You stand on a straight desert road at night and observe a vehicle approaching. This vehicle...

You stand on a straight desert road at night and observe a vehicle approaching. This vehicle is equipped with two small headlights that are 0.621m apart. At what distance, in kilometers, are you marginally able to discern that there are two headlights rather than a single light source? Take the wavelength of the light to be 559nm and your pupil diameter to be 4.69mm. Answer in km!

Expert Solution

The two point light sources (distant headlights) each form concentric-ring diffraction patterns on the retina due to the circular aperture of the eye pupil .. these two neighbouring patterns overlap on the retina making it difficult to resolve two close individual sources.

According to the Rayleigh criterion the two sources are just resolved when the central maxima of one pattern overlaps the first minima of the other pattern .. in other words the two patterns have the same angular separation as the first minima has from the peak of the central maxima.

For a circular aperture, the first minima (n=1) has an angular separation from the peak of the central maxima given by ..
Sinθ = 1.22 λ / d .. (d = aperture/pupil diameter)
Sinθ = 1.22 x 559×10^-9m / 4.69×10^-3m ..

Sinθ =1.4541×10^-4

For the headlights .. sinθ ≈ (light separation) / (distance from eye)

1.4541×10^-4= 0.621m / D
D = 0.621/ 1.4541×10^-4

►D = 0.427×10⁴ m = 4.27 km

genius_generous answered 9 months ago

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