The rms current in a series LCR circuit is \(I\). The average power dissipated in the circuit is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(I^{2}R\)
The average power in an ac circuit can be written as \( P=I^{2}Z\cos\phi \). For a series LCR circuit, the power factor is \( \cos\phi=R/Z \).
\( \begin{aligned} P &= I^{2}Z\cos\phi \\ &= I^{2}Z\times\frac{R}{Z} \\ &= I^{2}R \end{aligned} \)
Option (1) \( I^{2}Z \) — Incorrect. The power factor is missing here. \( I^{2}Z \) gives the power only when \( \cos\phi=1 \), that is, at resonance.
Option (2) \( I^{2}X_L \) — Incorrect. An inductor takes no average power over a complete cycle. So \( X_L \) does not appear in the power.
Option (3) \( I^{2}(X_L-X_C) \) — Incorrect. The inductor and the capacitor store energy and give it back. Their reactances add nothing to the average power.
Option (4) \( I^{2}R \) — Correct. Only the resistor uses up power in the circuit. So the average power is \( I^{2}R \).
NCERT (2026-27):
Power in AC Circuit: the Power Factor – “This can also be written as,”
\( \displaystyle P=I^{2}Z\cos\phi \)\( \qquad [7.30(b)] \)
Example 7.7 – “(b) Suppose in a circuit, current I lags the voltage by an angle φ. Then power factor cosφ =R/Z.”
Power in AC Circuit: the Power Factor – “… So, φ may be non-zero in a RL or RC or RCL circuit. Even in such cases, power is dissipated only in the resistor.”