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. 0.83

Explanation

To find the ratio of the surface temperatures of the Sun and the North Star using the given wavelengths, we can apply Wien's Displacement Law, which states that the product of the maximum wavelength (\(\lambda_m\)) of radiation emitted by a black body and its absolute temperature (T) is a constant.

### Step-by-Step Solution:

1. **Write down Wien's Displacement Law:**
\[ \lambda_m \cdot T = b \]
where \(b\) is Wien's displacement constant.

2. **Set up the equation for both stars:**
For the Sun:
\[ \lambda_{m, \text{Sun}} \cdot T_{\text{Sun}} = b \]
For the North Star:
\[ \lambda_{m, \text{North Star}} \cdot T_{\text{North Star}} = b \]

3. **Equate the two expressions:**
Since both expressions equal the same constant \(b\), we can set them equal to each other:
\[ \lambda_{m, \text{Sun}} \cdot T_{\text{Sun}} = \lambda_{m, \text{North Star}} \cdot T_{\text{North Star}} \]

4. **Rearrange to find the ratio of temperatures:**
\[ \frac{T_{\text{Sun}}}{T_{\text{North Star}}} = \frac{\lambda_{m, \text{North Star}}}{\lambda_{m, \text{Sun}}} \]

5. **Substitute the given wavelengths:**
Given:
- \(\lambda_{m, \text{Sun}} = 510 \, \text{nm}\)
- \(\lambda_{m, \text{North Star}} = 350 \, \text{nm}\)

Substitute these values into the equation:
\[ \frac{T_{\text{Sun}}}{T_{\text{North Star}}} = \frac{350 \, \text{nm}}{510 \, \text{nm}} \]

6. **Calculate the ratio:**
\[ \frac{T_{\text{Sun}}}{T_{\text{North Star}}} = \frac{350}{510} \approx 0.6863 \]

7. **Final Result:**
Thus, the ratio of the surface temperatures of the Sun to the North Star is approximately:
\[ \frac{T_{\text{Sun}}}{T_{\text{North Star}}} \approx 0.69 \]