Question

Two bodies, R1 and R2, radiate power at temperatures \( T_1 \) and \( T_2 \) respectively. What is the ratio \( R_1 : R_2 \) of their radiated powers?

• A. \( \frac{R_1^2 T_1^4}{R_2^2 T_2^4} \)
• B. \( \frac{R_1^2 T_1^3}{R_2^2 T_2^3} \)
• C. \( \frac{R_1^4 T_1^2}{R_2^4 T_2^2} \)
• D. \( \frac{R_1^4 T_1^4}{R_2^4 T_2^4} \)

Hint:

The power radiated by a body is proportional to its surface area and the fourth power of its temperature. For spherical bodies, the surface area is proportional to \( R^2 \), and the temperature affects the power via \( T^4 \).

Answer 1

The Correct Option is A

The radiated power \( P \) of a body is governed by the Stefan-Boltzmann law, which states that: \[ P = \sigma A T^4, \] where \( P \) is the power radiated, \( A \) is the surface area, \( T \) is the absolute temperature, and \( \sigma \) is the Stefan-Boltzmann constant. For spherical bodies, the surface area \( A \) is proportional to \( R^2 \), where \( R \) is the radius. Therefore, the ratio of radiated powers \( P_1 : P_2 \) is: \[ \frac{P_1}{P_2} = \frac{R_1^2 T_1^4}{R_2^2 T_2^4}. \] Thus, the ratio of the radiated powers depends on both the radii and the temperatures of the bodies.