In an $LCR$ series circuit, an inductor $30 \mathrm{mH}$ and a resistor $1 \mathrm{\Omega }$ are connected to an $\mathrm{AC}$ source of angular frequency $300 \mathrm{rad} {\mathrm{s}}^{-1}.$ The value of capacitance for which the current leads the voltage by $45°$ is $\frac{1}{x}\times {10}^{-3} \mathrm{F}$. Then the value of $x$ is

In a circuit consisting of a capacitance and a generator with alternating emf, $E_g=E_{go}sin\omega t, V_C$ and $I_{\mathit{C}}$ are the voltage and current. Correct phasor diagram for such circuit is

Match List-I with List-II.

Choose the correct answer from the options given below.

A circular conducting coil of radius $1 \mathrm{m}$ is being heated by the change of magnetic field $\overrightarrow{B}$ passing perpendicular to the plane in which the coil is laid. The resistance of the coil is $2 \mathrm{\mu \Omega }$. The magnetic field is slowly switched off such that its magnitude changes in time as  

$B=\frac{4}{\pi }\times {10}^{-3} T\left(1-\frac{t}{100}\right)$

The energy dissipated by the coil before the magnetic field is switched off completely is $E=\_\_\_\_\_\_\mathrm{mJ}$.

Two circuits are shown in figure $\left(a\right)$ and $\left(b\right)$. At a frequency of _______ $\mathrm{rad} {\mathrm{s}}^{-1}$ the average power dissipated in one cycle will be the same in both the circuits.