$A$ small object is placed at the center of a large evacuated hollow spherical container. Assume that the container is maintained at $0 \ K$. At time $t = 0$,the temperature of the object is $200 \ K$. The temperature of the object becomes $100 \ K$ at $t = t_1$ and $50 \ K$ 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 $(t_2 / t_1)$ is:

The rectangular surface of area $8 \text{ cm} \times 4 \text{ cm}$ of a black body at a temperature of $127^{\circ}\text{C}$ emits energy at the rate of $E$ per second. If the length and breadth of the surface are each reduced to half of the initial value and the temperature is raised to $327^{\circ}\text{C}$,the rate of emission of energy will become

$A$ solid copper sphere of density $\rho$,specific heat capacity $C$ and radius $r$ is initially at $200 \ K$. It is suspended inside a chamber whose walls are at $0 \ K$. The time required (in $\mu s$) for the temperature of the sphere to drop to $100 \ K$ is ( $\sigma$ is Stefan's constant and all the quantities are in $SI$ units.)

$A$ black coloured solid sphere of radius $R$ and mass $M$ is inside a cavity with vacuum inside. The walls of the cavity are maintained at temperature $T_0$. The initial temperature of the sphere is $3T_0$. If the specific heat of the material of the sphere varies as $\alpha T^3$ per unit mass with the temperature $T$ of the sphere,where $\alpha$ is a constant,then the time taken for the sphere to cool down to temperature $2T_0$ will be ($\sigma$ is Stefan-Boltzmann constant).

Radiated energy at $T$ temperature is $E$ for a body of diameter $d$. If the temperature becomes $2T$ and the diameter becomes $d/4$,then the radiated energy will be:

$A$ star emits radiation with maximum intensity at a wavelength of $289.9 \, nm$. What is the intensity of the radiation emitted by the star? (Given: Stefan's constant $\sigma = 5.67 \times 10^{-8} \, W/m^2K^4$,Wien's constant $b = 2898 \, \mu m \cdot K$)