# 4.16 Wien's Law

The concept of thermal radiation is related to the ideas of energy and temperature. The key principles were discovered in 1898 by the German physicist Wilhelm Wien (pronounced Veen). Wien discovered that all bodies constantly emit thermal radiation. Thermal radiation is always concentrated at certain wavelengths; you can see the peak of the solar spectrum at the color yellow. Wien also found that the spectrum of thermal radiation does not depend on what the body is made of. He experimented with bodies of different temperatures and deduced the following principles, known collectively as Wien's law.

Wilhelm Wien. Click here for original source URL

&bull: All bodies emit thermal radiation spanning a broad range of wavelengths.

• The amount and peak wavelength of the radiation depends on the temperature of the body, but not on its composition.

• The higher the temperature, the more radiation is emitted and the shorter (or bluer) the wavelength of the bulk of the radiation.

Atoms and molecules are in ceaseless motion. This is true of all materials and for all states of matter. In a solid, the motions are purely vibrations. In a liquid or a gas, the motions also involve collisions of freely moving particles. We have seen that temperature is a measure of the kinetic energy of the particles in a material; motion only ceases at absolute zero on the Kelvin scale. Atoms and molecules emit radiation as a result of their motion. Wien discovered how this radiation depends on the temperature of the material.

Graphic representation of Wein's Law. Click here for original source URL

The mathematical form of Wien's law identifies the dominant wavelength, or color, of light coming from a body at a given temperature. It is surprisingly simple. Suppose we designate the temperature of the body as T, given in Kelvins. The wavelength at which the maximum amount of radiation is emitted can be called λ, given in meters. Using the variables T and λ, Wien's law can be expressed as:

λ = 0.0029 / T

The number 0.0029 is a constant of proportionality, and is the same in all applications of the law, as long as T is given in Kelvins and W in meters. This is an inverse relationship between wavelength and temperature. So the higher the temperature, the shorter or smaller the wavelength of the thermal radiation. The lower the temperature, the longer or larger the wavelength of the thermal radiation. For visible radiation, hot objects emit bluer light than cool objects.

For example, if the Sun has a surface temperature of 5700 K, what is the wavelength of maximum intensity of solar radiation? If we substitute 5700 K for T, we have λ = 0.0029 / 5700, or 5.1 × 10^{-7} meters. Knowing that violet light has a wavelength of about 4.0 × 10^{-7} meters, yellow about 5.6 × 10^{-7} meters, and red about 6.6 × 10^{-7} meters, what can we say about the color of the Sun's peak radiation? The peak wavelength of the Sun's radiation is at a slightly shorter wavelength than the color yellow, so it is a slightly greenish yellow. To see this greenish tinge to the Sun, you would have to look at it from space. It turns out that the Earth's atmosphere scatters some of the shorter waves of sunlight, which shifts its peak wavelength to pure yellow.

Remember that thermal radiation always spans a wide range of wavelengths; the mathematical form given above specifies the single wavelength that is the peak of the spectrum. So although the Sun appears yellowish-white, when you disperse sunlight with a prism you see radiation with all the colors of the rainbow. Yellow just represents the average wavelength of the emission.

What is the thermal radiation from an object much cooler than the Sun? For example, what is the wavelength and dominant type of radiation that your body gives off? As a hint, the body's "normal" temperature is 98.6° F, or about 37° C. First convert T to Kelvins. Since a Kelvin is a centigrade degree plus 273, T is about 310 K. Your radiating skin might be a bit cooler, say 305 K. Using the equation for Wien's law, we get λ = .0029 / 305 = 9.5 × 10^{-6} meters. The longest red wavelengths of visible light are around 7 × 10^{-7} meters, so this is far beyond the red end of the spectrum. The human body emits thermal radiation at wavelengths that are too long for the eye to detect.

Any object at a temperature of a few thousand degrees emits visible thermal radiation and any object at a few hundred degrees emits infrared thermal radiation. Another example of Wien's law at work is the technological development called "night goggles" or "infrared vision." These devices pick up infrared light and show it to us by converting it visible light, such as a TV or photo image. In this way we can "see" the infrared thermal radiation of cool objects.

Suppose a certain star has a noticeably red color to the naked eye. Given that deep red light has a wavelength around 700 nm, what temperature is the star? (A nanometer, abbreviated nm, is 10^{-9} meters.) Compare the answer to the 5700 K temperature of the Sun, mentioned above. If the wavelength of dominant radiation, W, is about 700 nm, or 7 × 10^{-7} meters, then Wien's law gives us 7 × 10^{-7} = 0.0029 / T. Solving for T, we have T = 4140 K. So the red star is around 1600 K cooler than the Sun.