Use of Stefan’s law to compare the power output, temperature and size of…

The idea

Stefan's law states that a black body radiates power P = σAT⁴, where σ is the Stefan constant, A is surface area and T is absolute temperature in kelvin. For a spherical star, A = 4πr², so P = 4πσr²T⁴. This lets you compare stars: if two stars have the same luminosity but different temperatures, the cooler one must be larger. Form a ratio to cancel σ and 4π, for example P₁/P₂ = (r₁²T₁⁴)/(r₂²T₂⁴). Rearranged, r ∝ √P / T². Always convert temperatures to kelvin, use radius in metres, and remember that doubling temperature increases power by a factor of sixteen. The law assumes the star behaves as a black-body radiator.

What earns the marks

• States Stefan's law as P = σAT⁴ and identifies each symbol, including T in kelvin and A as the emitting surface area.
• Uses A = 4πr² for a spherical star to obtain P = 4πσr²T⁴ and rearranges for radius, temperature or power as required.
• Forms a ratio for two stars so that σ and 4π cancel, for example P₁/P₂ = (r₁²T₁⁴)/(r₂²T₂⁴), and solves for the unknown quantity.
• Applies the fourth-power dependence correctly, for example recognising that doubling absolute temperature multiplies radiated power by sixteen at constant area.
• Compares stars of different temperature and size, explaining that a cooler star must have a larger radius to emit the same power as a hotter star.
• Converts temperatures from degrees Celsius to kelvin by adding 273 before substitution, and keeps consistent SI units throughout.
• States that the star is modelled as a black-body radiator, so the law gives the total power radiated across all wavelengths.

Common mistakes and how to put them right

• Substituting temperature in degrees Celsius instead of kelvin.

  Put it right: Add 273 to convert to kelvin before raising to the fourth power.

• Using the diameter or circumference in place of the radius when calculating surface area.

  Put it right: Use A = 4πr² with the radius, halving a diameter if necessary.

• Forgetting to square the radius when forming a ratio, or applying the fourth power to the radius.

  Put it right: Keep r² and T⁴ separate in the ratio expression.

• Treating σ as a variable that must be evaluated when comparing two stars.

  Put it right: Cancel it, along with 4π, in the ratio.

Exam tips

• Write the full equation P = 4πσr²T⁴ before substituting, then rearrange symbolically to reduce arithmetic errors.
• For comparison questions, set up a ratio and cancel constants; this is quicker and less error-prone than calculating absolute values.
• Check whether the question gives diameter or radius, and whether temperature is in kelvin, before starting the calculation.
• Give the final answer with an appropriate unit and a sensible number of significant figures, and comment on whether the result is physically reasonable.