A telescope of diameter `2m` uses light of wavelength `5000 Å` for viewing stars.The minimum angular separation between two stars whose is image just resolved by this telescope is

Options

A. `4xx10^(-4)rad`
B. `0.25xx10^(-6)rad`
C. `0.31xx10^(-4)rad`
D. `5.0xx10^(-3)rad`

Answer

To solve the problem of finding the minimum angular separation between two stars that can be just resolved by a telescope, we can use the formula derived from Rayleigh's criterion for resolution. Here’s a step-by-step solution:

### Step 1: Understand the formula for minimum angular separation
The minimum angular separation (θ) that can be resolved by a telescope is given by the formula:

\[ \theta = \frac{1.22 \lambda}{D} \]

where:
- \( \theta \) is the minimum angular separation in radians,
- \( \lambda \) is the wavelength of light used (in meters),
- \( D \) is the diameter of the telescope (in meters).

### Step 2: Convert the wavelength from angstroms to meters
Given the wavelength \( \lambda = 5000 \, \text{Å} \), we need to convert it to meters:

\[ \lambda = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} = 5 \times 10^{-7} \, \text{m} \]

### Step 3: Substitute the values into the formula
Now, we can substitute the values into the formula. The diameter \( D \) of the telescope is given as \( 2 \, \text{m} \).

\[ \theta = \frac{1.22 \times (5 \times 10^{-7})}{2} \]

### Step 4: Calculate the value of θ
Now, we perform the calculation:

\[ \theta = \frac{1.22 \times 5 \times 10^{-7}}{2} = \frac{6.1 \times 10^{-7}}{2} = 3.05 \times 10^{-7} \, \text{radians} \]

### Step 5: Final result
Thus, the minimum angular separation between two stars that can be just resolved by the telescope is:

\[ \theta \approx 3.05 \times 10^{-7} \, \text{radians} \]