A non-conducting thin disc of radius R charged uniformly over one side with surface density s rotates about its axis with an angular velocity `omega`. Find
(a) the magnetic induction at the centre of the disc,
(b) the magnetic moment of the disc.

Explanation

(a) Consider an element area of the disc of radius r and
thickness dr

`dq=sigma.2pir dr`
`:.` Current `dI=(dq)/T=sigma.2pir dr.omega/(2pi)`
`:. dB`=Magnetic field by this element
`=(mu_0dI)/(2r)`
or `B=intdB=(mu_0)/(4pi).2pisigmaomega int_0^R dr=(mu_0)/2 sigmaomegaR`
(b) Now dM=magnetic moment of element
`=dIpir^2=omega sigmardrpir^2`
`M=intdM=omegasigmapi int_0^Rr^3dr=1/4omegapisigmaR^4`