Question

Two spheres \( S_1 \) and \( S_2 \) have same radii but temperatures \( T_1 \) and \( T_2 \) respectively. Their emissive power is the same and emissivity is in the ratio 1:4. Then the ratio of \( T_1 \) to \( T_2 \) is

• A. \( \sqrt{2} : 1 \)
• B. \( 1 : 2 \)
• C. \( 2 : 1 \)
• D. \( 1 : \sqrt{2} \)

Hint:

In the Stefan-Boltzmann law, emissive power is proportional to the fourth power of temperature. Use the given emissivity ratio to solve for the temperature ratio.

Answer 1

The Correct Option is A

Step 1: Understanding the Stefan-Boltzmann law.
The emissive power \( E \) of a body is given by the Stefan-Boltzmann law: \[ E = \sigma \epsilon T^4 \] where \( \sigma \) is the Stefan-Boltzmann constant, \( \epsilon \) is the emissivity, and \( T \) is the temperature of the body.
Step 2: Relating the emissive power and temperature.
Given that the emissive power is the same for both spheres, we can write: \[ \sigma \epsilon_1 T_1^4 = \sigma \epsilon_2 T_2^4 \] Since the emissivity ratio is 1:4, we substitute \( \epsilon_1 = 1 \) and \( \epsilon_2 = 4 \) into the equation: \[ T_1^4 = 4 T_2^4 \]
Step 3: Solving for the temperature ratio.
Taking the fourth root of both sides: \[ T_1 = \sqrt{4} T_2 = 2 T_2 \] Thus, the ratio of \( T_1 \) to \( T_2 \) is \( \sqrt{2} : 1 \), which corresponds to option (A).