Two stars distant two light years are just resolved by a telescope. The diameter of the telescope lens is `0.25 m`. If the wavelength of light used is `5000 A^(0)`, then the minimum distance between the stars is

Options

A. `1.22 xx 10^(11) m`
B. `2.44 xx 10^(11) m`
C. `3.66 xx 10^(10) m`
D. `4.88 xx 10^(10) m`

Answer

To solve the problem of finding the minimum distance between two stars that are just resolved by a telescope, we can follow these steps:

### Step 1: Understand the given data
- Distance between the stars: \(2\) light years
- Diameter of the telescope lens: \(D = 0.25 \, \text{m}\)
- Wavelength of light used: \(\lambda = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m}\)

### Step 2: Convert light years to meters
1 light year is approximately \(10^{16} \, \text{m}\). Therefore, \(2\) light years is:
\[ x = 2 \times 10^{16} \, \text{m} \]

### Step 3: Use the formula for angular resolution
The formula for the angular resolution (\(\alpha\)) of a telescope is given by:
\[ \alpha = \frac{1.22 \lambda}{D} \]

### Step 4: Substitute the values into the resolution formula
Substituting the values of \(\lambda\) and \(D\):
\[ \alpha = \frac{1.22 \times (5000 \times 10^{-10})}{0.25} \]

### Step 5: Calculate \(\alpha\)
Calculating the above expression:
\[ \alpha = \frac{1.22 \times 5000 \times 10^{-10}}{0.25} = \frac{6100 \times 10^{-10}}{0.25} = 24400 \times 10^{-10} = 2.44 \times 10^{-6} \, \text{radians} \]

### Step 6: Relate angular resolution to the distance between stars
Using the small angle approximation, we have:
\[ \alpha \approx \frac{a}{x} \]
where \(a\) is the minimum distance between the stars.

### Step 7: Rearrange the equation to find \(a\)
Rearranging gives:
\[ a = \alpha \cdot x \]

### Step 8: Substitute the values to find \(a\)
Substituting the calculated \(\alpha\) and the distance \(x\):
\[ a = (2.44 \times 10^{-6}) \cdot (2 \times 10^{16}) \]
Calculating this gives:
\[ a = 4.88 \times 10^{10} \, \text{m} \]

### Conclusion
The minimum distance between the stars is:
\[ \boxed{4.88 \times 10^{10} \, \text{m}} \]