Question

A black body has maximum wavelength \( \lambda_m \) at temperature 2000 K. Its corresponding wavelength at temperature 3000 K will be

• A. \( \frac{4\lambda_m}{9} \)
• B. \( \frac{2\lambda_m}{3} \)
• C. \( \frac{3\lambda_m}{2} \)
• D. \( \frac{9}{4}\lambda_m \)

Hint:

Wien’s law: \(\lambda_{\text{max}} \propto \frac{1}{T}\). Higher temperature shifts the peak to shorter wavelengths. Remember to use absolute temperature (Kelvin) only.

Answer 1

The Correct Option is B

Step 1: Apply Wien's law.
The peak wavelength times temperature stays fixed: $\lambda_m T = \text{constant}$. So a hotter body peaks at a shorter wavelength.

Step 2: Set up the ratio.
\[ \lambda_1 T_1 = \lambda_2 T_2 \Rightarrow \lambda_2 = \lambda_1\frac{T_1}{T_2}. \]

Step 3: Put in temperatures.
With $T_1 = 2000\ \text{K}$ and $T_2 = 3000\ \text{K}$, $\lambda_2 = \lambda_m\cdot\tfrac{2000}{3000} = \tfrac{2}{3}\lambda_m$.

Step 4: State the new peak.
\[ \boxed{\tfrac{2\lambda_m}{3}} \]