In a series RLC circuit at resonance
(A) Impedance is maximum
(B) \( X_C = X_L \)
(C) \( Z = R \)
(D) \( \omega = 1/\sqrt{LC} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B), (C) and (D) only
The impedance of a series RLC circuit is
\( \displaystyle Z = \sqrt{R^{2} + (X_L - X_C)^{2}} \)
Resonance occurs when the inductive and capacitive reactances are equal, \( X_L = X_C \), so the reactive term vanishes.
(A) Impedance is maximum. — Incorrect. At resonance the impedance is the smallest it can be, \( Z = R \), and the current is the largest.
(B) \( X_C = X_L \) — Correct. This is the condition for resonance.
(C) \( Z = R \) — Correct. With \( X_L - X_C = 0 \), the formula above gives \( Z = R \).
(D) \( \omega = 1/\sqrt{LC} \) — Correct. Putting \( X_L = X_C \) gives \( \omega L = 1/(\omega C) \), hence \( \omega^{2} = 1/(LC) \).
Statements (B), (C) and (D) are correct, which is Option (4).
NCERT (2026-27):
Resonance – “… So if ω is varied, then at a particular frequency ω0, Xc = XL, and the impedance is minimum \( \left(Z = \sqrt{R^2 + 0^2} = R\right) \). This frequency is called the resonant frequency:
\( \displaystyle X_c = X_L \text{ or } \frac{1}{\omega_0 C} = \omega_0 L \text{ or } \omega_0 = \frac{1}{\sqrt{LC}} \qquad (7.28) \)”