R.m.s. values and equal heating

Key idea: R.m.s. current gives the same mean heating power as steady D.C. in the same resistor. For a sinusoid, the r.m.s. value is the peak divided by √2.

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GCE A-Level H2 Physics 2027

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Averaged over a complete cycle, a sinusoidal current is zero: it flows one way for half the time and the other way for the other half. Yet it heats a kettle element perfectly well, because the heating does not depend on the direction of the current. To say how large an alternating current is, we need a value that measures its heating effect.

Root-mean-square value

In a resistor of constant resistance R, the power at any instant is I²R. This is positive whichever way the current flows. The root-mean-square (r.m.s.) current Iᵣₘₛ is the steady direct current that would produce the same mean power in the same resistor:

Iᵣₘₛ²R = ⟨I²⟩ R ⇒ Iᵣₘₛ = square root of (⟨I²⟩)

where ⟨I²⟩ is the mean of I² over a complete cycle. The name gives the order of operations: square the current, take the mean, then the square root. The r.m.s. voltage is defined in the same way.

For a complete sinusoid, the squared normalised current has cycle mean 0.5. Its positive square root gives Iᵣₘₛ/I₀ = 1/√2. Both current directions contribute to heating.

Figure data: Square before averaging
Time, t (ms)	Squared current	Cycle mean
0.0	0.0000000	0.5
0.1	0.0009866	0.5
0.2	0.0039426	0.5
0.3	0.0088564	0.5
0.4	0.0157084	0.5
0.5	0.0244717	0.5
0.6	0.0351118	0.5
0.7	0.0475865	0.5
0.8	0.0618467	0.5
0.9	0.0778360	0.5
1.0	0.0954915	0.5
1.1	0.1147434	0.5
1.2	0.1355157	0.5
1.3	0.1577264	0.5
1.4	0.1812880	0.5
1.5	0.2061074	0.5
1.6	0.2320866	0.5
1.7	0.2591232	0.5
1.8	0.2871104	0.5
1.9	0.3159377	0.5
2.0	0.3454915	0.5
2.1	0.3756551	0.5
2.2	0.4063093	0.5
2.3	0.4373334	0.5
2.4	0.4686047	0.5
2.5	0.5000000	0.5
2.6	0.5313953	0.5
2.7	0.5626666	0.5
2.8	0.5936907	0.5
2.9	0.6243449	0.5
3.0	0.6545085	0.5
3.1	0.6840623	0.5
3.2	0.7128896	0.5
3.3	0.7408768	0.5
3.4	0.7679134	0.5
3.5	0.7938926	0.5
3.6	0.8187120	0.5
3.7	0.8422736	0.5
3.8	0.8644843	0.5
3.9	0.8852566	0.5
4.0	0.9045085	0.5
4.1	0.9221640	0.5
4.2	0.9381533	0.5
4.3	0.9524135	0.5
4.4	0.9648882	0.5
4.5	0.9755283	0.5
4.6	0.9842916	0.5
4.7	0.9911436	0.5
4.8	0.9960574	0.5
4.9	0.9990134	0.5
5.0	1.0000000	0.5
5.1	0.9990134	0.5
5.2	0.9960574	0.5
5.3	0.9911436	0.5
5.4	0.9842916	0.5
5.5	0.9755283	0.5
5.6	0.9648882	0.5
5.7	0.9524135	0.5
5.8	0.9381533	0.5
5.9	0.9221640	0.5
6.0	0.9045085	0.5
6.1	0.8852566	0.5
6.2	0.8644843	0.5
