Wien's Law Calculator – Peak Wavelength from Temperature
Determine the peak wavelength of blackbody radiation from temperature using Wien's displacement law.
Enter the temperature in Kelvin to calculate the peak wavelength (λmax), frequency, and radiation category.
Wien's Law Calculator – Peak Wavelength from Temperature
Determine the peak wavelength of blackbody radiation from temperature using Wien's displacement law.
Wien's Law Examples
Common temperature values and their peak wavelengths of blackbody radiation.
| Temperature | Peak Wavelength | Context |
|---|---|---|
| 5778 K (Sun's surface) | ≈ 501.5 nm (visible green) | The peak is in the visible green region, explaining why the human eye evolved peak sensitivity near 550 nm. |
| 2800 K (incandescent bulb) | ≈ 1035 nm (near infrared) | Most energy is radiated as infrared heat, making incandescent bulbs only about 5% efficient for visible light. |
| 310 K (human body) | ≈ 9348 nm (mid infrared) | Human body heat peaks deep in the mid-infrared, invisible to the naked eye but detectable by thermal cameras. |
| 2.725 K (cosmic background) | ≈ 1.06 mm (microwave) | The Big Bang afterglow — discovered in 1964 — is a near-perfect blackbody at 2.725 K, peaking in the microwave band. |
About the Wien's Law Calculator
Wien's displacement law is a fundamental relationship in thermodynamics and thermal radiation that describes the wavelength at which a blackbody emitter radiates most intensely. Formulated by German physicist Wilhelm Wien in 1893, the law states that the peak wavelength of thermal radiation is inversely proportional to the absolute temperature of the emitting body.
The mathematical expression is λmax = b / T, where λmax is the peak wavelength in metres, T is the absolute temperature in Kelvin, and b is Wien's displacement constant equal to 2.897771955 × 10⁻³ m·K. This elegant inverse relationship has profound implications: as a body gets hotter, it emits radiation at shorter (higher-energy) wavelengths. A cool body radiates in the infrared, a warm body glows red, a hot body glows white or blue-white.
The law emerges from Planck's law of blackbody radiation by differentiating the spectral radiance with respect to wavelength and setting the derivative to zero. The result is a transcendental equation whose solution gives the constant b. Planck's more complete quantum theory, developed in 1900, supersedes Wien's approximation for the full spectral distribution, but Wien's displacement law for the peak remains exactly valid as a special case.
Astronomical applications of Wien's law are particularly striking. The surface temperature of the Sun is approximately 5778 K, corresponding to a peak wavelength of about 502 nm — green light. The human visual system evolved to be most sensitive near this wavelength. Cooler red giant stars (3000–4000 K) peak in the near-infrared; hotter blue-white stars (20,000–50,000 K) peak in the ultraviolet. By measuring the peak wavelength of a star's spectrum, astronomers can determine its surface temperature with high precision.
In everyday life, Wien's law governs the appearance of heated metal. Steel glows faintly red at around 800–900 K, bright orange-red at 1100 K, and yellow-white at 1500 K. Incandescent light bulb filaments operate at about 2700–3000 K, producing warm yellow-white light that peaks in the near-infrared — which is why incandescent bulbs are relatively inefficient: most of their energy is emitted as heat rather than visible light.
Infrared thermography and remote sensing rely on Wien's law to infer temperatures from measured peak wavelengths. Medical infrared cameras detect body temperature variations (normal body temperature ≈ 310 K, λmax ≈ 9.3 μm, deep mid-infrared). Industrial kilns, furnaces, and steel-processing equipment use optical pyrometers and infrared sensors calibrated using Wien's law to measure temperatures contactlessly. The cosmic microwave background radiation, the thermal relic of the Big Bang, has a perfect blackbody spectrum with a peak corresponding to T ≈ 2.725 K — far into the microwave region, just as the name suggests.
How to Use the Wien's Law Calculator
- Enter the temperature of the blackbody emitter in Kelvin (K). Kelvin = Celsius + 273.15.
- Click Calculate. The calculator applies λmax = b / T using Wien's displacement constant b = 2.898 × 10⁻³ m·K.
- Read the peak wavelength in nm, μm, or cm depending on magnitude, plus the approximate frequency.
- The radiation type panel tells you whether the peak falls in the gamma, X-ray, UV, visible, infrared, or microwave region.
- Use the example buttons to load common temperatures (Sun, incandescent bulb, human body) for quick reference.
Wien's Law FAQ
What is Wien's displacement law?
Wien's displacement law states that the peak wavelength of thermal (blackbody) radiation is inversely proportional to the absolute temperature: λmax = b / T, where b = 2.898 × 10⁻³ m·K is Wien's displacement constant. As temperature increases, the peak wavelength shortens — hotter objects emit bluer (higher-energy) light. The law was derived by Wilhelm Wien in 1893 and is confirmed by Planck's complete quantum theory of blackbody radiation.
Why does the Sun peak in the green but appear yellow-white?
The Sun's photosphere at ~5778 K has a peak wavelength around 501–502 nm (green). However, the Sun emits across the full visible spectrum in roughly equal amounts near its peak, so the integrated colour appears white or pale yellow. The yellow appearance is partly due to atmospheric scattering that preferentially removes blue light at low angles, and partly due to the non-uniform spectral sensitivity of the human eye.
What is the Wien's displacement constant b?
Wien's displacement constant b = 2.897771955 × 10⁻³ m·K (metres times Kelvin). It can be derived from fundamental constants: b = hc / (x·kB), where h is Planck's constant, c is the speed of light, kB is Boltzmann's constant, and x ≈ 4.965 is the solution to the transcendental equation x·e^x/(e^x − 1) = 5. The NIST value is 2.897771955 × 10⁻³ m·K.
How does Wien's law relate to Planck's law?
Planck's law gives the complete spectral distribution of blackbody radiation: B(λ,T) = 2hc²/λ⁵ × 1/(e^(hc/λkT) − 1). Wien's law is derived by differentiating this with respect to λ and finding where it is maximised. Wien's law provides only the peak wavelength; Planck's law is needed for the full spectrum. Planck's law reduces to Wien's approximation at short wavelengths where hc/λkT ≫ 1.
Can Wien's law be applied to non-blackbody sources?
Wien's law strictly applies to ideal blackbody radiators. Real objects are 'grey bodies' with emissivity less than 1, which reduces the total emission but does not shift the peak wavelength. The peak wavelength relationship still holds if the emissivity is spectrally flat (grey body). For sources with strongly wavelength-dependent emissivity, Wien's law gives only an approximate guide to peak emission.
How do astronomers use Wien's law to measure star temperatures?
Astronomers measure the spectral energy distribution of a star and locate the wavelength of maximum flux. Applying λmax = b / T and solving for T gives the effective surface temperature. For the Sun, λmax ≈ 502 nm gives T ≈ 5778 K. For Betelgeuse (~3500 K), λmax ≈ 828 nm (near-infrared), explaining its red colour. For hot blue stars like Rigel (~12000 K), λmax ≈ 242 nm (ultraviolet), making them appear blue-white in visible light.