Wien's Displacement Law
Wien's Displacement Law relates the peak emission wavelength of a blackbody to its temperature. Hotter objects peak at shorter (bluer) wavelengths; cooler objects peak at longer (redder) wavelengths. The constant b = 2.8978 × 10⁻³ m·K.
λ_max = b / T
How It Works
Wien's Displacement Law (λ_max = b/T) tells you the wavelength at which a hot object radiates most intensely. Every object above absolute zero emits thermal radiation across a spectrum of wavelengths, but the peak shifts toward shorter (bluer) wavelengths as temperature rises. The constant b = 2.8978 × 10⁻³ m·K links temperature directly to that peak. Use the Solve For selector above to find either the peak wavelength from a known temperature or the temperature from a measured peak wavelength.
Example Problem
The Sun's surface temperature is approximately 5,778 K. At what wavelength does it emit most strongly?
- Identify the known values: temperature T = 5,778 K and Wien's displacement constant b = 2.8978 × 10⁻³ m·K.
- Determine what we are solving for: the peak emission wavelength λ_max.
- Write Wien's displacement law: λ_max = b / T.
- Substitute the known values: λ_max = 2.8978 × 10⁻³ / 5,778.
- Compute the result: λ_max ≈ 5.015 × 10⁻⁷ m = 501.5 nm.
- Interpret: 502 nm falls in the green-yellow part of the visible spectrum, which is why sunlight appears white (it peaks near the center of the visible range).
A simpler example: at 310 K (human body temperature), λ_max ≈ 9.35 μm — deep in the infrared, invisible to the eye but detectable by thermal cameras.
When to Use Each Variable
- Solve for Peak Wavelength — when you know the temperature, e.g., finding what color a star or heated object emits most strongly.
- Solve for Temperature — when you know the peak emission wavelength, e.g., determining a star's surface temperature from its observed spectrum.
Key Concepts
Wien's Displacement Law is a consequence of Planck's radiation law. It states that the product of peak wavelength and absolute temperature is a constant (b = 2.8978 × 10⁻³ m·K). This inverse relationship means that doubling the temperature halves the peak wavelength, shifting emission toward the blue/ultraviolet end of the spectrum. Temperature must be in kelvin (absolute scale) — using Celsius or Fahrenheit will produce incorrect results.
Applications
- Astronomy: determining stellar surface temperatures from spectral observations
- Infrared thermography: selecting the correct detector wavelength band for thermal imaging cameras
- Incandescent lighting: understanding why tungsten filaments at 3,000 K peak in the infrared and emit mostly heat
- Pyrometry: measuring furnace and molten metal temperatures from their thermal radiation color
- Climate science: analyzing Earth's blackbody emission peak (~10 μm) for greenhouse effect studies
Common Mistakes
- Using Celsius instead of Kelvin — Wien's Law requires absolute temperature; using Celsius gives wildly incorrect wavelengths
- Assuming the peak wavelength is the only wavelength emitted — blackbodies emit across all wavelengths; the peak is just where emission is strongest
- Confusing Wien's Law with the Stefan-Boltzmann Law — Wien's gives the peak wavelength, while Stefan-Boltzmann gives the total radiated power
- Forgetting to convert wavelength units — the formula gives meters; divide by 10⁻⁹ to get nanometers or by 10⁻⁶ for micrometers
Frequently Asked Questions
How can you determine a star's temperature from its color?
Divide Wien's displacement constant (b = 2.8978 × 10⁻³ m·K) by the object's absolute temperature in kelvin. The result is the peak emission wavelength in meters. For example, an object at 3,000 K has a peak wavelength of 2.8978 × 10⁻³ / 3,000 ≈ 966 nm (near infrared).
What is the relationship between temperature and peak wavelength?
Wien's displacement law formula is λ_max = b / T, where λ_max is the peak emission wavelength in meters, T is the absolute temperature in kelvin, and b = 2.8977719 × 10⁻³ m·K is Wien's displacement constant. Rearranging gives T = b / λ_max to find temperature from a known peak wavelength.
Why do hot objects change color?
As temperature increases, Wien's law shows that the peak emission wavelength shifts to shorter wavelengths. At ~800 K objects glow dull red, at ~3,000 K they appear orange-yellow, and above ~6,000 K they look blue-white. The color change reflects the shifting peak of the blackbody spectrum through the visible range.
What is the Wien's displacement constant?
Wien's displacement constant (b) equals 2.8977719 × 10⁻³ m·K (approximately 0.0029 m·K). It is derived from Planck's radiation law by finding the wavelength that maximizes the spectral radiance function. The constant links an object's temperature directly to its peak emission wavelength.
Does Wien's law apply to non-blackbodies?
Wien's law strictly applies only to ideal blackbody radiators. Real objects have emissivity less than 1 and may have spectral features that shift the apparent peak. However, for thermal emitters like stars, incandescent filaments, and molten metals, Wien's law provides an excellent approximation of the peak emission wavelength.
What is a blackbody?
A blackbody is an idealized object that absorbs all incoming radiation and re-emits it in a characteristic spectrum that depends only on its temperature. Stars, molten metal, and the cosmic microwave background approximate blackbodies. The Sun, for example, closely follows the blackbody curve with a surface temperature of about 5,778 K.
How is Wien's Law used in astronomy?
Astronomers measure the peak wavelength of a star's spectrum and use Wien's Law to determine its surface temperature. For example, a star peaking at 290 nm (ultraviolet) has T ≈ 10,000 K, while a red giant peaking at 700 nm has T ≈ 4,100 K. This technique is fundamental to stellar classification.
Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.
Wien's Displacement Law Formula
Wien's displacement law relates the peak emission wavelength of a blackbody to its absolute temperature:
λmax = b / T
Where:
- λmax — peak emission wavelength, measured in meters (m)
- T — absolute temperature, measured in kelvin (K)
- b — Wien's displacement constant = 2.8977719 × 10−3 m·K
The formula shows an inverse relationship: as temperature increases, the peak wavelength decreases. This is why hotter objects emit bluer light and cooler objects emit redder light. The law applies to ideal blackbody radiators and is a good approximation for stars, incandescent objects, and thermal emitters.
Worked Examples
Astrophysics
The Sun's surface temperature is 5,778 K. At what wavelength does it emit most strongly?
Using Wien's displacement law with the Sun's effective surface temperature:
- λ = b / T
- λ = 2.8978 × 10−3 / 5,778
- λ ≈ 5.015 × 10−7 m ≈ 502 nm
- λ ≈ 502 nm (green-yellow visible light)
This explains why sunlight appears white (it peaks near the center of the visible spectrum). Our eyes evolved to be most sensitive to this wavelength range.
Infrared Sensing
A human body at 310 K (37°C). What is the peak emission wavelength?
Thermal cameras detect infrared radiation from the human body. Using Wien's law:
- λ = b / T
- λ = 2.8978 × 10−3 / 310
- λ ≈ 9.35 × 10−6 m
- λ ≈ 9.35 μm (far infrared)
This is why thermal imaging cameras for people operate in the 8–14 μm atmospheric transmission window, centered on the human body's peak emission.
Materials Science
Molten steel glows at about 1,500 K. What is its peak wavelength?
Pyrometers measure furnace temperatures by analyzing thermal radiation color:
- λ = b / T
- λ = 2.8978 × 10−3 / 1,500
- λ ≈ 1.93 × 10−6 m
- λ ≈ 1.93 μm (near infrared)
Although the peak is in the near infrared, enough radiation spills into the visible red end of the spectrum to produce the characteristic orange-red glow of molten steel.
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Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.