At resonance of a series LCR circuit.
A. Phase difference between \( X_L \) and \( X_C \) is 90°
B. Source frequency = \( \displaystyle\frac{1}{2\pi\sqrt{LC}} \)
C. Peak current acquires maximum value
D. Inductive reactance is equal in magnitude to Capacitive reactance.
E. The total impedance of the circuit is maximum
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → B, C, D
A series LCR circuit is in resonance when the inductive and capacitive reactances are equal, \( X_L = X_C \), so that the impedance \( Z = \sqrt{R^{2} + (X_L - X_C)^{2}} \) takes its least value R.
(A) Phase difference between \( X_L \) and \( X_C \) is 90° — Incorrect. The voltage across the inductor leads the current by \( 90^\circ \) and that across the capacitor lags by \( 90^\circ \), so the two reactive effects are \( 180^\circ \) apart and cancel.
(B) Source frequency = \( \displaystyle\frac{1}{2\pi\sqrt{LC}} \) — Correct. \( \omega L = 1/\omega C \) gives \( \omega_0 = 1/\sqrt{LC} \), that is \( f_0 = 1/(2\pi\sqrt{LC}) \).
(C) Peak current acquires maximum value — Correct. With Z at its minimum R, \( I_0 = V_0/R \) is the largest possible.
(D) Inductive reactance is equal in magnitude to Capacitive reactance — Correct. This is the condition for resonance.
(E) The total impedance of the circuit is maximum — Incorrect. The impedance is minimum, equal to R, at resonance.
Shortcut: Statements A and E are both false, and Option (2) is the only option that contains neither of them.
NCERT (2026-27):
Resonance – “… So if ω is varied, then at a particular frequency ω0, Xc = XL, and the impedance is minimum
\( \displaystyle \left(Z = \sqrt{R^2 + 0^2} = R\right) \)
This frequency is called the resonant frequency:
\( \displaystyle X_c = X_L \quad \text{or} \quad \frac{1}{\omega_0 C} = \omega_0 L \quad \text{or} \quad \omega_0 = \frac{1}{\sqrt{LC}} \qquad (7.28) \)
At resonant frequency, the current amplitude is maximum; im = vm/R.”