Question 53: AC fundamentals

Question

The circuit shown in the figure contains an inductor $L$, a capacitor $C_0$, a resistor $R_0$ and an ideal battery. The circuit also contains two keys $\mathrm{K}_1$ and $\mathrm{K}_2$. Initially, both the keys are open and there is no charge on the capacitor. At an instant, key $\mathrm{K}_1$ is closed and immediately after this the current in $R_0$ is found to be $I_1$. After a long time, the current attains a steady state value $I_2$. Thereafter, $\mathrm{K}_2$ is closed and simultaneously $\mathrm{K}_1$ is opened and the voltage across $C_0$ oscillates with amplitude $V_0$ and angular frequency $\omega_0$.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

List-I
List-II

(P) The value of $I_1$ in Ampere is
(1) $0$

(Q) The value of $I_2$ in Ampere is
(2) $2$

(R) The value of $\omega_0$ in kilo-radians/s is
(3) $4$

(S) The value of $V_0$ in Volt is
(4) $20$

(5) $200$

Accepted Answer

(P) When $$K_1$$ is closed current in $$R_0$$ is $$I_1$$
At $$\mathrm{t}=0$$; circuit will be
$$\begin{aligned}
& \mathrm{I}_1=0 \
& \mathrm{P} ightarrow(1)
\end{aligned}$$
(Q) After long time inductor behave as a wire so $$I_2$$
$$\begin{aligned}
& \mathrm{I}_2=\frac{20}{5}=4 \mathrm{~A} \
& \mathrm{Q} ightarrow(3)
\end{aligned}$$
$$	ext { (R) When } K_2 	ext { is closed and } K_1 	ext { open }$$
$$\begin{aligned}
& \omega_0=\frac{1}{\sqrt{\mathrm{LC}}} \
& \omega_0=\frac{1}{\sqrt{25 	imes 10^{-3} 	imes 10 	imes 10^{-6}}}=\frac{1}{5 	imes 10^{-4}} \
& \omega_0=2 	imes 10^3 \mathrm{rad} / \mathrm{s} \
& \omega_0=2 \mathrm{kilo}-\mathrm{radian} / \mathrm{s} \
& \mathrm{R} ightarrow(2)
\end{aligned}$$
(S) Now $$K_2$$ is closed and $$K_1$$ open
$$\begin{aligned}
& \frac{1}{2} \mathrm{LI}_2^2=\frac{1}{2} \mathrm{CV}_0^2 \
& 25 	imes 10^{-3} 	imes(4)^2=10 	imes 10^{-6} 	imes \mathrm{V}_0^2 \
& \mathrm{~V}_0^2=2500 	imes 16 \
& \mathrm{~V}_0=50 	imes 4=200 \mathrm{~V} \
& \mathrm{~S} ightarrow(5)
\end{aligned}$$

Answer

$\mathrm{P} \rightarrow 1 ; \mathrm{Q} \rightarrow 3 ; \mathrm{R} \rightarrow 2 ; \mathrm{S} \rightarrow 5$

Answer

$\mathrm{P} \rightarrow 1 ; \mathrm{Q} \rightarrow 2 ; \mathrm{R} \rightarrow 3 ; \mathrm{S} \rightarrow 5$

Answer

$\mathrm{P} \rightarrow 1 ; \mathrm{Q} \rightarrow 3 ; \mathrm{R} \rightarrow 2 ; \mathrm{S} \rightarrow 4$

Answer

$\mathrm{P} \rightarrow 2 ; \mathrm{Q} \rightarrow 5 ; \mathrm{R} \rightarrow 3 ; \mathrm{S} \rightarrow 4$

AC fundamentals

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