Consider a series LCR circuit connected to an ac source of angular frequency \( \omega \) and identify the correct statements among the following:
(A) At resonance the circuit is purely resistive.
(B) When \( \omega^{2} \gt 1/(LC) \), the circuit is capacitive.
(C) When \( \omega^{2} \lt 1/(LC) \), the circuit is inductive.
(D) When \( \omega^{2} \gt 1/(LC) \), the circuit is inductive.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (D) only
The nature of a series LCR circuit is decided by comparing the inductive reactance \( X_L = \omega L \) with the capacitive reactance \( X_C = 1/\omega C \):
\( \displaystyle \omega^{2} \gt \frac{1}{LC} \;\Rightarrow\; \omega L \gt \frac{1}{\omega C} \;\Rightarrow\; X_L \gt X_C \)
(A) At resonance the circuit is purely resistive. — Correct. At resonance \( X_L = X_C \), the two reactances cancel, the impedance is just R and the current is in phase with the voltage.
(B) When \( \omega^{2} \gt 1/(LC) \), the circuit is capacitive. — Incorrect. Above resonance \( X_L \gt X_C \), so the net reactance is inductive, not capacitive.
(C) When \( \omega^{2} \lt 1/(LC) \), the circuit is inductive. — Incorrect. Below resonance \( X_C \gt X_L \), so the circuit is capacitive and the current leads the voltage.
(D) When \( \omega^{2} \gt 1/(LC) \), the circuit is inductive. — Correct. Above resonance \( X_L \gt X_C \), so the current lags the voltage.
NCERT (2026-27):
Phasor-Diagram Solution – “… If \( X_C \gt X_L \), \( \phi \) is positive and the circuit is predominantly capacitive. Consequently, the current in the circuit leads the source voltage. If \( X_C \lt X_L \), \( \phi \) is negative and the circuit is predominantly inductive. Consequently, the current in the circuit lags the source voltage.”
Resonance – “… So if \( \omega \) is varied, then at a particular frequency \( \omega_0 \), \( X_c = X_L \), and the impedance is minimum
\( \displaystyle Z = \sqrt{R^2 + 0^2} = R \)
This frequency is called the resonant frequency:
\( \displaystyle X_c = X_L \;\text{or}\; \frac{1}{\omega_0 C} = \omega_0 L \;\text{or}\; \omega_0 = \frac{1}{\sqrt{LC}} \qquad (7.28) \)”