Question

A star radiates heat as a black body at temperature ' (T) '. The total radiant energy per unit area received at a distance ' (R) ' from the centre of a star of radius ' (r) ' is : ((\sigma =) Stefan's constant)

• A. (\frac{\sigma r^2 T^4}{R^2})
• B. (\frac{\sigma r^2 T^4}{4\pi R^2})
• C. (\frac{\sigma r^2 T^4}{R^4})
• D. (\frac{4\pi \sigma r^2 T^4}{R^2})

Hint:

Intensity follows the inverse square law ($I \propto 1/R^2$).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
A star acts as a spherical black body radiator. We need to find the intensity of radiation (energy per unit area per unit time) at a distance \(R\) from its center.
Step 2: Key Formula or Approach:
1) Power radiated by a black body (Stefan-Boltzmann Law): \(P = \sigma A T^4 = \sigma (4\pi r^2) T^4\).
2) Intensity at distance \(R\): \(I = \frac{P}{4\pi R^2}\).
Step 3: Detailed Explanation:
The star is a sphere of radius \(r\). Its surface area is \(4\pi r^2\).
According to Stefan's Law, the total power \(P\) radiated is:
\[ P = \sigma \cdot (4\pi r^2) \cdot T^4 \]
This energy spreads out uniformly in all directions over the surface of a larger imaginary sphere of radius \(R\).
The radiant energy per unit area (Intensity \(I\)) at distance \(R\) is:
\[ I = \frac{P}{\text{Area of sphere with radius } R} \]
\[ I = \frac{\sigma (4\pi r^2) T^4}{4\pi R^2} \]
Cancel \(4\pi\) from the numerator and denominator:
\[ I = \frac{\sigma r^2 T^4}{R^2} \]
Step 4: Final Answer:
The radiant energy per unit area is \(\frac{\sigma r^2 T^4}{R^2}\).