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 internal energy per unit volume\(E=\) \(\frac{U}{V} \propto {T^4}\) and pressure \(P = \frac{1}{3}\left( {\frac{U}{V}} \right)\) If the shell now undergoes an adiabatic expansion the relation between \(T\) and \(R\) is

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(B) \(\;T \propto \frac{1}{R}\)

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\(As,\,P = \frac{1}{3}\left( {\frac{U}{V}} \right)\)

\(But\frac{U}{V} = K{T^4}\)

\(So,P = \frac{1}{3}K{T^4}\)

\(or\,\frac{{uRT}}{V} = \frac{1}{3}K{T^4}\,\,\,\left[ {As\,PV = u\,RT} \right]\)

\(\frac{4}{3}\pi {R^3}{T^3} = constant\)

\(Therfore,T \propto \frac{1}{R}\)

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