Wien's Displacement Law

Wien’s displacement law relates the observed wavelength of light from a star to its surface temperature, it states:

The black body radiation curve for different temperatures peaks at a wavelength which is inversely proportional to the temperature

This relation can be written as:

$λ_{m a x} \propto \frac{1}{T}$

Where
- $λ_{m a x}$ = the maximum wavelength emitted by the star at the peak intensity (m)
- $T$ = thermodynamic temperature at the surface of the star (K)

A black body is an object which:
- absorbs all the radiation that falls on it, and is also a good emitter
- does not reflect or transmit any radiation
A black-body is a theoretical object, however, stars are the best approximation there is
The radiation emitted from a black-body has a characteristic spectrum that is determined by the temperature alone

Wiens Law Graph, downloadable AS & A Level Physics revision notes

The intensity-wavelength graph shows how thermodynamic temperature links to the peak wavelength for four different stars

The constant of proportionality in Wien's law is given by

$λ_{m a x} T = 2.9 \times 10^{- 3} m K$

This equation tells us the higher the temperature of a body:
- the shorter the wavelength at the peak intensity
- the greater the intensity of the radiation at each wavelength

This means that hotter stars tend to be white or blue and cooler stars tend to be red or yellow

Table to compare surface temperature and star colour

Colour of star	Temperature / K
blue	> 33 000
blue-white	10 000 – 30 000
white	7500 – 10 000
yellow-white	6000 – 7500
yellow	5000 – 6000
orange	3500 – 5000
red	< 3500

Worked example

The spectrum of the star Rigel in the constellation of Orion peaks at a wavelength of 263 nm, while the spectrum of the star Betelgeuse peaks at a wavelength of 828 nm.

Which of these two stars is cooler, Betelgeuse or Rigel?

Answer:

Step 1: Write down Wien’s displacement law

$λ_{m a x} T = 2.9 \times 10^{- 3} m K$

Step 2: Rearrange for temperature T

$T = \frac{2.9 \times 10^{- 3}}{λ_{m a x}}$

Step 3: Calculate the surface temperature of each star

Rigel: $T = \frac{2.9 \times 10^{- 3}}{λ_{m a x}} = \frac{2.9 \times 10^{- 3}}{263 \times 10^{- 9}} = 11 026 = 11 000 K$

Betelgeuse: $T = \frac{2.9 \times 10^{- 3}}{λ_{m a x}} = \frac{2.9 \times 10^{- 3}}{828 \times 10^{- 9}} = 3502 = 3500 K$

Step 4: Write a concluding sentence

Betelgeuse has a surface temperature of 3500 K, therefore, it is much cooler than Rigel

Wiens Law Orion, downloadable AS & A Level Physics revision notes

The Orion Constellation; cooler stars, such as Betelgeuse, appear red or yellow, while hotter stars, such as Rigel, appear white or blue

Examiner Tip

Note that the temperature used in Wien’s Law is in Kelvin (K). Remember to convert from ^oC if the temperature is given in degrees in the question before using the Wien’s Law equation.

Wien's Displacement Law (CIE A Level Physics)

Revision Note