Stefan-Boltzmann Law (DP IB Physics: SL): Revision Note

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:

$P=\sigma A{T}^4$

• Where:

  • P = total power emitted across all wavelengths (W)

  • σ = the Stefan-Boltzmann constant

  • A = surface area of the body (m²)

  • 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 (m)

• Substituting the above for area, A, the Stefan-Boltzmann equation then becomes:

$L=4\mathrm{\pi }{r}^2\sigma T^4$

• 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)