Wien's Displacement Law
Wien's displacement law: lambda_max = b/T.
Find the peak wavelength of thermal radiation emitted by a blackbody at any temperature.
The Formula
Wien's displacement law describes the relationship between the temperature of a blackbody and the wavelength at which it emits the most radiation. As an object gets hotter, its peak emission wavelength shifts to shorter (higher-energy) wavelengths. The law was formulated by German physicist Wilhelm Wien in 1893, and it explains one of the most intuitive phenomena in physics: why objects change color as they heat up. A piece of iron heated in a forge first glows dull red, then bright orange, then yellow-white as the temperature rises — the peak emission moves through the visible spectrum.
The constant b in the formula is Wien's displacement constant, equal to 2.898 × 10⁻³ m·K (meters times Kelvin). Temperature T must be expressed in Kelvin, not Celsius or Fahrenheit. The result λmax is given in meters, but is often expressed in nanometers (nm) for visible light or micrometers (µm) for infrared radiation.
The law has profound applications in astronomy. By measuring the peak wavelength of light from a star, astronomers can calculate its surface temperature with great precision. The Sun's surface temperature of approximately 5,778 K produces a peak wavelength around 502 nm — right in the middle of the green portion of the visible spectrum. Our eyes evolved to be most sensitive near this wavelength for good reason. Cooler stars emit mostly red and infrared light, while hotter stars emit blue and ultraviolet light.
Wien's displacement law is a direct consequence of Planck's law of blackbody radiation. It represents the wavelength at which the derivative of the Planck function with respect to wavelength equals zero — the mathematical peak of the spectral distribution curve.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| λmax | Peak wavelength — wavelength of maximum emission | m (or nm) |
| b | Wien's displacement constant = 2.898 × 10⁻³ | m·K |
| T | Absolute temperature of the blackbody | K (Kelvin) |
Example 1
What is the peak emission wavelength of the Sun, whose surface temperature is approximately 5,778 K?
λmax = 2.898 × 10⁻³ / 5,778
λmax = 5.015 × 10⁻⁷ m = 501.5 nm
Peak wavelength ≈ 502 nm (green-yellow visible light)
Example 2
The human body has a surface temperature of about 37°C (310 K). In what part of the spectrum does it radiate most strongly?
λmax = 2.898 × 10⁻³ / 310
λmax = 9.35 × 10⁻⁶ m = 9,350 nm = 9.35 µm
Peak wavelength ≈ 9.35 µm (mid-infrared — invisible to the naked eye, detected by thermal cameras)
When to Use It
Apply Wien's displacement law when:
- Estimating the surface temperature of a star from its observed color or peak wavelength
- Designing infrared sensors and thermal cameras calibrated to specific temperature ranges
- Calculating the emission spectrum of industrial furnaces, kilns, or incandescent light sources
- Understanding why hotter objects appear bluer and cooler objects appear redder
- Selecting materials and coatings for radiation shielding at specific wavelengths
- Astronomy and astrophysics research involving stellar classification