Question
Our 230V household supply is labelled by its RMS value. But the actual peak voltage is around 325V — nearly 100V higher. Why do we use RMS instead of average or peak? And why does V_rms = V₀/√2?
Solution — Step by Step
AC voltage varies as V = V₀ sin(ωt) — it swings between +V₀ and -V₀ symmetrically. If we took a simple average over a full cycle, positive and negative halves cancel out perfectly, giving zero. That’s useless for comparing heating effect.
The power dissipated in a resistor is P = V²/R. Notice the square — both positive and negative voltages produce positive heating. So we average V² over one full cycle, not V itself. This is the mean square value.
Using the identity sin²(ωt) = (1 - cos 2ωt)/2, the cosine term averages to zero over a full cycle. We’re left with:
RMS = Root of the Mean of the Square. So:
For household supply, V_rms = 230V, so V₀ = 230 × √2 ≈ 325V. That’s your peak voltage.
The RMS value of AC produces the same heating effect as an equal DC voltage. A 230V AC supply heats a resistor exactly as much as a 230V DC battery would.
Why This Works
The whole point of RMS is to make AC comparable to DC for power calculations. When you write P = V²_rms / R, you get the average power dissipated — same formula, same meaning as DC.
The factor of 1/√2 comes purely from the mathematics of averaging sin² over a full cycle. The mean value of sin² is exactly 1/2 — that half is what gives us the √2 in the denominator. This is a fixed geometric fact, not an approximation.
This is why your geyser, iron box, and bulb ratings all use RMS values. Manufacturers specify the RMS voltage because that’s what determines how much heat their appliance produces, which is what actually matters for the consumer.
Alternative Method — Using the Form Factor
There’s a shortcut worth knowing for objective questions:
For a sinusoidal waveform, the form factor is defined as V_rms / V_avg, and its value is π/(2√2) ≈ 1.11.
The average value of |V₀ sin(ωt)| (rectified half-wave average) is:
Multiply by the form factor: V_rms = 1.11 × V_avg. You’ll reach the same V₀/√2.
For JEE MCQs on waveforms that are not sinusoidal (square wave, triangular wave), the 1/√2 formula does NOT apply. You must integrate V² directly. A square wave of amplitude V₀ has V_rms = V₀ exactly — this is a classic trap in JEE Main.
Common Mistake
The most common error: students confuse average value and RMS value. The average of a full sine wave is zero; the average of a half-rectified sine wave is V₀/π; but the RMS is V₀/√2 regardless of which half you consider (for a full sinusoid). Writing V_rms = V₀/2 or V_rms = V₀/π on a board exam will cost you full marks — the √2 in the denominator is non-negotiable for sinusoidal AC.
The confusion happens because students memorise V_avg = 2V₀/π and V_rms = V₀/√2 as two separate facts without understanding that they come from completely different operations (simple averaging vs. square-then-average-then-root). Once you see the derivation in Step 3, you’ll never mix them up again.