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

Answer

D

Explanation

As `phi = BA cos theta = B ((pi^(2))/(2)) cos omega t`
and `e = - (d phi)/(dt) = - (d)/(dt) ((B pi r^(2) cos omega t)/(2))`
`= - (B pi r^(2) omega sin omega t)/(2)`
Also , `P = (e^(2))/(R ) = ((B pi r^(2) omega)^(2))/(4 R) sin^(2) omega t`
But `lt sin^(2) omega t gt = (1)/(2)`
`:. P = ((B pi r^(2) omega)^(2))/(8 R)`