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Formula Wien's Displacement Law (Radiation Law) Temperature    Wavelength   

Formula
Formula: Wien's Displacement Law (Radiation Law)
Radiant exitance of the sun

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.