Stefan-Boltzmann Radiation Formula
Thermal radiation: P = εσT⁴.
Calculate heat radiated by any object.
Used in climate science, astrophysics, and engineering.
The Formula
Every object with a temperature above absolute zero radiates electromagnetic energy. The Stefan-Boltzmann Law gives the total power radiated across all wavelengths. Power P (in watts) depends on the emissivity ε of the surface, the Stefan-Boltzmann constant σ, the surface area A, and the absolute temperature T raised to the fourth power.
The Stefan-Boltzmann constant σ = 5.67 × 10⁻⁸ W/(m²·K⁴).
Emissivity ε ranges from 0 (perfect mirror, reflects all radiation) to 1 (perfect blackbody, absorbs and emits maximally). Most non-metallic surfaces: ε ≈ 0.9–0.95. Polished metals: ε ≈ 0.02–0.1. Human skin: ε ≈ 0.98 regardless of color.
The T⁴ dependence makes radiation extremely sensitive to temperature. Double the temperature → 16× the radiated power. This is why very hot objects (stars, furnaces) radiate enormously more than cool objects.
Net radiation between two objects: P_net = εσA(T_hot⁴ − T_cold⁴). This formula is central to climate science — Earth's energy balance depends on incoming solar radiation vs. outgoing thermal radiation.
Wien's Displacement Law tells you the peak wavelength: λ_max = 2.898×10⁻³/T. At room temperature (300 K), objects radiate in the infrared (~10 μm). The Sun (5,778 K) peaks in visible light (~500 nm).
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| P | Radiated power | Watts (W) |
| ε | Emissivity (0 to 1) | Dimensionless |
| σ | Stefan-Boltzmann constant (5.67×10⁻⁸) | W/(m²·K⁴) |
| A | Surface area | m² |
| T | Absolute temperature | Kelvin (K) |
Example 1
A human body (area 1.8 m², T = 310 K, ε = 0.98) — how much does it radiate?
P = 0.98 × 5.67×10⁻⁸ × 1.8 × (310)⁴
310⁴ = 9.235×10⁹; P = 0.98 × 5.67×10⁻⁸ × 1.8 × 9.235×10⁹
P ≈ 924 W (but the room also radiates back ~850 W, so net loss ≈ 74 W)
Example 2
A furnace wall at 800°C (1073 K), area 2 m², ε = 0.85.
P = 0.85 × 5.67×10⁻⁸ × 2 × (1073)⁴ = 0.85 × 5.67×10⁻⁸ × 2 × 1.325×10¹²
P ≈ 128,000 W = 128 kW radiated
When to Use It
- Climate science and Earth energy balance models
- Thermal imaging and infrared camera design
- Furnace and kiln heat loss calculations
- Spacecraft thermal control engineering