Every object above absolute zero radiates. A cast-iron pan, a person, a planet and a star are all doing the same thing, differing only in temperature and in how efficiently their surfaces couple to the electromagnetic field. The blackbody is the idealisation where that coupling is perfect: a surface that absorbs everything landing on it and, in thermal equilibrium, emits the maximum any body at that temperature can.
This blackbody radiation calculator from Arb Digital runs the three results that follow from that idealisation. Stefan-Boltzmann gives the total flux. Wien gives where the spectrum peaks. Planck gives the whole curve, so you can ask how much power comes out at one specific wavelength rather than in total. The constants are the exact CODATA values, which since the 2019 SI redefinition are defined rather than measured, so the arithmetic has no experimental uncertainty in it at all.
What This Blackbody Radiation Calculator Does
You enter a temperature in kelvin, Celsius or Fahrenheit, an emissivity, a radiating area and a wavelength you care about. The headline is the total radiant exitance in watts per square metre, which is what the surface loses to radiation across the whole spectrum.
The grid breaks that down. Peak wavelength is where the spectral curve is highest, from Wien's displacement law. Power from the area is the exitance multiplied by the area you entered, which is the number you want when sizing a radiator or estimating how much a hot component sheds. Spectral exitance at your chosen wavelength comes from the Planck function and is quoted per nanometre, because that is the unit spectrometer output usually arrives in. Photon energy at the peak converts the peak wavelength into electronvolts, which is the natural currency for anything involving detectors, band gaps or photochemistry.
The note underneath names the band the peak falls in, which is the fastest sanity check available. A room-temperature object peaks deep in the infrared, a heating element in the near infrared, and only above about 3,000 K does the peak climb far enough to make the object look properly white rather than orange.
How to Use It
- Enter the surface temperature, not the internal one. Radiation depends on the temperature of the emitting face. A furnace at 1,200 K with a 400 K casing radiates as a 400 K body from the outside.
- Set emissivity honestly. Leave it at 1 for a theoretical blackbody; drop it to a real material value when you want a real answer, because the flux scales with it linearly.
- Put in the area only if you want total power. The exitance figure is per square metre regardless.
- Choose a wavelength that matters to your problem. A sensor's band, a filter's centre, or an absorption line — the spectral exitance answers "how much is available here" rather than "how much in total".
- Read the band note. If the peak is in a part of the spectrum you did not expect, check the temperature scale before you check anything else.
The Three Laws, and a Worked Example
Stefan-Boltzmann gives the total: M = εσT⁴, where σ is the Stefan-Boltzmann constant. NIST lists it as exactly 5.670374419… × 10⁻⁸ W m⁻² K⁻⁴, exact because it is built from the defined values of the Planck and Boltzmann constants and the speed of light. The fourth power is the whole story of thermal radiation: doubling the absolute temperature multiplies the radiated flux by sixteen.
Wien's displacement law gives the peak: λmax = b/T, where b is the Wien wavelength displacement constant, 2.897771955… × 10⁻³ m K. Hotter means shorter, and the relationship is a simple inverse.
Planck's law gives the shape. Written as spectral exitance per unit wavelength it is Mλ = 2πhc² / [λ⁵(e^(hc/λkT) − 1)], with h the Planck constant, exactly 6.62607015 × 10⁻³⁴ J Hz⁻¹. Integrating it over all wavelengths reproduces Stefan-Boltzmann exactly, and differentiating it and setting the result to zero reproduces Wien. The two simpler laws are consequences of this one.
Work the default through. At T = 5,772 K with ε = 1, Wien gives λmax = 2.897772 × 10⁻³ / 5,772 = 5.020 × 10⁻⁷ m, or 502 nm — green-yellow, right in the middle of where human vision is most sensitive, which is not a coincidence. Stefan-Boltzmann gives M = 5.670374 × 10⁻⁸ × 5,772⁴ = 6.294 × 10⁷ W/m².
That second figure is checkable against something everybody knows. Spread the Sun's surface flux over the sphere at Earth's distance by multiplying by the square of the ratio of solar radius to orbital radius: 6.294 × 10⁷ × (6.957 × 10⁸ / 1.496 × 10¹¹)² = 1,361 W/m². That is the solar constant, measured at the top of the atmosphere, to four figures. The 5,772 K effective temperature is in fact defined so that this works out.
The Peak Moves If You Plot Against Frequency Instead
This is the single most common trap in blackbody work, and it catches physics graduates as often as anyone. The Wien constant above gives the peak of the spectrum plotted per unit wavelength. Plot the same radiation per unit frequency and the peak lands somewhere else entirely — at a wavelength about 1.76 times longer.
Nothing physical has changed. The problem is that "power per unit wavelength" and "power per unit frequency" are different densities over differently stretched axes. Converting between them requires a Jacobian factor of c/λ², and multiplying a curve by something that varies steeply across the axis moves where its maximum sits. Asking "what wavelength does a 5,772 K body peak at" therefore has two correct answers: 502 nm in the wavelength form and about 882 nm in the frequency form.
The practical rule is to state which form you mean and never mix them. This tool uses the wavelength form throughout, both for the peak and for the spectral exitance, because that is what optical and thermal engineering conventionally uses and what spectrometers report. If you are reading a radio-astronomy or spectroscopy paper working in frequency units, expect the other convention and check before comparing numbers.
Emissivity Is the Input That Is Not Physics
Everything else on this page is exact. Emissivity is not. It is a property of a real surface, it varies with wavelength, with temperature, with viewing angle and with how oxidised or dirty the surface happens to be, and the single number you type here is an average that hides all of that.
