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, we can use Wien's Displacement Law, which states that the wavelength of maximum intensity (\( \lambda_m \)) emitted by a black body is inversely proportional to its absolute temperature (T). The law is mathematically expressed as:

\[ \lambda_m \cdot T = b \]

where \( b \) is Wien's displacement constant, approximately equal to \( 2898 \, \text{µm K} \).

### Step 1: Write down the known wavelengths
We know:
- Wavelength of maximum intensity for the Sun, \( \lambda_{m1} = 510 \, \text{nm} \)
- Wavelength of maximum intensity for the North Star, \( \lambda_{m2} = 350 \, \text{nm} \)

### Step 2: Apply Wien's Displacement Law
According to Wien's Displacement Law, we can express the temperatures of the Sun and the North Star as follows:

\[ T_1 = \frac{b}{\lambda_{m1}} \quad \text{(for the Sun)} \]
\[ T_2 = \frac{b}{\lambda_{m2}} \quad \text{(for the North Star)} \]

### Step 3: Find the ratio of the temperatures
To find the ratio of the surface temperatures \( \frac{T_1}{T_2} \), we can substitute the expressions for \( T_1 \) and \( T_2 \):

\[ \frac{T_1}{T_2} = \frac{\frac{b}{\lambda_{m1}}}{\frac{b}{\lambda_{m2}}} = \frac{\lambda_{m2}}{\lambda_{m1}} \]

### Step 4: Substitute the values of \( \lambda_{m1} \) and \( \lambda_{m2} \)
Now, substituting the values:

\[ \frac{T_1}{T_2} = \frac{350 \, \text{nm}}{510 \, \text{nm}} = \frac{350}{510} \]

### Step 5: Simplify the ratio
Calculating the ratio:

\[ \frac{T_1}{T_2} = \frac{350}{510} = \frac{35}{51} \approx 0.686 \]

Thus, the ratio of the surface temperatures of the Sun to the North Star is approximately \( 0.686 \).

### Final Answer:
The ratio of the surface temperatures of the Sun and the North Star is \( \frac{T_1}{T_2} \approx 0.686 \).

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