Two spheres of radii $8 \ cm$ and $2 \ cm$ are cooling. Their temperatures are $127^{\circ} C$ and $527^{\circ} C$ respectively. Find the ratio of energy radiated by them in the same time.

According to Stefan's law of radiation,a black body radiates energy $\sigma T^4$ from its unit surface area every second,where $T$ is the surface temperature of the black body and $\sigma = 5.67 \times 10^{-8} \, W m^{-2} K^{-4}$ is known as Stefan's constant. $A$ nuclear weapon may be thought of as a ball of radius $0.5 \, m$. When detonated,it reaches a temperature of $10^6 \, K$ and can be treated as a black body.
$(a)$ Estimate the power it radiates.
$(b)$ If the surroundings have water at $30 \, ^\circ C$,how much water can $10 \%$ of the energy produced evaporate in $1 \, s$? $[S_W = 4186 \, J kg^{-1} K^{-1}$ and $L_v = 22.6 \times 10^5 \, J kg^{-1}]$
$(c)$ If all this energy $U$ is in the form of radiation,the corresponding momentum is $p = U/c$. How much momentum per unit time does it impart on a unit area at a distance of $1 \, km$?

For two identical cylindrical emitters,the ratio of their radii is $1:4$ and the ratio of their temperatures is $2:1$. The ratio of the amount of heat emitted by them is ....... (Assume length is proportional to radius for the cylinders).

$A$ spherical black body of radius $r$ radiates power $P$ and its rate of cooling is $R$. Which of the following relations are correct?
$(i) \ P \propto r$
$(ii) \ P \propto r^2$
$(iii) \ R \propto r^2$
$(iv) \ R \propto \frac{1}{r}$

Two spheres of the same material have radii $r_1$ and $r_2$ and surface temperatures $T_1$ and $T_2$ respectively. If their emissive power (total power radiated) is the same,find the ratio of their radii $r_1/r_2$.

Obtain the equation for the rate of emission of heat for a perfect black body by using the Stefan-Boltzmann law.