Question

The total radiant energy per unit area, normal to the direction of incidence, received at a distance R from the center of a star of radius r, whose outer surface radiates as a black body at a temperature T Kelvin is given by: (Where σ is Stefan's Constant)

A. σr²T⁴/R²
B. σr²T⁴/4πr²
C. σr²T⁴/r⁴
D. 4σr²T⁴/R²

Answer 1

The total radiant energy per unit area received at a distance R from a star is given by σr^2T^4/R^2, according to the Stefan-Boltzmann law.

Step-by-step explanation:

The total radiant energy per unit area received at a distance R from the center of a star of radius r, with the star's surface radiating as a black body at temperature T Kelvin, can be calculated using the Stefan-Boltzmann law. The law states that the total power radiated per unit area of a black body is proportional to the fourth power of the temperature, which can be mathematically represented as σT^4, where σ is the Stefan-Boltzmann constant. By considering the star's surface as a sphere, we have the surface area A as 4πr^2. The luminosity L, which is the total power radiated by the star, is therefore L = σA(T^4) = 4πr^2σT^4. To find the radiant energy per unit area at a distance R, we have to consider how this luminosity is spread over a sphere with radius R, which has surface area 4πR^2. Hence, the radiant energy per unit area is L divided by 4πR^2, which simplifies to σr^2T^4 / R^2, corresponding to answer choice A: σr^2T^4/R^2.