Physics of the Human Body

Irving P. Herman

Chapter 11 Light, Eyes, and Vision - all with Video Answers

Educators

Chapter Questions

Problem 1

Let us explore our blind spot a bit further.
(a) Why does the experiment demonstrating the blind spot in your left eye (using Fig. 11.4) prove it is "nasal?"
(b) Assume the eye acts as a lens in air with a 17 mm focal length (Standard eye model) and determine how far your blind spot is from your fovea (in mm and degrees)? Do you confirm the blind spot is $13-18^{\circ}$ away from the fovea? (c) Repeat the experiment described in Fig. 11.4 using your right eye, with your left eye closed. How do you explain your findings? How can you modify the experiment so you can determine the blind spot in that eye? Does this prove that the blind spot in your right eye is also nasal and located $13-18^{\circ}$ away from the fovea. Why?

Carlos Henrique De Lima

Numerade Educator

11:34

Problem 2

(a) Derive Newton's relation for a thin lens in air that forms a real image, $x x^{\prime}=f^2$, where $x=d_1-f$ (the distance the source is in front of the focal point to the left of the lens) and $x^{\prime}=d_2-f$ (the distance the image is after the focal point). (Hint: Draw a diagram from a source with a ray that is parallel to the optic axis before the lens and one that is parallel to it after the lens, and consider similar triangles.)
(b) If a source is 20 cm in front of (to the left of) the primary focal point and the image is 5 cm to the right of the secondary focal point, find the focal length and power of the lens and also confirm that (11.3) is satisfied.

Ameer Said

Numerade Educator

01:22

Problem 3

Snell's Law, $n_1 \sin \theta_1=n_2 \sin \theta_2$ (11.5), predicts that rays will undergo total internal reflection (and not be transmitted) when $n_1>n_2$ for incident angles $\theta_1>\arcsin \left(n_2 / n_1\right)$ (at the local planar interface). Show that light rays inside the eye that hit the cornea at angles exceeding $49^{\circ}$ cannot leave the eye (assuming the medium in the eye has refractive index 1.33) (Fig. 11.59a). (This total internal reflection can affect the visual inspection of eyes requiring these large angles, except if a goniolens is used (Fig.11.59b). These largeangle rays are transmitted through the eye into a saline/contact lens assembly in this goniolens and then they are reflected to more shallow angles for inspection.)

Abhishek Kumar

Numerade Educator

05:43

Problem 4

A thin lens in air is composed of material with an index of refraction of 1.062. For each posterior radius of curvature $R_{23}$ given in (a)-(c), find the anterior radius of curvature $R_{12}$ needed so this lens has a focal length of 50 mm . Also, for each case sketch the shape of the lens and describe it as being either biconvex, planoconvex, or positive meniscus.
(a) -0.0060 m
(b) $\infty$
(c) +0.0060 m .
(The parameters in this problem are equivalent to those for the eye crystalline lens, with the ratio of refractive indices of the lens to the humors being $1.42 / 1.337=1.062$.)

Zachary Warner

Numerade Educator

02:55

Problem 5

Show the equations for a thick lens in air, (11.42) and (11.43), reduce to the Lensmaker equation for a thin lens, (11.17), under appropriate conditions.

Salamat Ali

Numerade Educator

09:17

Problem 6

A glass lens in air has $n=1.5, R_{12}=200 \mathrm{~cm}$, and $R_{23}=-200 \mathrm{~cm}$.
(a) Assume it is a thin lens and find its focal length.
(b) Now assume that the lens has a thickness $t=5 \mathrm{~mm}$. Find its effective and back focal lengths.
(c) Will the rays cross the axis first, i.e., focus first, for the lens described in (a) or (b)? (Compare the focal length for the lens in (a) to the distances in the lens in (b) that incident parallel rays focus beyond the (i) back surface, (ii) middle plane, and (iii) front surface of the lens.)

Sandeep Kumar Dhania

Numerade Educator

04:15

Problem 7

(a) Estimate the loss of light at each of the four ocular interfaces. Assume that each interface is flat and that light hits each at normal incidence. (Use the optical data presented in Table 11.1 for Schematic eye 1.)
(b) Does this change significantly with wavelength?

