People with normal vision cannot focus their eyes underwater if they aren\'t wea

Question

People with normal vision cannot focus their eyes underwater if they aren't wearing a face mask or goggles and there is water in contact with their eyes. In a simplified model of the human eye, the aqueous and vitreous humors and the lens all have a refractive index of 1.40, and all the refraction occurs at the cornea, whose vertex is 2.60 cm from the retina.

With the simplified model of the eye, what corrective lens (specified by focal length as measured in air) would be needed to enable a person underwater to focus an infinitely distant object? (Be careful-the focal length of a lens underwater is not the same as in air! Assume that the corrective lens has a refractive index of 1.62 and that the lens is used in eyeglasses, not goggles, so there is water on both sides of the lens. Assume that the eyeglasses are 2.02 cm in front of the eye.)

Explanation / Answer

let n=refractive index of air=1

n1=refractive index of water=1.333

n2=refractive index of eye lens=1.4

to find the radius of curvature of cornea ,R:

(n/s)+(n2/s')=(n2-n)/R

s=40 cm

s'=2.6 cm

then (1/40)+(1.4/2.6)=(1.4-1)/R

==>R=0.71 cm

for refraction at cornea (to find object distance from cornea so that image will be at retina)

(n1/u)+(n2/v)=(n2-n1)/R

distance from the cornea to the retina=2.6 cm

R=0.71 cm

hence

(1.333/u)+(1.4/2.6)=(1.4-1.333)/0.71

==>u=-3 cm

that is the object for the cornea must be 3 cm behind the cornea.

given that glasses are 2.02 cm in front of the eye.

so for the lens,

object distance=u=infinity

image distance=v=2.02 cm + 3 cm=5.02 cm

using the lens formula:

(1/u)+(1/v)=1/f

==>(1/infinity)+(1/5.02)=1/f

==>f=5.02 cm

this is focal length in water.

then focal length in air=f*(n-n1)/(n1*(n-1))

where n=refractive index of material of the lens

focal length in aire=5.02*(1.62-1.333)/(1.333*(1.62-1))

=1.743266 cm

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