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
JEE MAIN 2015, Diffcult
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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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