2 sigma omega`
D. `(piR^(4))/4 sigma omega`

Answer

D

Explanation

Let us consider the disc to be made up of large number of concentric elementary rings.
Consider one such ring of radius x and thickness dx.
Charge on this elementary ring.
`dl=(dq)/(dt)=dqxxf=sigma omega dx`[ `:'` f is frequency and `omega=2pif`]

Magnetic moment of this elementary ring.
`dm=d//pix^(2)=pi sigma omega x^(3) dx`
`:.` Magnetic moment of the entire disc `M=int_(0)^(R)dM=pi sigma omega int_(0)^(R)x^(3)dx=1/4pi R^(2)sigma omega`