What is the farthest distance a spherical object of radius r = 5 mm can easily be seen when viewed by our naked eye? You may assume $\lambda$= 550 nm and that the diameter of the human pupil is d = 10.0 mm.

Question

What is the farthest distance a spherical object of radius r = 5 mm can easily be seen when viewed by our naked eye? You may assume $\lambda$= 550 nm and that the diameter of the human pupil is d = 10.0 mm.

Quizplus · Accepted Answer

The farthest distance at which a spherical object can easily be seen by the naked eye can be estimated using the Rayleigh criterion for resolution, which is given by the formula $ \theta = 1.22 \frac{\lambda}{D} $, where $ \theta $ is the angular resolution, $ \lambda $ is the wavelength of light, and $ D $ is the diameter of the aperture (in this case, the human pupil). The angular size of the object as seen from the eye must be greater than or equal to this angular resolution for the object to be resolvable. Given $ \lambda = 550 $ nm (or $ 550 \times 10^{-9} $ m) and $ D = 10.0 $ mm (or $ 10 \times 10^{-3} $ m), we can calculate $ \theta $. The actual distance $ L $ at which the object can be resolved can then be related to its physical size $ 2r $ (the diameter of the spherical object, which is $ 2 \times 5 $ mm or $ 10 \times 10^{-3} $ m) and the angular size $ \theta $ by the small angle approximation formula $ \theta \approx \frac{2r}{L} $. Solving for $ L $ gives the maximum distance at which the object can be resolved. Given the parameters, the calculation leads to a distance of approximately 150 meters, making option A correct.

What Is the Farthest Distance a Spherical Object of Radius λ\lambdaλ

What Is the Farthest Distance a Spherical Object of Radius $\lambda$