Stefan-Boltzmann Law
The Stefan-Boltzmann law P = σAT⁴ calculates the total power radiated by a blackbody.
Learn thermal radiation with worked examples.
The Formula
The Stefan-Boltzmann law tells us how much total energy a hot object radiates per second. It applies to idealized objects called blackbodies — perfect absorbers and emitters of radiation.
The remarkable feature of this law is the fourth-power dependence on temperature. Doubling the temperature of an object increases its radiated power by a factor of 16 (2⁴ = 16).
This formula was discovered experimentally by Josef Stefan in 1879 and derived theoretically by Ludwig Boltzmann in 1884. It is essential in astrophysics for calculating the luminosity of stars and in engineering for thermal design.
Variables
| Symbol | Meaning |
|---|---|
| P | Total power radiated (measured in watts, W) |
| σ | Stefan-Boltzmann constant = 5.67 × 10⁻⁸ W/(m²·K⁴) |
| A | Surface area of the radiating object (measured in square meters, m²) |
| T | Absolute temperature of the surface (measured in kelvin, K) |
For Real Objects
Real objects are not perfect blackbodies, so we include emissivity (ε):
Emissivity ranges from 0 to 1, where 1 is a perfect blackbody and lower values represent shinier, more reflective surfaces.
Example 1
The Sun has a surface temperature of about 5,778 K and a radius of 6.96 × 10⁸ m. What is its total radiated power (luminosity)?
Calculate surface area: A = 4πr² = 4π × (6.96 × 10⁸)²
A = 4π × 4.844 × 10¹⁷ = 6.087 × 10¹⁸ m²
Apply the formula: P = σAT⁴ = 5.67 × 10⁻⁸ × 6.087 × 10¹⁸ × (5,778)⁴
T⁴ = (5,778)⁴ = 1.115 × 10¹⁵
P = 5.67 × 10⁻⁸ × 6.087 × 10¹⁸ × 1.115 × 10¹⁵
P ≈ 3.85 × 10²⁶ W (the Sun's luminosity)
Example 2
A steel sphere with radius 0.1 m is heated to 800 K. Its emissivity is 0.6. How much power does it radiate?
Surface area: A = 4πr² = 4π × (0.1)² = 0.1257 m²
T⁴ = (800)⁴ = 4.096 × 10¹¹
P = εσAT⁴ = 0.6 × 5.67 × 10⁻⁸ × 0.1257 × 4.096 × 10¹¹
P = 0.6 × 5.67 × 10⁻⁸ × 5.149 × 10¹⁰
P ≈ 1,751 W (about 1.75 kW)
When to Use It
Use the Stefan-Boltzmann law for thermal radiation problems.
- Calculating the luminosity and temperature of stars
- Designing thermal insulation and heat shields
- Estimating heat loss from furnaces and ovens
- Determining the equilibrium temperature of planets
- Engineering infrared heating systems