13. For a coil having $L = 2 mH$, current flow through it is $I = t^2 e^{-t}$ then the time at which emf become zero: (2001)
Solution:
Induced emf is $e = -L \frac{dI}{dt}$. For $e = 0$, we must have $\frac{dI}{dt} = 0$.
Differentiating $I$: $\frac{dI}{dt} = \frac{d}{dt}(t^2 e^{-t}) = 2t e^{-t} - t^2 e^{-t} = t e^{-t}(2 - t)$.
Setting $\frac{dI}{dt} = 0$ gives $t = 2 s$ (for $t > 0$).
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