A black body has maximum wavelength `lambda_(m)` at temperature `2000 K`. Its corresponding wavelength at temperature 3000 will be

Options

A. `(2)/(3) lambda`
B. `(16)/(81) lambda`
C. `(81)/(16) lambda`
D. `(4)/(3) lambda`

Answer

To solve the problem, we will use Wien's Displacement Law, which states that the product of the maximum wavelength (\( \lambda_m \)) and the absolute temperature (T) of a black body is a constant. This can be mathematically expressed as:

\[ \lambda_m T = \text{constant} \]

### Step-by-Step Solution:

1. **Identify the given values:**
- Maximum wavelength at temperature \( T_1 = 2000 \, K \) is \( \lambda_m \).
- We need to find the maximum wavelength \( \lambda_2 \) at temperature \( T_2 = 3000 \, K \).

2. **Apply Wien's Displacement Law:**
According to the law, we can write:
\[ \lambda_m T_1 = \lambda_2 T_2 \]

3. **Substitute the known values:**
Substitute \( T_1 \) and \( T_2 \) into the equation:
\[ \lambda_m \cdot 2000 = \lambda_2 \cdot 3000 \]

4. **Rearrange the equation to solve for \( \lambda_2 \):**
To isolate \( \lambda_2 \), we rearrange the equation:
\[ \lambda_2 = \frac{\lambda_m \cdot 2000}{3000} \]

5. **Simplify the expression:**
This can be simplified to:
\[ \lambda_2 = \lambda_m \cdot \frac{2000}{3000} = \lambda_m \cdot \frac{2}{3} \]

6. **Conclusion:**
Thus, the corresponding wavelength at temperature \( 3000 \, K \) is:
\[ \lambda_2 = \frac{2}{3} \lambda_m \]