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Q.
If current in inductor of inductance $5 mH$ is varying as $I = t ^{2} e ^{-2} t$ then find time after which voltage drop across inductor becomes becomes zero.
Relation between voltage and current in inductance circuit is given by
$V_{0}\quad L \frac{dI}{dl}\,Now\,I = t^{2}\,e^{-2t}$
$V_{0}=-L \frac{d^{\left(t^2e^{-2t}\right)}}{dl}$
According to question
$0\quad L \frac{d}{dt}^{\left(t^2e^{\,2t}\right)}\,=-\left(2te^{-2t}+\left(-2\right)e^{-2t}t^{2}\right)$
$=2e^{-2t}\,t\left(1-t\right)=0\,\Rightarrow \,t = 1s$