A circuit element $X$ when connected to an a.c. supply of peak voltage $100\,V$ gives a peak current of $5\,A$ which is in phase with the voltage. A second element $Y$ when connected to the same a.c. supply also gives the same value of peak current which lags behind the voltage by $\frac{\pi}{2}$. If $X$ and $Y$ are connected in series to the same supply, what will be the rms value of the current in ampere?

A $100 \;\mu \,F$ capacitor in series with a $40\; \Omega$ resistance is connected to a $110\; V , 12\; k\,Hz$ supply.

$(a)$ What is the maximum current in the circuit?

$(b)$ What is the time lag between the current maximum and the voltage maximum?

Hence, explain the statement that a capacitor is a conductor at very high frequencies. Compare this behaviour with that of a capacitor in a $dc$ circuit after the steady state.

At time $t =0$, terminal $A$ in the circuit shown in the figure is connected to $B$ by a key and alternating current $I ( t )= I _0 \cos (\omega t$,$) , with I _0=1 A$ and $\omega=500 \ rad s ^{-1}$ starts flowing in it with the initial direction shown in the figure.

At $t=\frac{7 \pi}{6 \omega}$, the key is switched from $B$ to $D$. Now onwards on ly $A$ and $D$ are connected. $A$ total charge $Q$ flows from the battery to charge the capacitor fully. If $C =20 \mu, R =10 \Omega$ and the battery is ideal with emf of $50 \ V$, identify the correct statement $(s)$

$(A)$ Magnitude of the maximum charge on the capacitor before $t=\frac{7 \pi}{6 \omega}$ is $1 \times 10^{-3} C$.

$(B)$ The current in the left part of the circuit just before $t=\frac{7 \pi}{6 \omega}$ is clockwise

$(C)$ Immediately after $A$ is connected to $D$. the current in $R$ is $10 A$.

$(D)$ $Q =2 \times 10^{-3} C$.

An inductance coil has a reactance of $100\, \Omega$. When an $AC$ signal of frequency $1000\, Hz$ is applied to the coil, the applied voltage leads the current by $45^{\circ}$. The self- inductance of the coil is

A resistance of $400\,\Omega$ is connected in series with an inductor of $(3 / \pi)$ $H$ and this combination is connected across an $A.C.$ source as shown in figure.