IBPhysicsHLRevision NotesThe Particulate Nature of MatterThermal Energy TransfersStefan-Boltzmann Law

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First teaching 2023

First exams 2025

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Stefan-Boltzmann Law (HL IB Physics)

Revision Note

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Physics

Stefan-Boltzmann Law

  • The total power P radiated by a perfect black body depends on two factors:
    • Its absolute 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

  • The Stefan-Boltzmann Law can be calculated using:

P space equals space sigma A T to the power of 4

  • Where:
    • P = total power emitted across all wavelengths (W)
    • σ = the Stefan-Boltzmann constant
    • A = surface area of the body (m)
    • T = absolute temperature of the body (K)
  • The Stefan-Boltzmann law is often used to calculate the luminosity of celestial objects, such as stars
    • Stars can be approximated as black bodies, as almost all radiation incident on a star is absorbed
    • The power emitted across all wavelengths, P, for a star is just its luminosity, L
  • The surface area of a star (or other spherical object) is equal to A = 4πr2
    • Where r = radius of the star
  • Substituting the above for area, A, the Stefan-Boltzmann equation then becomes:

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 (m)
    • σ = the Stefan-Boltzmann constant
    • T = surface temperature of the star (K)

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 solar radii. 

Solar radius R☉ = 6.96 × 108 m

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
  • Radius of the Sun, R☉ = 6.96 × 108 m

Step 2: Write down the Stefan-Boltzmann equation and rearrange for radius r

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

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 3: 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

Radius of Proxima Centauri:  R = 1.061 × 108 m

Step 4: Divide the radius of Proxima Centauri by the radius of the Sun

R over R subscript ☉ space equals space fraction numerator 1.061 space cross times space 10 to the power of 8 over denominator 6.96 space cross times space 10 to the power of 8 end fraction space equals space 0.152 space R subscript ☉

  • Proxima Centauri has a radius which is about 0.152 times that of the Sun
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Katie M

Author: Katie M

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.