$A$ black sphere of radius $R$ radiates power $P$ at a certain temperature $T$. If the temperature is doubled and the radius is also doubled,the new power radiated will be: (in $P$)

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 $d_2/d_1$?

What will be the ratio of the rate of radiation of a metal sphere at two different temperatures $T_1 = 527^\circ C$ and $T_2 = 127^\circ C$ (in $:$)?

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.

Find the radiant energy emitted per second in $Js^{-1}$ by a lamp filament at $2000 K$. The surface area is $5.0 \times 10^{-5} m^{2}$,the relative emissivity is $0.85$,and $\sigma = 5.7 \times 10^{-8} W m^{-2} K^{-4}$.

The rate of radiation by a black body is $R$ at temperature $T$. Another body has the same area but an emissivity of $0.2$ and a temperature of $3T$. Its rate of radiation is: (in $R$)