Question

Two spherical black bodies of radius \( r_1 \) and \( r_2 \) with surface temperature \( T_1 \) and \( T_2 \) respectively, radiate the same power, then \( \frac{r_1}{r_2} \) is

• A. \( \left( \frac{T_2}{T_1} \right)^2 \)
• B. \( \left( \frac{T_1}{T_2} \right)^4 \)
• C. \( \left( \frac{T_1}{T_2} \right)^2 \)
• D. \( \left( \frac{T_2}{T_1} \right)^4 \)

Hint:

In problems involving power and temperature, the Stefan-Boltzmann law helps to relate the power radiated to the temperature and surface area of the body.

Answer 1

The Correct Option is A

Step 1: 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: \[ P = \sigma A T^4 \] where \( P \) is the power, \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area, and \( T \) is the temperature of the body.
Step 2: Surface area of spherical bodies.
The surface area of a sphere is \( A = 4\pi r^2 \), where \( r \) is the radius of the sphere.
Step 3: Equating power radiated.
For both bodies to radiate the same power: \[ \sigma A_1 T_1^4 = \sigma A_2 T_2^4 \] \[ 4\pi r_1^2 T_1^4 = 4\pi r_2^2 T_2^4 \] Simplifying, we get: \[ r_1^2 T_1^4 = r_2^2 T_2^4 \]
Step 4: Conclusion.
The ratio \( \frac{r_1}{r_2} \) is: \[ \left( \frac{T_2}{T_1} \right)^2 \] Hence, the correct answer is (A).