A line charge `lambda` per unit length is pasted uniformly on to the wire of a wheel of mass `m` and radius `R`. The wheel has light non-conducting spokes and is free to rotate about a vertical axis as shown in `Fig. 3.166`. A uniform magnetic field extends over a radial rigion of radius `r` given by `B = -B_(0)hat(k)(r le a, alt R) = 0` (otherwise). What is the angular velocity of the wheel this field is suddenly switched off?

Options

A. (a) `(-2B_(0)pia^(2)r)/(mR)hat(k)`
B. (b) `(-2B_(0)pia^(2)r)/(3mR)hat(k)`
C. ( c) `(B_(0)pia^(2)lambda)/(mR)hat(k)`
D. ( d) `(-B_(0)pia^(2)lambda)/(mR)hat(k)`

Answer

D

Explanation

(d) Induced
`E = (a^(2))/(2R)(dB)/(dt)`
`Iomega = int tau dt`
`rarr` `mR^(2)omega = int qERdt = int(qa^(2))/(2R)(dB)/(dt)Rdt = (lambda2piR)/(2R)a^(2)RintdB`
`rarr` `omega = (B_(0)pia^(2)lambda)/(mR)`
`rarr vec(omega) = -(B_(0)pia^(2)lambda)/(mR)hat(k)` (because clockwise sense)