Consider a spherical shell of radius R at temperature T. The black body radiation inside it can be considered as an ideal gas of photons with…

Consider a spherical shell of radius $\displaystyle R$ at temperature T. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume $\displaystyle \mathrm{u}=\frac{\mathrm{U}}{\mathrm{V}} \propto \mathrm{T}^4$ and pressure $\displaystyle \mathrm{p}=\frac{1}{3}\left(\frac{\mathrm{U}}{\mathrm{V}}\right)$. If the shell now undergoes an adiabatic expansion the relation between T and R is :