2007 AIPMT-PRE Thermal Properties of MatterStefan–Boltzmann law Medium Question 3 of 5 in Stefan–Boltzmann law

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.

A $\dfrac{r^2\sigma(t + 273)^4}{4\pi R^2}$
B $\dfrac{16\pi^2 r^2\sigma t^4}{R^2}$
C $\dfrac{r^2\sigma(t + 273)^4}{R^2}$
D $\dfrac{4\pi r^2\sigma t^4}{R^2}$

The answer is C.

Solution

Absolute temperature of the sun: $T = (t + 273)$ K
Power radiated per unit area of a black body $= \sigma T^4$
Surface area of the sun $= 4\pi r^2$
Total power radiated by the sun at $t^\circ C$ $= \sigma(t + 273)^4 \times 4\pi r^2$
At a distance $R$ this power is spread uniformly over a sphere of area $4\pi R^2$.
Power received by a unit surface $= \dfrac{\sigma(t + 273)^4\,4\pi r^2}{4\pi R^2}$
$= \dfrac{r^2\sigma(t + 273)^4}{R^2}$

Concept Solar constant from Stefan's law

Stefan–Boltzmann law in past papers

5 questions from this chapter have appeared across 3 exam years.