Question

If T is surface temperature of sun, R is the radius of sun, r is radius of earth’s orbit and S is solar constant, then total radiant energy of sun per unit time from the sphere of radius r, then
(A) \[\pi {r^2}S\]
(B) \[4\pi {r^2}S\]
(C) \[\sigma \dfrac{4}{3}\pi {r^3}{T^4}\]
(D) \[\sigma 4\pi {r^2}{T^4}\]

Answer 1

Hint: Use Stefan-Boltzmann law to express the solar radiant energy per unit time of the sun. We know that area of the sphere of radius R is \[4\pi {R^2}\]. Now consider the same sphere whose radius is equal to the radius of orbit of earth and express the total radiant energy per unit time.

Formula used:
The total energy output per unit time of the blackbody is,
\[P = A\sigma {T^4}\]
Here, A is the surface area of the sun, \[\sigma \] is Boltzmann’s constant and T is the surface temperature of the sun.

Complete step by step answer:
We need to calculate the total energy output per unit time of the sun whose radius is R using Stefan-Boltzmann law as follows,
\[P = A\sigma {T^4}\]
Here, A is the surface area of the sun, \[\sigma \] is Boltzmann’s constant and T is the surface temperature of the sun.
We have assumed that the sun is a perfectly black body and radiates energy in all the directions symmetrically. We know that surface area of sphere of radius R is expressed as,
\[A = 4\pi {R^2}\]
Therefore, we can express the total radiant energy of the sun per unit time as,
\[P = 4\pi {R^2}\sigma {T^4}\]
Now, we have given that the radius of the sphere is r and we have to determine the total radiant energy per unit time for the sphere of radius r. Therefore, we can simply substitute r for R in the above equation.
\[P = 4\pi {r^2}\sigma {T^4}\]

So, the correct answer is option (D).

Note:Most of the time, questions of this kind ask you to calculate the total radiant energy per unit surface area per unit time. Therefore, students have to divide the total radiant energy per unit surface area per unit time by the surface area of the sun. We did not take the term solar constant in our solution as the solar constant is the total solar radiation power per unit surface area of earth’s orbit and we want to calculate the total radiant energy per unit time.