question_answer

Assuming the sun to have a spherical outer surface of radius r, radiating like a black body at temperature \[t{{\,}^{o}}C,\]the power received by a unit surface, (normal to the incident rays) at a distance R from the centre of the sun is

A) \[\frac{4\pi {{r}^{2}}\sigma {{t}^{4}}}{{{R}^{2}}}\]

B)  \[\frac{{{r}^{2}}\sigma {{(t+273)}^{4}}}{4\pi {{R}^{2}}}\]

C)  \[\frac{16{{\pi }^{2}}{{r}^{2}}\sigma {{t}^{4}}}{{{R}^{2}}}\]

D)  \[\frac{{{r}^{2}}\sigma {{(t\,+273)}^{4}}}{{{R}^{2}}}\]

Correct Answer: D

Solution :

 From Stefan's law, the rate at which energy is radiated by sun at its surface is \[P=\sigma \,\times \,\,4\pi {{r}^{2}}{{T}^{4}}\]  [Sun is a perfectly black body as it emits radiations of all wavelengths and so for it e = 1] The intensity of this power at earth's surface (under the assumption\[R>>{{r}_{0}}\]) is \[I=\frac{P}{4\pi {{R}^{2}}}\] \[=\frac{\sigma \times 4\pi {{r}^{2}}{{T}^{4}}}{4\pi {{R}^{2}}}\] \[=\frac{\sigma {{r}^{2}}{{T}^{4}}}{{{R}^{2}}}\] \[=\frac{\sigma {{r}^{2}}{{(t+273)}^{4}}}{{{R}^{2}}}\]   where \[\,\sigma \]is the Stefan's constant.