College Physics

Raymond A. Serway, Jerry S. Faughn, Chris Vuille

Chapter 25

Optical Instruments - all with Video Answers

Educators

Chapter Questions

Problem 1

A lens has a focal length of $28 \mathrm{~cm}$ and a diameter of $4.0 \mathrm{~cm}$. What is the $f$ -number of the lens?

Netra Sharma

Netra Sharma

University of Wisconsin - Milwaukee

Problem 2

A certain camera has f-numbers that range from $1.2$ to 22. If the focal length of the lens is $55 \mathrm{~mm}$, what is the range of aperture diameters for the camera?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 3

A photographic image of a building is $0.0920 \mathrm{~m}$ high. The image was made with a lens with a focal length of $52.0 \mathrm{~mm}$. If the lens was $100 \mathrm{~m}$ from the building when the photograph was made, determine the height of the building.

Eduard Sanchez

Numerade Educator

Problem 4

The image area of a typical $35 \mathrm{~mm}$ slide is $23.5 \mathrm{~mm}$ by $35.0 \mathrm{~mm}$. If a camera's lens has a focal length of $55.0 \mathrm{~mm}$ and forms an image of the constellation Orion, which is $20^{\circ}$ across, will the full image fit on a 35 -mm slide?

Eduard Sanchez

Numerade Educator

Problem 5

A camera is being used with the correct exposure at an f-number of $4.0$ and a shutter speed of $1 / 32 \mathrm{~s}$. To "stop" a fast-moving subject, the shutter speed is changed to $1 / 256 \mathrm{~s}$. Find the new $f$ -number that should be used to maintain satisfactory exposure, assuming no change in lighting conditions.

Sachin Rao

Numerade Educator

Problem 6

(a) Use conceptual arguments to show that the intensity of light (energy per unit area per unit time) reaching the film in a camera is proportional to the square of the reciprocal of the $f$ -number as
$$I \propto \frac{1}{(f / D)^{2}}$$
(b) The correct exposure time for a camera set to $f / 1.8$ is $(1 / 500)$ s. Calculate the correct exposure time if the $f$ -number is changed to $f / 4$ under the same lighting conditions. Note: "f/4," on a camera, means "an $f$ -number of $4 . "$

Ajay Singhal

Numerade Educator

Problem 7

A certain type of film requires an exposure time of $0.010 \mathrm{~s}$ with an $f / 11$ lens setting. Another type of film requires twice the light energy to produce the same level of exposure. What $f$ -number does the second type of film need with the $0.010$ -s exposure time?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 8

A certain camera lens has a focal length of $175 \mathrm{~mm}$. Its position can be adjusted to produce images when the lens is between $180 \mathrm{~mm}$ and $210 \mathrm{~mm}$ from the plane of the film. Over what range of object distances is the lens useful?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 9

The near point of a person's eye is $60.0 \mathrm{~cm}$. To see objects clearly at a distance of $25.0 \mathrm{~cm}$, what should be the focal length and power of the appropriate corrective lens? (Neglect the distance from the lens to the eye.)

Eduard Sanchez

Numerade Educator

Problem 10

A patient can't see objects closer than $40.0 \mathrm{~cm}$ and wishes to clearly see objects that are $20.0 \mathrm{~cm}$ from his eye. (a) Is the patient nearsighted or farsighted? (b) If the eye-lens distance is $2.00 \mathrm{~cm}$, what is the minimum object distance $p$ from the lens? (c) What image position with respect to the lens will allow the patient to see the object?
(d) Is the image real or virtual? Is the image distance $q$ positive or negative? (e) Calculate the required focal length. (f) Find the power of the lens in diopters. (g) If a contact lens is to be prescribed instead, find $p, q$, and $f$, and the power of the lens.

