Physics quiz question: hyperopia contact lens prescription

Physics 205B Quiz 2, spring semester 2018
Cuesta College, San Luis Obispo, CA

A Physics 205B student with hyperopia has an uncorrected near point of 0.65 m. The refractive power of the contact lens used to correct this student's vision is:
(A) +0.40 D.
(B) +0.65 D.
(C) +1.5 D.
(D) +2.5 D.

Correct answer (highlight to unhide): (D)

The Physics 205B student is farsighted, and d_o = 0.65 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/f₁),

where f₁ 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 student's (accommodated) cornea/lens, f₂ is the focal length of the contact 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.

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 f₁ 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/f₁)]
                                                                   

    (1/d_o') – (1/d_o) = (1/f₂),

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

P₂ = (1/f₂),

P₂ = (1/d_o') – (1/d_o),

P₂ = (1/(0.25 m)) – (1/(0.65 m)) = + 2.4615384615... m^–1 = +2.5 D,

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

(Response (A) is (0.65 m – 0.25 m); response (C) is the absolute value of the inverse of the student's near point (0.65 m), which if it was –1.5 D would be the prescription for myopia with a far point of 0.65 m. Note that an incorrect calculation (0.65 m – 0.25 m)^–1 would also result in the correct response, to two significant figures.)

Section 30882, 30883
Exam code: quiz02P0wR
(A) : 4 students
(B) : 4 students
(C) : 21 students
(D) : 5 students

Success level: 15%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.22

Physics quiz question: glasses for farsighted student later developing presbyopia

Physics 205B Quiz 2, spring semester 2016
Cuesta College, San Luis Obispo, CA

A Physics 205B student currently wears glasses to correct for farsightedness. If this Physics 205B student later develops presbyopia, __________ glasses will be prescribed.
(A) converging.
(B) diverging.
(C) bifocal.
(D) (Not enough information is given.)

Correct answer (highlight to unhide): (A)

The student is currently farsighted, and wears glasses to see nearby things. This means that the student's glasses are currently converging lenses, in oder to take an object at the nominal reading distance of 25.0 cm, and create an upright, virtual image at the more distant uncorrected near point for this student.

When the student later develops presbyopia, this is the loss of the ability to accommodate in order to focus on nearby objects, and thus the student will at best still wear the same prescription converging lens glasses, or may require a stronger prescription converging lens glasses.

Sections 30882, 30883
Exam code: quiz02f3MA
(A) : 17 students
(B) : 8 students
(C) : 16 students
(D) : 0 students

Success level: 41%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.42

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

Presentation: cameras and eyes

After qualitatively and quantitatively analyzing images generated by lenses, we now apply these concepts to how cameras and eyes work.

First, the similarities between cameras and eyes.

A basic, camera consists of a single converging lens, and here projects a real image onto a ground glass plate. (How do you know that this is a real image?) A film negative, or a charge-coupled device (CCD) can also be placed at this location to capture this image for posterity.

Similarly an eye can be modeled as consisting of a single converging lens, projecting a real image on the retina, on the back of the eye. (How do you know that this is a real image?) Rod and cone cells at this location capture this image and sends it to the brain.

Second, key differences between cameras and eyes.

A camera would need to focus on objects at different d_o distances, here by default focused on a distant object (large d_o). With a fixed focal length lens, f is constant on the right side of the thin lens equation, such that in order to focus on nearby objects, decreasing d_o on the left side of the equation must increase the image distance d_i, which means the lens of camera must move outwards to increase the lens-to-image distance. Thus a camera initially focused on a distant object must move its lens outwards in order to focus on a nearby object.

An eye would also need to focus on objects at different d_o distances, here by default focused on a distant object (large d_o). However, the image distance d_i must remain constant (as the size of the eye cannot change). In order to focus on nearby objects, decreasing d_o on the left side of the thin lens equation means that on the right side of the equation, the focal length f must also decrease! The eye can do this by accommodation, where the ciliary muscles contract, changing the curvature of the lens, decreasing its focal length f. Thus a relaxed eye focused on a distant object must "squish" its lens in order to focus on a nearby object. You can feel the effort the ciliary muscles in your eye exert during accommodation if you force yourself to focus extremely close-up.

Third, common vision defects.

Normal vision consists of being able to focus on different object distances. Nominally the far point value--the farthest object distance that can be sharply focused by a relaxed eye--is infinity (or far away enough that the quantity 1/d_o in the thin lens equation can be considered close enough to zero).

The nominal near point value--the closest object distance that can be sharply focused by an accommodated eye--is 25 cm.

Myopia or "nearsightedness" is the diagnosis for having a normal near point (thus being able to see near), but having some measurably finite far point (thus not being able to see far), due to a defect in the curvature (and focal length) of the eye.

Hyperopia or "farsightedness" is the diagnosis for having a normal far point (thus being able to see far), but having a near point greater than 25 cm (thus not being able to see near), due to a defect in the curvature (and focal length) of the eye. Keep in mind that 25 cm is an arbitrary "reading distance," and young children with flexible lenses and strong ciliary muscles can "squish" and accommodate their eyes to focus on extremely nearby objects, and can have near points less than 10 cm.

Similar to the symptoms of hyperopia is presbyopia or "elderly vision," which is the gradual loss with age of the ability to accommodate and shorten the focal length of the eye to focus on objects as close as 25 cm. This is a natural consequence of aging, and people with normal vision will all eventually experience the loss of accommodation, and must hold reading material farther and farther away as their near points grow longer and longer.

So how would we correct for these vision defects?

Since myopia and hyperopia are both caused by defects in the curvature (and focal length) of the eye, then a radical solution would be to surgically reshape the curvature of the eye.

A more mundane solution to correct for vision defects would be prescribing glasses or contacts, which we will discuss further in the next presentation.

Physics midterm problem: bifocal prescription

Physics 205B Midterm 1, spring semester 2011
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 24.24

"Image of an eyeglass prescription"
Dpbsmith

http://en.wikipedia.org/wiki/File:Specrx-prescription2.jpg

Shown above is an (edited) copy of an eyeglass prescription, where the far and near corrective optics are prescribed to be –3.25 diopters and +2.00 diopters, respectively, for the right and left eyes. Determine (a) the uncorrected far point for the right eye, and (b) the uncorrected near point for the left eye of this patient. (Neglect the distance between these glasses and eyes.) Show your and explain your reasoning using properties of lenses and vision.

Solution and grading rubric:

• p:
  Correct. Uses the thin lens equation 1/p + 1/q = 1/f = P, where p is the far point F or near point N, and q = –∞ or –0.25 m, respectively, and solves for F and N. Minor negative sign errors okay.
  
• r:
  Nearly correct, but includes minor math errors.
  
• t:
  Nearly correct, but approach has conceptual errors, and/or major/compounded math errors. Applies F = –1/D correctly to find far point, but incorrectly applies N = 1/P to find near point; or may use N = 1/(P – 1/(0.25 m)) correctly to find near point, but incorrectly applies it to find the far point.
  
• v:
  Implementation of right ideas, but in an inconsistent, incomplete, or unorganized manner.
  
• x:
  Implementation of ideas, but credit given for effort rather than merit.
  
• y:
  Irrelevant discussion/effectively blank.
  
• z:
  Blank.
  

Grading distribution:
Section 30882
Exam code: midterm01g74S
p: 1 student
r: 0 students
t: 7 students
v: 0 students
x: 0 students
y: 0 students
z: 0 students

A sample "p" response (from student 2180) with a relatively minor sign error: