$A$ planet of radius $R_{p}$ is revolving around a star of radius $R^{*}$ which is at temperature $T^{*}$. The distance between the star and the planet is $d$. If the planet's temperature is $f T^{*}$, then $f$ is proportional to

Two spheres of the same material have radii $1 \ m$ and $4 \ m$ and temperatures $4000 \ K$ and $2000 \ K$ respectively. The ratio of the energy radiated by them is .....

Assuming the sun to have a spherical outer surface of radius $r$,radiating like a black body at temperature $t^{\circ} C$,the power received by a unit surface (normal to the incident rays) at a distance $R$ from the centre of the sun is,where $\sigma$ is the Stefan's constant.

$A$ hot metallic sphere of radius $r$ radiates heat. Its rate of cooling is

$A$ 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.)

Two spheres of radii $4 \, m$ and $1 \, m$ have temperatures $2000 \, K$ and $4000 \, K$ respectively. Find the ratio of their emitted energy.