This Stefan-Boltzmann law calculator finds how much power a surface radiates purely from being hot — P = εσAT⁴ — with emissivity, area and temperature, the T⁴ law that governs stars, furnaces and building heat loss alike.

–

Heat You Can Count in Light

Josef Stefan found the rule empirically in 1879 and Boltzmann derived it from thermodynamics five years later: a surface radiates power P = εσAT⁴, where σ = 5.670374×10⁻⁸ W/(m²·K⁴), A is area, ε is emissivity (1 for a perfect blackbody) and T is absolute temperature. The fourth power makes the numbers theatrical. One square metre of the Sun’s surface (5,778 K, ε ≈ 1) radiates 63.2 MW — multiply by the Sun’s full surface and you get its luminosity, 3.83×10²⁶ W, from a single line of arithmetic. A human body at 310 K radiates ~525 W/m² gross — over 900 W for a whole skin area, a figure that startles until you remember the surroundings radiate nearly as much back, leaving the familiar ~100 W net that a resting metabolism replaces. And a room-temperature wall at 293 K still emits ~420 W/m²: the “cold” world glows substantially in infrared, which is why radiative exchange is a first-class term in building physics, spacecraft thermal design (radiators are the ONLY way to dump heat in vacuum) and climate science, where Earth’s energy balance is quite literally an equation between absorbed sunlight and σT⁴ exhaust.

Used well, the law is a chain of honest choices: kelvin, always (a 25 °C input typed as “25” instead of 298 K understates the radiation by a factor of ~20,000); emissivity from the actual material (skin 0.97–0.98, oxidized metal 0.6–0.85, polished aluminium 0.03–0.05 — the variable that most often wrecks estimates); and net vs gross kept straight — engineering loss is εσA(T⁴ − Tₛ⁴), with Tₛ the surroundings’ temperature, and at small temperature differences the net is a small difference of two large T⁴ numbers, where input sloppiness explodes. The law’s T⁴ also explains stellar demographics at a glance: luminosity classes spread over factors of millions mainly through T⁴ and R², and the hottest main-sequence stars outshine the Sun per square metre by (40,000/5,778)⁴ ≈ 2,300×. Combined with Wien’s law — which supplies the spectrum’s peak — Stefan-Boltzmann completes the blackbody’s quantitative portrait: how much light, and where in the spectrum, from temperature alone.

Worked Examples (Verified)

  • Sun’s surface: 5,778 K, ε 1, 1 m² → flux 63.20 MW/m².
  • Earth’s surface average: 288 K → 390 W/m² gross upward radiation (the greenhouse story starts here).
  • Human skin: 307 K, ε 0.97 → ~489 W/m² gross; net vs a 20 °C room ≈ 83 W/m².
  • Furnace wall: 1,200 K → 117.6 kW/m² — why furnace doors are small and shielded.

Stefan-Boltzmann Discipline

  • Kelvin, non-negotiable: T⁴ amplifies unit errors monstrously. 300 K vs “300 °C” (573 K) differ 13-fold in output.
  • Emissivity is a material AND surface property: the same aluminium is 0.04 polished, 0.85 anodized/oxidized. Guess it and the answer is fiction.
  • Gross vs net: against cold space, gross ≈ net; in a room, net (T⁴ − Tₛ⁴) is the physically meaningful loss.
  • Small ΔT danger: when T ≈ Tₛ, net radiation is a difference of near-equal giants — measure temperatures carefully or the net flips sign.
  • The law prices THERMAL radiation only; reflected sunlight or lamp light bouncing off the surface is additional, not included.

Related Tools

Frequently Asked Questions

How do you calculate radiated power with Stefan-Boltzmann?