6.3	0.8422736	0.5
6.4	0.8187120	0.5
6.5	0.7938926	0.5
6.6	0.7679134	0.5
6.7	0.7408768	0.5
6.8	0.7128896	0.5
6.9	0.6840623	0.5
7.0	0.6545085	0.5
7.1	0.6243449	0.5
7.2	0.5936907	0.5
7.3	0.5626666	0.5
7.4	0.5313953	0.5
7.5	0.5000000	0.5
7.6	0.4686047	0.5
7.7	0.4373334	0.5
7.8	0.4063093	0.5
7.9	0.3756551	0.5
8.0	0.3454915	0.5
8.1	0.3159377	0.5
8.2	0.2871104	0.5
8.3	0.2591232	0.5
8.4	0.2320866	0.5
8.5	0.2061074	0.5
8.6	0.1812880	0.5
8.7	0.1577264	0.5
8.8	0.1355157	0.5
8.9	0.1147434	0.5
9.0	0.0954915	0.5
9.1	0.0778360	0.5
9.2	0.0618467	0.5
9.3	0.0475865	0.5
9.4	0.0351118	0.5
9.5	0.0244717	0.5
9.6	0.0157084	0.5
9.7	0.0088564	0.5
9.8	0.0039426	0.5
9.9	0.0009866	0.5
10.0	0.0000000	0.5
10.1	0.0009866	0.5
10.2	0.0039426	0.5
10.3	0.0088564	0.5
10.4	0.0157084	0.5
10.5	0.0244717	0.5
10.6	0.0351118	0.5
10.7	0.0475865	0.5
10.8	0.0618467	0.5
10.9	0.0778360	0.5
11.0	0.0954915	0.5
11.1	0.1147434	0.5
11.2	0.1355157	0.5
11.3	0.1577264	0.5
11.4	0.1812880	0.5
11.5	0.2061074	0.5
11.6	0.2320866	0.5
11.7	0.2591232	0.5
11.8	0.2871104	0.5
11.9	0.3159377	0.5
12.0	0.3454915	0.5
12.1	0.3756551	0.5
12.2	0.4063093	0.5
12.3	0.4373334	0.5
12.4	0.4686047	0.5
12.5	0.5000000	0.5
12.6	0.5313953	0.5
12.7	0.5626666	0.5
12.8	0.5936907	0.5
12.9	0.6243449	0.5
13.0	0.6545085	0.5
13.1	0.6840623	0.5
13.2	0.7128896	0.5
13.3	0.7408768	0.5
13.4	0.7679134	0.5
13.5	0.7938926	0.5
13.6	0.8187120	0.5
13.7	0.8422736	0.5
13.8	0.8644843	0.5
13.9	0.8852566	0.5
14.0	0.9045085	0.5
14.1	0.9221640	0.5
14.2	0.9381533	0.5
14.3	0.9524135	0.5
14.4	0.9648882	0.5
14.5	0.9755283	0.5
14.6	0.9842916	0.5
14.7	0.9911436	0.5
14.8	0.9960574	0.5
14.9	0.9990134	0.5
15.0	1.0000000	0.5
15.1	0.9990134	0.5
15.2	0.9960574	0.5
15.3	0.9911436	0.5
15.4	0.9842916	0.5
15.5	0.9755283	0.5
15.6	0.9648882	0.5
15.7	0.9524135	0.5
15.8	0.9381533	0.5
15.9	0.9221640	0.5
16.0	0.9045085	0.5
16.1	0.8852566	0.5
16.2	0.8644843	0.5
16.3	0.8422736	0.5
16.4	0.8187120	0.5
16.5	0.7938926	0.5
16.6	0.7679134	0.5
16.7	0.7408768	0.5
16.8	0.7128896	0.5
16.9	0.6840623	0.5
17.0	0.6545085	0.5
17.1	0.6243449	0.5
17.2	0.5936907	0.5
17.3	0.5626666	0.5
17.4	0.5313953	0.5
17.5	0.5000000	0.5
17.6	0.4686047	0.5
17.7	0.4373334	0.5
17.8	0.4063093	0.5
17.9	0.3756551	0.5
18.0	0.3454915	0.5
18.1	0.3159377	0.5
18.2	0.2871104	0.5
18.3	0.2591232	0.5
18.4	0.2320866	0.5
18.5	0.2061074	0.5
18.6	0.1812880	0.5
18.7	0.1577264	0.5
18.8	0.1355157	0.5
18.9	0.1147434	0.5
19.0	0.0954915	0.5
19.1	0.0778360	0.5
19.2	0.0618467	0.5
19.3	0.0475865	0.5
19.4	0.0351118	0.5
19.5	0.0244717	0.5
19.6	0.0157084	0.5
19.7	0.0088564	0.5
19.8	0.0039426	0.5
19.9	0.0009866	0.5
20.0	0.0000000	0.5

