Question

Two spherical black bodies radiate the same amount of heat per second. If their temperatures are \( T_1 \) and \( T_2 \), and their radii are \( R_1 \) and \( R_2 \), respectively, find the relation between their temperatures and radii.

• A. \( T_1 = \sqrt{2} T_2 \)
• B. \( T_1 = 2 T_2 \)
• C. \( T_1 = \frac{T_2}{\sqrt{2}} \)
• D. \( T_1 = \sqrt{3} T_2 \)

Hint:

When comparing the radiation of heat by two bodies, use the Stefan-Boltzmann law and equate the radiated power to solve for the unknown variable. Make sure to apply the correct formulas for the surface area of spherical bodies.

Answer 1

The Correct Option is A

We are given that two black bodies radiate the same amount of heat per second. The amount of heat radiated by a body is given by the Stefan-Boltzmann law: \[ P = \sigma A T^4 \] where: - \( P \) is the power radiated (amount of heat radiated per second), - \( \sigma \) is the Stefan-Boltzmann constant, - \( A \) is the surface area of the body, - \( T \) is the temperature of the body. For a spherical body, the surface area is: \[ A = 4 \pi R^2 \] Now, let the power radiated by the first body be \( P_1 \) and the second body be \( P_2 \). According to the problem, \( P_1 = P_2 \), so we have: \[ \sigma 4 \pi R_1^2 T_1^4 = \sigma 4 \pi R_2^2 T_2^4 \] Canceling common terms: \[ R_1^2 T_1^4 = R_2^2 T_2^4 \] Now, take the ratio of both sides: \[ \left( \frac{R_1}{R_2} \right)^2 = \left( \frac{T_2}{T_1} \right)^4 \] Taking the square root of both sides: \[ \frac{R_1}{R_2} = \frac{T_2}{T_1}^2 \] Finally, solving for \( T_1 \): \[ T_1 = \sqrt{2} T_2 \] Thus, the relation between the temperatures and radii is \( T_1 = \sqrt{2} T_2 \).