Julie Farhm

Numerade Educator

03:31

Problem 8

Show that the thick lens equations for arbitrary media, (11.40) and (11.41), reduce to results for a thick lens in air, (11.42) and (11.43), when $n_1=n_3=1$ and $n_2=n$.

Arun Bana

Numerade Educator

04:46

Problem 9

Use the parameters in the text for Schematic eye 1 to determine how far past the anterior surface of an optical element incident parallel rays converge to the optic axis to form a focus when the element is assumed to have a finite thickness and compare this to that assuming the thin lens approximation. Do this for the:
(a) Cornea.
(b) Crystalline lens.
(c) Does this suggest that the correction for a thick lens is significant?

Bruce Edelman

Numerade Educator

02:17

Problem 10

Model the tears on the cornea as a $7 \mu \mathrm{~m}$ thick layer with refractive index 1.33, with anterior and posterior radii of curvature the same as that of the anterior surface of the cornea for Schematic eye 1.
(a) What is the optical power of this tear layer?
(b) What does it correspond to in D?
(c) Do tears affect imaging significantly?

Ummatul Choudary

Numerade Educator

05:34

Problem 11

Thomas Young, famous for seminal experiments and interpretations in optics almost 200 years ago, showed that ocular accommodation was due to the crystalline lens and not the cornea because he could not focus on nearby objects when he immersed his eye in water. We know that his conclusion was right, but was his reasoning faulty? Use (11.48) and (11.56), and what you (should) know about accommodation.

Eduard Sanchez

Numerade Educator

03:15

Problem 12

Equation 11.63 shows that we need 4 D of accommodation for our eyes to image at all distances $\geq 25 \mathrm{~cm}$. Can the similar expression, $1 / f_{\text {eye, } \mathrm{NP}}-$ $1 / f_{\text {eye, FP }}$, which uses the focal lengths of a person's own near and far points, be used to determine that person's own accommodation? (Hint: Consider a very young person. Also, consider a person with more than 4D accommodation who still needs corrective lenses and perhaps also bifocals.)

Nicole Krahulik

Numerade Educator

01:42

Problem 13

Estimate the change in the radii of curvature of the crystalline lens surfaces needed to achieve 4 D of accommodation (so a person viewing a source at her far point can then view a source at her near point). Use (11.50) and (11.51) and alternately assume (a) that the curvatures of the anterior and posterior surfaces change by the same percentage or (b) that only the anterior surface changes. (Because accommodation is usually cited for an effective imaging element in air, the amount of accommodation required within the eye is larger by the refractive index of the humors, so $\simeq 4 \mathrm{D} \times 1.34=5.5 \mathrm{D}$ is needed.)

Ranjeet Singh

Numerade Educator

04:15

Problem 14

In a model of the crystalline lens, the lens has a refractive index of 1.42 and the humors before and after it have a refractive index of 1.337, and the magnitudes of the radii of curvature of the two lens surfaces are the same (but they have different signs). Find the radii of curvatures of the anterior and posterior surfaces if the total refractive power of the crystalline lens is the same as in (11.53) (ignoring the thickness of the lens).

Julie Farhm

Numerade Educator

11:12

Problem 15

A crystalline lens is modeled here as in Fig. 11.19, except it is biconvex with radii of curvature that always have equal magnitudes. This magnitude is 7.54 mm when a person views a source at her far point. (The crystalline lens still has an overall refractive power of 22.10 D , ignoring the thickness of the lens.) Find the radius of curvature magnitude when the lens adapts to give 4 D of accommodation (to then allow the person to see her near point), which is equivalent to 5.5 D in the eye (as is explained in Problem 11.13).

Dading Chen

Numerade Educator

01:22

Problem 16

The argument is made that extremely little ultraviolet light with wavelengths from 300 to 400 nm reaches the retina, so it does not cause retinal damage and consequently this light does not damage eyes. Discuss the merits of this argument. Also, where is this light absorbed in the eye?

Hast Aggarwal

Numerade Educator

01:26

Problem 17

The light from a red laser pointer has a wavelength near 650 nm . Where in the eye is most of this wavelength absorbed?

Narayan Hari

Numerade Educator

00:46

Problem 18

How well is the transmission spectrum of the eye matched to the spectral responses of rods and cones?

