A telescope has an objective lens of diameter 5 inch, determine the minimum angular separation between two distant objects for light of wavelength `6,000 Å`

Answer

To determine the minimum angular separation between two distant objects observed through a telescope, we can use Rayleigh's criterion. The formula for the minimum angular separation (θ) is given by:

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

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

### Step 1: Convert the diameter from inches to meters
The diameter of the objective lens is given as 5 inches. We need to convert this to meters.

\[ D = 5 \text{ inches} \times 0.0254 \text{ m/inch} = 0.127 \text{ m} \]

### Step 2: Convert the wavelength from angstroms to meters
The wavelength of light is given as 6000 Å (angstroms). We need to convert this to meters.

\[ \lambda = 6000 \text{ Å} \times 10^{-10} \text{ m/Å} = 6.0 \times 10^{-7} \text{ m} \]

### Step 3: Substitute the values into Rayleigh's criterion formula
Now that we have both \( \lambda \) and \( D \) in SI units, we can substitute these values into the formula for angular separation.

\[ \theta = \frac{1.22 \times (6.0 \times 10^{-7} \text{ m})}{0.127 \text{ m}} \]

### Step 4: Calculate the angular separation
Now we perform the calculation:

\[ \theta = \frac{1.22 \times 6.0 \times 10^{-7}}{0.127} \approx 5.76 \times 10^{-6} \text{ radians} \]

### Conclusion
The minimum angular separation between two distant objects for light of wavelength 6000 Å is approximately \( 5.76 \times 10^{-6} \) radians.

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