Question

The aperture diameter of a telescope is 5 m. The separation between the moon and the ear this $4 \times 10^5$ km. With light of wavelength of 5500 $?$, the minimum separation between objects on the surface of moon, so that they are just resolved, is close to :

• A. 600 m
• B. 20 m
• C. 60 m
• D. 200 m

Answer 1

The Correct Option is C

To determine the minimum separation between objects on the surface of the moon that can be resolved by the telescope, we need to use the Rayleigh criterion. The Rayleigh criterion for a circular aperture states that two point sources are just resolvable when the following condition is met:

\(\theta = 1.22 \frac{\lambda}{D}\)

where:

• \(\theta\) is the angular resolution (in radians),
• \(\lambda\) is the wavelength of light (meters),
• \(D\) is the diameter of the aperture (meters).

For this problem:

• The wavelength, \(\lambda = 5500 \ Å = 5500 \times 10^{-10} \, m\),
• The diameter of the telescope, D = 5 \, m.

First, calculate the angular resolution:

\(\theta = 1.22 \frac{5500 \times 10^{-10}}{5}\)

Calculating this gives:

\(\theta \approx 1.22 \times 1.1 \times 10^{-6} \, rad\)

The minimum resolvable separation on the moon's surface, s, is then found using:

s = \theta \times R

where R is the distance to the moon:

• R = 4 \times 10^5 \, km = 4 \times 10^8 \ m

Substitute the values into the formula:

s = 1.22 \times 1.1 \times 10^{-6} \times 4 \times 10^8 \, m

Calculating this gives:

s \approx 60 \, m

Thus, the minimum separation between objects on the surface of the moon, so that they are just resolved by this telescope, is 60 m.