$A$ sphere is at temperature $600 \ K$. In an external environment of $200 \ K$,its cooling rate is $R$. When the temperature of the sphere falls to $400 \ K$,then the cooling rate $R'$ will become:

$A$ black rectangular surface of area $A$ emits energy $E$ per second at $27^{\circ} C$. If length and breadth are reduced to $1/3$ of their initial values and the temperature is raised to $327^{\circ} C$,then the energy emitted per second becomes:

Two bodies $A$ and $B$ at temperatures $T_1 \ K$ and $T_2 \ K$ respectively have the same dimensions. Their emissivities are in the ratio $1: 3$. If they radiate the same amount of heat per unit area per unit time,then the ratio of their temperatures $(T_1: T_2)$ is

The unit of Stefan-Boltzmann's constant $(\sigma)$ is:

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 $1 \, km$?

Find the rate of energy radiated per minute by an incandescent lamp at $2000 \ K$. The surface area is $5 \times 10^{-5} \ m^{2}$,the emissivity is $0.85$,and $\sigma = 5.7 \times 10^{-8} \ W \ m^{-2} \ K^{-4}$. (in $J$)