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You stand on a straight desert road at night and observe a vehicle approaching. This vehicle is equipped with two small headlights that are 0.673 m 0.673 m 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 543 nm 543 nm and your pupil diameter to be 5.37 mm.

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To solve this problem we will apply trigonometric and optical concepts that allow us to obtain the minimum distance required. The resolution of the eye is given under the following condition,

[tex]\theta = \frac{1.22\lambda}{D}[/tex]

Here,

[tex]\lambda = Wavelength[/tex]

[tex]D = Diameter[/tex]

With the values we have that the diameter will be,

[tex]\theta = \frac{1.22(534nm)}{5.37mm}[/tex]

[tex]\theta = 1.213*10^{-4}[/tex]

The relation between the distance of the lights and the distance from the eye to the lamp is given under the function,

[tex]sin\theta = \frac{d}{L}[/tex]

For small angles [tex]sin\theta = \theta[/tex], then

[tex]\theta = \frac{d}{L}[/tex]

Here,

d = Distance between lights

L = Distance from eye to lamp

[tex]1.213*10^{-4} = \frac{0.673m}{L}[/tex]

[tex]L = \frac{0.673m}{1.213*10^{-4}}[/tex]

[tex]L = 5548.22m[/tex]

Therefore the distance will be 5.5km

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