JEE Main · 2006 · 30 Apr 2006 · Offline · Heat & Thermodynamics

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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

σ

is Stefan's constant.

(A) $\frac{R ^{2} σ T ^{4}}{r ^{2}}$
(B) $\frac{4 π r _{0}^{2} R ^{2} σ T ^{4}}{r ^{2}}$
(C) $\frac{π r _{0}^{2} R ^{2} σ T ^{4}}{r ^{2}}$ ✓ Correct
(D) $\frac{r _{0}^{2} R ^{2} σ T ^{4}}{4 π r ^{2}}$

Answer: (C)

Solution

The Sun radiates as a black body, emitting a total power

P = σ T^{4} \times 4 π R^{2}

from its surface. By the time this reaches the Earth's orbit it is spread over a sphere of area

4 π r^{2}

, so the intensity there is

\frac{σ T ^{4} 4 π R ^{2}}{4 π r ^{2}} = \frac{σ T ^{4} R ^{2}}{r ^{2}}

. The Earth intercepts this flux over its disc of area

π r_{0}^{2}

, so the power falling on it is

\frac{π r _{0}^{2} R ^{2} σ T ^{4}}{r ^{2}}

JEE Main · 2006 · 30 Apr 2006 · Offline · Single correct option

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JEE Main · 2006 · 30 Apr 2006 · Offline · Heat & Thermodynamics

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