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51. (AIEEE 2004 )

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 an angular frequency ω . The axis of rotation is perpendicular to the field. If the total resistance of the circuit is R , the mean power generated per period of rotation is

A. ( B π r ω ) 2 2 R

B. ( B π r 2 ω ) 2 8 R

C. B π r 2 ω 2 R

D. ( B π r ω 2 ) 2 8 R

Correct option is (B)

ϕ = B → . A → ; ϕ = B A cos ω t

ε = − d ϕ d t = ω B A sin ω t ;

i = ω B A R sin ω t

P i n s t = i 2 R = ( ω B A R ) 2 × R sin 2 ω t

p a v g = ∫ 0 T P i n s t × d t ∫ 0 T d t

= ( ω B A ) 2 R ∫ 0 T sin 2 ω t d t ∫ 0 T d t

= 1 2 ( ω B A ) 2 R

∴ P a b g = ( ω B π r 2 ) 2 8 R [ A = π r 2 2 ]