Question

Assuming the sun to be a spherical body of radius R at a temperature of T K, evaluate the total radiant power, incident on earth, at a distance r from the sun : where $r_{0}$ is the radius of the earth and $\sigma$ 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. $R^{2} \sigma T^{4} / r^{2}$

Answer 1

The Correct Option is B

From Stefan?s law, the rate at which energy is radiated by sun at its surface is $P=\sigma \times4\pi R^{2} \times T^{4}$ [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 $I=\frac{p}{4 \pi r^{2}}=\frac{\sigma\times4\pi R^{2}T^{4}}{4 \pi r^{2}}=\frac{\sigma R^{2} T^{4}}{r^{2}}$ The area of earth which receives this energy is only one half of total surface area of earth, whose projection would be $\pi r_{0}^{2}.$ $\therefore$ Total radiant power as received by earth $=\pi r_{0}^{2} \times I$ $=\frac{\pi r_{0}^{2}\times\sigma R^{2}T^{4}}{r^{2}}$ $=\frac{\pi r_{0}^{2} R^{2}\sigma T^{4}}{r^{2}}$