A small object is placed at the centre of a large evacuated hollow spherical container. Assume the container…

Using Stefan-Boltzmann law,

$P=\frac{\mathrm{d}Q}{\mathrm{d}t}=\sigma eA\left[T_B^4-T_S^4\right], \left(\because T_S=0 \mathrm{K}\right)$

So,

$\frac{\mathrm{d}Q}{\mathrm{d}t}=e\sigma A{T}^4, \left(\text{consider}, T=T_B\right)$

also,    

$\frac{\mathrm{d}Q}{\mathrm{d}t}=-ms\frac{\mathrm{d}T}{\mathrm{d}t}=e\sigma A{T}^4$

$\underset{200}{\overset{T}{-\int }}\frac{\mathrm{d}T}{T^4}={\int }_0^{t}k\mathrm{d}t, \text{where} k=\frac{\sigma A}{ms}$

$\Rightarrow \frac{1}{T^3}-\frac{1}{{\left(200\right)}^3}=3kt$

Now, 

For case $\left(1\right)$

$\frac{1}{{\left(100\right)}^3}-\frac{1}{{\left(200\right)}^3}=3k{t}_1 ...\left(1\right)$

For case $\left(2\right)$

$\frac{1}{{\left(50\right)}^3}-\frac{1}{{\left(200\right)}^3}=3k{t}_2 ...\left(2\right)$

Divide equation $\left(2\right)$ by $\left(1\right)$,

$\frac{\frac{1}{{\left(50\right)}^3}-\frac{1}{{\left(200\right)}^3}}{\frac{1}{{\left(100\right)}^3}-\frac{1}{{\left(200\right)}^3}}=\frac{t_2}{t_1}$

So, $\frac{t_2}{t_1}=9$