Question

A black body has maximum wavelength ' (\lambda_{m}) ' at temperature (2000 K). Its maximum wavelength at (3000 K) will be

• A. (\frac{3}{2} \lambda_{m})
• B. (\frac{16}{81} \lambda_{m})
• C. (\frac{81}{16} \lambda_{m})
• D. (\frac{2}{3} \lambda_{m})

Hint:

Peak wavelength is inversely proportional to the absolute temperature.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question relates the wavelength corresponding to maximum intensity (\(\lambda_{m}\)) of a black body with its absolute temperature (\(T\)).
This relationship is described by Wien's Displacement Law.
Step 2: Key Formula or Approach:
Wien's Displacement Law states that:
\[ \lambda_{m} \cdot T = \text{constant} \implies \lambda_{m} \propto \frac{1}{T} \]
Therefore, \(\frac{\lambda_{2}}{\lambda_{1}} = \frac{T_{1}}{T_{2}}\).
Step 3: Detailed Explanation:
Given:
Initial temperature \(T_{1} = 2000 \text{ K}\)
Initial maximum wavelength \(\lambda_{1} = \lambda_{m}\)
Final temperature \(T_{2} = 3000 \text{ K}\)
Let the new maximum wavelength be \(\lambda_{2}\).
Applying the ratio formula:
\[ \lambda_{2} = \lambda_{1} \cdot \frac{T_{1}}{T_{2}} \]
\[ \lambda_{2} = \lambda_{m} \cdot \frac{2000}{3000} \]
\[ \lambda_{2} = \frac{2}{3} \lambda_{m} \]
Step 4: Final Answer:
The maximum wavelength at \(3000 \text{ K}\) is \(\frac{2}{3} \lambda_{m}\).