Question

The two headlights of an approaching automobile are $1.4m$ apart. At what
(a) angular separation and
(b) maximum distance will the eye resolve them? Assume that the pupil diameter is $5.0mm,$ and use a wavelength of $550nm$ for the light. Also assume that diffraction effects alone limit the resolution so that Rayleigh's criterion can be applied.

Answer 1

(a) We use the Rayleigh criteria.
Thus, the angular separation (in radians) of the sources must be at least ${\theta }_R=1.22\lambda /d,$ where $\lambda$ is the wavelength and $d$ is the diameter of the aperture.
For the headlights of this problem,
${\theta }_R=\frac{1.22(550\times {10}^{-9}m)}{5.0\times {10}^{-3}m}=1.34\times {10}^{-4}rad$
or $1.3\times {10}^{-4}$ rad , in two significant figures
(b) If $L$ is the distance from the headlights to the eye when the headlights are just resolvable and $D$ is the separation of the headlights, then $D=L{\theta }_R$, where the small angle approximation is made.
This is valid for ${\theta }_R$ in radians.
Thus $L=\frac{D}{{\theta }_R}=\frac{1.4m}{1.34\times {10}^{-4}rad}=1.0\times {10}^{4}m=10km$