Eduard Sanchez

Numerade Educator

Problem 11

The accommodation limits for Nearsighted Nick's eyes are $18.0 \mathrm{~cm}$ and $80.0 \mathrm{~cm}$. When he wears his glasses, he is able to see faraway objects clearly. At what minimum distance is he able to see objects clearly?

Mayukh Banik

Numerade Educator

Problem 12

A certain child's near point is $10.0 \mathrm{~cm}$; her far point (with eyes relaxed) is $125 \mathrm{~cm}$. Each eye lens is $2.00 \mathrm{~cm}$ from the retina. (a) Between what limits, measured in diopters, does the power of this lens-cornea combination vary? (b) Calculate the power of the eyeglass lens the child should use for relaxed distance vision. Is the lens converging or diverging?

Eduard Sanchez

Numerade Educator

Problem 13

An individual is nearsighted; his near point is $13.0 \mathrm{~cm}$ and his far point is $50.0 \mathrm{~cm}$. (a) What lens power is needed to correct his nearsightedness? (b) When the lenses are in use, what is this person's near point?

Eduard Sanchez

Numerade Educator

Problem 14

A patient has a near point of $45.0 \mathrm{~cm}$ and far point of $85.0 \mathrm{~cm} .$ (a) Can a single lens correct the patient's vision? Explain the patient's options. (b) Calculate the power lens needed to correct the near point so that the patient can see objects $25.0 \mathrm{~cm}$ away. Neglect the eye-lens distance. (c) Calculate the power lens needed to correct the patient's far point, again neglecting the eye-lens distance.

Vishal Gupta

Numerade Educator

Problem 15

An artificial lens is implanted in a person's eye to replace a diseased lens. The distance between the artificial lens and the retina is $2.80 \mathrm{~cm} .$ In the absence of the lens, an image of a distant object (formed by refraction at the cornea) falls $2.53 \mathrm{~cm}$ behind the retina. The lens is designed to put the image of the distant object on the retina. What is the power of the implanted lens? Hint: Consider the image formed by the cornea to be a virtual object.

Eduard Sanchez

Numerade Educator

Problem 16

A person is to be fitted with bifocals. She can see clearly when the object is between $30 \mathrm{~cm}$ and $1.5 \mathrm{~m}$ from the eye. (a) The upper portions of the bifocals (Fig. P25.16) should be designed to enable her to see distant objects clearly. What power should they have? (b) The lower portions of the bifocals should enable her to see objects comfortably at $25 \mathrm{~cm}$. What power should they have?

Eduard Sanchez

Numerade Educator

Problem 17

A nearsighted woman can't see objects clearly beyond $40.0 \mathrm{~cm}$ (her far point). If she has no astigmatism and contact lenses are prescribed, what power and type of lens are required to correct her vision?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 18

A person sees clearly wearing eyeglasses that have a power of $-4.00$ diopters when the lenses are $2.00 \mathrm{~cm}$ in front of the eyes. (a) What is the focal length of the lens? (b) Is the person nearsighted or farsighted? (c) If the person wants to switch to contact lenses placed directly on the eyes, what lens power should be prescribed?

Keshav Singh

Numerade Educator

Problem 19

A stamp collector uses a lens with $7.5-\mathrm{cm}$ focal length as a simple magnifier. The virtual image is produced at the normal near point $(25 \mathrm{~cm}) .$ (a) How far from the lens should the stamp be placed? (b) What is the expected angular magnification?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 20

A magnifier has a maximum angular magnification of $+6.0 .$ Find the (a) focal length of the lens and (b) angular magnification when the eye is relaxed.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 21

A biology student uses a simple magnifier to examine the structural features of an insect's wing. The wing is held $3.50 \mathrm{~cm}$ in front of the lens, and the image is formed $25.0 \mathrm{~cm}$ from the eye. (a) What is the focal length of the lens? (b) What angular magnification is achieved?

