question_answer

                Assuming the sun to have a spherical outer surface of radius r, radiating like a black body at temperature \[{{t}^{\text{o}}}C,\] 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 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                 \[I=\frac{P}{4\pi \,{{R}^{2}}}\]                 \[=\frac{\sigma \times 4\pi \,{{r}^{2}}{{T}^{4}}}{4\pi {{R}^{2}}}\]                 \[=\frac{\sigma \,{{r}^{2}}{{T}^{4}}}{{{R}^{2}}}\]                 \[=\frac{\sigma {{r}^{2}}\,{{(t+273)}^{2}}}{{{R}^{2}}}\]