An elasticized conducting band is around a spherical ballon . Its plane through the center of the balloon. A uniform nmagnetic field of magnitude `0.04T` is directed perpendicular to the plane of the band. Air is let out of the balloon at `100 cm^(3) s^(-1)` at an instant when the radius of the balloon is `10 cm`. The induced emf in the band is

Options

A. (a) `15mu v`
B. (b) `25mu v`
C. ( c) `10mu v`
D. (d) `20mu v`

Answer

D

Explanation

(d) Volume of the balloon at any instant, when rdius is `r`,
`V = (4)/(3)pir^(3)`
Time rate of change of volume,
`(dV)/(dt) = 4pir^(2)(dr)/(dt)`
Time rate of change of radius of balloon,
`(dr)/(dt) = (1)/(4pir^(2))(dV)/(dt)`
Flux through rubber band at the given instant,
`phi = B(pir^(2))`
`Induced emf = -(dphi)/(dt) = -(d)/(dt)(Bpir^(2)) = -2pirB(dr)/(dt)`
`= -2pirB.((1)/(4pir^(2))(dV)/(dt)) = -(B)/(2r)(dV)/(dt)`

As volume of the balloon is decreasing, `(dV)/(dt)` is negative.
`E_(induced) = -((0.04))/(2 xx 10 xx 10^(-2)) xx (-100 xx 10^(-6)) = 20mu V`