Question

The total radiant energy per unit area normal to the direction of incidence, received at a distance R from the centre of a star of radius r, whose outer surface radiates as a black body at a temperature 'T' K 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. \(\frac{\sigma r^{4}T^{4}}{R^{4}}\)
• D. \(\frac{4\pi \sigma r^{2}T^{4}}{R^{2}}\)

Hint:

Find the total power radiated by the star and spread it over a sphere of radius R.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
By Stefan's law a black body radiates \(\sigma T^{4}\) per unit area per second.

Step 2: Total Power of the Star
\(P=\sigma T^{4}\times 4\pi r^{2}\).

Step 3: Spread Over a Sphere
At distance \(R\) the energy spreads over area \(4\pi R^{2}\).

Step 4: Calculation
\(I=\frac{4\pi r^{2}\sigma T^{4}}{4\pi R^{2}}=\frac{\sigma r^{2}T^{4}}{R^{2}}\), option (A).

Final Answer:
The energy received per unit area is sigma r squared T to the fourth over R squared. \[ \boxed{\text{(A) }\dfrac{\sigma r^{2}T^{4}}{R^{2}}} \]