Alexander Fufaev
My name is Alexander FufaeV and here I will explain the following topic:

Wien's Displacement Law: Temperature of a Black Body

Formula

Formula: Wien's Displacement Law (Radiation Law)
Radiant exitance of the sun
What do the formula symbols mean?

Temperature

Unit
Absolute temperature of the glowing body (i.e. the radiation source). This can be for example the sun. If you know the wavelength \( \lambda_{\text{max}} \), then you can determine the temperature of the sun directly from the earth! Here \( 2897.8 \,\cdot\, 10^{-6} \, \mathrm{mK} \) is the Wien constant. Note: \( \text{mK}\) is NOT "millikelvin", but the unit "meter times kelvin".

The sun has its radiation maximum at about a wavelength of \( \class{blue}{\lambda_{\text{max}}} = 500 \, \mathrm{nm} \) (nanometer: \(10^{-9} \, \mathrm{m}\)). With this wavelength we can estimate the temperature of the sun: \begin{align} T &~=~ \frac{2897.8 \,\cdot\, 10^{-6} \, \mathrm{m} \mathrm{K}}{ 500 \, \mathrm{nm} } \\\\ &~=~ \frac{2897.8 \,\cdot\, 10^{-6} \, \mathrm{m} \mathrm{K}}{ 500 \cdot 10^{-9} \, \mathrm{m} } \\\\ &~=~ 5796 \, \mathrm{K} \end{align}

The surface of the sun has a temperature of \( 5796 \, \mathrm{K} \). That is approximately \( 5523^{\circ} \, \mathrm{C}\).

Wavelength

Unit
The glowing body emits the light of different wavelength. The light of wavelength \( \class{blue}{ \lambda_{\text{max}} \) has the highest intensity. Exactly this wavelength is used in this formula. Usually you can determine it from the given spectral intensity distribution of the glowing body.

The surface of the star Sirius in the constellation 'big dog' has approximately a temperature of \( 10 \, 000 \, \mathrm{K} \). With Wien's radiation law we can determine the wavelength of the radiation which is emitted most by Sirius: \begin{align} \class{blue}{ \lambda_{\text{max}}} &~=~ \frac{2897.8 \,\cdot\, 10^{-6} \, \text{m} \text{K}}{ 10 \, 000 \, \mathrm{K} } \\\\ &~=~ 2.89 \cdot 10^{-7} \, \mathrm{m} \\\\ &~=~ 289 \, \mathrm{nm} \end{align}

So Sirius emits the most electromagnetic radiation, which has the wavelength 289 nanometers. This corresponds to extreme UV radiation.

Wien's constant

Wien's constant comes from the derivation using Planck's law of radiation.
Explanation

Video

Wien's displacement law describes the relationship between the temperature of a black body (an idealized object that absorbs and emits all rays that hit it) and the wavelength of the maximum of its spectral radiation. The law was formulated by Wilhelm Wien in 1893 and can be expressed as a formula as follows:

Formula anchor

Here, \( \class{blue}{\lambda_{\mathrm{max}}} \) stands for the wavelength of the maximum of the spectral intensity distribution, \( T \) for the absolute temperature of the black body in Kelvin, and \( 2897.8 \,\cdot\, 10^{-6} \, \mathrm{m} \mathrm{K} \) is the Wien constant. The Wien constant comes from the derivation using Planck's radiation law.

The interpretation of this law states that at higher temperatures the black body emits radiation with shorter wavelengths, while at lower temperatures the emitted radiation has wavelengths in the longer range of the spectrum. This means that the color of the emitted radiation changes with increasing temperature.

Note that Wien's displacement law is only an approximation formula for short wavelengths!

How to Determine the Temperature of the Sun without Touching it?

This lesson is also available as a YouTube video: Temperature of the Sun: how to determine

To calculate the temperature \( T \) of the Sun from Earth, one must first find out what light the Sun emits. The Sun emits a polychromatic light, that is, infrared light, visible light, and even X-rays and gamma rays. You need to figure out the wavelengths \( \lambda \) of each of these radiations and with what intensity ("how bright"?) they are emitted. If you plot all wavelengths with their intensities in a diagram, you get a spectral intensity distribution of the sun. From this distribution, you have to read off the wavelength \( \class{blue}{\lambda_{\text{max}}} \) which has the highest intensity (maximum of the function).

fufaev.org Example of Radiant Exitance of the Sun
Radiation maximum of the sun occurs at the blue light.

Then you can calculate the temperature of the sun with the Wien's displacement law. In the case of the sun, the maximum is within the blue light spectrum at the wavelength \( \lambda_{\text{max}} = 490 \, \mathrm{nm} = 490 \cdot 10^{-9}\, \mathrm{m} \). This results in a temperature of \( T = 5900 \, \mathrm{K}\) or \( 5630\, ^{\circ}\mathrm{C} \).