Stefan-Boltzmann Law Calculator
Radiated power from a hot surface with P = εσAT⁴.
Power per unit area: 56.704 kW/m². Emissivity ε = 1.
A 1 m² black body at 1000 K radiates P = εσAT⁴ = 56,704 W (about 56.7 kW). Because power grows with the fourth power of temperature, doubling T multiplies radiated power by 2⁴ = 16 — so the same surface at 2000 K pours out roughly 907 kW.
Radiation from a hot surface
Every object above absolute zero radiates thermal energy. The Stefan-Boltzmann law gives the total power leaving a surface as P = εσAT⁴ — emissivity times the Stefan-Boltzmann constant σ times area times absolute temperature to the fourth power. The T⁴ term makes radiation extraordinarily sensitive to temperature, which is why a glowing filament or a star sheds so much more energy than a warm body.
P = radiated power (W), ε = emissivity (0–1), σ = 5.670374×10⁻⁸ W·m⁻²·K⁻⁴, A = area (m²), T = temperature (K)
Worked example
How much power does a 1 m² black body (ε = 1) radiate at 1000 K?
- 1 Convert the temperature to kelvin. The law needs absolute temperature: here T = 1000 K already.
- 2 Write the Stefan-Boltzmann law. P = ε × σ × A × T⁴ with σ = 5.670374×10⁻⁸ W·m⁻²·K⁻⁴.
- 3 Raise T to the fourth power. 1000⁴ = 1×10¹² K⁴.
- 4 Multiply everything together. P = 1 × 5.670374×10⁻⁸ × 1 × 1×10¹² = 56,704 W ≈ 56.7 kW.
Radiated power of a 1 m² black body
Perfect emitter (ε = 1, A = 1 m²). Power scales with T⁴, so it climbs fast.
| Temperature (K) | Radiated power | Rough context |
|---|---|---|
| 300 | 459 W | Near room temperature |
| 500 | 3.54 kW | A hot oven surface |
| 1000 | 56.7 kW | Dull red heat |
| 2000 | 907 kW | Bright orange-white |
| 3000 | 4.59 MW | Incandescent bulb region |
| 5778 | 63.2 MW | The Sun’s surface |
Reading the result
Temperature must be in kelvin. Because the law uses T⁴, feeding it Celsius gives nonsense — always convert first (K = °C + 273.15). A small error in T is magnified fourfold in the power.
The T⁴ dependence dominates. Radiated power is proportional to T⁴, so power rises far faster than temperature: raising T by 19 % nearly doubles the output, and doubling T multiplies it by 16.
Emissivity scales the whole result. A perfect black body has ε = 1; real surfaces sit below that (polished metal near 0.05, skin near 0.98). The radiated power is simply the black-body value multiplied by ε.