A wire in the form of semi-circle of radius r rotates about its diameter with angular velocity in a magnetic field B . The axis of rotation is perpendicular to the field. The total resistance of the circuit is R . If the mean power generated per period of rotation is (B r^2( )^2 . x R , then the value of x is

Step-by-step solution

aligned & (8) = B A =B A =B A t,( = t) & Now, emf (e)=- d d t =- d d t (B A t) & = B A t & i= e R = B A ^t R & P_ ins =i^2 R= ( ^2 B^2 A^2 R^2 ^2 t ) R & = ^2 B^2 A^2 R ^2 t aligned aligned P_ avg & = ₀^T P_ ins d t ₀^T d t & = ^2 B^2 A^2 R ₀^T ^2 t d t ₀^T d t & = 1 2 ^2 B^2 A^2 R ( ^2 t _ t=T = 1 2 ) & = (B r^2 )^2 8 R ( A= r^2 2 ) x & =8 aligned