Stefan-Boltzmann Radiation Calculator

The Stefan-Boltzmann Law describes how much thermal radiation (heat in the form of electromagnetic waves) an object emits based on its temperature. It was first proposed by Slovenian physicist Josef Stefan in 1879 and later derived theoretically by Ludwig Boltzmann in 1884.

The Formula

For a perfect blackbody: P = σ × A × T⁴

For a real object (with emissivity): P = ε × σ × A × T⁴

Where:

• P = Power radiated (Watts)
• ε = Emissivity (0 to 1; 1 = perfect blackbody)
• σ = Stefan-Boltzmann constant = 5.670 × 10⁻⁸ W·m⁻²·K⁻⁴
• A = Surface area (m²)
• T = Absolute temperature (Kelvin)

The T⁴ Relationship

The most important feature of this law is the fourth-power dependence on temperature. Double the absolute temperature and the radiated power increases by 2⁴ = 16 times. This is why hotter objects radiate so much more intensely. A star at 6,000 K radiates sixteen times as hard per square metre as one at 3,000 K.

Emissivity Reference

Low-emissivity (low-e) coatings on windows exploit this. By reducing emissivity they reduce radiated heat loss in winter, which is worth real money on a big south-facing window.

Real-World Examples

• The Sun (surface ~5,778 K) radiates about 3.85 × 10²⁶ watts
• Incandescent bulbs at 2,700 K radiate mostly infrared, not visible light, which is exactly why they are inefficient
• Thermal cameras detect the infrared radiation predicted by this law

Working the human-body case properly

This is the example everyone reaches for, and it is the one most often quoted wrongly. Use skin temperature, not core temperature: skin sits near 33°C (306 K), not 37°C. With ε = 0.97 and a surface area of 1.7 m², the gross emission is

P = 0.97 × 5.670 × 10⁻⁸ × 1.7 × 306⁴ ≈ 820 W

That figure alone is absurd. Nobody radiates the output of eight light bulbs and stays warm. The missing half of the calculation is that you are standing inside a room that radiates back at you. A 20°C (293 K) room delivers about 690 W onto the same skin. The net is

820 − 690 ≈ 130 W

which is the right order for a resting adult, whose whole metabolic output is around 100 W. For a person at rest in still air, radiation is the largest single channel of heat loss, ahead of convection and well ahead of evaporation.

The practical consequence is that walls matter as much as air. Cold walls chill you even when the thermostat reads a comfortable number, because the absorbed term shrinks while the emitted term does not. That is the whole argument for radiant floor heating: warm surfaces let you feel comfortable at a lower air temperature, because the 690 W side of the equation goes up.