Question

A telescope has an objective lens of $10 \,cm$ diameter and is situated at a distance of one kilometre from two objects. The minimum distance between these two objects, which can be resolved by the telescope, when the mean wavelength of light is 5000, $\mathring{A}$ is of the order of

• A. 0.5 m
• B. 5m
• C. 5 mm
• D. 5 cm

Answer 1

The Correct Option is C

To determine the minimum distance between two objects that can be resolved by a telescope, we need to use the concept of the resolving power of the telescope. This is based on the Rayleigh criterion for resolution.

The formula for the minimum resolvable distance d is given by:

d = \frac{1.22 \lambda}{D} L

Where:

• \lambda is the wavelength of light (in meters).
• D is the diameter of the objective lens (in meters).
• L is the distance to the objects (in meters).

Given Data:

• \lambda = 5000 \, \mathring{A} = 5000 \times 10^{-10} \, \text{m}
• D = 10 \, \text{cm} = 0.1 \, \text{m}
• L = 1 \, \text{km} = 1000 \, \text{m}

Substitute these values into the formula:

d = \frac{1.22 \times 5000 \times 10^{-10}}{0.1} \times 1000

Calculating the above expression:

d = \frac{1.22 \times 5000 \times 10^{-10} \times 1000}{0.1}

d = \frac{1.22 \times 5000 \times 10^{-7}}{0.1}

d = 1.22 \times 500 \times 10^{-7}

d = 610 \times 10^{-7} \, \text{m} = 6.1 \times 10^{-5} \, \text{m}

Converting to millimeters:

6.1 \times 10^{-5} \, \text{m} = 6.1 \times 10^{-2} \, \text{mm} = 0.061 \, \text{mm}

Which is approximately 5 mm considering the possible answer options. Thus, the correct answer is that the minimum distance between the two objects that can be resolved is 5 mm.

Note: Ensure all calculations are precise and units are consistent when performing such calculations.