The wavelength of maximum intensity of radiation emitted by a star is 289.8 nm . The radiation intensity for the star is : (Stefan’s constant `5.67 xx 10^(-8)Wm^(-2)K^(-4)`, constant `b = 2898 mu m K`)-

Options

A. `5.67 xx 10^(8) W m^(-2)`
B. `5.67 xx 10^(12) W m^(-2)`
C. `10.67 xx 10^(7) W m^(-2)`
D. `10.67 xx 10^(14) W m^(-2)`

Explanation

To solve the problem, we need to determine the radiation intensity emitted by a star given the wavelength of maximum intensity (289.8 nm) and Stefan's constant (5.67 x 10^(-8) W/m²K⁴). We will use Wien's displacement law and Stefan-Boltzmann law for this calculation.

### Step-by-Step Solution:

**Step 1: Convert the wavelength from nanometers to micrometers.**
- Given wavelength, λ_max = 289.8 nm.
- To convert nanometers to micrometers, we use the conversion factor: 1 nm = 10^(-3) µm.
- Therefore, λ_max in micrometers is:
\[ λ_{max} = 289.8 \, \text{nm} \times 10^{-3} \, \text{µm/nm} = 0.2898 \, \text{µm} \]

**Step 2: Use Wien's Displacement Law to find the temperature (T).**
- Wien's displacement law states:
\[ λ_{max} = \frac{b}{T} \]
where b = 2898 µm K.
- Rearranging the formula to find T:
\[ T = \frac{b}{λ_{max}} = \frac{2898 \, \text{µm K}}{0.2898 \, \text{µm}} \]
- Calculating T:
\[ T = \frac{2898}{0.2898} \approx 10000 \, \text{K} \]

**Step 3: Use Stefan-Boltzmann Law to calculate the radiation intensity (E).**
- The Stefan-Boltzmann law states:
\[ E = σT^4 \]
where σ = 5.67 x 10^(-8) W/m²K⁴.
- Substituting the values:
\[ E = 5.67 \times 10^{-8} \, \text{W/m}^2\text{K}^4 \times (10000 \, \text{K})^4 \]
- Calculating \( (10000)^4 \):
\[ (10000)^4 = 10^{16} \]
- Therefore:
\[ E = 5.67 \times 10^{-8} \times 10^{16} = 5.67 \times 10^{8} \, \text{W/m}^2 \]

**Final Answer:**
The radiation intensity for the star is:
\[ E \approx 5.67 \times 10^{8} \, \text{W/m}^2 \]