Hubert Agamasu

Numerade Educator

Problem 22

A jeweler's lens of focal length $5.0 \mathrm{~cm}$ is used as a magnifier. With the lens held near the eye, determine (a) the angular magnification when the object is at the focal point of the lens and (b) the angular magnification when the image formed by the lens is at the near point of the eye $(25 \mathrm{~cm}) .$ (c) What is the object distance giving the maximum magnification?

Eduard Sanchez

Numerade Educator

Problem 23

A leaf of length $h$ is positioned $71.0 \mathrm{~cm}$ in front of a converging lens with a focal length of $39.0 \mathrm{~cm}$. An observer views the image of the leaf from a position $1.26 \mathrm{~m}$ behind the lens, as shown in Figure P25.23. (a) What is the magnitude of the lateral magnification (the ratio of the image size to the object size) produced by the lens? (b) What angular magnification is achieved by viewing the image of the leaf rather than viewing the leaf directly?

Eduard Sanchez

Numerade Educator

Problem 24

A lens having a focal length of $25 \mathrm{~cm}$ is used as a simple magnifier. (a) What is the angular magnification obtained when the image is formed at the normal near point $(q=-25 \mathrm{~cm}) ?$ (b) What is the angular magnification produced by this lens when the eye is relaxed?

Mayukh Banik

Numerade Educator

Problem 25

The desired overall magnification of a compound microscope is $140 \times$. The objective alone produces a lateral magnification of $12 \times$. Determine the required focal length of the eyepiece.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 26

The distance between the eyepiece and the objective lens in a certain compound microscope is $20.0 \mathrm{~cm} .$ The focal length of the objective is $0.500 \mathrm{~cm}$, and that of the eyepiece is $1.70 \mathrm{~cm}$. Find the overall magnification of the microscope.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 27

Your instructor asks you to design and construct a refracting telescope having a magnification of 45 using an available eyepiece of focal length $4.0 \mathrm{~cm}$. How long should you make the telescope tube?

Narayan Hari

Numerade Educator

Problem 28

A microscope has an objective lens with a focal length of $16.22 \mathrm{~mm}$ and an eyepiece with a focal length of $9.50 \mathrm{~mm}$. With the length of the barrel set at $29.0 \mathrm{~cm}$, the diameter of a red blood cell's image subtends an angle of $1.43$ mrad with the eye. If the final image distance is $29.0 \mathrm{~cm}$ from the eyepiece, what is the actual diameter of the red blood cell?

Eduard Sanchez

Numerade Educator

Problem 29

The length of a microscope tube is $15.0 \mathrm{~cm} .$ The focal length of the objective is $1.00 \mathrm{~cm}$, and the focal length of the eyepiece is $2.50 \mathrm{~cm}$. What is the magnification of the microscope, assuming it is adjusted so that the eye is relaxed? Hint: To solve this question, go back to basics and use the thin-lens equation.

Surjit Tewari

Numerade Educator

Problem 30

Find an equation for the length $L$ of a refracting telescope in terms of the focal length of the objective $f_{0}$ and the magnification $m$. (b) A knob adjusts the eyepiece forward and backward. Suppose the telescope is in focus with an eyepiece giving a magnification of $50.0 .$ By what distance must the eyepiece be adjusted when the eyepiece is replaced, with a resulting magnification of $1.00 \times 10^{2}$ ? Must the eyepiece be adjusted backward or forward? Assume the objective lens has a focal length of $2.00 \mathrm{~m}$.

Eduard Sanchez

Numerade Educator

Problem 31

The lenses of an astronomical telescope are $92 \mathrm{~cm}$ apart when adjusted for viewing a distant object with minimum eyestrain. The angular magnification produced by the telescope is $45 .$ Compute the focal length of each lens.

