Wien's Displacement Law Calculator
Enter a blackbody's temperature to find the wavelength where it emits the most radiation (or enter a peak wavelength to find the temperature). Results include unit conversions, the spectral band, and the peak frequency.
Find the Peak Wavelength or Temperature of a Hot Object
Wien's Displacement Law Formula
Where:
- \(\lambda_{\max}\) — peak emission wavelength, the wavelength at which the object radiates the most power per unit wavelength (m)
- \(T\) — absolute temperature of the object (K)
- \(b\) — Wien's displacement constant, \(b = 2.897771955 \times 10^{-3}\) m·K (CODATA 2018 value)
The formula expresses Wien's displacement law: the hotter an object, the shorter the wavelength at which it shines most brightly. The product \(\lambda_{\max} \times T\) is the same constant for every blackbody, so doubling the temperature halves the peak wavelength.
Wien's Displacement Law – Calculation Example
Example: the Sun's surface behaves roughly like a blackbody at 5800 K. At what wavelength does it emit most of its radiation?
\[ \lambda_{\max} = \frac{b}{T} = \frac{2.897771955 \times 10^{-3}\ \text{m·K}}{5800\ \text{K}} = 4.9962 \times 10^{-7}\ \text{m} \]
\[ \lambda_{\max} = 499.62\ \text{nm} \approx 0.4996\ \mu\text{m} \]
Result: the peak wavelength is 499.62 nm, which lies in the green part of the visible spectrum — close to the middle of the range the human eye can see. This is no coincidence: our eyes evolved to make the most of sunlight.
How to Use the Calculator
- Open the "I want to find" dropdown and choose whether you want the peak wavelength from a temperature, or the temperature from a peak wavelength. Only the relevant input field is shown.
- To find the wavelength: enter the surface temperature in "Blackbody temperature (T)" and pick its unit — K, °C or °F. Values in °C and °F are converted to kelvin automatically.
- To find the temperature: enter the wavelength in "Peak wavelength (λmax)" and choose nm, µm, mm or m.
- Read the result: the main quantity plus conversions, the Wien product, the peak frequency and the spectral band the peak falls in.
- Try the typical values shown in the field hints: the Sun ≈ 5800 K, an incandescent filament ≈ 2800 K, the human body ≈ 310 K.
What Is Wien's Displacement Law?
Every object above absolute zero emits electromagnetic radiation due to the thermal motion of its charged particles. An idealised emitter that absorbs and radiates perfectly at every wavelength is called a blackbody, and its spectrum has a characteristic bell shape described by Planck's law. Wien's displacement law, discovered by Wilhelm Wien in 1893, tells you where the top of that bell sits: the wavelength of maximum emission is inversely proportional to the absolute temperature.
The law explains phenomena you see every day. As a piece of metal heats up, its peak emission moves from the invisible infrared into the visible range — the metal first glows dull red, then orange, then yellow-white as \(\lambda_{\max}\) shifts to shorter wavelengths. It is also a fundamental tool in astrophysics: by measuring the spectrum of a star and locating its peak, astronomers can determine the star's surface temperature without ever touching it. Hot blue stars peak at short wavelengths (a 20,000 K star peaks near 145 nm, in the ultraviolet), while cool red stars like Betelgeuse peak well into the infrared.
Interpreting the result is straightforward: a peak below about 380 nm means most radiation is ultraviolet or beyond; between 380 and 750 nm the object radiates mainly visible light; above 750 nm it is dominated by infrared — heat radiation. Objects near room temperature (about 300 K) peak near 10 µm, which is why thermal imaging cameras are built to work in the long-wave infrared.
Typical Temperatures and Peak Wavelengths
| Object | Temperature | Peak wavelength |
|---|---|---|
| Very hot blue star | 20,000 K | 144.9 nm (ultraviolet) |
| Sun's surface | 5,800 K | 499.6 nm (green light) |
| Incandescent filament | 2,800 K | 1,034.9 nm (infrared) |
| Candle flame | 1,500 K | 1,931.8 nm (infrared) |
| Molten lava | 1,200 K | 2,414.8 nm (infrared) |
| Boiling water | 373 K | 7,768.5 nm (infrared) |
| Human body | 310 K | 9,347.7 nm (infrared) |
| Room-temperature wall | 293 K | 9,890.0 nm (infrared) |
Assumptions and Limitations
- Blackbody assumption. The law applies exactly only to perfect blackbodies. Real materials have an emissivity below 1 that varies with wavelength, which shifts the actual peak slightly. Stars, furnaces and lamp filaments are close enough for most purposes; shiny metals are poorer approximations.
