Question

A black disc has a radius 'R' and wavelength corresponding to the maximum intensity is '\(λ\)'. The emissive power (E) for the different radii and maximum wavelengths is directly proportional to

• A. \(R,λ\)
• B. \(R^2,λ^{-4}\)
• C. \(R^2,λ^{-2}\)
• D. \(R^2,λ\)

Hint:

Stefan law gives T^4, Wien law gives T proportional to 1/lambda.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A black body radiates energy at a rate given by Stefan's law: the power radiated per unit area is \(\sigma T^4\). For a disc of radius \(R\), the radiating area is proportional to \(R^2\).

Step 2: Link temperature to wavelength:
Wien's displacement law says \(\lambda_mT=\text{constant}\), so \(T\propto\dfrac1\lambda\). Hence \(T^4\propto\lambda^{-4}\).

Step 3: Combine:
The emitted power is
\[ E\propto(\text{area})\times T^4\propto R^2\lambda^{-4} \]

Step 4: Choose:
Option (B). The other options have a wrong power of \(\lambda\) or the wrong power of \(R\).

Final Answer:
E is proportional to R squared and lambda to the power -4. \[ \boxed{E\propto R^2,\ \lambda^{-4}} \]