A wire shaped as a semi-circle of radius a rotates about sn sxis `OO'` with an angular velocity `omega` in a unifrom magnetic field of induction `B'` (Fig). The rotation axis is perpendicular to the field direction. The total resistance of the circuit is equal to `R`. NEglecting the magnetic field of the induced current, find the meand amount of thermal power being generated in the loop fueing a rotation period.

Explanation

Flux of `vec(B)`, at an orbitarty moment of time `t` :
`Phi_(t) = vec(B) . vec(S) = B (pi a^(2))/(2) cos omega t`,
From Faraday's law, induced `e.m.f., xi_(I n) = (d Phi)/(dt)`
`= - (d(B pi (a^(2))/(2) cos omega t))/(dt) = (B pi a^(2) omega)/(2) sin omega t`.
and induced current, `i_(i n) = (xi_(i n))/(R) = (B pi a^(2))/(2R) omega sin omega t`.
Now thermal power, generated in the circuit, at the moment `t = t` :
`p(t) = xi_(i n) xx i_(i n) = ((B pi a^(2) omega)/(2))^(2) (1)/(R) sin^(2) omega t`
and mean thermal power generated,
`lt P gt = ([(B pi a^(2) omega)/(2)]^(2) (1)/(R) int_(0)^(T) sin^(2) omega t dt)/(int_(0)^(T) dt) = (1)/(2pi ((pi a^(2) B)/(2))^(2))`