The maximum wavelength of radiation emitted at 2000 K is `4mu m` . What will be the maximum wavelength of radiation emitted at 2400K

Options

A. `3.3 mum`
B. `0.66 mum`
C. 1 m
D. `1 mum`

Answer

To solve the problem, we will use Wien's Displacement Law, which states that the maximum wavelength of radiation emitted by a black body is inversely proportional to its temperature. This can be mathematically expressed as:

\[ \lambda_{\text{max}} \cdot T = b \]

where \( \lambda_{\text{max}} \) is the maximum wavelength, \( T \) is the absolute temperature in Kelvin, and \( b \) is Wien's displacement constant.

### Step-by-Step Solution:

1. **Identify the given values:**
- Maximum wavelength at \( T_1 = 2000 \, K \) is \( \lambda_1 = 4 \, \mu m \).
- We need to find the maximum wavelength \( \lambda_2 \) at \( T_2 = 2400 \, K \).

2. **Apply Wien's Displacement Law:**
According to the law, we can write:
\[ \lambda_1 \cdot T_1 = \lambda_2 \cdot T_2 \]

3. **Substitute the known values into the equation:**
\[ 4 \, \mu m \cdot 2000 \, K = \lambda_2 \cdot 2400 \, K \]

4. **Rearrange the equation to solve for \( \lambda_2 \):**
\[ \lambda_2 = \frac{4 \, \mu m \cdot 2000 \, K}{2400 \, K} \]

5. **Calculate \( \lambda_2 \):**
\[ \lambda_2 = \frac{8000 \, \mu m \cdot K}{2400 \, K} = \frac{8000}{2400} \, \mu m \]
\[ \lambda_2 = \frac{8000 \div 800}{2400 \div 800} \, \mu m = \frac{10}{3} \, \mu m \]
\[ \lambda_2 \approx 3.33 \, \mu m \]

6. **Final answer:**
The maximum wavelength of radiation emitted at \( 2400 \, K \) is approximately \( 3.33 \, \mu m \).