Wien's law and the Stefan–Boltzmann law Edexcel International A Level Physics revision
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In plain words
Heat a poker and it glows red, then orange, then white: its colour tells you its temperature. The same goes for stars, and blue ones are hotter than red ones. Put a star's temperature together with how much light it gives out, and you can work out how big it is without ever going there.
4 things to know
- A black body absorbs all the radiation that falls on it, and it gives out a continuous spectrum that depends only on its temperature. Stars are very nearly black bodies. A hotter one gives out more at every wavelength, and its peak moves to a shorter wavelength.
- Wien's displacement law: the wavelength of the peak is inversely proportional to the temperature. λmax × T = 2.90 × 10⁻³ m K.
- Stefan–Boltzmann law: L = σAT⁴. For a spherical star of radius r this is L = 4πσr²T⁴, where σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴.
- So the peak wavelength gives the temperature, and the luminosity and the temperature together give the radius.
Worked example
The spectrum of a star peaks at 500 nm, and its luminosity is 3.9 × 10²⁶ W. Find its surface temperature and its radius.
- T = 2.90 × 10⁻³ ÷ 5.00 × 10⁻⁷ = 5800 K.
- r² = L ÷ 4πσT⁴ = 3.9 × 10²⁶ ÷ (4π × 5.67 × 10⁻⁸ × 5800⁴) = 4.8 × 10¹⁷ m².
- r = 7.0 × 10⁸ m.
Tips and tricks
- Temperatures must be in kelvin, and T is raised to the fourth power: doubling the temperature gives sixteen times the power.
- A shorter peak wavelength means a hotter star. Blue stars are hot and red stars are cool.
It lands in your notebook with its questions as flashcards.
Wien's law and the Stefan–Boltzmann law: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
A star is hotter than the Sun. How does the peak wavelength of its spectrum compare?
The peak wavelength is inversely proportional to temperature.
How does the luminosity of a star of fixed size depend on its surface temperature T?
The Stefan–Boltzmann law.
What is a black body?
It is also the best possible emitter at every wavelength.
Which colour of star is the hottest?
Blue light has the shortest wavelength.
The radius of a star doubles while its surface temperature stays the same. What happens to its luminosity?
Its surface area is four times bigger.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
4 questions, 9 marks. Write your answers on paper, then check them.
Wien's law and the Stefan–Boltzmann law
Edexcel International A Level Physics WPH · 9 marks · papermunch.org
Name ______________________________ Date ______________
A star has a surface temperature of 3000 K. Find the wavelength at which its spectrum peaks.[2]
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9.7 × 10⁻⁷ m (970 nm, in the infrared).
A star has a radius of 1.0 × 10⁹ m and a surface temperature of 4000 K. Find its luminosity.[3]
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1.8 × 10²⁶ W. 4π × 5.67 × 10⁻⁸ × (1.0 × 10⁹)² × 4000⁴.
Two stars are the same size. The surface temperature of star X, in kelvin, is twice that of star Y. Compare their luminosities.[2]
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X is 16 times as luminous as Y.
Describe how the radiation curve of a black body changes as it gets hotter.[2]
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The curve is higher at every wavelength, so more power is radiated in total, and the peak moves to a shorter wavelength.
Answers: Wien's law and the Stefan–Boltzmann law
- 1. 9.7 × 10⁻⁷ m (970 nm, in the infrared).
- 2. 1.8 × 10²⁶ W. 4π × 5.67 × 10⁻⁸ × (1.0 × 10⁹)² × 4000⁴.
- 3. X is 16 times as luminous as Y.
- 4. The curve is higher at every wavelength, so more power is radiated in total, and the peak moves to a shorter wavelength.