A stiff wire bent into a semicircle of radius a is rotated with a frequency in a uniform magnetic field, as suggested in the figure. We have to find (a) the frequency and (b) the amplitude of the emf induced in the loop.

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(a) and (b)

Let us assume that at an instant the plane of the loop makes an angle with respect to the vertical. The unit vector perpendicular to the plane of the loop at that instant will be

As the magnetic field in which the semicircular loop of radius a is revolving is

the flux enclosed by the loop will be

By Faraday’s law the emf due to induction will be given by

As

,

The induced emf will therefore be given by the expression

The frequency of the induced emf will therefore be and its amplitude will be .