Question

With a telescope with an objective of diameter \(12.0 \mathrm{~cm}\), how close can two features on the Moon be and still be resolved? Take the wavelength of the light to be \(550 . \mathrm{nm}\), near the center of the visible spectrum.

Short answer

Answer: The minimum distance between two features on the Moon that can be resolved by a telescope with an objective diameter of 12.0 cm and using light with a wavelength of 550 nm is approximately 2.152 meters.

Step-by-step solution

Given information

We are given the following information: - Objective diameter (D) = 12.0 cm - Wavelength of light (λ) = 550 nm First, we need to convert wavelength to meters and objective diameter to meters. - λ = 550 * 10^(-9) m - D = 12 * 10^(-2) m

Apply Rayleigh's criterion

Rayleigh's criterion states that the minimum angular separation (θ) between two objects that can be resolved is given by: θ = 1.22 * λ / D We can now substitute the given values into this equation to find the minimum angular separation. θ = 1.22 * (550 * 10^(-9))/ (12 * 10^(-2)) θ ≈ 5.595 * 10^(-6) radians

Convert the angular separation to distance

Now that we have the minimum resolvable angle, we can use the distance from the Earth to the Moon (R) to find the minimum resolvable distance (d) between two features on the Moon. The average distance between the Earth and the Moon is approximately 384,400 km. The formula relating angular separation, distance, and radius is given by: d = θ * R First, we need to convert the distance to the Moon to meters. R = 384,400 * 10^3 m Next, we can substitute the values of θ and R into the formula to find the minimum resolvable distance. d = (5.595 * 10^(-6)) * (384,400 * 10^3) d ≈ 2.152 m

Present the final answer

The minimum distance between two features on the Moon that can be resolved by a telescope with an objective diameter of 12.0 cm and using light with a wavelength of 550 nm is approximately 2.152 meters.