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

Answer

C

Explanation

By Stefan's law, total energy radiated by the sun per unit time (or power radiated by the sun)
`P_(s)=sigmaAT^(4)`
`P_(s)=sigma[4piR^(2)]T^(4)`
where A = Surface area of the sun = `4piR^(2)`
This is incident on the earth's surface. As the earth is at a distance r from the sun.

`therefore` Power received per unit area of earth's surface = `P_(s)/(4pir^(2))`
`therefore` Total power incident as the earth's surface
`P_(E)=P_(s)/(4pir^(2))xx4pir_(0)^(2)`
where `r_(0)` is the radius of the earth.
`thereforeP_(E)=(sigma[4piR^(2)T^(4)]r_(0)^(2))/r^(2)`
`thereforeP_(E)=(4piR^(2)r_(0)^(2)(sigmaT^(4)))/r^(2)`