A very long solenoid of radius $R$ is carrying current $I(t) = kte^{-\alpha t}$ ($k > 0$), as a function of time ($t \ge 0$). Counterclockwise current is taken to be positive. A circular conducting coil of radius $2R$ is placed in the equatorial plane of the solenoid and concentric with the solenoid. The current induced in the outer coil is correctly depicted, as a function of time, by
Answer: (B) see figure
The field of a long solenoid is confined inside it, so the flux through the outer coil is $\Phi = \mu_0n\pi R^2I(t)$, proportional to $I$. The induced current is proportional to
$$-\frac{dI}{dt} = -k(1 - \alpha t)e^{-\alpha t}$$
- At $t = 0$ it is $-k$: negative (clockwise) and non-zero.
- It becomes zero at $t = 1/\alpha$, when $I$ is maximum.
- For $t > 1/\alpha$ it is positive, reaches a maximum at $t = 2/\alpha$ and then decays to zero.
This is graph (2).
Solution by Sreeraj P, M.Sc Physics