The intensity of radiation emitted by the sun has its maximum value at a wavelength of 510 nm and that emitted by the North star has the maximum value at 350 nm. If these stars behave like black bodies, then the ratio of the surface temperatures of the sun and the north star is

Options

A. `1.46`
B. `0.69`
C. `1.21`
D. `0.083`

Answer

To find the ratio of the surface temperatures of the Sun and the North Star using Wien's Displacement Law, we can follow these steps:

### Step-by-Step Solution:

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. Mathematically, it can be expressed as:
\[ T \lambda_{max} = b \]
where \( b \) is a constant (Wien's displacement constant).

2. **Set up the relationship**:
From Wien's law, we can express the temperatures of the Sun (T_s) and the North Star (T_n) in terms of their respective maximum wavelengths (λ_s for the Sun and λ_n for the North Star):
\[ T_s \cdot \lambda_s = T_n \cdot \lambda_n \]

3. **Rearranging the equation**:
We can rearrange the equation to find the ratio of the temperatures:
\[ \frac{T_s}{T_n} = \frac{\lambda_n}{\lambda_s} \]

4. **Substituting the given values**:
We know from the problem that:
- λ_n (North Star) = 350 nm
- λ_s (Sun) = 510 nm

Substituting these values into the equation gives:
\[ \frac{T_s}{T_n} = \frac{350 \, \text{nm}}{510 \, \text{nm}} \]

5. **Calculating the ratio**:
Now, we can calculate the ratio:
\[ \frac{T_s}{T_n} = \frac{350}{510} \approx 0.6863 \]

6. **Final answer**:
Rounding this value gives us approximately:
\[ \frac{T_s}{T_n} \approx 0.69 \]

### Conclusion:
The ratio of the surface temperatures of the Sun to the North Star is approximately 0.69.