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A non-conducting thin disc of radius R c...

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.

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(a) Consider an elementary ring of the disc of radius x and thickness `dx` shown by shaded portion in figure. Charge on the elementary ring, `dq=sigma2pixdx`. Current due to this element is

`dI=(sigma2pixdx)/(T)=(sigma2pixdx)(omega)/(2pi)=sigmaomegaxdx`
Magnetic field induction at the centre, due to this element is
`dB=(mu_0)/(4pi)(2pidI)/(x)=(mu_0)/(2)(dI)/(x)=(mu_0)/(2)(sigmaomegaxdx)/(x)=(mu_0)/(2)sigmaomegadx`
Magnetic field induction at the centre of the disc will be, `B=int_(0)^(R) (mu_0)/(2)sigmaomegadx=(mu_0)/(2)sigmaomegaR`
(b) Magnetic dipole moment of teh elementary ring, `dM=dIxxpix^2=(sigmaomegaxdx)xxpix^2`
`=sigmapiomegax^3dx`
Magnetic dipole moment of the whole disc will be
`M=sigmapiomega int_(0)^(R)x^3dx=1/4sigmapiomegaR^4`
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