The maximum resolution of the eye depends on the diameter of the opening of the pupil (a diffraction effect) and the size of the retinal cells. The size of the retinal cells (about 5.0 mm in diameter) limits the size of an object at the near point (25 cm) of the eye to a height of about 50 mm. (To get a reasonable estimate without having to go through complicated calculations, we shall ignore the effect of the fluid in the eye). A) Given that the diameter of the human pupil is about 2.0 mm, does the Rayleigh criterion allow us to resolve a 50 mm tall object at 25 cm from the eye with light of wavelength 550 nm?
B) According to the Rayleigh criterion, what is the shortest object we could resolve at the 25.0 cm near point with light of wavelength 550 nm?
C) What angle would the object in part B subtend at the eye?
D) Compare the value, found in part C, with the experimental value of about 1.00 min. Which of the following is correct?
1. The found value is very close to the experimental value.
2. The found value essentially differs from the experimental value.
E) Which effect is more important in limiting the resolution of our eyes, diffraction or the size of the retinal cells?

A)  θ = 3.36 10⁻⁴ rad

B) y = 8.4 10⁻⁵ m

C)  θ = 3.36 10⁻⁴ rad

D) the correct answer is 1 the value found is close to the experimental one

E)  the size of the cells is the one that most limits the minimum size of the object that we can see.  

Explanation:

A) The Rayleigh criterion determines what is the size of two objects that can be resolved due to diffraction, it establishes that the bodies are resolved if the maximum diffraction of one coincides with the first minimum of diffraction of the other, based on this it is get

for a slit      θ = λ / D

for a circular aperture, solving in polar coordinates gives the expression

                  θ = 1.22 λ / D

in our case the diameter of the pupil is D = 2.00 mm = 0.002 m                    

they tell us to use lam = 550 nm

              θ = 1.22 550 10⁻⁹ / 0.002

              θ = 3.36 10⁻⁴ rad

The angle that the object covers is

           tan θ' = y / x

           θ' = tan⁻¹ (50 10⁻³/25 10⁻²)

           θ' = 0.197

Since the angle supported by the object is greater than the minimum angle, it is resolved or is visible

B) The smallest size of the visible or resolved object

           y = x tan θ

           y = 0.25 tan 3.36 10⁻⁴

           y = 8.4 10⁻⁵ m

C) θ = 3.36 10⁻⁴ rad

D)  θ_exp= 1 min

we reduce to degre

     θ_exp= 1 min (1 ° / 60 min) = 1/60 °

we reduce to radians

     θ_exp  = 1/60 ° (π rad / 180 °)

     θ_exp = 2.9 10⁻⁴ rad

When comparing these two values the order of magnitude is the same, even though the real value is a little smaller.

the correct answer is 1 the value found is close to the experimental one

E) The angle supported by the size of the retinal cells is 0.197 rad calculated in part A

the angle of resolution by diffraction is 3.36 10⁻⁴ radians

When comparing these two values we see that the size of the cells is the one that most limits the minimum size of the object that we can see.

the tires are rotating at 0.434 rpmlinear speed=45 m/sdiameter=0.550 mradius=0.275 mnow v= rωω= v/rω=45/0.275ω=163.6 rad/sω=163.6 rad/s (1rev/2π rad) (60s/1m)ω=1563 rpm