Which of the following statements are correct?
(A) In ac circuit containing only resistance, power consumption is \( I^{2}R \)
(B) In an LCR circuit, there is resonance at zero phase difference between \( X_C \) and \( X_L \) and the power consumption is \( I(\omega X_C + \omega X_L)^{1/2} \)
(C) In an LCR circuit, at resonance, impedance is minimum.
(D) In an LCR circuit, there is resonance at zero phase difference between \( X_C \) and \( X_L \) and the power consumption is zero.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
The average power in an a.c. circuit is \( P = V_{\mathrm{rms}} I_{\mathrm{rms}} \cos\phi \); only the resistance consumes power. At resonance \( X_L = X_C \), the voltage and current are in phase and the impedance is \( Z = R \).
(A) In ac circuit containing only resistance, power consumption is \( I^{2}R \) — Correct. For a pure resistor \( \phi = 0 \), so \( P = I^{2}R \) with \( I \) the rms current.
(B) In an LCR circuit, there is resonance at zero phase difference between \( X_C \) and \( X_L \) and the power consumption is \( I(\omega X_C + \omega X_L)^{1/2} \) — Incorrect. At resonance the power is \( I^{2}R \); the expression given does not even have the units of power.
(C) In an LCR circuit, at resonance, impedance is minimum. — Correct. \( Z = \sqrt{R^{2} + (X_L - X_C)^{2}} \) is least when \( X_L = X_C \), where \( Z = R \).
(D) In an LCR circuit, there is resonance at zero phase difference between \( X_C \) and \( X_L \) and the power consumption is zero. — Incorrect. At resonance the current is maximum and \( \cos\phi = 1 \), so the power consumed is the largest, \( I^{2}R \), not zero.
NCERT (2026-27):
Resonance – “… So if ω is varied, then at a particular frequency ω0, Xc = XL, and the impedance is minimum
\( \displaystyle Z = \sqrt{R^2 + 0^2} = R \)”
Power in AC Circuit: The Power Factor – “… Case (i) Resistive circuit: If the circuit contains only pure R, it is called resistive. In that case φ = 0, cos φ = 1. There is maximum power dissipation.”
Power in AC Circuit: The Power Factor – “… Case (iv) Power dissipated at resonance in LCR circuit: At resonance Xc – XL = 0, and φ = 0. Therefore, cosφ = 1 and P = I2Z = I2R. That is, maximum power is dissipated in a circuit (through R) at resonance.”