A coil of area `A` lies in a uniform magnetic field `B` with its plane perpendicular to the field. In this position the normal to the coil makes an angle `0^(@)` with a field. The coil rotates at a uniform rate about its diameter to complete one rotation in time `T` . Find the average induced e.m.f. in the coil during the interval when coil rotates from:
(a) `0^(@)` to `90^(@)`
(b) `90^(@)` to `180^(@)`
(c) `180^(@)` to `270^(@)`
(d) `270^(@)` to `360^(@)`

Explanation

(i) On rotating the coil from `0^(@)` to `90^(@)` .
Initial flux through the coil, `phi_(1) - BA cos 0^(@) = BA`
Flux after rotating it through `90^(@)` , `phi_(2) - BA cos 90^(@) = 0`
Time taken to complete this rotation, `T= (90^(@))(360^(@)) xx t=(t)/(4)`
`:.` Average emf induced
`epsilon=-(N(phi_(2)-phi_(1)))/(t)`
`= (-N(0-BA))/((t)/(4))=(4NBA)/(t)`
(ii) On rotating the coil from `90^(@)` to `180^(@)`,
`phi_(1) = BA cos 90^(@) = 0`
`phi_(2) = BA cos 180^(@) = -BA`, `T = t // 4`
`:.` Average emf induced is given by
`epsilon=(-N(phi_(2)-phi_(1)))/(T)`
`=(N(-BA-0))/((t)/(4))=(4NBA)/(t)`
(iii) On rotating the coil from `180^(@)` to `270^(@)`,
`phi_(1) = BA cos 180^(@) = -BA`
`phi_(2) = BA cos 270^(@) = 0`, `T=t // 4`
`:. epsilon=(-N (phi_(2)-phi_(1)))/(T)`
`=(-N(BA-0))/((t)/(4))=(4NBA)/(t)`
(iv) On rotating the coil from `270^(@)` to `360^(@)`
`phi_(1)=BA cos 270^(@)=0`
`phi_(2)=BA cos360^(@)=BA` , `T=(t)/(4)`
`epsilon=(-N(phi_(2)-phi_(1)))/(T)`
`=(-N(BA-0))/((t)/(4))=(-4NBA)/(t)`