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 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?

Given:

Uniform magnetic field of induction = B

Radius = r

Frequency = ω

The axis of rotation is perpendicular to the field

Total resistance of the circuit = R

Diameter of the circle = ?

Mean power generated per period of rotation = ?

The flux associated with coil of area A and magnetic induction B is Q = BA cosθ = 1/2 Bπr<sup>2</sup> cosωt

E<sub>induced</sub> = -dQ/dt = -d/dt(1/2 Bπr<sup>2</sup> cosωt) = 1/2Bπr<sup>2</sup>ω sinωt

Power = p = e<sub>induced</sub>/R = B<sup>2</sup>π<sup>2</sup>r<sup>4</sup>ω<sup>2</sup>/4R.1/2 = (Bπr<sup>2</sup>ω)<sup>2</sup>/8R

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