Question

If the radius of a star is \( R \) and it acts as a black body, what would be the temperature of the star, in which the rate of energy production is \( Q \)?

• A. \( \frac{Q}{4 \pi R^2 \sigma} \)
• B. \( \frac{Q}{4 \pi R^2} \)
• C. \( (4 \pi R^2 Q)^{1/4} \)
• D. \( (Q / 4 \pi R^2 \sigma)^{1/4} \)

Hint:

The temperature of a star can be found using the Stefan-Boltzmann law, considering it as a black body.

Answer 1

The Correct Option is D

Step 1: Use the Stefan-Boltzmann law.
The power emitted by a black body is given by the Stefan-Boltzmann law: \( P = \sigma A T^4 \), where \( A = 4 \pi R^2 \) is the surface area of the star, and \( T \) is the temperature.
Step 2: Solve for \( T \).
Rearranging the equation for \( T \), we get \( T = \left( \frac{Q}{4 \pi R^2 \sigma} \right)^{1/4} \). Final Answer: \[ \boxed{(Q / 4 \pi R^2 \sigma)^{1/4}} \]