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 cmfrom 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 1.78cm in front of the eye.)

From the given data [tex]n_a=1.333[/tex] and [tex]n_b=1.40[/tex]. The distance from the cornea to the retina in this model of the eye is [tex]2.60cm[/tex], [tex]R=0.71cm[/tex]. Substituting these values:

[tex]\frac{1.333}{S}+\frac{1.40}{2.6}=\frac{1.40-1.33}{0.71}\\\\=\frac{1.333}{S}=\frac{1.40-1.33}{0.71}-\frac{1.40}{2.6}=-3.02cm[/tex]

This means that the object for the cornea must be 3.02cm behind the cornea. Now assume that the glasses are 2.00cm in front of the eye, so then:

[tex]S'=2.00cm+S=5.02cm[/tex]

From thin lens equation:

[tex]\frac{1}{S}+\frac{1}{S'}=\frac{1}{f'_1}\\\\=\frac{1}{infinity}+\frac{1}{5.02}=\frac{1}{f'_1}=f'_1=5.00cm[/tex]

This is the focal length in water but to get that of air, we use :

[tex]f_1=f'_1[\frac{n-n_{liq}}{n_{liq}(n-1)}]\\\\=5.02[\frac{1.62-1.333}{1.333(1.62-1)}]=1.74cm[/tex]

A converging lens is needed.