Daniel Alva

Numerade Educator

Problem 32

A certain telescope has an objective of focal length $1500 \mathrm{~cm} .$ If the Moon is used as an object, a $1.0-\mathrm{cm}$ -long image formed by the objective corresponds to what distance, in miles, on the Moon? Assume $3.8 \times 10^{8} \mathrm{~m}$ for the Earth-Moon distance.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 33

Astronomers often take photographs with the objective lens or mirror of a telescope alone, without an eyepiece. (a) Show that the image size $h^{\prime}$ for a telescope used in this manner is given by $h^{\prime}=f h /(f-p)$, where $h$ is the object size, $f$ is the objective focal length, and $p$ is the object distance. (b) Simplify the expression in part (a) if the object distance is much greater than the objec?ive focal length. (c) The "wingspan" of the International Space Station is $108.6 \mathrm{~m}$, the overall width of its solar panel configuration. When it is orbiting at an altitude of $407 \mathrm{~km}$, find the width of the image formed by a telescope objective of focal length $4.00 \mathrm{~m}$.

Eduard Sanchez

Numerade Educator

Problem 34

An elderly sailor is shipwrecked on a desert island, but manages to save his eyeglasses. The lens for one eye has a power of $+1.20$ diopters, and the other lens has a power of $+9.00$ diopters. (a) What is the magnifying power of the telescope he can construct with these lenses? (b) How far apart are the lenses when the telescope is adjusted for minimum eyestrain?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 35

A person decides to use an old pair of eyeglasses to make some optical instruments. He knows that the near point in his left eye is $50.0 \mathrm{~cm}$ and the near point in his right eye is $100 \mathrm{~cm} .$ (a) What is the maximum angular magnification he can produce in a telescope? (b) If he places the lenses $10.0 \mathrm{~cm}$ apart, what is the maximum overall magnification he can produce in a microscope? Hint: Go back to basics and use the thin-lens equation to solve part (b).

James Kiss

Numerade Educator

Problem 36

Galileo devised a simple terrestrial telescope that produces an upright image. It consists of a converging objective lens and a diverging eyepiece at opposite ends of the telescope tube. For distant objects, the tube length is the objective focal length less the absolute value of the eyepiece focal length. (a) Does the user of the telescope see a real or virtual image? (b) Where is the final image? (c) If a telescope is to be constructed with a tube of length $10.0 \mathrm{~cm}$ and a magnification of $3.00$, what are the focal lengths of the objective and eyepiece?

Eduard Sanchez

Numerade Educator

Problem 37

A converging lens with a diameter of $30.0 \mathrm{~cm}$ forms an image of a satellite passing overhead. The satellite has two green lights (wavelength $500 \mathrm{~nm}$ ) spaced $1.00 \mathrm{~m}$ apart. If the lights can just be resolved according to the Rayleigh criterion, what is the altitude of the satellite?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 38

The pupil of a cat's eye narrows to a vertical slit of width $0.500 \mathrm{~mm}$ in daylight. What is the angular resolution for a pair of horizontally separated mice? (Use $500-\mathrm{nm}$ light in your calculation.)

Mayukh Banik

Numerade Educator

Problem 39

To increase the resolving power of a microscope, the object and the objective are immersed in oil $(n=1.5)$. If the limiting angle of resolution without the oil is $0.60$ \murad, what is the limiting angle of resolution with the oil? Hint: The oil changes the wavelength of the light.

Eduard Sanchez

Numerade Educator

Problem 40

Calculate the limiting angle of resolution for the eye, assuming a pupil diameter of $2.00 \mathrm{~mm}$, a wavelength of $500 \mathrm{~nm}$ in air, and an index of refraction for the eye of $1.33$. (b) What is the maximum distance from the eye at which two points separated by $1.00 \mathrm{~cm}$ could be resolved?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 41

A vehicle with headlights separated by $2.00 \mathrm{~m}$ approaches an observer holding an infrared detector sensitive to radiation of wavelength $885 \mathrm{~nm}$. What aperture diameter is required in the detector if the two headlights are to be resolved at a distance of $10.0 \mathrm{~km}$ ?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 42

A helium-neon laser emits light that has a wavelength of $632.8 \mathrm{~nm}$. The circular aperture through which the beam emerges has a diameter of $0.200 \mathrm{~cm}$. Estimate the diameter of the beam $3.00 \mathrm{~km}$ from the laser.

