Physics - EMI/AC Question with Solution | TestHub

Figure shows a purely inductive circuit, in which a sinusoidal emf is applied. The induced emf across the inductor is - $L\frac{di}{dt}$ so that from Kirchhoff's law, $$\begin{array}{rl}& {\epsilon }_0\sin \omega t-L\frac{di}{dt}=0\\ & \Rightarrow \frac{di}{dt}=\frac{{\epsilon }_0}{L}\sin \omega t\\ & \text{ or }i=-\frac{{\epsilon }_0}{\cos }\cos \omega t+C\end{array}$$ The constant $C$ can be shown to be zero from the fact that, the emf being sinusoidal, the current should also be so, and the average current over one time period has to be zero. $$\therefore i=-\frac{{\epsilon }_0}{\omega L}\cos \omega \mathrm{t}\text{ or }i=\frac{{\epsilon }_0}{\omega L}\sin (\omega \mathrm{t}-\pi /2)$$ which shows that the current lags behind the emf by a phase of $\pi /2$