If Wien's constant b = 0.3 cm K, then the temperature of the Sun having a maximum intensity of radiation at `5000Å` wavelength is

Options

A. 5000 K
B. 6000 K
C. 4000 K
D. 7000 K

Answer

To find the temperature of the Sun using Wien's Law, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Wien's Law**:
Wien's 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 relationship is given by the formula:
\[ b = \lambda_{\text{max}} \cdot T \]
where \(b\) is Wien's constant.

2. **Identify Given Values**:
From the problem, we have:
- Wien's constant \(b = 0.3 \, \text{cm K}\)
- Maximum wavelength \(\lambda_{\text{max}} = 5000 \, \text{Å}\)

3. **Convert Units**:
We need to convert the wavelength from angstroms to centimeters for consistency with the units of Wien's constant.
\[ 1 \, \text{Å} = 10^{-10} \, \text{m} = 10^{-8} \, \text{cm} \]
Therefore,
\[ \lambda_{\text{max}} = 5000 \, \text{Å} = 5000 \times 10^{-8} \, \text{cm} = 5 \times 10^{-5} \, \text{cm} \]

4. **Apply Wien's Law**:
Substitute the values into Wien's Law:
\[ b = \lambda_{\text{max}} \cdot T \]
Rearranging for temperature \(T\):
\[ T = \frac{b}{\lambda_{\text{max}}} \]

5. **Calculate Temperature**:
Substitute \(b = 0.3 \, \text{cm K}\) and \(\lambda_{\text{max}} = 5 \times 10^{-5} \, \text{cm}\):
\[ T = \frac{0.3 \, \text{cm K}}{5 \times 10^{-5} \, \text{cm}} = \frac{0.3}{5 \times 10^{-5}} = \frac{0.3 \times 10^{5}}{5} = \frac{30}{5} \times 10^{5} = 6 \times 10^{5} \, \text{K} \]

6. **Final Result**:
The temperature of the Sun is:
\[ T = 6000 \, \text{K} \]