AC RMS Value and Power
On this page
Direct answer
The rms (root mean square) value of an alternating current is the steady direct current that produces the same heat in a resistor over one full cycle: for a sine wave I = I0 sin ωt, I_rms = I0/√2 ≈ 0.707 I0, and likewise V_rms = V0/√2. Average power delivered to a circuit is P = V_rms × I_rms × cos φ, where cos φ is the power factor; only a pure resistor consumes the full product V_rms I_rms (φ = 0). NCERT treats the Indian mains of 220 V, 50 Hz as an rms figure — the peak is 220√2 ≈ 311 V. Over a complete cycle the signed average of a sinusoidal current is zero, which is exactly why AC instruments are built to read rms instead.
What you must remember
- RMS definition and value: I_rms = I0/√2 = 0.707 I0 for sinusoidal AC, from ⟨I²⟩ = I0²/2 over a cycle; heat produced H = I_rms² R t.
- Mean values: average current over a full cycle = 0; average of the rectified wave |I| = 2I0/π ≈ 0.637 I0 — NEET has tested both numbers in one question.
- Power formula: P = V_rms I_rms cos φ; pure resistor: cos φ = 1, P = V_rms I_rms; pure inductor or capacitor: cos φ = 0, current flows but no average power is drawn (wattless current).
- Mains convention: Indian domestic supply 220 V, 50 Hz (NCERT Chapter 7); V0 = 311 V; equipment ratings quote rms, never peak.
- NCERT's benchmark example: a 100 W bulb rated at 220 V has R = 484 Ω and draws I_rms = 100/220 ≈ 0.45 A.
- Instrument logic: an ordinary moving-coil galvanometer reads zero for AC (average torque cancels); hot-wire ammeters work on I²R heating and read rms directly.
How to run an AC power numerical
Take a heater of resistance 20 Ω on 220 V, 50 Hz mains. Work in rms throughout: I_rms = 220/20 = 11 A, and P = V_rms I_rms = 2420 W, cross-checked as I_rms²R = 121 × 20 = 2420 W. Heat in ten minutes: H = Pt = 2420 × 600 = 1.45 × 10^6 J, or in board units, 2420 W × (1/6) h ≈ 0.4 kWh. If instead the question hands you the peak current I0 = 11 A, the power becomes ½ I0² R = 1210 W — exactly half. That factor of two, between squaring before or after dividing by √2, is the most reliably profitable slip in the whole chapter, and the half-value answer always sits among the options.
Where NEET sets the trap
The paper rarely asks for a definition; it asks for discrimination. A stem gives the peak current and asks which direct current produces the same heating — the options carry 0.5 I0, 0.637 I0 and 0.707 I0, so a student who memorised numbers without their meanings picks the rectified average. The second construction is the phase angle: with a series LCR circuit the power factor cos φ = R/Z, and questions ask how power changes when only L or C is varied — the answer tracks R/Z, not the current alone. Third, the assertion-reason favourite: an ideal inductor draws current yet consumes no average power over a cycle. NCERT states this in one line, and it converts directly into a marks-earning statement.
Frequently asked questions
What is the rms value of an alternating current?
It is the direct current that produces the same heat in the same resistor over one full cycle; for a sinusoidal wave I_rms = I0/√2 ≈ 0.707 I0.
Why is the average of an AC over a full cycle zero?
A sinusoidal current is positive for half the cycle and negative for the other half, so the signed average cancels; heating depends on I², which never goes negative, hence the rms convention.
What is the power factor of a purely inductive AC circuit?
Zero: the current lags the voltage by 90°, cos 90° = 0, so the circuit draws no average power over a cycle — the wattless current of NCERT's chapter.
A bulb is rated 100 W at 220 V AC. What current does it draw?
I_rms = P/V_rms = 100/220 ≈ 0.45 A; both the rating and the mains voltage are rms values, so no √2 conversion is needed anywhere.
Which instrument measures alternating current correctly?
A hot-wire instrument, because its deflection depends on I²R heating and therefore reads rms directly; a plain moving-coil meter averages to zero.