Question

A small object is placed at the center of a large evacuated hollow spherical container. Assume that the container is maintained at $0K$. At time $t=0$, the temperature of the object is $200K$. The temperature of the object becomes $100K$ at $t=t_1$ and $50K$ at $t=t_2$. Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio $\left(t_2/t_1\right)$ is______.

Answer 1

Answer: 9

$-C\frac{dT}{dt}=\left(T^4-T_s^4\right)$
$\underset{200}{\overset{100}{\int }}\frac{dT}{T^4-T_s^4}=\underset{0}{\overset{t}{\int }}-\frac{1}{c}dt$
$-\frac{1}{3}{\left[\frac{1}{T^3}\right]}_{200}^{100}=-\frac{1}{c}\left(t_1\right)$
$\Rightarrow \left[\frac{1}{(100{)}^3}-\frac{1}{(200{)}^3}\right]=\frac{3}{c}{t}_1$
Similarly, $\left[\frac{1}{(50{)}^3}-\frac{1}{(200{)}^3}\right]=\frac{3}{c}{t}_2$
$\frac{t_2}{t_1}=\frac{\left[\frac{1}{(50{)}^3}-\frac{1}{(200{)}^3}\right]}{\left[\frac{1}{(100{)}^3}-\frac{1}{(200{)}^3}\right]}=9$