Import Question JSON

$\text{Hint: } \varepsilon = -\frac{d\phi}{dt}$ $\text{Step: Identify the graph between the current induced in the outer coil as a function of time.}$ $\text{The current in the solenoid is given by:}$ $I(t) = kte^{-\alpha t}(k > 0),$ $\text{The magnetic field } B \text{ inside a long solenoid carrying current } I \text{ is given by:}$ $\Rightarrow B = \mu_0 n I$ $\text{Since the solenoid is long and the current varies with time, we substitute } I(t);$ $B(t) = \mu_0 n (kte^{-\alpha t})$ $\text{The induced emf is defined by:}$ $\varepsilon = \frac{d\phi}{dt} = 4\pi \mu_0 n R^2 \left( ke^{-\alpha t} - kate^{-\alpha t} \right)$ $= 4\pi \mu_0 n R^2 ke^{-\alpha t} (1 - \alpha t)$ $\Rightarrow \varepsilon = -4\pi \mu_0 n R^2 ke^{-\alpha t} (1 - \alpha t)$ $\text{The induced current is given by:}$ $I_{\text{ind}} = \frac{\varepsilon}{R_c} = -\frac{4\pi \mu_0 n R^2 ke^{-\alpha t} (1 - \alpha t)}{R_c}$ $\text{At } t = 0 \text{ the induced current is given by:}$ $I_{\text{ind}}(0) = -\frac{4\pi \mu_0 n R^2 k}{R_c}$ $\text{At } t = \frac{1}{\alpha} \text{ (where } 1 - \alpha t = 0) \text{ the induced current is given by:}$ $I_{\text{ind}}\left( \frac{1}{\alpha} \right) = 0$ $\text{As } t \rightarrow \infty \text{ the induced current is given by:}$ $\Rightarrow I_{\text{ind}} \rightarrow 0$ $\text{The induced current starts at a negative value becomes zero at } t = \frac{1}{\alpha}, \text{ and approaches zero as } t \text{ increases.}$ $\text{The correct depiction of the induced current as a function of time is represented by a curve that starts negative, crosses zero, and approaches zero again.}$ $\text{Hence, option (2) is the correct answer.}$