Question

A red star having radius \( r_R \) at a temperature \( T_R \) and a white star having radius \( r_W \) at a temperature \( T_W \), radiate the same total power. If these stars radiate as perfect black bodies, then

• A. \( r_R>r_W \) and \( T_R>T_W \)
• B. \( r_R<r_W \) and \( T_R>T_W \)
• C. \( r_R>r_W \) and \( T_R<T_W \)
• D. \( r_R<r_W \) and \( T_R<T_W \)

Hint:

In Stefan-Boltzmann's law, the radiated power depends on both the temperature and the surface area, which is proportional to the square of the radius.

Answer 1

The Correct Option is C

Step 1: Understanding Stefan-Boltzmann Law.
According to the Stefan-Boltzmann law, the power radiated by a black body is proportional to the fourth power of its temperature and the surface area. The total power radiated \( P \) by a star is given by: \[ P = \sigma A T^4 = \sigma (4\pi r^2) T^4 \] where \( \sigma \) is the Stefan-Boltzmann constant, \( r \) is the radius of the star, and \( T \) is the temperature.
Step 2: Equating the power of both stars.
Since both stars radiate the same total power: \[ P_R = P_W \quad \Rightarrow \quad \sigma (4\pi r_R^2) T_R^4 = \sigma (4\pi r_W^2) T_W^4 \] Simplifying, we get: \[ r_R^2 T_R^4 = r_W^2 T_W^4 \]
Step 3: Solving for the relationship between \( r_R \) and \( r_W \), and \( T_R \) and \( T_W \).
Taking the square root of both sides: \[ r_R T_R^2 = r_W T_W^2 \] Thus, \( r_R \) and \( r_W \) are inversely related to \( T_R^2 \) and \( T_W^2 \). For \( r_R \) to be less than \( r_W \), \( T_R \) must be greater than \( T_W \).
Step 4: Conclusion.
The correct relationship is \( r_R<r_W \) and \( T_R>T_W \), so the correct answer is (B).