Question

You stand on a straight desert road at night and observe a vehicle approaching. This vehicle is equipped with two small headlights that are 0.627 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 541 nm and your pupil diameter to be 4.55 mm.

Answer 1

To solve the problem it is necessary to apply the concepts related to angular resolution or spatial resolution, that is, it refers to the power of an instrument to separate two objects from an image.

The optical limit due to diffraction can be empirically calculated from the Rayleigh cemetery, where

`sin\theta = 1.22(\lambda)/(D)`

Where,

D = Diameter
`\rightarrow` Eye

`\lambda`= Wavelength

Replacing our values then we have to

`sin\theta = 1.22(\frac{541*10^(-9)}{4.55*10{-3}})`

`sin\theta = 0.000145059`

For small angles we have that

`sin\theta \approx \theta`

From the trigonometric definition of Sin? We have to

`sin\theta = (d)/(L)`

Where

d = Distance between lights

L = Length

Then,

`sin\theta = (d)/(L) \approx = \theta`

`\theta = (d)/(L)`

`0.000145059  = (0.627m)/(L)`

`L = 4322.28m ((1km)/(1000m))`

`L = 4.322km`

Therefore the distance, in kilmeters, you are able to discern that there are two headlights rather than a single light source is 4.322km