RMS is the DC equivalent for power — not the average

Alternating Currents — AQA A-Level Physics

Key Definition

Root mean square (RMS) value — The RMS value of an alternating current is the value of direct current that would produce the same heating effect (same power dissipation) in a given resistor.

Think of it this way: if you connect a 230 V RMS AC supply to a kettle, and then a 230 V DC supply to an identical kettle, both kettles heat the water at the same rate. That's what RMS means. It's the DC equivalent for powerThe rate of energy transfer. Measured in watts (W). transfer.
The word 'root mean square' describes how you calculate it: square all the values, take the mean of the squares, then take the square root. For a sine wave, the maths simplifies to:

V_{\text{rms}} = \frac{V_0}{\sqrt{2}}

$V_{\text{rms}}$: RMS voltageThe energyThe capacity to do work. Measured in joules (J). transferred per unit chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). between two points. Measured in volts (V). Informal term for potential difference. (V)
$V_0$: peak voltageThe energyThe capacity to do work. Measured in joules (J). transferred per unit chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). between two points. Measured in volts (V). Informal term for potential difference. (V)

I_{\text{rms}} = \frac{I_0}{\sqrt{2}}

$I_{\text{rms}}$: RMS currentThe rate of flow of chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C).. Measured in amperes (A). (A)
$I_0$: peak currentThe rate of flow of charge. Measured in amperes (A). (A)

The factor 1/√2 (approximately 0.707) applies specifically to sinusoidal AC. A square wave would have a different factor (actually 1, since it's always at peak).
When someone says '230 V mains', that's the RMS value. The actual peak voltageThe energyThe capacity to do work. Measured in joules (J). transferred per unit charge between two points. Measured in volts (V). Informal term for potential difference. is 230 × √2 = 325 V. The voltage in your wall socket swings between +325 V and -325 V, 50 times every second.
Similarly, if a fuse is rated at 13 A, that's 13 A RMS. The peak currentThe rate of flow of charge. Measured in amperes (A). through it reaches 13 × √2 = 18.4 A.

Common Mistake MEDIUM

Students often: Confusing RMS with the average value of AC.
Instead: The average of a complete AC sine cycle is zero (the positive and negative halves cancel exactly). RMS is not an average of the voltage — it's the DC equivalent for power. V_rms = V₀/√2 ≈ 0.707 × V₀. The mean value of a rectified sine wave is 2V₀/π ≈ 0.637 × V₀, which is different again.

Worked Example

A signal generator produces a sinusoidal output with a peak voltage of 12.0 V. Calculate the RMS voltage.

Show Solution

List known values

- Peak voltage: $V_0 = 12.0 \text{ V}$

Write the RMS equation

$$V_{\text{rms}} = \frac{V_0}{\sqrt{2}}$$

Substitute and evaluate

$$V_{\text{rms}} = \frac{12.0}{\sqrt{2}} = \frac{12.0}{1.414}$$ $$= 8.49 \text{ V}$$

Answer

$V_{\text{rms}} = 8.5$ V (2 s.f.)