AC Power & Power FactorJEE Main

Real, reactive and apparent power — interactive Physics simulation for IIT-JEE.

Concept

In AC circuits voltage and current can be out of step by phase φ. Only the in-phase part delivers average power: $P = V_{rms}I_{rms}\cos\phi$ . The factor $\cos\phi$ is the power factor; the out-of-phase component shuttles energy back and forth uselessly ('wattless current') while still heating the wires.

Key formula

P = V_{rms}I_{rms}\cos\phi, \qquad S = V_{rms}I_{rms}, \qquad Q = S\sin\phi, \qquad V_{rms} = \frac{V_0}{\sqrt2}

Derivation

Instantaneous power $p = V_0\sin\omega t \cdot I_0\sin(\omega t - \phi)$ . Product-to-sum:

$ $p = \tfrac12 V_0I_0[\cos\phi - \cos(2\omega t - \phi)]$ $

The oscillating term averages to zero over a cycle, leaving $\bar p = \tfrac12 V_0I_0\cos\phi = V_{rms}I_{rms}\cos\phi$ . Pure L or C: φ = ±90°, average power ZERO.

Scenarios to explore

AC Power & Power Factor — Real, reactive & apparent power — P = VI·cos φ.
Purely resistive load — Phase 0: power factor 1, and all the apparent power is real power.
Wattless current — At 90° the circuit draws current but uses no average power.

Real-world applications

Utilities bill industry for poor power factor — capacitor banks correct it.
Transformer & generator ratings in VA, not W (they must carry S).
RMS is what multimeters read and what heats resistors.

JEE exam tips

Choke coil: limits AC current with (ideally) zero power loss — that's why fans hum, not heat.
At LCR resonance φ = 0, PF = 1, current & power both maximum.
Wattless component = I sin φ; power component = I cos φ.

Common mistakes

Using peak values in power formulas without the ½.
P = VI in AC without cos φ.
Thinking an ideal inductor/capacitor consumes power — average is zero.

Exam traps to avoid

230 V mains is RMS — the peak is 325 V.
cos φ = R/Z in a series LCR circuit — connects this to impedance triangles.