Jesse Kooistra

Numerade Educator

04:07

Problem 19

How well are the spectral responses of the rods and cones matched to the spectrum of solar light? How well are they matched to the spectrum of solar light that actually reaches sea level?

Keenan Mintz

University of Miami

01:44

Problem 20

A nanowatt of light, equally spread across the wavelengths $400-$ 700 nm , is incident on your eye. Approximately what power impinges on your retina?

Keshav Singh

Numerade Educator

02:27

Problem 21

Which can cause more retinal damage, light that is highly spatially coherent that can focus according to (11.67) or light that is less coherent that cannot be focused as well. Why?

Mahendra K

Numerade Educator

02:20

Problem 22

A laser enters your eye. Is more damage done if you turn your eye to look at the laser source (which is a reflex) or if you do not stare into the beam? Why?

Khoobchandra Agrawal

Numerade Educator

01:58

Problem 23

In a Snellen chart a letter is 8.8 mm high in the $20 / 20$ line. How high is it in the 20/200 line?

Dading Chen

Numerade Educator

01:58

Problem 24

In the text it was said that a 8.8 mm letter in the $20 / 20$ line of the Snellen chart subtended a 5 min arc at the eye of a person standing 20 ft from the chart. This 20/20 (i.e., $20 \mathrm{ft} / 20 \mathrm{ft}$ ) line is also known as the $6 / 6$ (i.e., $6 \mathrm{~m} / 6 \mathrm{~m}$ ) line. Does this 5 min arc angle more nearly refer to a person standing 20 ft or 6 m from the chart?

Dading Chen

Numerade Educator

01:58

Problem 25

(a) The letters in the $20 / 20$ line of the Snellen chart are 8.8 mm high and wide. Show that this letter is about $25 \mu \mathrm{~m}$ high and wide on the retina. (b) Explain why this is considered excellent vision.

Dading Chen

Numerade Educator

02:50

Problem 26

If you use an illuminance much greater than 480 lux in conducting a Snellen eye test, your pupil diameter begins to decrease. Discuss how this could affect the visual acuity you are trying to measure in this test, given the effect of pupil diameter on diffraction and lens aberrations. (Most people think you see better in bright light.)

Mayukh Banik

Numerade Educator

04:44

Problem 27

(a) Show that the 8.8 mm high letters in the $20 / 20$ (or $6 / 6$ ) line of the Snellen chart subtend an angle of 1.47 mrad when you are 6 m away (where $1 \mathrm{mrad}=1 \mathrm{milliradian}=1 \times 10^{-3} \mathrm{rad}$ ).
(b) The Rayleigh criterion for barely resolving objects separated by an angle $\theta$ is that $\theta$ needs to be at least $\theta_{\mathrm{R}}=1.22 \lambda / d$, where $d$ is the diameter of the lens (or that of the aperture limiting the lens). Show how this criterion is related to the Airy diameter (11.67).
(c) This Rayleigh criterion is a strict limit imposed by diffraction. Show that for 500 nm light, this angle is 0.122 mrad for an aperture diameter of 0.5 cm . How many times the Rayleigh criterion limit is the arc angle needed for 20/20 vision?
(d) The resolution limit for most people is 0.5 mrad , and for the most acute vision under optimum conditions it is 0.2 mrad . How many times the Rayleigh criterion limit are these?

Khoobchandra Agrawal

Numerade Educator

04:56

Problem 28

How small would the pupil diameter need to be to affect the ultimate diffraction limit of the eye lensing system? Given the sizes of cones, would this change make a practical difference?

Mayukh Banik

Numerade Educator

04:56

Problem 29

(a) Assume for the moment that the rods and cones are $0.1 \mu \mathrm{~m}$ in diameter, that they are tightly packed (with no space between them), and that each is individually connected to the brain by a single neuron. Ignoring optical aberrations, would you see images sharper in the blue or the red? Why?
(b) Ignore the assumptions in part (a), returning to the normal conditions, and address the same questions again.

