Wien's Displacement Law Calculator
Solve the classic Wien displacement relation λmax = b / T for any of its three quantities — peak emission wavelength, Wien's displacement constant, or absolute temperature — with step-by-step math, live blackbody spectrum curves, and companion thermal-radiation metrics.
Temperature sets the colour
Wien's law is the mathematical statement of a fact everyone knows intuitively: hot metal glows orange, then yellow, then white as it heats up. Each successive stage is a shorter peak wavelength. The same physics sets the colour of stars: red dwarfs at ~3,000 K peak in the near infrared, the Sun at 5,778 K peaks in the green, blue supergiants at 30,000 K peak in the far ultraviolet. The relationship is inverse — double the temperature and you halve the peak wavelength.
The law is a straight line on log–log axes
Because λmax = b/T, taking logarithms gives log λmax = log b − log T — a straight line of slope −1 on a log–log plot. This is why Wien's law is so easy to work with: every decade increase in temperature shifts the peak by exactly one decade toward shorter wavelengths. Experimentalists use this linearity to extract temperatures from measured spectra without ever fitting a full Planck curve.
Astronomers use it as a thermometer
Measuring a star's peak wavelength — or equivalently its colour index — gives its surface temperature directly via T = b/λmax. This is the basis of the entire spectral classification system (O, B, A, F, G, K, M) that underlies modern astronomy. Even without a full spectrum, a simple ratio of blue to red filter measurements yields a reliable temperature, which then feeds into estimates of radius, luminosity, and distance.
It also drives remote sensing
Wien's law tells engineers where to put detectors. Earth-observing satellites for land-surface temperature use ~10 μm (mid-infrared) channels because 300 K bodies peak there. Space telescopes for CMB studies operate in the mm band because 2.725 K radiation peaks at ~1 mm. Solar observatories look in the visible. Fire-detection satellites use mid-infrared because flames at 800–1,200 K peak between 2.4 and 3.6 μm. Every choice of wavelength band is ultimately a Wien's law calculation.
How to Use This Calculator
Wien's law links three quantities: λmax, b, and T. Pick which one you want the tool to solve, then fill in the other two.
1. Choose what to solve for
Pick λmax, b, or T from the toggle. The calculator rearranges λmax = b / T automatically — including T = b / λmax when you have a measured spectrum.
2. Use realistic presets
Presets cover objects from the cosmic microwave background (2.725 K, 1.06 mm) through room temperature (300 K, 9.66 μm) and the Sun (5,778 K, 501.5 nm) up to blue supergiants (30,000 K, 96.6 nm).
3. Match the units carefully
Wavelength is entered in nanometres and temperature in kelvin, so the constant b is in nm·K. If you are working in μm·K or m·K, convert first. The unit converter card further down the page handles this automatically.
4. Read the result and verdict
The result card shows the peak wavelength in nm plus its equivalent in μm and m, the photon energy at that wavelength, and a colour swatch showing the actual colour for visible-light results. A colour-coded badge classifies the band from gamma through visible to radio.
5. Check the schematic and charts
The diagram plots three blackbody curves (T/2, T, 2T) over a log-wavelength axis, with each peak marked and the visible spectrum shown as a coloured band underneath. The chart plots the same three curves as normalized spectral radiance.
6. Switch to Advanced for depth
Advanced mode adds a log–log plot showing the slope −1 line, a family of curves for different values of b, a sensitivity check on a +1% input change, and companion metrics including Wien frequency, photon energy and momentum, and the Stefan–Boltzmann flux.
What This Tool Offers
Built around the classic Wien displacement relation λmax = b / T, the fundamental link between temperature and the peak of a thermal spectrum.
Three-way solving
Solve for peak wavelength, Wien's constant, or absolute temperature. The calculator rearranges λmax = b / T for whichever variable you choose.
Live blackbody spectrum diagram
Three normalized Planck curves at T/2, T, and 2T on a log-wavelength axis, with peak markers labelled in real units and a visible-spectrum colour band beneath.
Spectral-band verdict
Each result is classified from gamma through X-ray, UV, the six visible colours, near/mid/far infrared, submillimetre, and radio — with a coloured badge and a live colour swatch.
Derived hints everywhere
Each field shows the value converted to related units — μm and m for wavelength; °C, °F, and eV for temperature; m·K and μm·K for b.
