In a uniform magneitc field of induced B a wire in the form of a semicircle of radius r rotates about the diameter of hte circle with an angular frequency `omega`. The axis of rotation is perpendicular to hte field. If the total resistance of hte circuit is R, the mean power generated per period of rotation is

Options

A. `((B pi omega)^(2))/(2R)`
B. `((B pi r omega^(2))^(2))/(2R)`
C. `(B pi r^(2) omega)/(2R)`
D. `((B pi r^(2) omega)^(2))/(8R)`

Answer

D

Explanation

The magnetic flux linked with the rotating coil is
`phi = BA cos omega t = B cdot ((pi r^(2))/(2)) cos omega t`
`:.` The induced e.m.f. (e) is
`:. e = -(d phi)/(dt) = -(B cdot pir^(2))/2 omega(-sin omega t)`
`1/2 Bpi r^(2)omega sin omega t`
`:.` Power = `(e^2)/R = ((B pir^(2)omega)^(2)sin^(2)omega t)/(4R)`
But the mean value of `sin^(@)omega t =1/2`
`:.` Mean power generated per cycle (period) of rotation
`=((B pir^(2)omega)^(2))/(4R) xx 1/2`
`P_("mean") = ((Bpir^(2)omega)^(2))/(8R)`.