Mayukh Banik

Numerade Educator

04:33

Problem 30

(a) Reference [561] (as cited in [560]) used experimental data measured using a human eye with 3.0 mm pupil diameter to obtain an analytic fit to the line spread function: $I(i)=0.47 \exp \left(-3.3 i^2\right)+0.53 \exp (-0.93|i|)$. The distance on the retina from the fovea is $i$, in arc min. Show that Fig. 11.30 accurately plots this function.
(b) Plot on this same set of axes this same function after each point has been broadened by 1 arc min , and compare them. (One relatively crude, way to do this is divide the $x$-axis into bins that are 0.1 arc min wide, so you have a set of rectangles that have heights $I(i)$ and widths 0.1 arc min centered at $i=0.0, \pm 0.1, \pm 0.2, \pm 0.3, \ldots$ arc min. Replace these by rectangles with widths of 1.0 arc min . Sum the contributions at each $i$ and then normalize all points by the value at $i=0$. This can be made less crude by using a smaller width or a broadening function that is smoother than a rectangle.)

Sana Riaz

Numerade Educator

09:25

Problem 31

In Schematic eye 1 and other models of the eye the radius of curvature of the anterior surface of the (biconvex) crystalline lens is larger in magnitude than that of the posterior surface. Use Fig. 11.27 to explain qualitatively why this asymmetry helps lessen spherical aberration in the eye.

Katie Mcalpine

Numerade Educator

06:22

Problem 32

Dispersion in glass is usually presented by the refractive index at the center of the yellow line doublet $(D)$ from a sodium lamp at $589.3 \mathrm{~nm}, n(D)$, and the dispersion constant (or Abbe number or V -number or constringence) $V=(n(D)-1) / \Delta n$, where $\Delta n=n(F)-n(C)$ is the difference in the indices in the blue $(F, 486.1 \mathrm{~nm})$ and red ( $C, 656.3 \mathrm{~nm}$ ) lines from a hydrogen lamp. The refractive index is assumed to vary linearly between the red and the blue. For borosilicate crown glass BSC-2: $n(D)=1.517$ and $V=64.5$ and for dense flint glass DF-2: $n(D)=1.617$ and $V=36.6$. (Chromatic aberration can be minimized in a doublet lens (and achromat), with the two component lens made of two glasses, such as these, with very different dispersion.)
(a) If the focal length of a simple single-component lens composed of either type of glass is designed to be 17 mm in the yellow, what is its focal length in the red and the blue?
(b) If the screen for the image is 17 mm away from the lens (as for the retina), what is the size of the image size (blur) for each range of color? (Ignore the limitations of diffraction. Assume the source is at infinity.)

Sean Dougherty

Numerade Educator

02:11

Problem 33

The dispersion constants for ocular media in Table 11.3 use a revised definition of dispersion that differs slightly from the earlier definition used in Problem 11.32. Now $V=(n(d)-1) / \Delta n$, where $\Delta n=n\left(F^{\prime}\right)-n\left(C^{\prime}\right)$ is the difference in the indices in the blue ( $F^{\prime}, 480.0 \mathrm{~nm}$ ) and red ( $C, 643.8 \mathrm{~nm}$ ) lines from a cadmium lamp and $n(d)$ is that at the center of the yellow line doublet (d) from a sodium lamp at 587.6 nm . Find the refractive powers at each of the four interfaces in the eye, and the sum of these refractive powers, at each of these three wavelengths. (Assume the refractive index of the cornea is at 587.6 nm and that dispersion constant of the cornea is 51.5 .)

Khoobchandra Agrawal

Numerade Educator

09:50

Problem 34

Use (11.68) (describing chromatic aberration in the eye in diopters) to estimate the change in index of refraction for the eye crystalline lens in the red $(630 \mathrm{~nm})$ and blue $(470 \mathrm{~nm})$ relative to the yellow ( 578 nm ). Compare this to that of glass in Problem 11.33

Kelley Commeford

Numerade Educator

04:15

Problem 35

Spectral dispersion in water is characterized by $n(D)=1.333, V=55$ (see Problem 11.32 for definitions). Model the air/anterior cornea interface instead as an air/water interface. Evaluate the chromatic aberration by determining the change in focal length across the visible (in D). How does this compare to the $\sim 2 \mathrm{D}$ of chromatic aberration in the eye?