P = εσAT⁴: emissivity × 5.670374×10⁻⁸ × area in m² × (temperature in kelvin)⁴. A perfect blackbody square metre at the Sun’s 5,778 K radiates 6.32×10⁷ W. This calculator returns both the flux (per square metre) and the total power for your area, with emissivity adjustable from polished metal (~0.05) to skin and matte surfaces (~0.95). For NET loss against surroundings, subtract their σTₛ⁴ term — the FAQ below explains when that matters.

Why does temperature enter to the fourth power?

Two compounding effects: as T rises, each photon carries more energy (the spectrum shifts up, Wien’s law) AND the number of photons emitted per second rises steeply — Planck’s law integrated over all wavelengths yields exactly T⁴. Boltzmann’s 1884 derivation got it from pure thermodynamics (radiation pressure in an expanding cylinder), one of the great pre-quantum results; Planck’s later quantum law contains it as a corollary. Practically: 2× hotter means 16× the radiation — the steepest common scaling in everyday physics, and the reason high-temperature engineering is radiation engineering.

What is emissivity?

The fraction of blackbody radiation a real surface emits at the same temperature: ε = 1 for a perfect blackbody, 0.95–0.98 for human skin and water, ~0.9 for brick and wood, 0.6–0.85 for oxidized metals, 0.03–0.05 for polished silver or aluminium. By Kirchhoff’s law, good absorbers are good emitters at the same wavelength — which is why black surfaces both heat fastest in sun and radiate best at night, and why spacecraft wrap in shiny low-ε foil to hold heat. Emissivity is the input that most determines whether a Stefan-Boltzmann estimate is engineering or astrology.

How much power does a human body radiate?

Gross: skin at ~307 K with ε ≈ 0.97 radiates about 489 W/m², roughly 880 W over the body’s ~1.8 m² — comparable to a small space heater, and true right now. But the room radiates back: walls at 293 K emit ~418 W/m², of which the skin absorbs most, so the NET radiative loss shrinks to roughly 80 W per square metre — and with clothing moving the radiating surface cooler, the body’s total heat loss settles near resting metabolic output (~100 W), split between radiation, convection and evaporation. Clothing works mostly by moving the radiating surface to a cooler temperature (and trapping air against convection), not by changing emissivity — your coat’s outer surface, not your skin, sets the T in its T⁴.

How is Stefan-Boltzmann used in climate science?

It IS Earth’s thermostat equation. The planet must radiate to space what it absorbs from the Sun (~240 W/m² averaged): solving σT⁴ = 240 gives an effective emission temperature of 255 K — colder than the 288 K surface, because much of the radiation to space leaves from high, cold atmosphere. Greenhouse gases raise the altitude (and lower the temperature) of that effective emission, so the surface warms until the books balance again. Every climate model, however elaborate, is bookkeeping wrapped around this T⁴ exhaust pipe.

What is the difference between gross and net radiation?

Gross is what the surface emits: εσT⁴, always on, in all directions. Net is emission minus absorbed incoming radiation — for a surface facing surroundings at Tₛ, approximately εσ(T⁴ − Tₛ⁴). The distinction decides real problems: a satellite panel facing deep space (Tₛ ≈ 3 K) loses essentially its gross output; a person in a warm room loses only the small net; a roof under a cold clear sky can net-radiate below air temperature (that’s how dew and frost form on clear nights). Match the variant to the problem before trusting the number.

Can Stefan-Boltzmann find a star’s temperature or size?

Yes — it’s the working tool of stellar astrophysics. Measure a star’s luminosity L (from brightness and distance) and its temperature T (from spectrum/Wien), and the radius follows from L = 4πR²σT⁴ — this is how we know Betelgeuse spans ~700 solar radii while a white dwarf packs similar heat into an Earth-sized body. Run it forward, as here, and a measured T and R predict L. The trio L, R, T bound by σ is the Hertzsprung–Russell diagram’s quantitative skeleton.

Is this Stefan-Boltzmann law calculator free?

Yes — free, instant, no sign-up. Enter temperature in kelvin, area and emissivity for radiant flux and total radiated power.