Question 80 of 276

A very long solenoid of radius $R$ is carrying current $I\left(t\right)=kt{e}^{-\alpha t}\left(k\gt 0\right),$ as a function of time $\left(t\geq 0\right).$ Counterclockwise current is taken to be positive. A circular conducting coil of radius $2R$ is placed in the equitorial 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:

ABCD

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Answer: (B)

${\phi }_{outer}=\left({\mu }_{0}nKt{e}^{-\alpha t}\right)4\pi {R}^{2}$
$\epsilon =\dfrac{-d\phi }{dt}=-{\mu }_{0}nK4\pi {R}^{2}({e}^{-\alpha t}\left[1-\alpha t\right])$
${i}_{induced}=\dfrac{-c{e}^{-\alpha t}\left[1-\alpha t\right]}{Resistance}$
At $t=0,$ ${I}_{induced}=-ve$
From the above equation and initial value of current we can say the correct graph is B.