- Wavelength vs frequency peak. The constant \(b\) applies to the spectrum plotted per unit wavelength. If the same spectrum is plotted per unit frequency, the peak occurs at a different place — the calculator shows this frequency peak using its own constant, \(5.879 \times 10^{10}\) Hz/K. The two peaks do not satisfy \(\lambda_{\max} = c/\nu_{\max}\).
- Temperature in kelvin. The law requires absolute temperature. If you enter °C or °F, the calculator converts them; a value at or below absolute zero is rejected.
- Surface temperature only. The result is the effective surface temperature of the emitting region. It says nothing about the interior temperature of the object.
- Peak ≠ colour. \(\lambda_{\max}\) tells you where the spectrum peaks, not the perceived colour. The Sun peaks in the green yet appears white because it radiates strongly across the whole visible range.
If you are working with other thermal physics problems, you may also find our Carnot Efficiency Calculator useful for heat engines, or the Speed of Sound in Air Calculator, which uses temperature in a different way — sound speed in a gas depends directly on the gas temperature.
Frequently Asked Questions
How do I calculate Wien's law by hand without a calculator?
Divide Wien's constant by the absolute temperature in kelvin: \(\lambda_{\max} = b/T\) with \(b \approx 2.9 \times 10^{-3}\) m·K. For a quick mental estimate, use \(\lambda_{\max} \approx 2898/T\) with the answer in micrometres. Example: at 300 K, \(2898/300 \approx 9.7\) µm. To go the other way, divide 2898 by the wavelength in µm to get kelvin.
What is a 'normal' peak wavelength for everyday objects?
Anything near room temperature (roughly 293–310 K) peaks at 9–10 µm in the long-wave infrared — this includes walls, furniture and the human body. That is why thermal cameras are designed around the 8–14 µm band. Hot filament lamps peak near 1 µm (still mostly infrared), while the Sun's 5800 K surface peaks around 500 nm, in visible green light.
Can I enter the temperature in Celsius or Fahrenheit?
Yes. The calculator converts °C and °F to kelvin automatically, because Wien's law only works with absolute temperature. If you compute by hand, convert first: \(T = T_{\degree C} + 273.15\) or \(T = (T_{\degree F} + 459.67) \times 5/9\). A common mistake is plugging Celsius straight into the formula — 100 °C is 373 K, not 100 K, and the difference changes the answer nearly fourfold.
Why doesn't the peak frequency match c divided by the peak wavelength?
This surprises many people. Planck's spectrum can be plotted per unit wavelength or per unit frequency, and the two curves peak at different places because the conversion between them is non-linear. The wavelength peak uses \(b = 2.898 \times 10^{-3}\) m·K, while the frequency peak uses \(\nu_{\max} = 5.879 \times 10^{10} \cdot T\) Hz. So \(\lambda_{\max} \neq c/\nu_{\max}\) — the calculator shows both peaks separately.
What is the difference between the peak wavelength and the color I actually see?
The peak only marks the top of the spectrum; the object still radiates across a broad range of wavelengths. The Sun peaks in green but looks white because it emits strongly in every visible color, which our eyes combine. Similarly, a star peaking in the infrared can still appear visibly red. For perceived color, the whole spectrum matters, not just the maximum.
Does Wien's law work for real objects, like a flame or a light bulb?
Only approximately. The law is exact for ideal blackbodies. Real materials have emissivity below 1 that varies with wavelength, which can shift the true emission peak somewhat. Stars, furnace interiors and lamp filaments are close to blackbodies, so the law works well for them. Shiny polished metals and gases with strong emission lines (like neon signs) deviate much more, so treat results as estimates there.
What happens if I enter a temperature of 0 K or below?
The calculator rejects it. Wien's law requires a temperature strictly above absolute zero (0 K, −273.15 °C, −459.67 °F), because dividing the constant by zero is undefined and negative absolute temperatures are unphysical in this context. In practice, every real object is warmer than 0 K — even deep space has a cosmic microwave background corresponding to about 2.7 K, peaking near 1.1 mm in the microwave band.