Two coils have a mutual inductance \(5 \times 10^{-3} \mathrm{H}\). The current changes in the first coil according to the equation \(\mathrm{I}_1=\mathrm{I}_0 \sin \omega \mathrm{t}\) where \(\mathrm{I}_0=10 \mathrm{~A}\) and \(\omega=100 \pi \mathrm{rad} / \mathrm{s}\). What is the value of the maximum e.m.f. in the coil?

\(\mathrm{e}=\frac{\mathrm{MdI}}{\mathrm{dt}}...(i)\)
\(\mathrm{e}=\mathrm{M} \frac{\mathrm{~d}}{\mathrm{dt}}\left(\mathrm{I}_0 \sin \omega \mathrm{t}\right)\)
Now, \(\frac{\mathrm{d}}{\mathrm{dt}}\left(\mathrm{I}_0 \sin \omega \mathrm{t}\right)=\mathrm{I}_0 \omega \cos \omega \mathrm{t}\)
For maximum value of emf, \(\frac{\mathrm{dl}}{\mathrm{dt}}\) is maximum
\(\begin{array}{ll}
& \Rightarrow \cos \omega \mathrm{t}=1 \\
\therefore \quad & \frac{\mathrm{dI}}{\mathrm{dt}}=\mathrm{I}_0 \omega ...(ii)\\
\therefore \quad & \mathrm{e}=0.005 \times 10 \times 100 \pi=5 \pi \mathrm{~V}
\end{array}\)
...(from (i) and (ii))