Consider an LC circuit, with inductance $L=0.1 \,\mathrm{H}$ and capacitance $C=10^{-3} \mathrm{~F}$, kept on a plane. The area of the circuit is $1 \mathrm{~m}^{2}$. It is placed in a constant magnetic field of strength $B_{0}$ which is perpendicular to the plane of the circuit. At time $t=0$, the magnetic field strength starts increasing linearly as $B=B_{0}+\beta t$ with $\beta=0.04 \,\mathrm{T\,s}^{-1}$. The maximum magnitude of the current in the circuit is _________ $m A$.

In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance C $$\mu$$F across a 200 V, 50 Hz supply. The power consumed by the lamp is 500 W while the voltage drop across it is 100 V. Assume that there is no inductive load in the circuit. Take rms values of the voltages. The magnitude of the phase-angle (in degrees) between the current and the supply voltage is $$\varphi $$. Assume, $$\pi$$$$\sqrt 3 $$ $$ \approx $$ 5.

The value of C is ____________.

In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance C $$\mu$$F across a 200 V, 50 Hz supply. The power consumed by the lamp is 500 W while the voltage drop across it is 100 V. Assume that there is no inductive load in the circuit. Take rms values of the voltages. The magnitude of the phase-angle (in degrees) between the current and the supply voltage is $$\varphi $$. Assume, $$\pi$$$$\sqrt 3 $$ $$ \approx $$ 5.

The value of $$\varphi$$ is ____________.

The inductors of two LR circuits are placed next to each other, as shown in the figure. The values of the self-inductance of the inductors, resistors, mutual-inductance and applied voltages are specified in the given circuit. After both the switches are closed simultaneously, the total work done by the batteries against the induced EMF in the inductors by the time the currents reach their steady state values is _________ mJ.