$A$ black rectangular surface of area $A$ emits energy $E$ per second at $127^\circ C$. If length and breadth are reduced to half of their initial values and the temperature is raised to $527^\circ C$, then the energy emitted becomes:

The adjoining diagram shows the spectral energy density distribution $E_\lambda$ of a black body at two different temperatures. If the areas under the curves are in the ratio $16 : 1$,the value of temperature $T$ is ......... $K$. (in $,000$)

$A$ black body,at a temperature of $227\,^{\circ}C$,radiates heat at a rate of $7\,cal\,cm^{-2}\,s^{-1}$. At a temperature of $727\,^{\circ}C$,the rate of heat radiated in the same units will be:

Assume that the solar constant is $1.4 \, kW/m^2$,the radius of the sun is $7 \times 10^5 \, km$,and the distance of the earth from the center of the sun is $1.5 \times 10^8 \, km$. Given Stefan's constant is $\sigma = 5.67 \times 10^{-8} \, W m^{-2} K^{-4}$,find the approximate temperature of the sun in $K$.

$Assertion :$ Bodies radiate heat at all temperatures.
$Reason :$ Rate of radiation of heat is proportional to the fourth power of absolute temperature.

Assuming the Sun to be a spherical body of radius $R$ at a temperature of $T \ K$,evaluate the total radiant power incident on Earth at a distance $r$ from the Sun. Where $r_0$ is the radius of the Earth and $\sigma$ is Stefan's constant.