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 0.659 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 539 nm and your pupil diameter as 4.77 mm.

Answer 1

Given that :

wavelength of light, $\lambda$ = 539 x 10^-9 m

pupil's diameter, D = 4.77 x 10^-3 m

separation distance between two headlights, d = 0.659 m

using an equation,           sin $\theta$ = $\lambda$ / D                                                                       { eq.1 }

inserting the value of ' $\lambda$ ' & 'd' in above eq.

sin $\theta$ = (539 x 10^-9 m) / (4.77 x 10^-3 m)

sin $\theta$ = 112.9 x 10^-6

sin $\theta$ = 1.13 x 10^-4

The distance from an eye which will be given as :

sin $\theta$ = d / y                                                                       { eq.2 }

inserting the values in eq.2,

y = (0.659 m) / (1.13 x 10^-4)

y = (0.583 x 10⁴ m)

y = 5.83 x 10³ m

y = 5.83 km