Calculate thermal radiation power using P = εσAT⁴. Find radiated power, net heat transfer, peak wavelength, and spectrum position for any temperature.
The **Stefan-Boltzmann Law Calculator** computes thermal radiation power using P = εσAT⁴, the relationship that describes how hot objects radiate energy. Because the output scales with the fourth power of absolute temperature, even a modest temperature change can have a large effect on radiated power.
The Stefan-Boltzmann constant links thermodynamics to electromagnetism, and the related Wien displacement law gives the approximate peak wavelength of the emitted spectrum. Together, they let you estimate both the strength and the color shift of thermal radiation.
This calculator reports total and net radiated power, peak emission wavelength, a spectrum view, and emissivity-based grey-body support so you can compare real surfaces with ideal blackbody behavior.
Thermal radiation becomes important whenever heat transfer happens without contact, especially for furnaces, spacecraft, infrared sensing, and hot surfaces in air or vacuum.
Seeing the temperature, emissivity, and surface-area terms together makes it easier to judge how strongly a surface radiates and how much the result changes when any one of those inputs changes.
P = εσAT⁴ Where: P = radiated power (W), ε = emissivity (0-1), σ = 5.670 × 10⁻⁸ W/(m²·K⁴), A = surface area (m²), T = absolute temperature (K) Wien displacement: λ_max = 2897.8 / T (μm) Net radiation: P_net = εσA(T⁴ − T_ambient⁴)
Result: 6.32 × 10⁷ W/m²
P/A = (1)(5.670e-8)(5778⁴) = 6.32 × 10⁷ W/m². The Sun radiates 63.2 MW per square meter of its surface. With the full solar surface area (~6.08 × 10¹⁸ m²), the total luminosity is 3.85 × 10²⁶ W. Peak wavelength = 2898/5778 = 502 nm (green-yellow visible light).
Josef Stefan discovered the T⁴ law experimentally in 1879; Ludwig Boltzmann derived it theoretically in 1884. The law was one of the first connections between thermodynamics and electromagnetic theory, preceding quantum mechanics by two decades.
The blackbody radiation spectrum described by Planck's law gives the intensity at each wavelength. Integrating over all wavelengths yields the Stefan-Boltzmann law. The spectrum peaks at a wavelength inversely proportional to temperature (Wien's law). Together, these relationships completely characterize thermal radiation.
**Stellar Classification:** Stars are classified by their surface temperature, which determines their color and luminosity. O-type stars (>30,000 K) appear blue-white and radiate >10⁵ times the Sun. M-type stars (<3,700 K) appear red and radiate <0.01 solar luminosities. The Hertzsprung-Russell diagram plots this luminosity-temperature relationship.
**Planetary Energy Balance:** Earth absorbs solar radiation (input = solar constant × cross-section × (1 - albedo)) and emits thermal radiation (output = εσ × surface area × T⁴). Setting input = output determines the equilibrium temperature — about 255 K (-18°C) without greenhouse gases, 288 K (15°C) with them.
**Furnace Design:** Industrial furnaces at 1,000-1,500°C transfer most of their heat by radiation. Furnace walls, burner geometry, and load placement are designed to optimize radiative heat transfer using view factors and emissivity data.
**Thermal Insulation:** Low-emissivity coatings (like aluminum foil) block radiative heat transfer. Multi-layer insulation (MLI) used on spacecraft consists of many thin aluminized Mylar layers separated by spacers, achieving effective thermal conductivities below 0.001 W/(m·K) — thousands of times better than fiberglass insulation.
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This comes from integrating the Planck blackbody spectrum over all wavelengths. The T⁴ dependence emerges because both the peak intensity and the width of the emission spectrum increase with temperature, compounding the effect.
Emissivity (ε) is the ratio of a surface's radiation to that of a perfect blackbody at the same temperature. Polished metals: ε ≈ 0.03-0.1. Oxidized metals: ε ≈ 0.4-0.8. Water, skin, most non-metals: ε ≈ 0.9-0.98.
The Sun's peak emission is at 502 nm (green), but it emits strongly across the entire visible spectrum. Our eyes perceive this broad-spectrum emission as white. Atmospheric scattering shifts the apparent color toward yellow at lower angles.
Radiation dominates at high temperatures (above ~500°C in air) because of the T⁴ dependence. At room temperature, convection typically dominates for surfaces in air. In vacuum, radiation is the only heat transfer mechanism.
Vantablack (ε > 0.999), carbon nanotube forests, and blackbody cavity simulators approach ideal behavior. Soot and matte black paint are good approximations (ε ≈ 0.95-0.97) for engineering calculations.
In space, with no air for conduction or convection, thermal radiation is the only way to reject heat. Spacecraft radiators must balance solar absorption and thermal emission to maintain safe operating temperatures.