The temperature (approx) of the body omitting wavelength `0.3 mum` corresponding to maximum intensity is

Options

A. `2xx10^(4)K`
B. `10^(4)K`
C. `2 xx 10^(3)K`
D. `10^(3)K`

Explanation

To find the temperature of a body emitting at a wavelength of 0.3 micrometers (μm) corresponding to maximum intensity, we can use Wien's Displacement Law. The law states that the wavelength at which the intensity of radiation is maximum (λ_max) is inversely proportional to the absolute temperature (T) of the black body. The formula is given by:

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

where:
- \( \lambda_{\text{max}} \) is the wavelength at maximum intensity,
- \( T \) is the absolute temperature in Kelvin,
- \( b \) is Wien's displacement constant, approximately \( 3 \times 10^{-3} \) m·K.

### Step-by-step Solution:

1. **Identify the Wavelength**:
The given wavelength is \( 0.3 \, \mu m \). We need to convert this to meters:
\[ 0.3 \, \mu m = 0.3 \times 10^{-6} \, m = 3 \times 10^{-7} \, m \]

2. **Use Wien's Displacement Law**:
According to Wien's law:
\[ \lambda_{\text{max}} \cdot T = b \]
Rearranging this gives:
\[ T = \frac{b}{\lambda_{\text{max}}} \]

3. **Substitute the Values**:
Substitute \( b = 3 \times 10^{-3} \, m \cdot K \) and \( \lambda_{\text{max}} = 3 \times 10^{-7} \, m \):
\[ T = \frac{3 \times 10^{-3} \, m \cdot K}{3 \times 10^{-7} \, m} \]

4. **Calculate the Temperature**:
Simplifying the equation:
\[ T = \frac{3 \times 10^{-3}}{3 \times 10^{-7}} = 10^{4} \, K \]

5. **Final Answer**:
Thus, the temperature of the body emitting at a wavelength of \( 0.3 \, \mu m \) corresponding to maximum intensity is:
\[ T = 10,000 \, K \]