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A flat dielectric disc of radius R carri...

A flat dielectric disc of radius `R` carries an exces charge on its surface. The surface charge density `sigma`. The disc rotastes about an axis perpendicular to its lane passing thrugh the centre with angulasr velocity `omega`. Find the toruque on the disc if it is placed in a uniform magnetic field `B` directed perpendicular to the rotation axis.

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Consider an anular ring of radius `r` and of thickness dr on this disc. Charge within this ring,

As ring rotates with angular velocity `omega`, the equivalent current is
`i=(dq)("frequency")`
`(sigma)(2pidr)(omega(2pi)) or i=sigmadrdr`
`=(sigma)(2pidr)(omega(2pi))or i=sigmardr`
Magnetic moment of this anulalr ring,
`M=iA=(sigmadr)(pir^2) ("along the axis of rotation")`
Torque on this ring
`dtau=MBsin90^@=(sigmapir^3B)dr`
`:.` Total torque on the disc is `tau=int_0^Rtau(sigmaomegapiB)int_0^Rr^3dr`
`=(sigmaomegaBR^4)/4`
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