Table of Contents
The Stefan-Boltzmann Law
The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body per unit time is proportional to the fourth power of the body's absolute temperature. Discovered experimentally by Josef Stefan in 1879 and derived theoretically by Ludwig Boltzmann in 1884, this law is fundamental to understanding thermal radiation, stellar luminosity, and heat transfer.
The law explains why doubling an object's temperature increases its radiated power by a factor of 16 (2^4). This extreme sensitivity to temperature is why stars, which have surface temperatures of thousands of Kelvin, emit enormous amounts of energy. The Sun at 5,778 K emits about 3.846 × 10^26 watts of radiant power.
Formulas
For net radiation between a body and its surroundings: P_net = εσA(T^4 - T_ambient^4). Wien's displacement law gives the peak emission wavelength: λ_max = 2.898 × 10^-3 / T meters.
Emissivity Values
| Surface | Emissivity (ε) |
|---|---|
| Perfect black body | 1.00 |
| Human skin | 0.98 |
| Water | 0.96 |
| Asphalt | 0.93 |
| Concrete | 0.92 |
| Oxidized steel | 0.79 |
| Polished aluminum | 0.04 |
| Polished gold | 0.02 |
Applications
- Astrophysics: Calculating stellar luminosity and surface temperature.
- Climate science: Modeling Earth's radiation budget and greenhouse effect.
- Engineering: Designing furnaces, heat shields, and thermal insulation.
- Infrared thermometry: Non-contact temperature measurement using emitted radiation.
Frequently Asked Questions
What is a black body?
A black body is an idealized object that absorbs all incident electromagnetic radiation and emits the maximum possible radiation at every wavelength for its temperature. No real object is a perfect black body, but many come close (soot, carbon black, cavities with small openings).
Why does power depend on T^4?
The T^4 dependence arises from integrating Planck's radiation law over all wavelengths. Each wavelength's emission depends on temperature, and when summed across the entire spectrum, the total scales as T^4. This was one of the great successes of early quantum theory.
How does emissivity affect the result?
Emissivity (ε) scales the radiation linearly. A surface with ε = 0.5 emits half the radiation of a black body at the same temperature. Highly polished metals have very low emissivity (0.02-0.05), which is why they are used as thermal radiation shields.