The total energy of a black body radiation source is collected for five minutes and used to heat water. The temperature of the water increases from $10.0^{\circ} C$ to $11.0^{\circ} C$. The absolute temperature of the black body is doubled and its surface area halved and the experiment repeated for the same time. Which of the following statements would be most nearly correct?

If the temperature of a perfectly black body is increased by $50\%$,find the percentage increase in the amount of radiation emitted from its surface.

If the temperature of a perfectly black body is reduced from $T$ to $T/2$,find the percentage decrease in the rate of emission.

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 $1 \, km$?

The surface of a black body is at a temperature $727^{\circ} C$ and its cross-section is $1 \,m^2$. Heat radiated from this surface in one minute in joules is (Stefan's constant $=5.7 \times 10^{-8} \,W / m^2 / K^4$ )

If the temperature of the sun were to be increased from $T$ to $2T$ and its radius from $R$ to $2R$,then the ratio of the radiant energy received on the earth to what it was previously will be