Reading:

Cutnell and Johnson, Chapter 26.

Problems for Review:

Reviewing these problems gives you good exposure to this week's main topics, in practice. All are worked in the Student Study Guide [SSG] or Student Solutions Manual [SSM]; solutions are posted on our website http://www.physics.emory.edu/Classes/142-003.html , for those without these texts.

Conceptual:

MCAT review problem, [SSG] p 513.

Quantitative:

Chapter 26, problems 9, 13, 29, 33, 41, 49, 53, 59, 65, 69, 77, 89, 97.

Problems to Hand In:

Conceptual:

Questions 1 through 7 refer to the situation below:

In the human eye, light is focused on the retina by the combined action of the cornea and lens. A simplified model of the eye, the reduced eye, ignores the lens by assuming the cornea is filled with a fluid of index of refraction 1.47 rather than the actual value of 1.33. For small angles, the object ($p$) and image ($q$) distances are related to the radius of curvature of the cornea by

$$\frac{n'}{p} + \frac{n}{q} = \frac{n-n'}{R} \ \ .$$

For $p$ very large, this gives the focal length of the reduced eye as $f = nR/(n-n')$, where $R$ is the radius of curvature of the cornea, $n$ is the index of refraction of the fluid inside the cornea ($n=1.47$), and $n'$ is the index of refraction of the medium outside the eye ($n' = 1.00$ for an object viewed in air). The distance from the cornea to the retina is 24 mm for a typical eye. The radius of curvature of the cornea, when viewing an object at very great distance, is 7.7 mm. You may wish to verify that the image of an object infinitely far away will be focused on the retina, by the cornea, in this model of the eye.

The retina consists of a large number of closely spaced, light-sensitive receptor cells. These cells vary slightly in size and spacing. In the most acute part of the eye (the fovea centralis), where the color sensitive cells are packed tightly, they are about one micrometer between centers. Outside this region they are 3 $\mu{\rm m}$ to 5 $\mu{\rm m}$ apart. Two points of light from an object will not be seen as two distinct points unless the images fall on non-adjacent receptors. In a typical eye, the distance between non-adjacent cells is 2 $\mu{\rm m}$ apart. Using the cornea-to-retina distance of 24 mm, the angular separation, in radians, of two object points which can just be resolved is $r = x/(24\ {\rm mm})$, where $x$ is the linear separation of non-adjacent receptor cells.

1. The speed of light in vacuum is $3.0 \ \times\ 10^8\ {\rm m/s}$. What is the speed of light in a medium with index of refraction $1.47$?
   1. $1.0 \ \times\ 10^8\ {\rm m/s}$
   2. $2.0 \ \times\ 10^8\ {\rm m/s}$
   3. $3.0 \ \times\ 10^8\ {\rm m/s}$
   4. $4.0 \ \times\ 10^8\ {\rm m/s}$

2. Light traveling through air strikes the surface of a medium with index of refraction $1.47$ at an angle of incidence of $5^o$. What is the angle of refraction, in radians?
   1. 0.059
   2. 0.087
   3. 0.128
   4. 0.243

3. For the reduced eye model, to what value must the radius of curvature of the cornea change if the image of an object at 250 mm in air is to fall on the retina 24 mm behind the cornea?
   1. 5.6 mm
   2. 7.2 mm
   3. 7.7 mm
   4. 8.2 mm

4. If an object of height 20 cm is viewed from a distance of 120 cm by a normal reduced eye (cornea to retina distance of 24 mm), what is the size of the image on the retina?
   1. 2 mm
   2. 3 mm
   3. 4 mm
   4. 5 mm

5. For receptor spacing of 1 $\mu{\rm m}$, how closely spaced may two dots be, if when viewed from a distance of 250 mm, they can just be resolved?
   1. 1.3 $\mu{\rm m}$
   2. 21 $\mu{\rm m}$
   3. 0.68 mm
   4. 73 mm

   
   Questions 6 and 7 deal with the following specific case:

   The focal length of the normal relaxed eye is 24 mm, corresponding to the cornea to retina distance. If the focal length of an abnormal relaxed eye is 23 mm,

6. Which of the statements below is true?
   1. The eye is nearsighted, with a far point of 23 mm.
   2. The eye is nearsighted, with a far point of 55 cm.
   3. The eye is farsighted, with a near point of 23 mm.
   4. The eye is farsighted, with a near point of 55 cm.

7. What must be the focal length of a corrective contact lens for this eye?
   1. 23 mm, a converging lens.
   2. 55 mm, a converging lens.
   3. -23 mm, a diverging lens.
   4. -55 mm, a diverging lens.

Quantitative:

Chapter 26, problems 12, 20, 24, 30, 36, 58, 66, 90, 96, 107.