Stefan-Boltzmann Law Calculator
Enter emissivity, surface area and temperature to compute blackbody radiated power. Human body ε=0.97, A=1.5 m², T=310 K → P≈744 W.
Input Data
Results
At a glance:The Stefan-Boltzmann law states that the total power radiated per unit area by a blackbody is proportional to the fourth power of its absolute temperature: j = σT⁴, and for a real surface with emissivity ε and area A the total radiated power is P = σεAT⁴, where σ = 5.670374419e-8 W/(m²·K⁴) is the Stefan-Boltzmann constant. The law was deduced empirically by Josef Stefan in 1879 from John Tyndall's measurements and derived theoretically by Ludwig Boltzmann in 1884 from thermodynamics. It underlies essentially all of thermal-radiation engineering. The fourth-power dependence means doubling the temperature increases the radiated power 16-fold: a body at 1000 K radiates about 123 times more power per area than at 300 K. The peak wavelength follows Wien's displacement law λ_max = b/T (b≈2.898e-3 m·K): a human at 310 K peaks near 9.35 μm (far infrared), while the Sun at 5778 K peaks at 0.50 μm (visible green).
Formula
Radiated power: P = σ·ε·A·T⁴
Radiant exitance (power density): j = σ·ε·T⁴
Peak wavelength: λ_max = b / T (b ≈ 2.898×10⁻³ m·K)
$$P = \sigma\,\varepsilon\,A\,T^{4}, \quad j = \sigma\,\varepsilon\,T^{4}, \quad \lambda_{\max} = \frac{b}{T}$$How to Use
- Enter the emissivity ε (0–1) of the surface.
- Enter the surface area A (m²) and the temperature T (K).
- The calculator returns radiated power P, power density j and the peak wavelength λ_max.
Case Studies
Human body radiation
Skin ε≈0.97, A≈1.5 m², T=310 K gives P = 5.67e-8 × 0.97 × 1.5 × 310⁴ ≈ 744 W.
In reality clothing and convection reduce the net loss, but the radiative component is large.
This is why thermal-imaging cameras see people clearly at room temperature.
Stellar and solar radiation
The Sun (T=5778 K, ε≈1) has a peak wavelength of about 0.50 μm, in the visible band.
Radiated power scales with T⁴, so a star only twice as hot radiates 16× more per unit area.
Earth's radiative balance uses the same law to estimate the effective radiating temperature of about 255 K.
FAQ
Why is radiated power proportional to T⁴?
Integrating Planck's blackbody spectrum u(λ,T)=8πhc/λ⁵·1/(e^(hc/λkT)-1) over all wavelengths gives P=σεT⁴. The T⁴ arises from (1) each photon's energy ∝kT and (2) the photon number density ∝T³, their product ∝T⁴. Boltzmann derived it thermodynamically; Planck gave the quantum microscopic basis in 1900.
What is emissivity ε?
Emissivity is the ratio of a real surface's radiating ability to that of an ideal blackbody (0–1). A blackbody has ε=1, a perfect mirror ε=0. By Kirchhoff's law, at thermal equilibrium absorptivity equals emissivity. Black objects both absorb and emit well (ε≈0.9); polished metals absorb and emit poorly (ε≈0.05). Human skin ε≈0.97 in the infrared regardless of color.
Why don't room-temperature objects glow?
At 300 K the radiated power density is about 460 W/m² but the peak wavelength is 9.66 μm (far infrared), invisible to the eye. Visible light spans 400–700 nm, corresponding to blackbody temperatures of 4140–7250 K, so an object needs to exceed about 800 K to start glowing dull red.
Why does fusion need extremely high temperature?
Deuterium-tritium fusion must overcome Coulomb repulsion, requiring plasma temperatures >1e8 K. At that temperature the plasma's radiative loss P=σεT⁴ grows enormously (bremsstrahlung, synchrotron), so if losses exceed fusion gain the plasma quenches. The Lawson criterion nTτ > threshold guarantees net energy; ITER and NIF pursue ignition.
How does Earth reach radiative balance?
Earth absorbs solar radiation S=1361 W/m² × πR² × (1-α) with albedo α=0.3 and radiates P=4πR²σεT⁴. Balance gives T=(S(1-α)/(4σ))^(1/4) ≈ 255 K (-18 °C). The actual surface is 288 K (+15 °C) because the greenhouse effect raises the effective radiating altitude. The greenhouse effect adds ΔT≈33 °C.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.