In a uniform magnetic field of induction B , a wire in the form of a semicircle of radius r rotates about the diameter of the circle with angular frequency ω . If the total resistance of the circuit is R , the mean power generated per period of rotation is

Step-by-step solution

The flux associated with coil of area A and magnetic induction B is ϕ = B A cos ⁡ θ = 1 2 B π r 2 cos ⁡ ω t ∵ A = 1 2 π r 2 ∴ e i n d u c e d = - d ϕ d t = - d d t 1 2 B π r 2 cos ⁡ ω t = 1 2 B π r 2 ω sin ⁡ ω t ∴ p o w e r p = e i n d u c e d 2 R = B 2 π 2 r 4 ω 2 sin 2 ⁡ ω t 4 R Hence, P m e a n = < p > = B 2 π 2 r 4 ω 2 4 R . 1 2 ∵ < sin ⁡ ω t > = 1 2 = B π r 2 ω 2 8 R