Question

Late one night on a highway, a car speeds by you and fadesinto the distance. Under these conditions the pupils of your eyes(average refractive index = 1.36) have diameters of about6.0 mm. The taillights of this car areseparated by a distance of 1.4 m andemit red light (wavelength = 660 nm in vacuum). How far away fromyou is this car when its taillights appear to merge into a singlespot of light because of the effects of diffraction?

Answer 1

Answer:

10432.19076 m

Explanation:

$\lambda$ = Wavelength = 660 nm

d = Diameter of pupils = 6 mm

s = Distance between lights = 1.4 m

L = Distance from observer

From Rayleigh's criteria we have

$1.22\lambda=dsin\theta$

As $\theta$ is small $sin\theta=\dfrac{s}{L}$

So, the equation becomes

$1.22\lambda=d\dfrac{s}{L}\\\Rightarrow L=\dfrac{ds}{1.22\lambda}\\\Rightarrow L=\dfrac{6\times 10^{-3}\times 1.4}{1.22\times 660\times 10^{-9}}\\\Rightarrow L=10432.19076\ m$

The car is 10432.19076 m from me