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Question

(A) The total power radiated by the Sun (a black body) is given by the Stefan-Boltzmann law: $P_{total} = \sigma A T^4$.Here,the surface area $A = 4 \

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$\frac{r^2 \sigma (t + 273)^4}{R^2}$

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(A) The total power radiated by the Sun (a black body) is given by the Stefan-Boltzmann law: $P_{total} = \sigma A T^4$.Here,the surface area $A = 4 \pi r^2$ and the absolute temperature $T = (t + 273) \, K$.So,$P_{total} = \sigma (4 \pi r^2) (t + 273)^4$.This power is distributed uniformly over a spherical surface of radius $R$ at a distance $R$ from the center of the Sun.The surface area of this sphere is $A' = 4 \pi R^2$.The power received per unit area (intensity $I$) at distance $R$ is given by $I = \frac{P_{total}}{A'} = \frac{\sigma (4 \pi r^2) (t + 273)^4}{4 \pi R^2}$.Simplifying this,we get $I = \frac{r^2 \sigma (t + 273)^4}{R^2}$.

Suppose the Sun is a spherical body of radius $r$ with a surface temperature of $t \, ^\circ C$. It radiates energy like a black body. The power received per unit area at a distance $R$ from the center of the Sun will be: ($\sigma$ is the Stefan-Boltzmann constant)

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