What is Wien's displacement law?

Wien's displacement law says the wavelength at which a blackbody emits most strongly is inversely proportional to its temperature. The peak wavelength equals the Wien constant, 2.898 thousandths of a metre-kelvin, divided by the temperature in kelvin. Hotter objects peak at shorter wavelengths, which is why a heated metal glows red, then orange, then white as it gets hotter.

How is total radiated power found?

The Stefan-Boltzmann law gives the radiant exitance, the power emitted per square metre of surface, as the emissivity times the Stefan-Boltzmann constant times temperature to the fourth power. Because of that fourth power, doubling the temperature multiplies the radiated power by sixteen, so temperature dominates the output.

A blackbody is an idealised object that absorbs all radiation falling on it and re-emits a smooth spectrum that depends only on its temperature. Real surfaces fall short of this, described by an emissivity between 0 and 1. Stars, heated filaments and even people approximate blackbodies well enough for these laws to give useful estimates.

Blackbody Radiation Calculator

Every object above absolute zero glows with thermal radiation, and this calculator turns a temperature into the numbers that describe that glow. Enter a temperature in kelvin, and optionally an emissivity, and you get the peak emission wavelength, the total radiant exitance, the peak frequency and the part of the spectrum the peak falls in. Two classic laws do the work. Wien's displacement law fixes the wavelength where emission is strongest: it is inversely proportional to temperature, so a cool object peaks deep in the infrared while a hot star peaks in the visible or ultraviolet. The Stefan-Boltzmann law gives the total power radiated per square metre, and because it depends on temperature to the fourth power, small rises in temperature produce large jumps in output. The default temperature of 5778 kelvin is the surface temperature of the Sun, which peaks in green light at around 500 nanometres, even though the Sun looks white because it emits across the whole visible band. You can drop the temperature to model a red-hot element at 1000 kelvin, a light-bulb filament near 2800 kelvin, or human skin near 305 kelvin, which peaks in the far infrared. Set an emissivity below one to model a real, grey surface rather than an ideal blackbody. Change the temperature and every figure updates immediately.

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Temperature

Emissivity (0 to 1)

501.5 nm

peak emission wavelength (Wien's law)

Radiant exitance (W/m²)6.32e+7

Peak frequency (Hz)3.40e+14

Spectral bandVisible (green)

Wien's law and the Stefan-Boltzmann law for an ideal blackbody, scaled by emissivity. Real spectra vary with surface and angle, so treat as an estimate.

How it works

The peak wavelength comes from Wien's displacement law: the Wien constant of 2.897771955 thousandths of a metre-kelvin divided by the temperature, with the result shown in nanometres. The peak frequency uses the frequency form of Wien's law, a constant of about 5.879 times ten to the tenth hertz per kelvin multiplied by the temperature, and it sits at a slightly different point in the spectrum than the wavelength peak because the spectrum is not symmetric. The radiant exitance uses the Stefan-Boltzmann law: emissivity times the Stefan-Boltzmann constant, 5.670374 times ten to the minus eight watts per square metre per kelvin to the fourth, times temperature to the fourth power. The spectral band is worked out by comparing the peak wavelength against the standard boundaries between infrared, visible, ultraviolet and beyond, with the visible range split into its colours.

Worked example

Set the temperature to 5778 kelvin, the Sun's surface, with an emissivity of 1. Wien's law gives a peak wavelength of 2.897771955 thousandths of a metre-kelvin divided by 5778, which is about 501.5 nanometres, in the green part of the visible band. The Stefan-Boltzmann law gives a radiant exitance of about 6.32e+7 watts per square metre, and the frequency form of Wien's law puts the peak frequency near 3.40e+14 hertz. The peak lands in the band shown as Visible (green). These match the defaults shown in the calculator above.

Related calculators

Wavelength Calculator: convert between wavelength, frequency and speed.
Photon Energy Calculator: energy of a photon from its wavelength.
Frequency Calculator: frequency from period or wavelength.
De Broglie Wavelength Calculator: the wavelength of a moving particle.
Kinetic Energy Calculator: energy of a moving mass.

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