Figure data: Square before averaging
Time, t (ms)	Squared current	Cycle mean
0.0	0.0000000	0.5
0.1	0.0009866	0.5
0.2	0.0039426	0.5
0.3	0.0088564	0.5
0.4	0.0157084	0.5
0.5	0.0244717	0.5
0.6	0.0351118	0.5
0.7	0.0475865	0.5
0.8	0.0618467	0.5
0.9	0.0778360	0.5
1.0	0.0954915	0.5
1.1	0.1147434	0.5
1.2	0.1355157	0.5
1.3	0.1577264	0.5
1.4	0.1812880	0.5
1.5	0.2061074	0.5
1.6	0.2320866	0.5
1.7	0.2591232	0.5
1.8	0.2871104	0.5
1.9	0.3159377	0.5
2.0	0.3454915	0.5
2.1	0.3756551	0.5
2.2	0.4063093	0.5
2.3	0.4373334	0.5
2.4	0.4686047	0.5
2.5	0.5000000	0.5
2.6	0.5313953	0.5
2.7	0.5626666	0.5
2.8	0.5936907	0.5
2.9	0.6243449	0.5
3.0	0.6545085	0.5
3.1	0.6840623	0.5
3.2	0.7128896	0.5
3.3	0.7408768	0.5
3.4	0.7679134	0.5
3.5	0.7938926	0.5
3.6	0.8187120	0.5
3.7	0.8422736	0.5
3.8	0.8644843	0.5
3.9	0.8852566	0.5
4.0	0.9045085	0.5
4.1	0.9221640	0.5
4.2	0.9381533	0.5
4.3	0.9524135	0.5
4.4	0.9648882	0.5
4.5	0.9755283	0.5
4.6	0.9842916	0.5
4.7	0.9911436	0.5
4.8	0.9960574	0.5
4.9	0.9990134	0.5
5.0	1.0000000	0.5
5.1	0.9990134	0.5
5.2	0.9960574	0.5
5.3	0.9911436	0.5
5.4	0.9842916	0.5
5.5	0.9755283	0.5
5.6	0.9648882	0.5
5.7	0.9524135	0.5
5.8	0.9381533	0.5
5.9	0.9221640	0.5
6.0	0.9045085	0.5
6.1	0.8852566	0.5
6.2	0.8644843	0.5
6.3	0.8422736	0.5
6.4	0.8187120	0.5
6.5	0.7938926	0.5
6.6	0.7679134	0.5
6.7	0.7408768	0.5
6.8	0.7128896	0.5
6.9	0.6840623	0.5
7.0	0.6545085	0.5
7.1	0.6243449	0.5
7.2	0.5936907	0.5
7.3	0.5626666	0.5
7.4	0.5313953	0.5
7.5	0.5000000	0.5
7.6	0.4686047	0.5
7.7	0.4373334	0.5
7.8	0.4063093	0.5
7.9	0.3756551	0.5
8.0	0.3454915	0.5
8.1	0.3159377	0.5
8.2	0.2871104	0.5
8.3	0.2591232	0.5
8.4	0.2320866	0.5
8.5	0.2061074	0.5
8.6	0.1812880	0.5
8.7	0.1577264	0.5
8.8	0.1355157	0.5
8.9	0.1147434	0.5
9.0	0.0954915	0.5
9.1	0.0778360	0.5
9.2	0.0618467	0.5
9.3	0.0475865	0.5
9.4	0.0351118	0.5
9.5	0.0244717	0.5
9.6	0.0157084	0.5
9.7	0.0088564	0.5
9.8	0.0039426	0.5
9.9	0.0009866	0.5
10.0	0.0000000	0.5
10.1	0.0009866	0.5
10.2	0.0039426	0.5
10.3	0.0088564	0.5
10.4	0.0157084	0.5
10.5	0.0244717	0.5
10.6	0.0351118	0.5
10.7	0.0475865	0.5
10.8	0.0618467	0.5
10.9	0.0778360	0.5
11.0	0.0954915	0.5
11.1	0.1147434	0.5
11.2	0.1355157	0.5
11.3	0.1577264	0.5
11.4	0.1812880	0.5
11.5	0.2061074	0.5
11.6	0.2320866	0.5
11.7	0.2591232	0.5
11.8	0.2871104	0.5
11.9	0.3159377	0.5
12.0	0.3454915	0.5
12.1	0.3756551	0.5
12.2	0.4063093	0.5
12.3	0.4373334	0.5
12.4	0.4686047	0.5
12.5	0.5000000	0.5
12.6	0.5313953	0.5
12.7	0.5626666	0.5
12.8	0.5936907	0.5
12.9	0.6243449	0.5
13.0	0.6545085	0.5
13.1	0.6840623	0.5
13.2	0.7128896	0.5
13.3	0.7408768	0.5
13.4	0.7679134	0.5
13.5	0.7938926	0.5
13.6	0.8187120	0.5
13.7	0.8422736	0.5
13.8	0.8644843	0.5
13.9	0.8852566	0.5
14.0	0.9045085	0.5
14.1	0.9221640	0.5
14.2	0.9381533	0.5
14.3	0.9524135	0.5
14.4	0.9648882	0.5
14.5	0.9755283	0.5
14.6	0.9842916	0.5
14.7	0.9911436	0.5
14.8	0.9960574	0.5
14.9	0.9990134	0.5
15.0	1.0000000	0.5
15.1	0.9990134	0.5
15.2	0.9960574	0.5
15.3	0.9911436	0.5
15.4	0.9842916	0.5
15.5	0.9755283	0.5
15.6	0.9648882	0.5
15.7	0.9524135	0.5
15.8	0.9381533	0.5
15.9	0.9221640	0.5
16.0	0.9045085	0.5
16.1	0.8852566	0.5
16.2	0.8644843	0.5
16.3	0.8422736	0.5
16.4	0.8187120	0.5
16.5	0.7938926	0.5
16.6	0.7679134	0.5
16.7	0.7408768	0.5
16.8	0.7128896	0.5
16.9	0.6840623	0.5
17.0	0.6545085	0.5
17.1	0.6243449	0.5
17.2	0.5936907	0.5
17.3	0.5626666	0.5
17.4	0.5313953	0.5
17.5	0.5000000	0.5
17.6	0.4686047	0.5
17.7	0.4373334	0.5
17.8	0.4063093	0.5
17.9	0.3756551	0.5
18.0	0.3454915	0.5
18.1	0.3159377	0.5
18.2	0.2871104	0.5
18.3	0.2591232	0.5
18.4	0.2320866	0.5
18.5	0.2061074	0.5
18.6	0.1812880	0.5
18.7	0.1577264	0.5
18.8	0.1355157	0.5
18.9	0.1147434	0.5
19.0	0.0954915	0.5
19.1	0.0778360	0.5
19.2	0.0618467	0.5
19.3	0.0475865	0.5
19.4	0.0351118	0.5
19.5	0.0244717	0.5
19.6	0.0157084	0.5
19.7	0.0088564	0.5
19.8	0.0039426	0.5
19.9	0.0009866	0.5
20.0	0.0000000	0.5

