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Directions : In the following question more than one of the answers given may be correct. Select the correct answer and mark it according to the code:

The solar constant for the earth is S. The surface temperature of the sun is T K. The sun subtends an angle\[\theta \]at the earth. (1) \[S\propto {{T}^{4}}\]                             (2) \[S\propto {{T}^{2}}\] (3) \[S\propto {{\theta }^{2}}\]                  (4) \[S\propto \theta \]

A)  1, 2 and 3 are correct

B)  1 and 2 are correct

C)   2 and 4 are correct

D)  1 and 3 are correct

Correct Answer: D

Solution :

                 If R is the radius of the sun and T its temperature, then the energy emitted by the sun per sec through radiation in accordance with Stefan's law will be given by \[P=A\sigma {{T}^{4}}=4\pi {{R}^{2}}\sigma {{T}^{4}}\]                 Energy received per unit area per second on the earth of radius r \[S=\frac{P}{4\pi {{r}^{2}}}=\frac{4\pi {{R}^{2}}\sigma {{T}^{4}}}{4\pi {{r}^{2}}}\] \[=(\sigma {{T}^{4}}){{\left( \frac{R}{r} \right)}^{2}}\] \[=\frac{1}{4}\sigma {{T}^{4}}{{\left( \frac{2R}{r} \right)}^{2}}\] Angle subtended by the sun at the earth, \[\theta =\left( \frac{2R}{r} \right)\] Or           \[S=\frac{1}{4}\sigma {{T}^{4}}{{\theta }^{2}}\] Hence, option  is correct.