Assuming the Sun to be a spherical body of radius R at a temperature of TK, evaluate the total radiant powered incident of Earth at a distance r from the sun
where `r_0` is the radius of the Earth and `sigma` is Stefan's constant.

Options

A. `(4pir_(0)^(2)R^(2)sigmaT^(4))/(r^(2))`
B. `(4pir_(0)^(2)R^(2)sigmaT^(4))/(r^(2))`
C. `(4pir_(0)^(2)R^(2)sigmaT^(4))/(4pir^(2))`
D. `(4pir_(0)^(2)R^(2)sigmaT^(4))/(r^(2))`

Answer

B

Explanation

Energy radiated per sec by the Sun in all possible directions (Assume the Sun as perfect black body)
`E =(4 pi R^(2))sigmaT^(4)`
Intensity (I) of the Sun on the Earth surface
`I=(sigma(4piR^(2))T^(4))/(4pir^(2))=sigma((R)/(r))^(2)T^(4)`
Total radiant energy per sec as received on earth
`=(pir_(0)^(2))I=pisigma((R)/(r))^(2)r_(0)^(2)T^(4)`
[ `:.` The area of the Earth which receives the energy is only `(1)/(4)` th of total surface area of the Earth `(r gt gt R)` whose disc has surface area equal to `pi r_(0)^(2)`w.r.t sun] .