The magnetic field B shown in is directed into the plane of the paper. ACDA is a semicircular conducting loop of radius r with the centre at O. The loop is now made ot rotate clockwise with a constant angular velocity `omega` about on axis passing through O and perpendicular to the plane of the paper. The resistance of the loop is R. Obtain an expression for the magnitude of the induced current in the loop. Plot a graph between the induced current i and `omegat`, for two periods of rotation.

Explanation

When the loop rotates through an angle `theta`, which is less then `pi, the area inside the field region is `A=(theta)/(pi) (pir^2)/(2) = (theta r^2)(2) = (omega t r^2)/(2)
The flux of the magnetic field at time t is `Phi = BA =B (omegatr^2)(2)
The induced emf` =- (d Phi)/(dt) =-(B omegar^2)/(2)`
The magnitude of the induced current will be `i = (B omegar^2)/(2R)`
As the flux is increasing, the direction of the induced current will be anticlockwise so that the field due to the induced current is opposite to the original field
Afte half a rotaion, the area in the field region will start decreasing and will be given by `A(t) = (pir^2)/(2) - (omegat r^2)/(2)`
Hence, the induced current will have the same magnitude but opposite sense
The plot for two time periods is shown n figure.
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(##HCV_VOL2_C38_S_013_S01##)