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Wien's Displacement Law (CIE A Level Physics)

Revision Note

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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:

lambda subscript m a x end subscript space proportional to space 1 over T

  • Where
    • lambda subscript m a x end subscript = 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

lambda subscript m a x end subscript T space equals space 2.9 cross times 10 to the power of negative 3 end exponent space straight m space straight 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

lambda subscript m a x end subscript T space equals space 2.9 space cross times space 10 to the power of negative 3 end exponent space straight m space straight K

Step 2: Rearrange for temperature T

T space equals space fraction numerator 2.9 space cross times space 10 to the power of negative 3 end exponent space over denominator lambda subscript m a x end subscript end fraction

Step 3: Calculate the surface temperature of each star

Rigel: T space equals space fraction numerator 2.9 space cross times 10 to the power of negative 3 end exponent space over denominator lambda subscript m a x end subscript end fraction space equals space fraction numerator 2.9 space cross times space 10 to the power of negative 3 end exponent space over denominator 263 space cross times space 10 to the power of negative 9 end exponent end fraction space equals space 11 space 026 space equals space 11 space 000 space straight K

Betelgeuse: T space equals space fraction numerator 2.9 space cross times 10 to the power of negative 3 end exponent space over denominator lambda subscript m a x end subscript end fraction space equals space fraction numerator 2.9 space cross times space 10 to the power of negative 3 end exponent space over denominator 828 space cross times space 10 to the power of negative 9 end exponent end fraction space equals space 3502 space equals space 3500 space straight 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.

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Ashika

Author: Ashika

Expertise: Physics Project Lead

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.