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Stefan's Law (OCR A Level Physics): Revision Note

Exam code: H556

Katie M

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Katie M

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Stefan's Law

  • An object's luminosity depends on:

    • its surface temperature

    • its surface area

  • The relationship between these is known as Stefan's law or the Stefan-Boltzmann law, which states:

The total energy emitted by a black body per unit area per second is proportional to the fourth power of the absolute temperature of the body

  • Stefan's law equation is given by:

L space equals space 4 straight pi r squared sigma T to the power of 4

  • Where:

    • L = luminosity of the star (W)

    • r = radius of the star

    • σ = the Stefan-Boltzmann constant

    • T = surface temperature of the star (K)

  • This equation shows that the luminosity of a star is directly proportional to:

    • the square of its radius L space proportional to space r squared

    • its surface area L space proportional to space 4 straight pi r squared

    • the fourth power of its surface temperature L space proportional to space T to the power of 4

Worked Example

The surface temperature of Proxima Centauri, the nearest star to Earth, is 3000 K, and its luminosity is 6.506 × 1023 W.

Calculate the radius of Proxima Centauri, in km.

Answer:

Step 1: List the known quantities

  • Surface temperature, T = 3000 K

  • Luminosity, L = 6.506 × 1023 W

  • Stefan's constant, σ = 5.67 × 10−8 W m−2 K−4

Step 2: Write down Stefan's law

L space equals space 4 straight pi r squared sigma T to the power of 4

Step 3: Rearrange the equation for radius r 

r space equals space square root of fraction numerator L over denominator 4 straight pi sigma T to the power of 4 end fraction end root

Step 4: Substitute the values into the equation 

r space equals space square root of fraction numerator 6.506 cross times 10 to the power of 23 over denominator 4 straight pi cross times left parenthesis 5.67 cross times 10 to the power of negative 8 end exponent right parenthesis cross times 3000 to the power of 4 end fraction end root space equals space 1.062 cross times 10 to the power of 8 space straight m

Step 5: Write the final answer in the correct unit

  • The radius of Proximal Centauri is 1.062 cross times 10 to the power of 5 space km or 106 space 200 space km (4 s.f.)

Examiner Tips and Tricks

Remember to convert temperatures into Kelvin.

Check the values you obtain in your calculations. Do they make sense? Do they fit in with the magnitudes of other stars/objects that you know about?

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    Katie M

    Author: Katie M

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    Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.