Wien's Law Calculator
Enter a blackbody temperature to find the wavelength and frequency of peak emission, or enter a peak wavelength to solve for temperature. Results appear instantly and update as you type. Switch between forward and reverse mode using the solve-for selector, and choose your preferred wavelength unit.
Formula
Worked example
The Sun has a surface temperature of 5,778 K. Plugging into Wien's law: lambda_max = 2.8978e-3 / 5778 = 5.016e-7 m = 501.6 nm. That sits squarely in the green-yellow part of the visible spectrum. The peak frequency is 5.8789e10 × 5778 = 3.398e14 Hz = 339.8 THz, corresponding to about 882 nm in wavelength terms - different from the wavelength peak because the two spectral density functions have different shapes.
What is Wien's displacement law?
Wien's displacement law, published by Wilhelm Wien in 1893, states that the peak wavelength of radiation emitted by a perfect blackbody is inversely proportional to its absolute temperature. Hotter objects emit most intensely at shorter wavelengths: a cool iron bar glows red, a hotter bar turns yellow, and at extreme temperatures objects appear blue-white. The law is expressed as lambda_max = b / T, where b = 2.8977719 × 10^-3 m K is Wien's displacement constant, derived from Planck's radiation law by differentiating the spectral radiance with respect to wavelength and solving for the maximum. The law is foundational in astrophysics, infrared thermometry, and any application involving thermal radiation.
Two forms of Wien's law: wavelength and frequency peaks
The wavelength peak and the frequency peak of a blackbody spectrum are not at the same photon energy. That counterintuitive fact arises because spectral radiance per unit wavelength (B_lambda) and spectral radiance per unit frequency (B_nu) are related by a Jacobian factor, so their maxima fall at different points. The wavelength form gives lambda_max = b / T with b = 2.8978 × 10^-3 m K, while the frequency form gives f_max = alpha × T with alpha = 5.8789 × 10^10 Hz/K. For the Sun (5,778 K), the wavelength peak is about 502 nm (green) while the frequency peak corresponds to about 880 nm (near-infrared). This calculator computes both, so you can choose the form appropriate to your measurement or instrument.
Applications: astronomy, thermometry, and engineering
Astronomers use Wien's law to estimate stellar surface temperatures from color or peak emission wavelength - it is one of the oldest and most reliable tools in observational astrophysics. Infrared thermometers and thermal cameras work on a closely related principle: the sensor detects the intensity or peak of radiation emitted by an object and converts it to temperature without contact. Industrial furnaces, blackbody calibration sources, and radiometric instruments all rely on an accurate model of blackbody emission. Wien's law also explains why objects at room temperature (about 300 K) emit in the mid-infrared around 10,000 nm, which is why thermal cameras can image people in complete darkness.
Limitations of Wien's law
Wien's law applies exactly only to an ideal blackbody - an object that absorbs all incident radiation and emits a smooth Planck spectrum. Real objects have an emissivity less than 1 and may emit selectively at certain wavelengths (like gases or selective coatings), making their spectra deviate from the ideal curve. The law also breaks down at very low frequencies (long wavelengths), where the full Planck function is needed. For temperature measurement of real surfaces, an emissivity correction is required. Despite these caveats, Wien's law is an excellent first approximation for many objects over a wide range of temperatures.
Wien's law: example blackbodies
| Source | Temperature (K) | Peak wavelength (nm) | Spectral region |
|---|---|---|---|
| Human skin | 308 | 9,410 | Mid-wave infrared |
| Incandescent bulb filament | 2,700 | 1,074 | Near-infrared |
| Red dwarf star (Proxima Centauri) | 3,042 | 952 | Near-infrared |
| M-type red giant (Betelgeuse) | 3,500 | 828 | Near-infrared |
| K-type orange star (Arcturus) | 4,286 | 676 | Red visible |
| G-type yellow star (Sun) | 5,778 | 502 | Green-yellow visible |
| A-type white star (Sirius) | 9,940 | 292 | Ultraviolet |
| B-type blue-white star (Rigel) | 12,100 | 239 | Ultraviolet |
| O-type blue star (Zeta Puppis) | 40,000 | 72 | Far ultraviolet |
Peak emission wavelengths for common natural and artificial blackbody sources. Human skin emits primarily in the mid-infrared. The Sun's peak is near 500 nm.
Frequently asked questions
What is Wien's displacement constant?
Wien's displacement constant b = 2.8977719 × 10^-3 m K is the proportionality constant in Wien's law. It can be derived from Planck's constant, the speed of light, and the Boltzmann constant: b = hc / (k_B × x_max), where x_max ≈ 4.965114 is the solution to x e^x / (e^x - 1) = 5. The value is fixed by fundamental physical constants and is known to eight significant figures from CODATA measurements.
Why does the wavelength peak and frequency peak differ?
The spectral radiance can be expressed per unit wavelength (B_lambda) or per unit frequency (B_nu). Because wavelength and frequency are reciprocally related, stretching one axis compresses the other, shifting the peak. The wavelength peak for a 5,778 K blackbody is about 502 nm; the corresponding frequency is 340 THz. But the frequency peak is at 340 THz only if you use B_nu - if you compute f_max = alpha × T you get 340 THz for B_nu, which corresponds to about 880 nm in wavelength. The two formulas answer two genuinely different questions about the same spectrum.
Can I use Wien's law to measure temperature without contact?
Yes - that is the operating principle behind pyrometers and thermal cameras. An infrared sensor measures the intensity or spectral distribution of radiation emitted by a surface. If the object behaves approximately like a blackbody, Wien's law (or the full Planck function) connects the measurement to temperature. Real surfaces have emissivity below 1, so most instruments apply an emissivity correction; many allow the user to set emissivity manually for accurate readings.
What temperature produces peak emission in the visible spectrum?
Visible light spans roughly 380 nm (violet) to 700 nm (red). Using lambda_max = b / T, the temperature for a 380 nm peak is about 7,625 K (a hot A-type star), and for 700 nm it is about 4,140 K (an orange K-type star). Our Sun, at 5,778 K, peaks around 502 nm in the green-yellow. So objects whose surfaces are in the range of 4,000-8,000 K radiate most intensely in visible light - which is why stars appear colored.
Does Wien's law apply to the cosmic microwave background?
Yes. The cosmic microwave background (CMB) is the most perfect blackbody spectrum ever measured, corresponding to a temperature of 2.725 K. Wien's law gives a peak wavelength of b / 2.725 ≈ 1.06 mm, in the microwave range, which is exactly where observations place the CMB peak. This agreement provides one of the strongest confirmations of Big Bang cosmology.