JEE PYQ: Electromagnetic Induction Question 17
Question 17 - 2019 (09 Apr Shift 2)
A very long solenoid of radius $R$ is carrying current $I(t) = kte^{-\alpha t}$ ($k > 0$), as a function of time ($t \ge 0$). Counter clockwise 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:
(1) Graph showing current starting negative, rising to positive peak, then decaying
(2) Graph showing current starting positive, then decaying with a sign change
(3) Graph showing current starting negative, crossing zero, positive peak, then oscillating
(4) Graph showing current starting at zero, rising then decaying
Show Answer
Answer: (1)
Solution
$\phi = BA = (\mu_0 n I)A = \mu_0 n A k t e^{-\alpha t}$. $e = -\frac{d\phi}{dt} = -\mu_0 n A k \frac{d}{dt}(te^{-\alpha t}) = -\mu_0 n A k[e^{-\alpha t}(1-\alpha t)]$. $i \propto -e^{-\alpha t}(1-\alpha t)$. At $t = 0$, $i$ is negative. The current changes sign at $t = 1/\alpha$. This matches graph (1).
BETA
