Physics 9702 · Paper 4

r.m.s. Values & Power

An alternating current averages to zero — yet it lights lamps. The average that matters is the average of the square, and it brings a factor of √2.

Root-mean-square

The of an alternating current is the steady (direct) current that delivers the same average power to a resistive load. It is the honest “effective” value — mains “230 V” is an r.m.s. figure.

Why r.m.s.: power goes as I², and I² (sampled below) is always positive with mean ½I₀². The steady current with that same mean-square value is I₀/√2 — the r.m.s. current, delivering the same average power.

I_{rms} = \dfrac{I_0}{\sqrt{2}} \qquad V_{rms} = \dfrac{V_0}{\sqrt{2}}

Symbols

$I_{rms},\ V_{rms}$ = root-mean-square values (sinusoidal a.c. only) (A, V)

Where √2 comes from: power goes as $I^2$ , and the figure shows $I^2$ averaging to exactly $\tfrac{1}{2}I_0^2$ . Square root of the mean of the square: $I_0/\sqrt{2} \approx 0.707\,I_0$ .
Both conversion directions are live: 2024 gave the peak 3.5 A and wanted r.m.s. 2.5 A (divide) (9702/42/O/N/24 Q8(b)(iii)); 2025 gave r.m.s. 6.0 V and wanted the peak 8.5 V (multiply) (9702/42/M/J/25 Q8(a)(ii)). Decide which way before touching √2.
With r.m.s. values, every d.c. power formula works unchanged: $P = I_{rms}^2 R = V_{rms}^2/R$ . Use peaks instead and the answer doubles — in fact mean power is exactly half the peak power, a show-that in 2024: peak $3.5^2 \times 680 = 8330$ W against mean $2.47^2 \times 680 = 4170$ W (9702/42/O/N/24 Q8(c)).
Power–time sketches are rubric-marked: a sinusoidal curve (at twice the supply frequency) whose troughs sit on the time axis — power never goes negative — with peaks placed and labelled (9702/41/M/J/24 Q7(b)(ii)).

When √2 is banned: non-sinusoidal a.c.

A 2023 square-wave current ran at

2I_0

for half each cycle and

I_0

(reversed) for the other half. The only safe route is first principles: power is

4I_0^2R

then

I_0^2R

, mean

\tfrac{5}{2}I_0^2R

, so

I_{rms} = \sqrt{5/2}\,I_0

. Writing

I_0/\sqrt{2}

scored zero —

\sqrt{2}

belongs to sine waves only (9702/41/O/N/23 Q7).

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