Suppose you wanted to be able to see astronauts on the moon. What is the smallest diameter of the objective lens required to resolve a 0.61 m object on the moon?
To determine the smallest diameter of the objective lens required to resolve a 0.61 m object on the moon, we can use a concept called the Rayleigh criterion. According to this criterion, two point sources can be considered resolved if the center of the Airy disk (the circular spot of light created by the diffraction of light passing through the lens) of one source falls on the first minimum (dark region) of the other source.
The formula for calculating the angular resolution, which is the minimum angle between two distinct objects that can be resolved, is given by:
θ = 1.22 * (λ / D)
where θ is the angular resolution (in radians), λ is the wavelength of light, and D is the diameter of the objective lens.
To resolve a 0.61 m object on the moon, we need to calculate the angular resolution required.
Assuming the light used is in the visible spectrum, with a wavelength of around 550 nm (nanometers), we can substitute these values into the formula:
θ = 1.22 * (550 nm / D)
Now, let's rearrange the formula to solve for D (diameter):
D = 1.22 * (550 nm / θ)
Since the object size is 0.61 m, we can convert it to angular size using the formula:
θ = 0.61 m / distance
The distance between the Earth and the moon varies, so for this example, let's assume it is approximately 384,000 km (kilometers), or 384,000,000 meters.
θ = 0.61 m / 384,000,000 m
Simplifying:
θ ≈ 1.59 × 10^⁻⁹ radians
Now, substitute this value into the previous formula to calculate the diameter:
D = 1.22 * (550 nm / 1.59 × 10^⁻⁹ radians)
Using a calculator, we find:
D ≈ 4.19 cm
So, the smallest diameter of the objective lens required to resolve a 0.61 m object on the moon is approximately 4.19 cm.