Tool · Units and conversions

Wien's Law Calculator

Where does a hot object’s spectrum peak, and how much power does it radiate? From a temperature, or from a peak wavelength, the calculator gives the Wien peak per unit wavelength, per unit frequency and in photon number, the Stefan-Boltzmann exitance and the share of the power inside a wavelength band, with the Planck spectrum drawn. Background: blackbody radiation and infrared bands.

Temperature

Start from

Temperature (K)

Peak wavelength (µm)

0 °C is 273.15 K. The peak wavelength is the maximum of the spectrum per unit wavelength.

Band and surface

Band from (µm)

Band to (µm)

Emissivity

The default band is the visible, 0.38 to 0.78 µm. The emissivity scales the exitance of a gray body; it does not move the peak.

Presets

Readouts

Planck spectrum

Spectral exitance per unit wavelength. The vertical line is λ_max; the shaded region is the chosen band.

Peak wavelength against temperature

Wien’s displacement law is a straight line of slope −1 on log scales. The dot is the current temperature.

Learn with it

Two short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. The displacement constants, the Stefan-Boltzmann law and the band fractions are compared with CODATA values and with the tabulated blackbody functions.

Check	Expected	Computed	Tolerance

The expected values are the CODATA 2018 Wien constants, σT⁴ with the CODATA σ, and the exact limits of the blackbody fractional function. The tolerance is the largest difference from Expected that still passes, relative to Expected or to 1, whichever is larger.

The model

A blackbody at temperature $T$ has the Planck spectral exitance

M_{λ} (λ, T) = \frac{2 π h c^{2}}{λ^{5}} \frac{1}{e^{h c / λ k T} - 1}

Setting its derivative to zero gives $x = 5 (1 - e^{- x})$ with $x = h c / λ k T$ , whose root $x = 4.965114$ is Wien’s displacement law,

λ_{\max} = \frac{b}{T}, b = \frac{h c}{4.965114 k} = 2.897771955 \times 10^{- 3} m K

The spectrum per unit frequency peaks where $x = 3 (1 - e^{- x})$ , at $ν_{\max} = 5.878925757 \times 10^{10} H z / K \times T$ , a wavelength 1.76 times longer than $λ_{\max}$ ; the photon-number spectrum per unit wavelength peaks at $x = 3.920690$ . The total exitance is $ε σ T^{4}$ and the share below a wavelength follows from the exact series

F_{0 \to λ} = \frac{15}{π^{4}} \sum_{n = 1}^{\infty} \frac{e^{- n x}}{n} (x^{3} + \frac{3 x^{2}}{n} + \frac{6 x}{n^{2}} + \frac{6}{n^{3}})

The emissivity $ε$ is taken as constant with wavelength, a gray body; real surfaces have emissivities that vary across the spectrum, which changes the band shares and can move the apparent peak.

Worked example

At the Sun’s effective temperature of 5772 K, $λ_{\max} = 2.897772 \times 10^{- 3} / 5772 = 502.04 n m$ , the exitance is 62.94 MW/m², and 46.50 % of it lies between 380 and 780 nm. Plotted per unit frequency, the same spectrum peaks at 339.33 THz, which is 883.5 nm. Skin at 307 K peaks at 9.439 µm and, with emissivity 0.98, radiates 493.6 W/m², 38.04 % of it in the 8 to 14 µm band of thermal cameras.

Related: the lumens to watts converter gives the luminous efficacy of blackbody light, and the wavelength to color converter shows how the peak wavelength looks.

References: M. Planck, “Ueber das Gesetz der Energieverteilung im Normalspectrum,” Annalen der Physik 309, 553–563 (1901). E. Tiesinga et al., “CODATA recommended values of the fundamental physical constants: 2018,” Rev. Mod. Phys. 93, 025010 (2021). J. R. Howell, M. P. Mengüç and R. Siegel, Thermal Radiation Heat Transfer, 6th ed., CRC Press (2015), chapter 1.