The spread is enormous. Polished aluminium sits near 0.05, so it radiates about five percent of what a blackbody at the same temperature would. Anodise or oxidise the same aluminium and it climbs above 0.8. Most non-metals — paint, ceramic, wood, skin, water — cluster between 0.9 and 0.98 regardless of colour, which is why visual colour tells you almost nothing about thermal emissivity. Human skin is close to 0.98 in the thermal infrared whatever its visible shade.
Two consequences follow. First, an infrared thermometer aimed at bare metal reads badly low unless you correct for emissivity, which is why technicians stick a patch of matt tape on shiny surfaces before measuring. Second, in a design where radiation matters, emissivity is a variable you get to choose. Spacecraft radiators are coated to push it towards 1; multi-layer insulation is aluminised to push it towards 0. Same physics, opposite goals.
Emission Is Not the Same as Net Loss
The number this tool returns is gross emission. A real object also absorbs radiation from whatever surrounds it, and what you feel or measure is the difference. For a body at temperature T in surroundings at Tsurr, the net radiative flux is εσ(T⁴ − Tsurr⁴).
The difference is often the entire answer. Skin at 306 K emits about 487 W/m² at ε = 0.98, which sounds like far more heat than a person produces. But a room at 293 K sends back about 410 W/m², so the net loss is only around 77 W/m². Over roughly 1.8 m² of body surface, of which perhaps 80 percent radiates freely, that is about 111 W — the right order for a resting adult. Take the gross figure alone and you would conclude a person radiates enough to power a kettle.
The same trap appears in building physics, satellite thermal design and cooking. A pizza oven at 700 K does not simply dump 13 kW/m² into the base; the base warms and radiates back, and the net rate falls as it does. Any time an answer from this page looks implausibly large, the missing term is almost always the surroundings.
Where This Sits Next to Our Other Tools
This page handles thermal emission across the whole spectrum. If you want the energy of a single photon at a given wavelength or frequency rather than a thermal distribution, the photon energy calculator is the direct route, and the wavelength calculator converts between wavelength, frequency and wave speed. For the discrete emission lines of an atom rather than a continuous thermal spectrum, the Bohr model calculator computes hydrogen-like energy levels and transition wavelengths.
On the practical side, the temperature converter handles scale changes independently, the energy converter moves between joules, electronvolts and calories, and the power converter deals with watts, horsepower and BTU per hour. If you arrived here to work out how much sunlight a roof receives, the solar panel calculator starts from irradiance rather than from the Sun's surface.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using Celsius in the fourth-power law — the T⁴ term needs absolute temperature. At room conditions, using 20 instead of 293 understates the flux by a factor of about 46,000.
- Quoting gross emission as heat loss — the surroundings radiate back, and the net flux is proportional to the difference of the fourth powers, not to one of them.
- Mixing the wavelength and frequency forms of Wien's law — they give peaks that differ by a factor of about 1.76, and both are correct for their own convention.
- Assuming visible colour predicts emissivity — white paint and black paint are both around 0.9 in the thermal infrared, while bright bare metal is near 0.05.
- Reading the peak as though most of the power sits there — the peak is where the density is highest, not where the energy is concentrated. Well over half the flux lies at wavelengths longer than the peak.
Related Free Tools From Arb Digital
Use the photon energy calculator for single-photon energies and the wavelength calculator to move between wavelength and frequency. The Bohr model calculator covers line spectra rather than continuous thermal ones. For units, the temperature converter, energy converter and power converter each handle one axis of the problem, and the solar panel calculator takes irradiance through to generated output. Everything is listed on the free online tools hub.
Frequently Asked Questions
It states that the total power radiated per unit area of a blackbody is the Stefan-Boltzmann constant multiplied by the fourth power of absolute temperature, with an emissivity factor for real surfaces. NIST gives the constant as exactly 5.670374419 times ten to the minus eight watts per square metre per kelvin to the fourth.
Divide the Wien wavelength displacement constant, 2.897771955 times ten to the minus three metre kelvin, by the absolute temperature. At 5,772 kelvin that gives 502 nanometres. The result is in metres, so multiply by a billion to read it in nanometres.
Because power per unit wavelength and power per unit frequency are densities over differently stretched axes, and converting between them introduces a factor that varies across the axis. The frequency-form peak sits at a wavelength about 1.76 times longer. Both are correct within their own convention, so always state which you mean.
Use 1 for a theoretical blackbody. For real surfaces, polished metals are around 0.05, oxidised or anodised metals around 0.8, and most non-metals including paint, ceramic, water and skin fall between 0.9 and 0.98. Visible colour is a poor guide, because white and black paint behave almost identically in the thermal infrared.
No, it gives gross emission. Net radiative loss is the emissivity times the Stefan-Boltzmann constant times the difference between the fourth power of the body's temperature and the fourth power of the surroundings' temperature. Skin emits around 487 watts per square metre but nets only about 77 in a normal room.
That is its effective temperature: the temperature a perfect blackbody would need to radiate the same total flux from the same surface area. Running the Stefan-Boltzmann law at 5,772 kelvin and scaling to Earth's orbital distance reproduces the measured solar constant of about 1,361 watts per square metre.
No. The peak marks where the spectral density is highest, not where the energy accumulates. The Planck curve falls steeply on the short-wavelength side and gently on the long side, so well over half the total flux lies at wavelengths longer than the peak.
This tool is provided for educational and reference use. It models an ideal grey body with a single wavelength-independent emissivity, which real surfaces only approximate, so treat the results as a first-order estimate rather than a substitute for measured thermal data in any safety-critical design.
