Chapter 46: Problem 17

Question: The two headlights of an approaching automobile are $d~=~1.42~{\rm m}$ apart. At what (a) angular separation and (b) maximum distance will the eye resolve them. Assume a pupil diameter of $a~=~5.00~{\rm mm}$, and a wavelength of $\lambda~=~562~{\rm nm}$. Also, assume that diffraction effects alone limit the resolution.

Solution: Recall that for s slit the first diffraction minmum occurs when $a\sin\theta~=~\lambda$, while for circular apertures it occurs when $a\sin\theta~=~1.22\lambda$, where a is the diameter of the circular aperture. If the angular separation between the two sources is less than the angular distance to the first diffraction minmum, then the two sources cannot be resolved (Rayleighs Criterion). So we have that the minimum angular separation of the two sources is given by $a\sin\theta~=~1.22\lambda$, and hence $\theta~=~1.37\times 10^{-4}~{\rm rad}$, which corresponds to a distance of $D~=~{d\over \theta}~\sim~10~{\rm km}$.