Stefan-Boltzmann Law
The Stefan-Boltzmann law calculates total radiant energy emitted by a black body using P = σAT⁴.
Learn with worked examples.
The Formula
The Stefan-Boltzmann law describes the total power radiated per unit time by a perfect black body. It shows that radiated power grows extremely rapidly with temperature, scaling with the fourth power.
This law was first discovered experimentally by Josef Stefan in 1879 in Austria. Ludwig Boltzmann later derived it theoretically in 1884 from thermodynamic principles. It is fundamental to understanding stellar luminosity, thermal radiation, and heat loss from hot objects.
The law applies to ideal black bodies that absorb all incident radiation. Real objects emit less radiation, which is accounted for by including an emissivity factor (ε) between 0 and 1. The modified form becomes P = εσAT⁴.
Variables
| Symbol | Meaning |
|---|---|
| P | Total radiated power (watts, W) |
| σ | Stefan-Boltzmann constant (5.67 × 10⁻⁸ W/m²K⁴) |
| A | Surface area of the radiating body (m²) |
| T | Absolute temperature of the body (kelvin, K) |
Example 1
A black body sphere with radius 0.1 m is heated to 500 K. What is its total radiated power?
Calculate the surface area: A = 4πr² = 4π(0.1)² = 0.1257 m²
Apply the formula: P = σAT⁴ = 5.67 × 10⁻⁸ × 0.1257 × 500⁴
P = 5.67 × 10⁻⁸ × 0.1257 × 6.25 × 10¹⁰
P ≈ 445.3 W
Example 2
The Sun has a surface temperature of about 5,778 K and a radius of 6.96 × 10⁸ m. What is its total luminosity?
Surface area: A = 4πr² = 4π(6.96 × 10⁸)² = 6.087 × 10¹⁸ m²
Apply the formula: P = σAT⁴ = 5.67 × 10⁻⁸ × 6.087 × 10¹⁸ × 5778⁴
T⁴ = 5778⁴ = 1.115 × 10¹⁵
P = 5.67 × 10⁻⁸ × 6.087 × 10¹⁸ × 1.115 × 10¹⁵
P ≈ 3.85 × 10²⁶ W (matching the measured solar luminosity)
When to Use It
Use the Stefan-Boltzmann law whenever you need to calculate thermal radiation from a hot object.
- Estimating the luminosity of stars from their surface temperature
- Calculating heat loss from furnaces, engines, or heated surfaces
- Determining the equilibrium temperature of planets
- Designing thermal insulation and radiative cooling systems