PART A. Traveling down a long, straight road late at night, a driver notices a l
ID: 1348806 • Letter: P
Question
PART A. Traveling down a long, straight road late at night, a driver notices a light approaching in the distance. After a bit the driver notices that the single light has split into two distinct headlights. Estimate the maximum distance that approaching car can be assuming that 1) the driver's pupils are 4 mm in diameter, 2) the approaching car's headlights are 5 ft apart, 3) the headlights emit primarily 580 nm wavelength light, and 4) that the driver's eyesight is diffraction limited.
PART B. After passing the oncoming car, the driver looks at the dashboard to check the speedometer. The driver notices that he can clarly read the numbers 0.8 m away if he pushes his head against the headrest, but they get blurry if he leans forward. What prescription eyeglasses (in Diopters) would this driver need to comfortably read a book 25 cm from his face? Assume the glasses would sit 2 cm in front of his eyes.
Explanation / Answer
as per rayleigh criteria,
angular resolution(in radian) =1.22*wavelength/diameter of aperture
here angular resolution=distance between headlight/maximum distance from which they can be resolved
let maximum distance be D ft.
then angular resolution=5/D
then as per rayleigh criteria,
5/D=1.22*580*10^(-9)/(0.004)
==>D=5*0.004/(1.22*580*10^(-9))=28.265 km
part B:
here this person has long sightedness.
so to resolve that, he has to be prescribed a convex lens so that object closer to the eye will appear at a distant which is convenient for the eye.
here object distance =-(25-2)=-23 cm
image distance=-(80-2)=-78 cm
let focal length be f.
then using thin lens equation:
(1/image distance)-(1/object distance)=1/focal length
==>(1/-78)-(1/-23)=1/focal length
==>focal length=32.618 cm=0.32618 m
hence lens power=1/focal length=3.0657 diopter
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