Consider a spherical shell of radius $\mathrm{R}$ at temperature $\mathrm{T}$. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume $u=\frac{U}{V} \propto T^{4}$ and pressure…

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

Aicharya · Accepted Answer

As, $\mathrm{P}=\frac{1}{3}\left(\frac{\mathrm{U}}{\mathrm{V}}ight)$
But $\frac{\mathrm{U}}{\mathrm{V}}=\mathrm{KT}^{4}$
So, $\mathrm{P}=\frac{1}{3} \mathrm{KT}^{4}$
or $\quad \frac{\mathrm{uRT}}{\mathrm{V}}=\frac{1}{3} \mathrm{KT}^{4}[\mathrm{As} \mathrm{PV}=\mathrm{u} \mathrm{RT}]$
$\frac{4}{3} \mathrm{pR}^{3} \mathrm{~T}^{3}=$ constant
Therefore, $\mathrm{T} \propto \frac{1}{\mathrm{R}}$ .

Consider a spherical shell of radius $\mathrm{R}$ at temperature $\mathrm{T}$. The black body radiation…

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