A conducting loop of finite resistance lies on the x-y plane. There is a constant magnetic field in the z direction. The area of the loop varies with time t, as A = A₀(1+sin t) in appropriate units. The figure that correctly indicates the qualitative behaviour of the power P dissipated in the loop as a function of time is: (a) (b) (c) (d)

Answer: (c). Flux φ = BA = BA₀(1+sin t), so e = -(dφ)/(dt) = -BA₀cos t. Power P = e²/R = (B²A₀²)/(R)cos ² t. At t=0, P is maximum; it falls to zero at t=π/2 and rises again (a cos ² t curve). So the answer is (c).

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Magnetic Flux & Faraday's Law

A conducting loop of finite resistance lies on the $x-y$ plane. There is a constant magnetic field in the $z$ direction. The area of the loop varies with time $t$ , as $A = A_0(1+\sin t)$ in appropriate units. The figure that correctly indicates the qualitative behaviour of the power $P$ dissipated in the loop as a function of time is:

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Answer: (c)

Flux $\phi = BA = BA_0(1+\sin t)$ , so $e = -\dfrac{d\phi}{dt} = -BA_0\cos t$ .
Power $P = \dfrac{e^2}{R} = \dfrac{B^2A_0^2}{R}\cos^2 t$ .
At $t=0$ , $P$ is maximum; it falls to zero at $t=\pi/2$ and rises again (a $\cos^2 t$ curve).
So the answer is (c).

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