Experimental investigations show that the intensity of solar radiation is maximum for a wavelength `480 nm` in the visible ragion. Estimate the surface temperature of sun. (Given Wien's constant `b = 2.88 xx 10^(-3) m K`).

Options

A. `4000 K`
B. `6000 K`
C. `8000 K`
D. `10^(6) K`

Explanation

To estimate the surface temperature of the Sun using the given wavelength of maximum intensity of solar radiation and Wien's constant, we can follow these steps:

### Step 1: Understand Wien's Displacement Law
Wien's Displacement 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:

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

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

### Step 2: Rearrange the Formula to Solve for Temperature
To find the temperature (T), we can rearrange the formula:

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

### Step 3: Convert Wavelength to Meters
The given wavelength is in nanometers (480 nm). We need to convert this to meters for consistency with Wien's constant, which is in meters:

\[ \lambda_{\text{max}} = 480 \, \text{nm} = 480 \times 10^{-9} \, \text{m} \]

### Step 4: Substitute the Values into the Formula
Now we can substitute the values of \( b \) and \( \lambda_{\text{max}} \) into the rearranged formula:

\[ b = 2.88 \times 10^{-3} \, \text{m K} \]

Substituting the values:

\[ T = \frac{2.88 \times 10^{-3}}{480 \times 10^{-9}} \]

### Step 5: Calculate the Temperature
Now we perform the calculation:

\[ T = \frac{2.88 \times 10^{-3}}{480 \times 10^{-9}} = \frac{2.88}{480} \times 10^{6} \]

Calculating \( \frac{2.88}{480} \):

\[ \frac{2.88}{480} \approx 0.006 \]

Thus,

\[ T \approx 0.006 \times 10^{6} = 6000 \, \text{K} \]

### Step 6: Conclusion
The estimated surface temperature of the Sun is approximately:

\[ T \approx 6000 \, \text{K} \]