Question

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 $u=\frac{U}{V}\propto T^4$ and pressure $p=\frac{1}{3}\left(\frac{U}{V}\right)$. If the now undergoes an adiabatic expansion the relation between $T$ and $R$ is

• A. $T\propto \frac{1}{R}$
• B. $T\propto {\frac{1}{R}}^3$
• C. $T\propto e^{-R}$
• D. $T\propto e^{-3R}$

Answer 1

The correct option is A $T\propto \frac{1}{R}$

$\text{Explanation:-}$

According to question,

$p=\frac{1}{3}\left(\frac{U}{V}\right)$

$\Rightarrow \frac{nRT}{V}=\frac{1}{3}\left(\frac{U}{V}\right)[\because pV=nRT]$ or

$\frac{nRT}{V}\propto \frac{1}{3}{T}^{4}orV{T}^3=constant$ or

$\frac{4}{3}\pi R^3{T}^3=constantorTR=constant$

$\Rightarrow T\propto \frac{1}{R}$

$\text{Hence the correct option is A}$