Q. A copper rod is bent into a semi-circle of radius a and at ends straight parts are bent along diameter of the semi circle and are passed through fixed, smooth, and conducting ring O and O’ as shown in figure. A capacitor having capacitance C is connected to the rings. The system is located in a uniform magnetic field of induction B such that axis of rotation OO’ is perpendicular to the field direction. At initial moment of time (t = 0), plane of semi-circle is set in rotation with constant angular velocity ω .Neglect the resistance and inductance of the circuit. The current flowing through the circuit as function of time is

(a) $\displaystyle \frac{1}{4} \pi\omega^2 a^2 CB cos\omega t $

(b) $ \displaystyle \frac{1}{2} \pi\omega^2 a^2 CB cos\omega t $

(c) $ \displaystyle \frac{1}{4} \pi\omega^2 a^2 CB sin\omega t $

(d) $ \displaystyle \frac{1}{4} \pi\omega^2 a^2 CB sin\omega t $

Ans: (b)

Sol: Flux $ \displaystyle \phi = B \frac{\pi a^2}{2} cos\omega t + BA $

$\displaystyle V = \frac{-d\phi}{dt}$

As , q = CV and I = dq/dt