Solar constant $(S)$ depends upon the temperature of the Sun $(T)$ as ........

Radiation from a black body at the thermodynamic temperature $T_1$ is measured by a small detector at distance $d_1$ from it. When the temperature is increased to $T_2$ and the distance to $d_2$,the power received by the detector is unchanged. What is the ratio $d_2/d_1$?

The ratio of energies of radiation emitted by a black body at $600 \ K$ and $900 \ K$ when the surrounding temperature is $300 \ K$ is: (in $/16$)

Two spheres $P$ and $Q$ have the same emissivity and radii $8 \ cm$ and $2 \ cm$ respectively. They are maintained at temperatures $127^{\circ}C$ and $527^{\circ}C$ respectively. Find the ratio of the radiant energy emitted by $P$ to that by $Q$.

$A$ black body of mass $34.38 \ g$ and surface area $19.2 \ cm^2$ is at an initial temperature of $400 \ K$. It is allowed to cool inside an evacuated enclosure kept at a constant temperature of $300 \ K$. The rate of cooling is $0.04 \ ^{\circ}C/s$. The specific heat of the body in $J \ kg^{-1} \ K^{-1}$ is (Stefan's constant $\sigma = 5.73 \times 10^{-8} \ W \ m^{-2} \ K^{-4}$)

If the temperature of the Sun gets doubled, the rate of energy received on the Earth will increase by a factor of