Julie Farhm

Numerade Educator

01:39

Problem 36

Apply simple theory (and ignore the distance between refractive surfaces) to the schematic eyes to find the total refractive power for:
(a) Schematic eye 2
(b) Schematic eye $2^{\prime}$ (accommodated version of part (a)).
(c) Schematic eye 3.
(d) How much refractive power is added in the accommodated version of the Schematic eye 2? Is this enough so these two models can accommodate for near and far vision?

Penny Riley

Numerade Educator

01:53

Problem 37

(advanced problem) Repeat Problem 11.36 using the more exact treatment of the schematic models (i.e., account for the distances between refractive surfaces).

Ajay Singhal

Numerade Educator

11:12

Problem 38

Find the refractive power of the cornea, crystalline lens, and the entire eye for the Schematic exact eye, ignoring the distances between refractive surfaces. How do these compare to the values with more exact treatment, $43.05 \mathrm{D}, 19.11 \mathrm{D}$, and 58.64 D , respectively?

Dading Chen

Numerade Educator

11:12

Problem 39

(advanced problem) Show that the refractive powers of the cornea and crystalline lens combine to give the overall refractive power of the eye for the Schematic exact eye. The first and second principal points of the cornea are at -0.0496 mm and -0.0506 mm , respectively, and those of the crystalline lens are at 5.678 mm and 5.808 mm , respectively. Also use the data provided in Problem 11.38 and Table 11.1. As part of this problem, show that the positions of the principal points of the overall system agree with those given in Table 11.1.

Dading Chen

Numerade Educator

Problem 40

Show that $\mathrm{PP}^{\prime}=\mathrm{NN}^{\prime}$ and $\mathrm{FP}=\mathrm{N}^{\prime} \mathrm{F}^{\prime}$ for each model in Table 11.1.

Check back soon!

02:16

Problem 41

Model an effective eye lens as a cornea thin lens with focal length 25 mm followed by a "crystalline lens" thin lens with focal length 50 mm , with both reduced lenses in air:
(a) What is their combined effective focal length if their separation is zero?
(b) What is their combined effective focal length if their separation is 9 mm (as in the eye)?
(c) What is this difference in diopters and is it significant?

Abhishek Jana

Numerade Educator

11:12

Problem 42

Another published set of refractive indices for the eye components has $n_{\text {cornea }}=1.376, n_{\text {aqueous }}=1.336, n_{\text {lens }}=1.40, n_{\text {vitreous }}=1.337$, and the four radii of curvature as (from anterior to posterior) $7.8 \mathrm{~mm}, 6.4 \mathrm{~mm}, 10.1 \mathrm{~mm}$, and -6.1 nm . How do the refractive power and focal length differ from those calculated earlier, using (11.54)-(11.55).

Dading Chen

Numerade Educator

01:39

Problem 43

Let us explore accommodation using the simplified schematic eye (such as Schematic eye 2) for two people, subjects H and M, both having about 9D of accommodation, by using the data in Table 11.6 [535].
(a) Estimate the ages of the subjects.
(b) Qualitatively, how do the subjects differ in how their eyes accommodate?
(c) Find the refractive power of each interface in each case.
(d) Use simple theory (and ignore the distance between the refractive surfaces) to find the refractive power of the eye in each case and the amount of accommodation in both subjects.
(e) (advanced problem) Accounting for the distance between the refractive surfaces, confirm the total refractive powers and the principal point locations of each eye, as listed in Table 11.6.

Penny Riley

Numerade Educator

04:15

Problem 44

Use simple theory (and ignore the distance between the refractive surfaces) to determine how the total refractive power of the eye would change if the refractive index of only one of the following optical elements were increased by $1 \%$ ?
(a) cornea
(b) aqueous humor
(c) Crystalline lens.
(d) Vitreous humor.

Start with the values for Schematic eye 1.

Julie Farhm

Numerade Educator

01:42

Problem 45

People from Planet X have eyes that are constructed much like ours. They have the same components and the refractive indices of each medium are the same as ours. One difference is that their eyeballs are twice as long as ours. Assume geometric optics theory and ignore the distances between eye components to explore in what other ways their eyes are different. Assume that the radii of curvature of their four refractive interfaces (anterior and posterior surfaces of the cornea and crystalline lens) are in the same proportion as ours.

