Question

A spherical body of radius ' \(r\) ' radiates power ' \(P\) ' at \(T\) kelvin. If the radius is halved and the temperature doubled the power radiated in the same time ' \(t\) ' will be

• A. \(\frac{P}{2}\)
• B. \(2 P\)
• C. \(4 P\)
• D. \(8 P\)

Hint:

Radiation: $P \propto r^2 T^4$

Answer 1

The Correct Option is C

Concept: Stefan-Boltzmann law: \[ P \propto A T^4 \]

Step 1: Area dependence. \[ A \propto r^2 \] \[ \text{New area} = \left(\frac{r}{2}\right)^2 = \frac{1}{4} \]

Step 2: Temperature effect. \[ T \rightarrow 2T \Rightarrow T^4 \rightarrow 16T^4 \]

Step 3: Combine. \[ P' = P \times \frac{1}{4} \times 16 = 4P \]

Step 4: Conclusion.
New power = $4P$ Final Answer: Option (C)