question_answer

Assuming the sun to be a spherical body of radius R at a temperature of TK, evaluate the total radiant power, incident on earth, at a distance from the sun. Where r0 is the radius of the earth and a is Stefan’s constant.

A) \[4\pi r_{0}^{2}\,{{R}^{2}}\,\sigma {{T}^{4}}/\,{{r}^{2}}\]

B) \[\pi r_{0}^{2}\,{{R}^{2}}\,\sigma {{T}^{4}}/\,{{r}^{2}}\]

C) \[r_{0}^{2}\,{{R}^{2}}\,\sigma {{T}^{4}}/4\pi \,{{r}^{2}}\]

D) \[{{R}^{2}}\,\sigma {{T}^{4}}/\,{{r}^{2}}\]

Correct Answer: B

Solution :

[b] From Stefan’s law, the rate at which energy is radiated by sun at its surface is

(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

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

The area of earth which receives this energy is only one-half of total surface area of earth, whose projection would be \[\pi r_{0}^{2}\].

\         Total radiant power as received by earth\[=\,\pi r_{0}^{2}\times \,1\]

\[=\frac{\pi r_{0}^{2}\times \,\sigma {{R}^{2}}{{T}^{4}}}{{{r}^{2}}}\,=\,\frac{\pi r_{0}^{2}{{R}^{2}}\,\sigma {{T}^{4}}}{{{r}^{2}}}\]