Question

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}}} $

Answer 1

Answer: (D)

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.