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

Explanation

Charge on the differntial circular
strip is
`dq=(2piR sin theta)(R d theta)sigma`
`implies dq=2pisigma R^2 sin theta d theta,`
`dI=(omegadq)/(2pi)=omegasigmaR^2 sin theta d theta`
(a) Magnetic field at the centre is
`dB=(mu_0dI(R^2sin^2 theta))/(2R^3)`
`(mu_0 omega sigmaRsin^3 theta d theta)/2`
`B=(mu_0omegasigmaR)/2 int_0^(pi)sin^3 theta d theta=mu_0omegasigmaR int_0^(pi//2) sin^3 theta d theta`
`implies vecB=2/3 mu_0omega sigmaR(hatk)` `[ :' sin^3 theta=1/4(3 sin theta-sin 3 theta)]`
(b) Magnetic moment due to elementary ring, `dmu=(dI) pir^2`
`mu=int(dI)pir^2=2int_0^(pi//2)(omega sigmaR^2 sin theta) pi R^2 sin^2 theta d theta`
`=2piR^4omegasigmaint_0^(pi//2) sin^3 theta d theta=4/3pisigmaR^4omega`