Resolving two point sources of light
A person is standing on a long straight road with a truck approaching in the distance. How far is the truck front the person when he/she can resolve the two headlights as separate sources of light?
To figure this out, we need to use the Rayleigh Criterion, which tells us the minimum angular separation 𝛳 between two sources such that they can be resolved from some distance L away.
The Rayleigh Criterion comes from the equation for the the minima in a circular aperture diffraction pattern, where 𝛳 is the angular distance from the center of the central bright spot to the first dark ring and D is the diameter of the aperture.
Without getting into too many of the details, the intensity pattern for a circular aperture far-field diffraction pattern (aka the Airy pattern) is modeled using a first order Bessel function (where k is the wavenumber and a is the aperture radius). The zeros of the Bessel function correspond to the minima in the Airy pattern (i.e. 0, 3.83, 7.02, 10.17…). For further information, see my mini project on pinhole diffraction.
Using the geometry of our diagram and the small angle approximation, we can rewrite the Rayleigh Criterion as an expression for our distance L, from the person’s eye to the truck headlights.
Now comes the estimation part. The aperture diameter in our equation is the diameter of the pupil. In the lab light, we estimated the pupil is about 3mm in diameter, so probably 5mm is a good approximation of its size at night. We estimated that the distance between the two headlights of the truck is probably 2m, and that the light they would be emitting would be yellowish-white - so 500 nm, in the middle of the visible spectrum.
With all these measurements considered, we found the distance L to be about 16 km, which is about 10 miles. Provided it's a straight road, clear night, and no obstructions, that’s a pretty far distance!