Check your understanding 1 Zero average, non-zero heating

A current is + 2.0 A for half a cycle and -2.0 A for the other half. Calculate its mean current and its r.m.s. current. Explain why a resistor still heats.

Mean current

Unit: A

Show the steps Optional working

Step 1: Mean squared current

Unit: A²

R.m.s. current

Unit: A

Show answer

The mean current is (2.0-2.0)/2 = 0. The mean square is (4.0 + 4.0)/2 = 4.0 A², giving Iᵣₘₛ = 2.0 A. Heating depends on I²R, which is positive in both directions.

Check your understanding 2 Explain the heating

Explain why zero mean current does not imply zero mean power. Compare averaging before and after squaring.

Your explanation of square and mean

Show answer

Compare ⟨I⟩²R with ⟨I²⟩ R. Opposite currents cancel in the mean of I; their squares do not cancel. A resistor heats in both directions.

R.m.s. value of a sinusoid

For I = I₀ sin ω t, I² = I₀² sin² ω t. Since sin² θ = (1/2)(1- cos 2θ) and cos 2θ averages to zero over a cycle, the mean of sin² ω t is 1/2. So ⟨I²⟩ = (1/2)I₀² and

Iᵣₘₛ = I₀/(square root of 2), Vᵣₘₛ = V₀/(square root of 2)

These hold only for a sinusoid. For any other waveform, go back to square, mean, root.

Mains voltage ratings are quoted as r.m.s. values. The 230 V mains has a peak of 230 square root of 2 = 325 V.

Problem

A sinusoidal current has an r.m.s. value of 3.00 A and a frequency of 60.0 Hz. It is zero and increasing at t = 0. Write an equation for the current.

Show full solution

Peak value
Method
Multiply the r.m.s. current by square root of 2.
Reason
The number in front of the sine must be the peak, not the r.m.s. value.
Working
Answer
I₀ = square root of 2 × 3.00 = 4.24 A
Angular frequency
Reason
ω = 2π f.
Working
Answer
ω = 2π × 60.0 = 120π rad s⁻¹
Equation
Working
Answer
I = 4.24 sin(120π t), with I in amperes and t in seconds

Quick check

The peak is larger than the r.m.s. value, as it must be: the current spends most of each cycle below its peak.

Problem

A sinusoidal voltage has a peak value of 20.0 V. Find its r.m.s. value, and state what that value means for a resistor connected to it.

