Question

Black sphere of radius \(\text{R}\) radiates power \(\text{P}\) at certain temperature \(T\). If the temperature is doubled, the radius gets doubled. Now the power radiated would be

• A. \(4\text{ P}\)
• B. \(8\text{ P}\)
• C. \(16\text{ P}\)
• D. \(64\text{ P}\)

Hint:

Radiation power \(\propto R^2 T^4\). Both area and temperature matter.

Answer 1

The Correct Option is D

Concept:
According to Stefan–Boltzmann law: \[ P = \sigma A T^4 \] where \(A = 4\pi R^2\) Step 1: Initial power. \[ P = \sigma (4\pi R^2) T^4 \]
Step 2: New conditions. Radius doubled: \[ A' = 4\pi (2R)^2 = 4 \times 4\pi R^2 = 4A \] Temperature doubled: \[ T' = 2T \Rightarrow (T')^4 = 16T^4 \]
Step 3: New power. \[ P' = \sigma (4A)(16T^4) = 64 \sigma A T^4 \] \[ P' = 64P \]
Step 4: Conclusion. \[ \text{Power} = 64P \]