Final answer:
Using the Rayleigh criterion, the maximum distance an observer can be to resolve two lamps 1.0 m apart with a pupil diameter of 4.5 mm is approximately 6250 meters.
Step-by-step explanation:
To determine the maximum distance an observer can be and still resolve the lamps as two separate sources of light affected only by the diffraction of light entering the eye, we can use the Rayleigh criterion formula:
The Rayleigh criterion is given by θ = 1.22 λ / D, where λ is the wavelength of the light, and D is the diameter of the pupil. Using the given values, λ = 589 nm = 589 x 10^-9 m and D = 4.5 mm = 4.5 x 10^-3 m, we calculate the angle θ.
θ = 1.22 x (589 x 10^-9 m) / (4.5 x 10^-3 m) = 1.60 x 10^-4 rad
We can now find the maximum distance using the small angle approximation, where θ ≈ sin(θ) for very small angles. Let d be the distance between the two lamps (1.0 m), and L be the maximum distance at which they can still be resolved. The Rayleigh criterion implies that d/L = θ.
L = d/θ = 1.0 m / (1.60 x 10^-4 rad) ≈ 6250 m
Therefore, the maximum distance at which an observer can resolve two lamps 1.0 m apart as separate sources of light, given the diameter of the pupil is 4.5 mm, is approximately 6250 meters.