Wien's law

Wien's displacement law describes the relationship between the temperature of a blackbody and the wavelength at which it emits radiation most intensely.

The relationship between temperature and the max wavelength is given below:

${\lambda }_{max}{\text{ T = constant = 2.9 x 10}}^{-3}\text{ m K}$

Where:

• ${\lambda }_{max}$ = peak wavelength (m)
• T = absolute temperature of blackbody(K)
• 2.9 x 10^-3^ m K = Wien's displacement constant

Worked example - Estimating the radius of Betelgeuse

Betelgeuse has a luminosity of 1.2 x 10³¹ W and a peak wavelength of 970 nm. Calculate the radius of Betelgeuse.

Step 1: Convert nm to m

to convert from. nm to m, multiply by 1 x 10^-9

970 nm = 970 x 10^-9 m

Step 2: Wien's law formula

${\lambda }_{max}T=2.9\times {10}^{-3}$

Step 3: Rearranged formula

$T=\frac{2.9\times {10}^{-3}}{{\lambda }_{max}}$

Step 4: Substitution and correct evaluation

$T=\frac{2.9\times {10}^{-3}}{970\times {10}^{-9}}=2989.7\text{ K}$

Step 5: Stefan’s law formula

$L=4\pi R^2\sigma T^4$

Step 6: Rearranged formula

$R=\sqrt{\frac{L}{4\pi \sigma T^4}}$

Step 7: Substitution and correct evaluation

$R=\sqrt{\frac{1.2\times {10}^{31}}{4\pi \times 5.67\times {10}^{-8}\times {(2989.7)}^4}}{\text{ = 4.6 x 10}}^{11}\text{ m}$