The spectral energy distribution of the sun (temperature = 6050 K ) has a maximum at `4753Å` The temperature of a star for which this maximum is at `9506 Å` is

Options

A. 6050 K
B. 3025 K
C. 12100 K
D. 24200 K

Answer

To solve the problem, we will use Wien's Displacement Law, which states that the wavelength at which the spectral energy distribution of a black body spectrum is maximized (λ_max) is inversely proportional to the temperature (T) of the black body. The relationship can be expressed as:

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

where \(b\) is Wien's displacement constant.

### Step-by-Step Solution:

1. **Identify the given values**:
- For the Sun:
- Temperature, \(T_{\text{sun}} = 6050 \, \text{K}\)
- Maximum wavelength, \(\lambda_{\text{sun}} = 4753 \, \text{Å} = 4753 \times 10^{-10} \, \text{m}\)
- For the star:
- Maximum wavelength, \(\lambda_{\text{star}} = 9506 \, \text{Å} = 9506 \times 10^{-10} \, \text{m}\)

2. **Apply Wien's Displacement Law**:
According to Wien's Displacement Law:
\[ \lambda_{\text{sun}} \cdot T_{\text{sun}} = \lambda_{\text{star}} \cdot T_{\text{star}} \]

3. **Rearranging the equation to find the temperature of the star**:
\[ T_{\text{star}} = \frac{\lambda_{\text{sun}} \cdot T_{\text{sun}}}{\lambda_{\text{star}}} \]

4. **Substituting the known values**:
\[ T_{\text{star}} = \frac{(4753 \times 10^{-10} \, \text{m}) \cdot (6050 \, \text{K})}{9506 \times 10^{-10} \, \text{m}} \]

5. **Calculating the temperature of the star**:
\[ T_{\text{star}} = \frac{4753 \cdot 6050}{9506} \]
\[ T_{\text{star}} = \frac{28755150}{9506} \approx 3025 \, \text{K} \]

6. **Final Result**:
The temperature of the star is approximately \(3025 \, \text{K}\).