In AC circuits, the relation that holds is \( Z = \displaystyle\frac{E_v}{I_v} \) where.
A. \( Z = \sqrt{R^{2} + X_L^{2}} \) in case of a.c. circuit containing R and L in series.
B. \( Z = \sqrt{R^{2} + X_C^{2}} \) in case of a.c. circuit containing R and C in series.
C. \( Z = X_L - X_C \) in case of a.c. circuit containing L and C in series.
D. \( Z = \sqrt{R^{2} + (X_L - X_C)^{2}} \) in case of ac circuit containing R, L and C in series
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A, B, C, D only
Note: The given answer is as per the official NTA Answer Key. However, statement (C) is true only when \( X_L \gt X_C \): for L and C in series the impedance is the magnitude \( |X_L - X_C| \). Strictly, the correct answer would be Option (3) A, B and D only, as explained below.
In a series a.c. circuit the voltage across R is in phase with the current, across L it leads by \( 90^\circ \) and across C it lags by \( 90^\circ \). The impedance is therefore the square root of the sum of the squares of the resistance and the net reactance, \( Z = E_v/I_v \).
(A) \( Z = \sqrt{R^{2} + X_L^{2}} \) in case of a.c. circuit containing R and L in series. — Correct. Only \( X_L \) is present.
(B) \( Z = \sqrt{R^{2} + X_C^{2}} \) in case of a.c. circuit containing R and C in series. — Correct. Only \( X_C \) is present.
(C) \( Z = X_L - X_C \) in case of a.c. circuit containing L and C in series. — Correct as per NTA; however, it is not correct as a general statement. With \( R = 0 \), \( Z = \sqrt{(X_L - X_C)^{2}} = |X_L - X_C| \), which equals \( X_L - X_C \) only when \( X_L \gt X_C \); for \( X_C \gt X_L \) it is \( X_C - X_L \).
(D) \( Z = \sqrt{R^{2} + (X_L - X_C)^{2}} \) in case of ac circuit containing R, L and C in series — Correct. This is the general result.
So strictly (A), (B) and (D) are correct, which is Option (3). It seems that NTA has assumed that \( X_L \gt X_C \) in statement (C).
NCERT (2026-27):
Phasor-Diagram Solution – “… By analogy to the resistance in a circuit, we introduce the impedance Z in an ac circuit:
\( \displaystyle i_m = \frac{v_m}{Z} \qquad [7.25(b)] \)
where
\( \displaystyle Z = \sqrt{R^2 + (X_C - X_L)^2} \qquad (7.26) \)”