Relate mean and maximum power in an AC resistor
Use the sinusoidal current model to explain why a resistive load's mean power is half its maximum power.
Section 21 · Alternating currents
A Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 21. These original explanations and calculations are study aids; full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use period, frequency, sine graphs, current, voltage, power, diodes, capacitors and SI units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
For a sinusoidal AC in a resistor, how does mean power compare with maximum power?
What you will learn
- Recall and use mean power = half maximum power for a sinusoidal AC in a resistive load
- Connect instantaneous resistive power to current squared
- Distinguish mean power from peak power
Resistive power follows the square of current
For a resistor R, instantaneous power is p = i²R (also p = v²/R). If i = I₀ sin ωt, then p = I₀²R sin²ωt. Power is never negative for an ideal resistor, even when the current reverses, because the current is squared.
The maximum occurs at |i| = I₀, so Pmax = I₀²R. Over a complete sinusoidal cycle, the mean value of sin²ωt is 1/2. Therefore the mean power in a resistive load is Pmean = Pmax/2. This stated half-maximum result applies to a sinusoidal alternating current and a resistive load.
Mean power is the cycle average
Mean power averages the changing instantaneous power over a full cycle. It is not the power at the zero crossings, and it is not the maximum. At current zero crossings, instantaneous resistive power is zero; near either current peak, it is greatest.
If Pmax is given, halve it for Pmean. If the mean is given, double it for Pmax. Check that the question concerns a resistive load and a sinusoidal AC before using this shortcut; a non-resistive load can have a different power relationship.
Pause and explain
A sinusoidal AC resistor has maximum instantaneous power 80 W. What is its mean power?
Worked example
A sinusoidal current in a resistor produces a maximum instantaneous power of 120 W. Find its mean power over a cycle.
Show the worked solution
- Identify the stated sinusoidal current and resistive load.
- Use Pmean = Pmax/2.
- Substitute the maximum power.
Answer Pmean = 120/2 = 60 W.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A sinusoidal current in a 5.0 Ω resistor has peak value 4.0 A. Calculate its maximum and mean powers.
Give me a hint
Use Pmax = I₀²R, then halve it for the cycle mean.
Compare my reasoning
- Pmax = (4.0)² × 5.0 = 80 W.
- Pmean = Pmax/2.
- Pmean = 40 W.
Maximum power = 80 W; mean power = 40 W.
Look for these in your work
- Peak power calculated
- Resistive sinusoidal condition applied
- Mean distinguished from maximum
A heater's resistive element receives a sinusoidal current and has mean power 75 W. What is the maximum instantaneous power?
Give me a hint
Reverse the relationship: Pmax = 2Pmean.
Compare my reasoning
- Pmean = Pmax/2.
- Rearrange to Pmax = 2Pmean.
- Pmax = 2 × 75 = 150 W.
Maximum instantaneous power = 150 W.
Look for these in your work
- Relationship recalled
- Equation rearranged
- Power stated
A student says an alternating current that averages to zero gives zero mean power in a resistor. Explain the error and state the mean-to-maximum relationship.
Give me a hint
Compare the time dependence of current and i²R.
Compare my reasoning
- The current changes sign, so its average over a cycle can be zero.
- Resistor power is p = i²R and is non-negative for either current direction.
- For sinusoidal current, mean power is half the maximum power.
The current average may be zero, but squaring it gives positive power during both half-cycles; Pmean = Pmax/2.
Look for these in your work
- Current and power distinguished
- Square dependence explained
- Required ratio stated
Common mistakes
- Averaging current first and then squaring it
- Allowing negative current to imply negative resistor power
- Calling peak power the cycle mean
- Applying the half-maximum result to any load without checking the resistive condition
What can you explain now?
For a sinusoidal AC in a resistor, how does mean power compare with maximum power?
Compare with the explanation
It is half the maximum
The cycle mean of sin² is 1/2.
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Make it stick
Why can a reversing current have positive mean power in a resistor?
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