Library / Electrical / ELE-24
Electrical
Joule heating
Power lost as heat in a resistance.
- Ref
- ELE-24
- Rev
- v1
- Field
- Electrical
- Result
- P [W]
I is the current through the resistance and R the resistance. The power appears as heat. For alternating current use the RMS value of the current. The same power can be written P = VI = V²/R, depending on which quantities are known.
Power lost as heat in a resistance. It grows with the square of current, so cables carrying twice the current lose four times the power.
Calculator
- Heat produced in one hour576 kJ
- Power at twice the current640 W
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Halving the current in a cable cuts the loss to a quarter. This is why power is moved at high voltage: the same power needs a smaller current.
- A 160 W loss in a 10 Ω element is enough to make it too hot to touch; in a sealed enclosure it must be removed as heat.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (W) | Local step | Change in result |
|---|---|---|---|---|---|---|
| I Current | 0.001 to 1,000 A | 1e-5 | 1e7 | 10,000,000 | +10% | +21 % |
| R Resistance | 0.001 to 100,000 Ω | 0.016 | 1,600,000 | 1,600,000 | +10% | +10 % |
Sweep of Current
| Current (A) | Heat power (W) |
|---|---|
| 0.001 | 1e-5 |
| 0.005623 | 3.162e-4 |
| 0.03162 | 0.01 |
| 0.1778 | 0.3162 |
| 1 | 10 |
| 5.623 | 316.2 |
| 31.62 | 10,000 |
| 177.8 | 316,200 |
| 1,000 | 1e7 |
Sweep one variable
Variables and how each one contributes
Heat power
What the equation gives you. Everything else feeds it.
Current
The current through the resistance. Squared: doubling the current quadruples the heat. A 10 % rise in I raises P by 21 %.
Resistance
The resistance that carries the current. Directly proportional at a fixed current: twice the resistance wastes twice the power.
Worked example with the values above
- Square the current: I² = (4)² = 16 A².
- Multiply by R: 16 × 10.
- P = 160 W.
Assumptions and limits
- Steady current, or the RMS value for AC.
- Constant resistance, although it rises as the conductor heats.
Common mistakes
- Forgetting to square the current.
- Using the peak value of an AC current in place of the RMS value. This overstates P by a factor of 2.
Statistics and uncertainty
P ∝ I²R, so a 1 % error in the current gives a 2 % error in the power and a 1 % error in R gives 1 %. The current is therefore the quantity to measure well. Resistance also changes with temperature, so a cold-resistance value understates the hot loss.
Where it is used
- Sizing cables, fuses and connectors.
- Estimating the cooling needed for a resistor or load bank.
Maths you need
- Ohm’s law: V = IR
- Power: energy per second