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In an inductor of self-inductance \(\mathrm{L}=2 \mathrm{mH}\), current changes with time according to relation \(\mathrm{i}=\mathrm{t}^2 \mathrm{e}^{-\mathrm{t}}\). At what time emf is zero?
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The correct answer is:
\(2 \mathrm{~s}\)
\(\mathrm{L}=2 \mathrm{mH}, \mathrm{i}=\mathrm{t}^2 \mathrm{e}^{-\mathrm{t}}\)
\(\mathrm{E}=-\mathrm{L} \frac{\mathrm{di}}{\mathrm{dt}}=-\mathrm{L}\left[-\mathrm{t}^2 \mathrm{e}^{-\mathrm{t}}+2 \mathrm{te} \mathrm{e}^{-\mathrm{t}}\right]\)
when \(\mathrm{E}=0\),
\(\begin{aligned}
& -\mathrm{e}^{-\mathrm{t}} \mathrm{t}^2+2 \mathrm{te}-\mathrm{t}=0 \\
& \text {or, } 2 \mathrm{t} \mathrm{e}^{-\mathrm{t}}=\mathrm{e}^{-\mathrm{t}} \mathrm{t}^2 \\
& \Rightarrow \mathrm{t}=2 \mathrm{sec}.
\end{aligned}\)
\(\mathrm{E}=-\mathrm{L} \frac{\mathrm{di}}{\mathrm{dt}}=-\mathrm{L}\left[-\mathrm{t}^2 \mathrm{e}^{-\mathrm{t}}+2 \mathrm{te} \mathrm{e}^{-\mathrm{t}}\right]\)
when \(\mathrm{E}=0\),
\(\begin{aligned}
& -\mathrm{e}^{-\mathrm{t}} \mathrm{t}^2+2 \mathrm{te}-\mathrm{t}=0 \\
& \text {or, } 2 \mathrm{t} \mathrm{e}^{-\mathrm{t}}=\mathrm{e}^{-\mathrm{t}} \mathrm{t}^2 \\
& \Rightarrow \mathrm{t}=2 \mathrm{sec}.
\end{aligned}\)
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