Stefan-Boltzmann Law Calculator
This calculator applies the Stefan-Boltzmann law to find the total power radiated by a surface given its temperature, area and emissivity, or to reverse-solve for the temperature that produces a known radiated power. Choose a material from the preset list to fill emissivity automatically, or enter your own. Switch between Kelvin, Celsius and Fahrenheit for temperature, and between metric and imperial for area and power output.
Formula
Worked example
The Sun has a radius of about 6.957 × 10^8 m, giving a surface area of about 6.08 × 10^18 m^2. Its effective surface temperature is 5778 K and its emissivity is approximately 1 (it behaves as a near-perfect black body). Applying the Stefan-Boltzmann law: P = 5.6704 × 10^-8 × 1 × 6.08 × 10^18 × 5778^4 ≈ 3.83 × 10^26 W. This matches the measured solar luminosity. The peak emission wavelength from Wien's law is 2,897,772 / 5778 ≈ 501 nm, which is green visible light.
What is the Stefan-Boltzmann law?
The Stefan-Boltzmann law describes how much thermal radiation a surface emits per unit time. It states that the total power radiated is proportional to the fourth power of the absolute temperature (in Kelvin) of the surface. This fourth-power relationship means radiation is extremely sensitive to temperature: doubling the temperature increases the emitted power by a factor of 16. The law was formulated experimentally by Josef Stefan in 1879 and derived theoretically by Ludwig Boltzmann in 1884 from thermodynamic principles. The constant of proportionality, the Stefan-Boltzmann constant (sigma), has the value 5.6704 × 10^-8 W/(m^2 K^4) and is a combination of fundamental constants of nature: the Boltzmann constant, the speed of light and Planck's constant.
Understanding emissivity
A perfect black body absorbs all radiation that strikes it and emits the maximum possible radiation for its temperature, giving it an emissivity of exactly 1. Real surfaces are described as gray bodies: they emit some fraction of the black body value, quantified by the emissivity (e), a dimensionless number between 0 and 1. Most non-metallic surfaces such as skin, concrete, water and asphalt have emissivities above 0.85, meaning they are efficient emitters. Polished metals are the opposite: their smooth surfaces reflect most radiation and have emissivities as low as 0.02-0.10. Emissivity depends on the material, its surface finish, and its temperature, and the values in the reference table above are approximate. When precise results are needed, consult a material-specific datasheet.
Wien's displacement law and peak wavelength
Wien's displacement law (a consequence of Planck's blackbody radiation formula) states that the wavelength at which a surface emits most intensely is inversely proportional to its temperature: lambda_max = b / T, where b = 2.898 × 10^-3 m K (or 2,897,772 nm K). At room temperature (about 300 K), this gives a peak around 9,660 nm, deep in the infrared, which is why ordinary objects glow in thermal cameras but not to the naked eye. At about 800 K, the peak shifts into the near-infrared and objects start to glow a dim red. The Sun (at about 5778 K) peaks at roughly 500 nm in the green part of the visible spectrum. This calculator shows the peak wavelength and identifies the spectral region automatically.
Applications in science and engineering
The Stefan-Boltzmann law has broad applications. In astrophysics, it is used to estimate the luminosity and effective surface temperatures of stars directly from observations. In thermal engineering, it is central to radiative heat transfer calculations for insulation, spacecraft thermal control, furnace design and building energy modeling. Infrared thermometers use it to infer a surface temperature from the power of emitted radiation, which is why emissivity must be set correctly on the instrument. Climate science uses it to model how much energy Earth radiates back to space and how the greenhouse effect alters the energy balance. In semiconductor fabrication, it governs how wafers heat and cool inside rapid thermal processing chambers.
Emissivity of common materials
| Material | Emissivity (e) | Notes |
|---|---|---|
| Black body (theoretical) | 1.00 | Perfect emitter |
| Human skin | 0.97-0.99 | Regardless of pigmentation |
| Water (liquid) | 0.95-0.97 | Nearly ideal emitter |
| Ice | 0.96-0.98 | Near black body |
| Concrete | 0.91-0.93 | Common building material |
| Brick | 0.90-0.95 | Varies with color and age |
| Glass | 0.84-0.92 | Ordinary window glass |
| Asphalt | 0.88-0.93 | Road surface |
| Wood | 0.85-0.92 | Varies by species and finish |
| Oxidized steel | 0.70-0.85 | Rust and scale increase emissivity |
| Polished steel | 0.05-0.10 | Mirror-like surface |
| Aluminum foil | 0.03-0.07 | Highly reflective |
Typical emissivity values at ordinary temperatures. These are approximate: surface finish, oxidation and temperature all affect the actual value.
Frequently asked questions
What is the Stefan-Boltzmann constant?
The Stefan-Boltzmann constant (sigma) is 5.6704 × 10^-8 W/(m^2 K^4). It sets the proportionality between a surface's absolute temperature to the fourth power and the power it radiates per unit area as a perfect black body. The constant is derived from more fundamental constants: sigma = (2 pi^5 k_B^4) / (15 h^3 c^2), where k_B is the Boltzmann constant, h is Planck's constant and c is the speed of light.
Why does the law use T^4 (temperature to the fourth power)?
The fourth-power dependence comes from integrating Planck's blackbody radiation formula over all wavelengths. Planck's law describes how much energy a perfect emitter radiates at each wavelength for a given temperature. When you sum this over the entire spectrum, the total power comes out proportional to T^4. Ludwig Boltzmann derived this from classical thermodynamics in 1884, and it is one of the first places that the fourth-power law appears naturally from first principles.
What is emissivity and how does it affect the calculation?
Emissivity (e) is a dimensionless number from 0 to 1 that measures how efficiently a surface emits thermal radiation compared to a perfect black body at the same temperature. A black body has e = 1; polished metals can have e as low as 0.03. The radiated power is multiplied by emissivity, so a surface with e = 0.5 radiates half as much power as a perfect black body at the same temperature. In this calculator, you can either select a material preset to fill emissivity automatically or enter a custom value.
How do I reverse-solve for temperature from radiated power?
Select "Temperature (T)" from the "Solve for" dropdown. Then enter the known radiated power, surface area and emissivity. The calculator rearranges the Stefan-Boltzmann law as T = (P / (sigma × e × A))^(1/4) and solves directly. This is the principle behind non-contact infrared thermometers: they measure the power of emitted radiation and back-calculate the surface temperature.
What is Wien's displacement law and why is it shown here?
Wien's displacement law relates the temperature of a body to the wavelength at which it emits most intensely: lambda_max = b / T, where b = 2.898 × 10^-3 m K. It is shown alongside the Stefan-Boltzmann result because they both characterize the same emitting body. The Stefan-Boltzmann law gives the total power; Wien's law tells you the color (spectral peak) of that emission. Together they explain why hot objects glow red, then orange, then white as they get hotter, and why stars of different temperatures appear different colors.
Does the calculator work for cooling or heat loss calculations?
Yes. The radiated power is the net emission from the surface at a given temperature. In many practical heat-transfer problems, you also need to subtract the radiation absorbed from the surroundings (which follows the same formula using the surroundings temperature). For the full net radiation heat transfer between a surface at T1 and surroundings at T2, use P_net = sigma × e × A × (T1^4 - T2^4). This calculator computes the gross emission from one surface; subtract the surroundings contribution separately for a net heat-loss figure.