Stefan-Boltzmann Law
Reference for Stefan-Boltzmann law P = σAT⁴ for blackbody radiation.
Covers σ = 5.67×10⁻⁸ W/m²K⁴, emissivity, and stellar luminosity applications.
The Formula
The Stefan-Boltzmann law states that the total energy radiated per unit time by a body is proportional to the fourth power of its absolute temperature. A small increase in temperature causes a dramatic increase in radiated power.
For a perfect blackbody, the emissivity ε = 1. Real objects have ε between 0 and 1 depending on their surface properties.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| P | Total radiated power | watts (W) |
| ε | Emissivity of the surface (0 to 1) | dimensionless |
| σ | Stefan-Boltzmann constant (5.67 × 10⁻⁸ W/m²K⁴) | W/m²K⁴ |
| A | Surface area of the radiating body | m² |
| T | Absolute temperature | Kelvin (K) |
Example 1
Calculate the power radiated by the Sun (radius ≈ 6.96 × 10⁸ m, T ≈ 5778 K)
Surface area A = 4π × (6.96 × 10⁸)² ≈ 6.09 × 10¹⁸ m²
Assume ε = 1 (blackbody approximation)
P = 1 × 5.67 × 10⁻⁸ × 6.09 × 10¹⁸ × (5778)⁴
T⁴ = (5778)⁴ ≈ 1.115 × 10¹⁵
P ≈ 3.85 × 10²⁶ W — matches the Sun's known luminosity
Example 2
A steel plate (ε = 0.6, area = 2 m²) at 500 K. How much power does it radiate?
P = 0.6 × 5.67 × 10⁻⁸ × 2 × (500)⁴
T⁴ = 500⁴ = 6.25 × 10¹⁰
P = 0.6 × 5.67 × 10⁻⁸ × 2 × 6.25 × 10¹⁰
P ≈ 4,253 W (about 4.25 kW)
When to Use It
- Estimating heat loss from hot surfaces by thermal radiation
- Calculating stellar luminosity from surface temperature
- Designing thermal insulation and radiative cooling systems
- Climate science — modeling Earth's energy balance
- Industrial furnace and kiln engineering
Key Notes
- Formula: P = εσAT⁴: P is radiated power (watts), ε is emissivity (0–1), σ = 5.67×10⁻⁸ W/(m²·K⁴) is the Stefan-Boltzmann constant, A is surface area, and T is absolute temperature in Kelvin. Temperature must be in Kelvin — this formula fails completely with Celsius.
- Fourth-power dependence: Doubling absolute temperature increases radiated power 2⁴ = 16-fold. The Sun (T ≈ 5,778 K) radiates ~10⁴× more power per unit area than Earth (T ≈ 255 K effective). Small temperature changes have dramatic effects at high temperatures.
- Emissivity ε: A perfect blackbody has ε = 1. Human skin: ε ≈ 0.97 (emits nearly as a blackbody in infrared). Polished aluminum: ε ≈ 0.05 (reflects most radiation — why space blankets are aluminized). Emissivity varies with wavelength; the formula uses the integrated value.
- Wien's displacement law (companion formula): λ_max = 2,898 µm·K / T: The peak wavelength of thermal radiation shifts to shorter (bluer) wavelengths at higher temperatures. The Sun peaks in visible yellow-green (~500 nm); room-temperature objects (~300 K) peak in mid-infrared (~10 µm) — invisible to the eye but detectable by thermal cameras.
- Applications: Stefan-Boltzmann law is used in stellar luminosity calculation (measuring star temperatures from colors), Earth's energy balance (radiative equilibrium), furnace and kiln design, infrared camera calibration, spacecraft thermal control, and pyrometry (non-contact temperature measurement).