Suzanne W.

Numerade Educator

12:58

Problem 46

Ophthalmologists and optometrists treat 20 ft (or 6 m ) as the "optical infinity." For the human eye, what is the difference in refractive power (in D) needed to see at this optical infinity vs. the real infinity?

Daniel Alva

Numerade Educator

03:33

Problem 47

In a Goldmann tonometer, the force needed to make a circular region of the cornea (with diameter 3.06 mm ) flat is measured. The resulting pressure needed to make it flat is equal to the intraocular pressure, and therefore this is a good test for glaucoma. Show that for an intraocular pressure of 15 mmHg , this applied force corresponds to the gravitational force of 1.5 g applied over this area.

Arun Bana

Numerade Educator

04:45

Problem 48

A person with myopia has a far point 60 cm from her eye. What corrective eyeglasses and contact lenses should she wear? (Give your answer in D.)

Ryan Kutayiah

Texas A&M University

01:09

Problem 49

A person with hyperopia has a near point 500 cm from his eye. What corrective eyeglasses and contact lenses should he wear? (Give your answer in D.)

Mirza Aslam Beig

Numerade Educator

01:43

Problem 50

Aside from corrections for aberrations, what are the potential advantages of using eyeglasses with positive or negative meniscus lenses, rather than biconvex/biconcave or planoconvex/planoconcave lenses?

Mayukh Banik

Numerade Educator

02:23

Problem 51

Must contact lenses always be meniscus lenses?

Farhanul Hasan

Numerade Educator

02:07

Problem 52

Prove that you can make positive or negative focal length contact lenses even though $R_{23}$ (in Fig 11.12) is always $>0$.

Mirza Aslam Beig

Numerade Educator

02:58

Problem 53

For each case depicted in Fig. 11.60, describe the status of the patient's vision and how it could be corrected, if necessary.

Khoobchandra Agrawal

Numerade Educator

04:43

Problem 54

Find the prescription in diopters (D) to correct the eyesight for:
(a) A myopic person with a far point of 2 m (using contact lenses).
(b) A hyperopic person with a near point of 1 m who wants to read material 25 cm away and who has very good crystalline lens accommodation (using eyeglasses).
(c) A person with perfect vision for far points who, because of poor accommodation (presbyopia), has a near point of 1 m , and who wants to read material 25 cm away (using eyeglasses).
(d) How do the people in parts (b) and (c) differ? (Could they both use their eyeglasses while attending a baseball game and working at a computer terminal?)
(e) What is the accommodation of the person in part (c) (assuming a standard 17 mm long eyeball for the Standard eye model, with air replacing the humors)?
(f) If the near point for the person in part (a) is 15 cm without contact lenses, what is it when the prescribed contact lenses are worn?

Netra Sharma

University of Wisconsin - Milwaukee

02:44

Problem 55

An ophthalmologist estimates the prescription for a patient based on the smallest lines she can read in the Snellen chart: for 20/20: 0D to -0.25 D , for $20 / 30:-0.50 \mathrm{D}$, for $20 / 40:-0.75 \mathrm{D}$, for $20 / 50:-1.00 \mathrm{D}$ to -1.25 D , for $20 / 100:-1.75 \mathrm{D}$ to -2.00 D , and for $20 / 200:-2.00 \mathrm{D}$ to -2.50 D . Find the far point for patients who are able to read each given Snellen chart line (and no better without eyeglasses), using average prescriptions when ranges are given. (Note that the optical infinity we are using in our analysis is not the 20 ft value assumed in the Snellen chart. See Problem 11.46.)

Dilip Paruchuri

Numerade Educator

01:02

Problem 56

A myopic person has an eyeglass prescription for -3.00 D . What is the appropriate prescription for contact lenses?

Narayan Hari

Numerade Educator

05:49

Problem 57

Explain the optical corrections in the following prescription for eyeglasses.

$$
\begin{array}{ll}
\text { O.D. } & -4.00+1.50 \times 90+2.3 \\
\text { O.S. } & -3.00 \\
+2.3
\end{array}
$$

Nathan Silvano

Numerade Educator