A non conducting disc of radius R, charge Q is rotating about an axis passing through its centre and perpendicular to its plane, with an angular velocity 'ω' .

The magnetic moment of the disc is =?

a) 1/4 qωR^2 b) 1/2 qωR^2. c) qωR. d) 1/2qωR^2

surface charge density σ = q/πr²
Now let us take a small cross section of length dR whose radius varies from R to R + dR
The charge on this annular region is ,
dq = 2πRσdR 
= 2πR x q/πr² x dR
= 2q/r² x RdR
This charge dq passes through a line once in a time, T = 2π/ω
So charge crossing per unit time i.e. current dI = dq/T
= dqω/2π
=2q/r² x RdR x ω/2π
The area enclosed by this current = πR²
Hence, magnetic moment due to current loop ,
dM = πR² dI
=πR² x  2q/r² x RdR x ω/2π
= ωq/r²  x R³dR
Taking integration on both the sides where limit of R varies from 0 to R we get,
M = ωq/r²  x 
=> M = ωq/r²  x r⁴/4
=> M = ωqr²/4
Hence the moment of inertia will be ωqr²/4