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  4. A very long solenoid of radius R is carrying current I(t) = kte- α t (k > 0), as a function of time (t le 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 :

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Q. A very long solenoid of radius $R$ is carrying current $I(t) = kte^{- \alpha t} (k > 0)$, as a function of time $(t \le 0)$. 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 :

JEE MainJEE Main 2019Electromagnetic Induction

Solution:

$\phi_{\text{outer}} \left(\mu_{0} nKte^{-\alpha t}\right) 4\pi R^{2}$
$ \varepsilon =\frac{-d\phi}{dt} =-Ce^{-\alpha t} \left[1-\alpha t\right]$
$ i_{\text{induced}} = \frac{-Ce^{-\alpha t} \left[1-at\right]}{\left(\text{Resistance}\right)} $
At $t=0 \,\,\,\,\,i_{\text{induced}} =-ve $