Question

Two black spheres $P$ and $Q$ have radii in the ratio $3:2$. The wavelengths of maximum intensity radiation are in the ratio $3:4$ respectively. The ratio of radiated power by $P$ to $Q$ is

• A. $\dfrac{74}{9}$
• B. $\dfrac{64}{9}$
• C. $\dfrac{16}{9}$
• D. $\dfrac{25}{9}$

Hint:

Black body power depends on both surface area and the fourth power of temperature.

Answer 1

The Correct Option is B

Step 1: Use Wien’s displacement law.
According to Wien’s law,
\[ \lambda_{\max} T = \text{constant} \] Hence,
\[ \frac{T_P}{T_Q} = \frac{\lambda_Q}{\lambda_P} = \frac{4}{3} \]

Step 2: Use Stefan–Boltzmann law.
Radiated power:
\[ P \propto A T^4 \propto r^2 T^4 \]

Step 3: Substitute ratios.
\[ \frac{P_P}{P_Q} = \left(\frac{3}{2}\right)^2 \left(\frac{4}{3}\right)^4 \] \[ = \frac{9}{4} \times \frac{256}{81} = \frac{64}{9} \]

Step 4: Conclusion.
The ratio of radiated power is $\dfrac{64}{9}$.