question_answer

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:                                                                                                                                     [AIPMT (S) 2007]

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

\[I=\frac{P}{4\pi \,{{R}^{2}}}\]

\[=\frac{\sigma \times 4\pi \,{{r}^{2}}{{T}^{4}}}{4\pi {{R}^{2}}}\]

\[=\frac{\sigma \,{{r}^{2}}{{T}^{4}}}{{{R}^{2}}}\]

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

where \[\sigma \] is the Stefan's constant.