Question

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

Answer 1

Answer: (D)

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}} $
$=\left(\sigma T^{4}\right)\left(\frac{R}{r}\right)^{2}$
$=\frac{1}{4} \sigma T^{4}\left(\frac{2 R}{r}\right)^{2}$
Angle subtended by the sun at the earth,
$\theta=\left(\frac{2 R}{r}\right)$
Or $S=\frac{1}{4} \sigma T^{4} \theta^{2}$