When the switch $S$ is closed at time $t = 0$,what will be the nature of the graphs for the induced $emf$ $e$ across the inductor $L$ and the current $i$ in the circuit?

In the circuit shown,an inductor $L$ and a capacitor $C$ are connected as shown. $A_1$ and $A_2$ are ammeters. When the key $K$ is pressed to complete the circuit,what will be the readings of $A_1$ and $A_2$ just after closing the key $K$?

An induction coil stores $32 \ J$ of magnetic energy and dissipates energy as heat at the rate of $320 \ W$ when a current of $4 \ A$ is passed through it. Find the time constant of the circuit when the coil is joined across a battery.

An inductor coil stores $64 \, J$ of magnetic field energy and dissipates energy at the rate of $640 \, W$ when a current of $8 \, A$ is passed through it. If this coil is joined across an ideal battery,find the time constant of the circuit in seconds.

In the figure below,the switches $S_1$ and $S_2$ are closed simultaneously at $t=0$ and a current starts to flow in the circuit. Both the batteries have the same magnitude of the electromotive force (emf) $V$ and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current $I$ in the middle wire reaches its maximum magnitude $I_{\max}$ at time $t=T$. Which of the following statements is (are) true?
$(A)$ $I_{\max}=\frac{V}{2R}$
$(B)$ $I_{\max}=\frac{V}{4R}$
$(C)$ $T=\frac{L}{R} \ln 2$
$(D)$ $T=\frac{2L}{R} \ln 2$

In an $L-R$ decay circuit,the initial current at $t = 0$ is $I$. The total charge that has flown through the resistor until the energy in the inductor has reduced to one-fourth of its initial value is: