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The total radiant energy per unit area, normal to the direction of incidence, received at a distance R from the centre of a star of radius r, whose outer surface radiates as a black body at a temperature T K, is given by (where $\sigma$ is Stefan's constant)
A
$\frac{4\pi\sigma r^2T^4}{R^2}$
B
$\frac{\sigma r^2T^4}{R^2}$
C
$\frac{\sigma r^2T^4}{4\pi r^2}$
D
$\frac{\sigma r^4T^4}{r^4}$
Detailed Solution
By Stefan's law, a black body radiates energy $\sigma T^4$ per unit area per second.
Total power radiated by the star: $P = \sigma T^4\times4\pi r^2$
At a distance R this power is spread uniformly over a sphere of area $4\pi R^2$.
Energy received per unit area per second (normal to the direction of incidence): $I = \frac{P}{4\pi R^2} = \frac{\sigma T^4\times4\pi r^2}{4\pi R^2}$
$I = \frac{\sigma r^2T^4}{R^2}$
Total power radiated by the star: $P = \sigma T^4\times4\pi r^2$
At a distance R this power is spread uniformly over a sphere of area $4\pi R^2$.
Energy received per unit area per second (normal to the direction of incidence): $I = \frac{P}{4\pi R^2} = \frac{\sigma T^4\times4\pi r^2}{4\pi R^2}$
$I = \frac{\sigma r^2T^4}{R^2}$