Stefan-Boltzmann Law
Calculate total radiated power from a blackbody using temperature.
Stefan-Boltzmann law explained with examples.
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