Question

A very long solenoid of radius $R$ is carrying current $I\left(t\right)=kt{e}^{-\alpha t}(k>0),$ 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-

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

Flux linked with the coil is,

$\varphi =BA=\left({\mu }_{0}nI\right)A={\mu }_{0}n\left(kt{e}^{-\alpha t}\right)A$

$\because e=-\frac{d\varphi }{dt}=-{\mu }_{0}nAk\frac{d}{dt}\left(t{e}^{-\alpha t}\right)$

$=-{\mu }_{0}nAk\left[t{e}^{-\alpha t}\right(-\alpha )+e^{-\alpha t}\times 1]$

$=-{\mu }_{0}nAk\left[e^{-\alpha t}\right(1-\alpha t\left)\right]$

$i_{ind}=\frac{e}{\text{Resistance}}=\frac{-{\mu }_{0}nAk}{\text{Resistance}}\left[e^{-\alpha t}\right(1-\alpha t\left)\right]$

At $t=0,i\Rightarrow -ve$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(A\right)$ is the correct answer.