Three linked charts
Normalized Planck curves at three temperatures, the log–log line of slope −1, and a family of λmax–T curves for three values of b.
Reference tables
A wavelength unit converter (nm, μm, mm, m, Å, eV), a λmax lookup grid across temperatures, and a table of common blackbody peaks from the CMB to neutron stars.
Reference Tables & Conversions
A wavelength unit converter, a λmax lookup grid, and a table of blackbody peaks for common objects.
Wavelength unit converter
Peak Wavelength Lookup Grid
Computed from λmax = b / T with b = 2,897,771.955 nm·K. Rows are the temperature, columns are the value of b (in nm·K). Values shown in nm.
| T (K) | b × 0.5 | b × 0.9 | b (standard) | b × 1.1 | b × 2 |
|---|
Common Blackbody Peaks
Representative temperatures and the corresponding Wien peak wavelengths. These span the full electromagnetic spectrum from radio to gamma rays.
| Object / source | T (K) | λmax | Spectral band |
|---|---|---|---|
| Cosmic microwave background | 2.725 | 1.063 mm | microwave / submillimetre |
| Liquid nitrogen | 77 | 37.6 μm | far infrared |
| Earth's surface (average) | 288 | 10.06 μm | mid infrared |
| Room temperature | 300 | 9.66 μm | mid infrared |
| Human body | 310 | 9.35 μm | mid infrared |
| Incandescent filament | 2,800 | 1,035 nm | near infrared |
| Red dwarf star (M-type) | 3,000 | 966 nm | near infrared |
| Sun's photosphere | 5,778 | 501.5 nm | visible (green) |
| Sirius A (A-type) | 9,940 | 291.6 nm | ultraviolet |
| Rigel (B-type supergiant) | 12,100 | 239.5 nm | ultraviolet |
| Blue supergiant (O-type) | 30,000 | 96.6 nm | far ultraviolet |
| Neutron star surface | 600,000 | 4.83 nm | soft X-ray |
| Accretion disc (inner) | 10,000,000 | 0.290 nm | hard X-ray |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| λmax | Peak emission wavelength | 0.1 nm – 10 mm | length |
| b | Wien's displacement constant | 2.8978 × 10⁻³ | m·K |
| T | Absolute temperature | 1 K – 10⁷ K | K |
| νmax | Peak frequency (Wien frequency) | — | Hz |
| Eγ | Photon energy at λmax | — | eV |
| F | Total radiant flux, σT⁴ (Stefan–Boltzmann) | — | W/m² |
| h | Planck constant | 6.62607015 × 10⁻³⁴ | J·s |
| c | Speed of light | 2.99792458 × 10⁸ | m/s |
| k | Boltzmann constant | 1.380649 × 10⁻²³ | J/K |
| σ | Stefan–Boltzmann constant | 5.670374419 × 10⁻⁸ | W/(m²·K⁴) |
What Does λmax = b / T Measure?
Wien's displacement law connects three quantities that every physicist and astronomer works with: the peak wavelength of a thermal spectrum (λmax), a fundamental constant of nature (b), and the absolute temperature of the radiating body (T). Together they answer one of the most intuitive questions in physics — what colour does a hot object glow? — with a precision that made it the foundation of quantitative stellar astronomy.
The law was formulated in 1893 by Wilhelm Wien, who received the Nobel Prize in Physics in 1911 for his work on thermal radiation. It is a direct corollary of Planck's radiation law (formulated in 1900), and its derivation is one of the most beautiful exercises in mathematical physics: differentiating the Planck distribution and setting the derivative to zero produces a transcendental equation whose solution is the constant b = hc/(k·x) with x ≈ 4.965.
