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You are given many resistances, capacitors and inductors. These are connected to a variable DC voltage source (the first two circuits) or an AC voltage source of $50 \mathrm{~Hz}$ frequency (the next three circuits) in different ways as shown in Column II. When a current $I$ (steady state for DC or rms for $A C$ ) flows through the circuit, the corresponding voltage $V_1$ and $V_2$ (indicated in circuits) are related as shown in Column I. Match the two
Options:
Solution:
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Verified Answer
The correct answer is:
(A) R,S,T, (B) Q,R,S,T, (C) P,Q, (D) Q,R,S,T
(A) R,S,T, (B) Q,R,S,T, (C) P,Q, (D) Q,R,S,T
In circuit (P), I can't be non zero in steady state.
In circuit (Q) :
$V_1=0$ and $V_2=2 I=V$ (also)
$$
\begin{aligned}
& \text { In circuit (R): } V_1=X_L I=(2 \pi f \mathrm{~L}) I \\
& \quad=\left(2 \pi \times 50 \times 6 \times 10^{-3}\right) I \\
& \quad=1.88 I \\
& V_2=2 I
\end{aligned}
$$
In circuit (S): $V_1=X_L I=1.88 I$
$$
\begin{array}{r}
V_2=X_C I=\left(\frac{1}{2 \pi f C}\right) I \\
=\left(\frac{1}{2 \pi \times 50 \times 3 \times 10^{-6}}\right) I=(1061) I
\end{array}
$$
In circuit (T):
$$
\begin{aligned}
& V_1=I R=(1000) I \\
& V_2=X_C I=(1061) I
\end{aligned}
$$
Therefore the correct options are as under
(A) $\rightarrow$ R, S, T $\quad$ (B) $\rightarrow$ Q, R, S, T
(C) $\rightarrow$ Q or $P, Q \quad$ (D) $\rightarrow$ Q, R, S, T
In circuit (Q) :
$V_1=0$ and $V_2=2 I=V$ (also)
$$
\begin{aligned}
& \text { In circuit (R): } V_1=X_L I=(2 \pi f \mathrm{~L}) I \\
& \quad=\left(2 \pi \times 50 \times 6 \times 10^{-3}\right) I \\
& \quad=1.88 I \\
& V_2=2 I
\end{aligned}
$$
In circuit (S): $V_1=X_L I=1.88 I$
$$
\begin{array}{r}
V_2=X_C I=\left(\frac{1}{2 \pi f C}\right) I \\
=\left(\frac{1}{2 \pi \times 50 \times 3 \times 10^{-6}}\right) I=(1061) I
\end{array}
$$
In circuit (T):
$$
\begin{aligned}
& V_1=I R=(1000) I \\
& V_2=X_C I=(1061) I
\end{aligned}
$$
Therefore the correct options are as under
(A) $\rightarrow$ R, S, T $\quad$ (B) $\rightarrow$ Q, R, S, T
(C) $\rightarrow$ Q or $P, Q \quad$ (D) $\rightarrow$ Q, R, S, T
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