Wien's constant

What is Wien's constant?

Wien's constant is a physical constant that is used in defining the relationship between the thermodynamic temperature of a black body (an object that radiates electromagnetic energy perfectly) and the wavelength at which the intensity of the radiation is the greatest. Wien's constant is typically represented by the lowercase letter b. It is equal to about 2.89777 ⋅ 10^-3 meter-kelvin (0.289777 centimeter-kelvin).

The Committee on Data of the International Science Council (CODATA) provides a more precise value: 2.897771955... ⋅ 10^-3. The trailing ellipsis indicates that the value continues beyond the number of decimal places shown here. The CODATA value for Wien's constant is based on the formula b = (h ⋅ c / k) / 4.965114231…, where:

• b is Wien's constant.
• h is Planck's constant.
• c is the speed of light in a vacuum.
• k is the Boltzmann constant.

What is Wien's constant best known for?

Wien's constant is best known for its role in Wien's displacement law, which states that the absolute temperature of a black body is inversely proportional to the electromagnetic wavelength at its peak radiation intensity. A black body is an ideal object that absorbs all electromagnetic radiation without reflecting any light but that emits thermal radiation. Wien's law is represented by the formula λ_max = b / T, where:

• λ_max is the wavelength at peak radiation intensity in meters.
• b is Wien's constant in meter-kelvin.
• T is the absolute temperature in kelvin.

With this formula, you can determine the wavelength of peak radiation intensity if you know the absolute temperature, or you can determine the temperature if you know the wavelength at peak intensity. For example, the sun is believed to have a surface temperature of about 5,780 kelvin. The following set of equations demonstrates how to calculate the wavelength at peak intensity (in meters):

λ_max = b / T
λ_max = 2.89777 ⋅ 10^-3 m ⋅ K / 5,780 K
λ_max = 0.0028977 m / 5,780
λ_max = 5.01332 ⋅ 10^-7 m

The sun's wavelength at peak radiation intensity is about 5.01332 ⋅ 10^-7 m. This comes to 501.33218 nanometers, or about 500 nm, which puts the wavelength in the green portion of the visible electromagnetic spectrum. However, the sun appears to emit white light because it also radiates strongly in the other portions of the visible spectrum. It simply radiates most strongly in the green portion.

Using Wien's law to determine surface temperature

You can also use Wien's law to find the surface temperature of an object based on its peak wavelength. For example, the following set of equations shows how to calculate the temperature of a star with a peak wavelength of about 400 nm (the violet end of the visible range):

λ_max = b / T
400 nm = 2.89777 ⋅ 10^-3 m ⋅ K / T
400 nm = 0.0028977 m ⋅ K / T
400 nm = 2,897,700 nm ⋅ K / T
T = 2,897,700 nm ⋅ K / 400 nm
T = 7,244.25 K

The surface temperature of a star with a peak wavelength of 400 nm is about 7,244.25 K, which means the star is hotter than our sun. This points to another important aspect of Wien's law: As a black body grows hotter, the wavelength of its peak energy output grows shorter, placing it higher on the electromagnetic spectrum.

See also: solar constant, passive solar, solar cooling, radiant energy, kinetic energy, heat and potential energy.