Question

A black body has maximum wavelength 1 m at 2000 K. Its corresponding wavelength at 3000 K will be

• A. $ \frac{3}{2} \lambda m $
• B. $ \frac{2}{3} \lambda m $
• C. $ \frac{16}{81} \lambda m $
• D. $ \frac{81}{16} \lambda m $

Answer 1

The Correct Option is B

To solve this question, we need to use Wien's Displacement Law, which states that the wavelength corresponding to the maximum intensity of radiation (denoted as \lambda_m) is inversely proportional to the temperature (T) of the black body. Mathematically, this is expressed as:

\lambda_m \cdot T = b

where b is Wien's constant.

Given two conditions for the black body:

1. At T_1 = 2000 \, K, the maximum wavelength \lambda_{m1} = 1 \, m.
2. At T_2 = 3000 \, K, we need to find the corresponding wavelength \lambda_{m2}.

Using Wien's law for both conditions, we have:

1. \lambda_{m1} \cdot T_1 = b
2. \lambda_{m2} \cdot T_2 = b

Since b is a constant, we equate the two expressions:

\lambda_{m1} \cdot T_1 = \lambda_{m2} \cdot T_2

Substituting the known values:

1 \cdot 2000 = \lambda_{m2} \cdot 3000

Solving for \lambda_{m2}:

\lambda_{m2} = \frac{2000}{3000} = \frac{2}{3} \, m

Therefore, the corresponding wavelength at 3000 K is \frac{2}{3} \, m.

This matches the option: \frac{2}{3} \lambda \, m.

Thus, the correct answer is \frac{2}{3} \lambda \, m.