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)  \[\frac{{{r}^{2}}\sigma {{(t+273)}^{4}}}{{{R}^{2}}}\] where \[\sigma \] is the Stefan's constant.

Correct Answer: D

Solution :

From Stefarfs 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                 \[I=\frac{P}{4\pi {{R}^{2}}}\]                    \[=\frac{\sigma \times 4\pi {{r}^{2}}{{T}^{4}}}{4\pi {{R}^{2}}}\]                    \[=\frac{\sigma {{r}^{2}}{{T}^{4}}}{{{R}^{2}}}\]                    \[=\frac{\sigma {{r}^{2}}{{(t+273)}^{4}}}{{{R}^{2}}}\]