Wien's Displacement Formula
The defining relationship is:
where the constant is:
and x ≈ 4.965114 is the unique positive root of x = 5(1 − e⁻ˣ). Rearranged to solve for the other two unknowns:
b = λmax · T
Related quantities that follow from the peak wavelength:
Photon momentum: p = h/λmax
Peak frequency (Wien frequency): νmax ≈ 5.879 × 10¹⁰ Hz/K · T (NOT c/λmax)
Total flux (Stefan–Boltzmann): F = σT⁴
Wien's Law Variables
| Symbol | Meaning | Notes |
|---|---|---|
| λmax | Peak emission wavelength | The wavelength at which B(λ, T) reaches its maximum |
| b | Wien's displacement constant | Derived from h, c, k; independent of temperature and material |
| T | Absolute temperature | Must be in kelvin; the law does not apply to grey bodies without correction |
| x | Root of x = 5(1 − e⁻ˣ) | x ≈ 4.9651142317; appears in the derivation of b |
| σ | Stefan–Boltzmann constant | Connects total flux to T⁴, not just to the peak |
How Each Variable Affects the Peak
Wien's law is striking in its simplicity: the peak wavelength is inversely proportional to absolute temperature. Double the temperature and the peak wavelength halves. Halve the temperature and it doubles. This inverse relationship is what makes the law so useful as a thermometer — a single measurement of λmax immediately gives T with no other inputs required.
On log–log axes, the law is a straight line of slope −1. This linearity makes the mathematical structure transparent: every decade of temperature corresponds to exactly one decade of wavelength, just in the opposite direction. It is also why the same formula, expressed in different units, retains the same shape — the log–log line just shifts vertically when b changes units.
The law applies rigorously only to ideal blackbodies. For real objects — stars, metals, planets — the peak wavelength shifts slightly from the ideal value because real surfaces have wavelength-dependent emissivity. Astronomers correct for this using the object's "effective temperature", defined as the temperature of the blackbody that would radiate the same total flux. For most applications the correction is small (a few percent) and Wien's law remains an excellent first approximation.
Note carefully that the peak frequency νmax is not equal to c/λmax. Because B(λ) and B(ν) are different functions, their maxima occur at different places. The Wien frequency is νmax ≈ 5.879 × 10¹⁰ Hz/K · T, and the product νmax · λmax ≈ 1.76 × 10⁸ m/s — not c. This is a common source of confusion in textbooks.
Wien's Law Calculation Examples
λmax = 2,897,771.955 / 5,778 = 501.5 nm (green visible light)
λmax = 2,897,771.955 / 300 = 9,659 nm = 9.66 μm (mid infrared, invisible to the eye)
λmax = 2,897,771.955 / 2.725 = 1,063,000 nm = 1.063 mm (microwave)
T = b / λmax = 2,897,771.955 / 400 = 7,244 K (F-type star)
T = 2,897,771.955 / 10,000 = 289.8 K ≈ 16.6 °C
b = λmax · T = 501.5 × 5,778 = 2,897,667 nm·K (0.004% from CODATA)
Frequently Asked Questions
It says that the wavelength at which a hot object radiates most strongly is inversely proportional to its absolute temperature. Hot objects glow at short wavelengths (blue, violet, ultraviolet); cool objects glow at long wavelengths (red, infrared, microwave). The law is "displacement" because it describes how the peak of the thermal spectrum is displaced — shifted — as temperature changes. It is a direct consequence of Planck's radiation law and was discovered empirically by Wilhelm Wien in 1893, several years before Planck's theoretical derivation.
Both effects happen simultaneously. As temperature rises, the total emitted power grows rapidly (as T⁴, per Stefan–Boltzmann), but the shape of the spectrum also changes: the peak moves to shorter wavelengths and the distribution narrows. The physics is that hotter bodies have more energy per photon on average, so the Maxwell–Boltzmann-like distribution of photon energies shifts to higher values, which corresponds to shorter wavelengths via E = hc/λ. The T⁴ growth and the λmax ∝ 1/T shift are two separate, complementary consequences of increasing temperature.
Approximately, yes, and often very well. Real objects emit with a wavelength-dependent emissivity ε(λ) that modifies the Planck curve, shifting the peak slightly. For metals, ε rises with wavelength in the infrared, which pushes the observed peak to slightly longer wavelengths than Wien would predict. For stars, the correction is small (typically a few percent), which is why stellar temperatures from Wien's law agree closely with those from full spectral fitting. For extreme cases — highly reflective surfaces, metamaterials, or emitters with engineered spectral selectivity — the law can be significantly violated and full radiative-transfer modelling is required.
Because B(λ) and B(ν) are different functions and their maxima occur at different places. If you differentiate B(λ) = (2hc²/λ⁵)/(e^(hc/λkT) − 1) with respect to λ and set it to zero, you get x ≈ 4.965 where x = hc/(λmaxkT). If you differentiate B(ν) = (2hν³/c²)/(e^(hν/kT) − 1) with respect to ν, you get a different equation with root y ≈ 2.821. The two peaks are at different points on the spectrum, and c/λmax is not the peak frequency. The Wien frequency is νmax = y·kT/h ≈ 5.879 × 10¹⁰ Hz/K · T. This subtlety is often missed in elementary treatments.