Find the r.m.s. voltage

R.m.s. voltage

Unit: V

A steady voltage equal to this r.m.s. value gives the same resistor

the same mean powerthe same peak powerzero mean power

Hints

Hint 1

meaning

The meaning is a comparison with a steady voltage across the same resistor.

Show full solution

Value and meaning
Working
Answer
Vᵣₘₛ = 20.0/(square root of 2) = 14.1 V. A steady 14.1 V across the same resistor would give the same mean power.

Learner claim

For I = 5.00 sin(100π t), in amperes, a student says the r.m.s. current is 5.00 A. What does 5.00 A represent, and what is the r.m.s. current?

Try this before viewing the solution

5.00 A is the

peak currentmean currentr.m.s. current

R.m.s. current

Unit: A

Show solution step by step

Read the peak
Method
Identify 5.00 A as the peak I₀.
Reason
sin(100π t) varies between -1 and + 1, so the current varies between -5.00 A and + 5.00 A.
Working
Answer
I₀ = 5.00 A
Convert
Working
Answer
Iᵣₘₛ = 5.00/(square root of 2) = 3.54 A

Common mistake

The coefficient in this sinusoidal equation is the peak value. A steady 5.00 A would heat a resistor twice as fast as this alternating current.

Problem

A repeating current is 4.0 A for one quarter of each cycle and zero for the other three quarters. Find its r.m.s. value.

Find the r.m.s. current

Show the steps Optional working

Step 1: Cycle mean of squared current

Unit: A²

R.m.s. current

Unit: A

Hints

Hint 1

not a sinusoid

I₀/square root of 2 does not apply. Weight each value of I² by the fraction of the cycle it lasts.

Show solution step by step

Mean square
Reason
The current is 4.0 A for a quarter of the time and zero otherwise.
Working
Answer
⟨I²⟩ = (1/4)(4.0)² + (3/4)(0)² = 4.0 A²
Root
Working
Answer
Iᵣₘₛ = square root of 4.0 = 2.0 A

Quick check

The ordinary mean current is 1/4 × 4.0 = 1.0 A, which is smaller: the r.m.s. value gives more weight to the large values, because heating depends on I².

Examination question

A sinusoidal mains supply is labelled 230 V, 50.0 Hz.

(a) Explain what is meant by the r.m.s. value of an alternating voltage. [2 marks]

(b) Calculate the peak voltage of the supply. [1 mark]

Write your answer before viewing the mark scheme

Your five-mark answer

Show solution step by step

(a) Meaning
2 marks
Method
Compare with a steady voltage across the same resistor.
Reason
The r.m.s. value is defined by equal heating.
Working
Answer
It is the steady (direct) voltage that would produce the same mean power in the same resistor.
(b) Peak
1 mark
Reason
Mains labels give r.m.s. values.
Working
Answer
V₀ = square root of 2 × 230 = 325 V
(c) Equation
2 marks
Working
Answer
ω = 2π × 50.0 = 100π rad s⁻¹, so V = 325 sin(100π t), in volts

Self-mark with the mark scheme

Tick each mark point that your answer includes.

Tick each point only if your answer states it clearly.

Mark points

1 mark: Compares with a steady (D.C.) voltage.
1 mark: Same mean power, or heating, in the same resistor.
1 mark: Peak voltage 325 V.
1 mark: ω = 100π rad s⁻¹.
1 mark: V = 325 sin(100πt), using the peak, not 230 V.

Common mistakes

Putting the r.m.s. value in front of the sine instead of the peak.
Treating the r.m.s. value as the average current; the average of a sinusoid over a cycle is zero.
Using I₀/square root of 2 for a waveform that is not a full sinusoid, such as a half-wave rectified or square wave.
Taking the square root before averaging.

Spotted an error? Report a correction

Syllabus and review details

GCE A-Level H2 Physics 2027 · 2027
Content Overview, PDF pages 9–10; Subject Content, PDF pages 11–30

Last reviewed: 2026-08-28

R.m.s. values and equal heating

Guided lesson

Guided steps

Reading settings

Root-mean-square value

Check your understanding 1 Zero average, non-zero heating

Check your understanding 2 Explain the heating

R.m.s. value of a sinusoid

From r.m.s. value to equation

Peak value

Angular frequency

Equation

What an r.m.s. voltage means

Value and meaning

Which number is the r.m.s. current?

Read the peak

Convert

A pulsed current

Mean square

Root

The mains supply

(a) Meaning

(b) Peak

(c) Equation

Common mistakes