Physics

Physics

Problem

Show that for an ideal blackbody the product λmax⁡4R\lambda_{\max}^{4}R is a constant, where RR is the total power radiated per unit area and λmax⁡\lambda_{\max} is the wavelength of maximum emission. Hence calculate the numerical value of this constant using σ=5.67×10−8 W m−2 K−4\sigma = 5.67\times10^{-8}\,\text{W\,m}^{-2}\,\text{K}^{-4} and b=2.898×10−3 m Kb = 2.898\times10^{-3}\,\text{m\,K}.
Intuitive explanation

Wien's displacement law tells you that λmax⁡\lambda_{\max} is inversely proportional to temperature, while the Stefan–Boltzmann law says total radiated power RR scales as T4T^4. The key insight is to eliminate temperature between these two relationships: multiply λmax⁡\lambda_{\max} (which contains 1/T1/T) by RR (which contains T4T^4) to see how the temperature dependence cancels, leaving only fundamental constants.

Intuitive explanation

Wien's displacement law tells you that λmax⁡\lambda_{\max} is inversely proportional to temperature, while the Stefan–Boltzmann law says total radiated power RR scales as T4T^4. The key insight is to eliminate temperature between these two relationships: multiply λmax⁡\lambda_{\max} (which contains 1/T1/T) by RR (which contains T4T^4) to see how the temperature dependence cancels, leaving only fundamental constants.

Browse the problem library