From the derivative of Planck's law. Writing x = hc/(λkT), Planck's law becomes B(x) ∝ x⁵/(eˣ − 1) times constants. Differentiating with respect to x and setting dB/dx = 0 gives 5(1 − e⁻ˣ) = x, a transcendental equation with the unique positive solution x ≈ 4.965114231744276. Substituting back gives λmaxT = hc/(kx) = b. Plugging in CODATA values of h, c, and k gives b = 2.897771955 × 10⁻³ m·K. The constant is exact given the SI definitions of h, c, and k (all now defined exactly since the 2019 SI redefinition), though x is irrational.
By measuring the star's spectrum and finding where it peaks — or, more practically, by measuring the ratio of flux in two well-separated photometric bands (e.g. B and V, or J and K) and calibrating that ratio against known temperatures. The B−V colour index is essentially a Wien's law measurement in disguise: blue stars have small B−V (short-wavelength peak), red stars have large B−V (long-wavelength peak). Once the temperature is known, it feeds into stellar models that give radius (from luminosity and temperature), mass (from the mass–luminosity relation), and age (from evolutionary tracks). Wien's law is therefore a foundational measurement of stellar astrophysics.
They are complementary. Wien's law tells you where the peak is (λmax = b/T); Stefan–Boltzmann tells you how much total power is radiated (F = σT⁴). Together they characterise a blackbody spectrum completely enough for many purposes. Integrating the Planck distribution over all wavelengths gives the Stefan–Boltzmann law; differentiating it with respect to wavelength gives Wien's law. Both are consequences of the same underlying physics, expressed in two different ways.
Yes, this is one of its most important practical applications. Infrared thermometers, thermal cameras, and satellite radiometers all work by measuring the radiance in one or more infrared bands and inferring temperature. In the simplest case (a true blackbody filling the field of view), a single band is enough: T = b/λpeak, or more commonly T is inferred from B(λ) directly using Planck's equation. For real surfaces with unknown emissivity, two-band or multi-band radiometry is used, exploiting the fact that the ratio of radiances at two wavelengths depends strongly on temperature but only weakly on emissivity. Earth-observing satellites use this to map sea-surface temperature, land-surface temperature, and cloud-top temperature globally.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| λ_max | Peak wavelength | m |
| b | Wien's displacement constant | m*K |
| T | Surface temperature | K |
What it means
Wien’s law states that the wavelength (λ_max) at which a blackbody emits maximum spectral radiance is inversely proportional to its absolute temperature T, with constant b (Wien’s displacement constant ≈ 2.898×10⁻³ m·K). This is used to estimate the temperature of stars from their colour (peak wavelength). It also applies to any thermal emitter, such as the Sun, planets, and even human bodies. Understanding this law helps astronomers classify stars and understand the physics of thermal radiation.
Worked example
Wien's Displacement Law – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| T (K) | 5778 |
| Parameter | Value |
|---|---|
| T | 25000 |
Common mistakes
- Wien’s displacement law: λ_max = b / T – peak wavelength of blackbody radiation.
- b: Wien’s constant = 2.897×10⁻³ m·K.
- T: Temperature in Kelvin – λ_max in metres.
- Hotter objects: Peak shifts to shorter wavelengths (bluer).
- Approximate: For stars, λ_max is in the visible or infrared.
Applications
Wien's displacement law, λ_max = b/T, gives the wavelength at which a blackbody emits most intensely, inversely proportional to its temperature. This is used to determine the temperature of stars, planets, and other thermal emitters. By measuring the peak wavelength in a spectrum, astronomers can estimate the effective temperature of a star. This law is also applied in thermal imaging and in the study of cosmic microwave background radiation. Understanding Wien's law helps in interpreting the colours and temperatures of objects across the universe.
- Stellar temperature estimation from spectral peaks
- Characterisation of exoplanet atmospheric temperatures
- Study of cosmic microwave background radiation
- Thermal imaging and remote sensing
- Education on blackbody radiation and colour‑temperature relations
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