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. (a)5000 K
B. (b)6000 K
C. (c)4000 K
D. (d)7000 K

Answer

To solve the problem, we will use Wien's Displacement Law, which states that the product of the maximum wavelength (λ_max) and the temperature (T) of a black body is a constant (Wien's constant, b). The formula is given by:

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

### Step-by-step Solution:

1. **Identify the Given Values:**
- Wien's constant, \( b = 0.3 \, \text{cm K} \)
- Wavelength, \( \lambda_{max} = 5000 \, \text{Å} \)

2. **Convert Units:**
- Convert Wien's constant from cm to meters:
\[ b = 0.3 \, \text{cm} = 0.3 \times 10^{-2} \, \text{m} = 0.003 \, \text{m K} \]
- Convert the wavelength from Ångströms to meters:
\[ \lambda_{max} = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} = 5 \times 10^{-7} \, \text{m} \]

3. **Apply Wien's Law:**
- Rearranging the formula to find temperature:
\[ T = \frac{b}{\lambda_{max}} \]
- Substitute the values into the equation:
\[ T = \frac{0.003 \, \text{m K}}{5 \times 10^{-7} \, \text{m}} \]

4. **Calculate Temperature:**
- Performing the division:
\[ T = \frac{0.003}{5 \times 10^{-7}} = \frac{0.003}{5} \times 10^{7} = 0.0006 \times 10^{7} = 6 \times 10^{3} \, \text{K} \]
- Therefore, the temperature of the Sun is:
\[ T = 6000 \, \text{K} \]

5. **Conclusion:**
- The temperature of the Sun, having a maximum intensity of radiation at \( 5000 \, \text{Å} \), is \( 6000 \, \text{K} \).

### Final Answer:
The temperature of the Sun is \( 6000 \, \text{K} \).