Physics quiz question: hyperopia contact lens prescription

Physics 205B Quiz 2, spring semester 2013
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 24.25

A Physics 205B student with hyperopia has an uncorrected near point of 0.45 m. The refractive power of the contact lens used to correct this student's vision is:
(A) +0.45 D.
(B) +1.8 D.
(C) +2.2 D.
(D) +5.0 D.

Correct answer (highlight to unhide): (B)

The Physics 205B student is farsighted, and d_o = 0.45 m is the nearest object distance that this student's unaided eye can see. The thin lens equation for this student's unaided eye is then:

(1/d_o) + (1/d_i) = (1/f₂),

where f₂ is the focal length of the student's (accommodated) cornea/lens, and d_i is the distance from the student's cornea/lens to the retina at the back of the eye, where the (inverted) real image is projected.

This student would like to see things at an object distance of d_o' = 0.25 m when wearing contact lenses, where the (') indicates this is the corrected farthest object distance. The thin lens equation for this student's eye with contacts is then:

(1/d_o') + (1/d_i) = (1/f₁) + (1/f₂),

where f₁ is the focal length of the contact lens (which is in front of the eye), f₂ is the focal length of the student's (accommodated) cornea/lens (which is behind the contacts), and d_i is the distance from the student's cornea/lens to the retina at the back of the eye, where the (inverted) real image is projected.

Since the student's cornea/lens to retina distance is constant, d_i is the same for both equations; similarly the students student's (relaxed) cornea/lens focal length f₂ is also constant. We can eliminate these two quantities from both equations by subtracting the first equation from the second:

    [(1/d_o') + (1/d_i) = (1/f₁) + (1/f₂)]
– [(1/d_o) + (1/d_i) =                (1/f₂)]
                                                                   

    (1/d_o') – (1/d_o) = (1/f₁),

such that the refractive power of the contact lens is then:

P₁ = (1/f₁),

P₁ = (1/d_o') – (1/d_o),

P₁ = (1/(0.25 m)) – (1/(0.45 m)) = +1.777... m^–1 = +1.8 D,

where the units of diopters (D) is equal to inverse meters (m^–1).

(Response (C) is the inverse of the student's near point (0.45 m), while response (D) is (0.45 m – 0.25 m)^–1.)

Section 30882
Exam code: quiz02hYp0
(A) : 2 students
(B) : 7 students
(C) : 24 students
(D) : 0 students

Success level: 21%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.33