Charles Carter

Numerade Educator

Problem 43

Suppose a 5.00-m-diameter telescope were constructed on the Moon, where the absence of atmospheric distortion would permit excellent viewing. If observations were made using $500-\mathrm{nm}$ light, what minimum separation between two objects could just be resolved on Mars at closest approach (when Mars is $8.0 \times 10^{7} \mathrm{~km}$ from the Moon)?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 44

A spy satellite circles Earth at an altitude of $200 \mathrm{~km}$ and carries out surveillance with a special high-resolution telescopic camera having a lens diameter of $35 \mathrm{~cm}$. If the angular resolution of this camera is limited by diffraction, estimate the separation of two small objects on Earth's surface that are just resolved in yellow-green light $(\lambda=550 \mathrm{~nm})$.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 45

A $15.0$ -cm-long grating has 6000 slits per centimeter, Can two lines of wavelengths $600.000 \mathrm{~nm}$ and $600.003 \mathrm{~nm}$ be separated with this grating? Explain.

Eduard Sanchez

Numerade Educator

Problem 46

The $\mathrm{H}_{\alpha}$ line in hydrogen has a wavelength of $656.20 \mathrm{~nm}$. This line differs in wavelength from the corresponding spectral line in deuterium (the heavy stable isotope of hydrogen) by $0.18 \mathrm{~nm}$. (a) Determine the minimum number of lines a grating must have to resolve these two wavelengths in the first order. (b) Repeat part (a) for the second order.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 47

Light of wavelength $550 \mathrm{~nm}$ is used to calibrate a Michelson interferometer. With the use of a micrometer screw, the platform on which one mirror is mounted is moved $0.180 \mathrm{~mm}$. How many fringe shifts are counted?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 48

Monochromatic light is beamed into a Michelson interferometer. The movable mirror is displaced $0.382 \mathrm{~mm}$, causing the central spot in the interferometer pattern to change from bright to dark and back to bright $N=1700$ times. (a) Determine the wavelength of the light. What color is it? (b) If monochromatic red light is used instead and the mirror is moved the same distance, would $N$ be larger or smaller? Explain.

Eduard Sanchez

Numerade Educator

Problem 49

An interferometer is used to measure the length of a bacterium. The wavelength of the light used is $650 \mathrm{~nm}$. As one arm of the interferometer is moved from one end of the cell to the other, 310 fringe shifts are counted. How long is the bacterium?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 50

Mirror $\mathrm{M}_{1}$ in Active Figure $25.15$ is displaced a distance $\Delta L$. During this displacement, 250 fringe shifts are counted. The light being used has a wavelength of $632.8 \mathrm{~nm}$. Calculate the displacement $\Delta L$.

Eduard Sanchez

Numerade Educator

Problem 51

A thin sheet of transparent material has an index of refraction of $1.40$ and is $15.0 \mu \mathrm{m}$ thick. When it is inserted in the light path along one arm of an interferometer, how many fringe shifts occur in the pattern? Assume the wavelength (in a vacuum) of the light used is $600 \mathrm{~nm}$. Hint: The wavelength will change within the material.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 52

The Michelson interferometer can be used to measure the index of refraction of a gas by placing an evacuated transparent tube in the light path along one arm of the device. Fringe shifts occur as the gas is slowly added to the tube. Assume $600-\mathrm{nm}$ light is used, the tube is $5.00 \mathrm{~cm}$ long, and 160 fringe shifts occur as the pressure of the gas in the tube increases to atmospheric pressure. What is the index of refraction of the gas? Hint: The fringe shifts occur because the wavelength of the light changes inside the gas-filled tube.

Eduard Sanchez

Numerade Educator

Problem 53

The Yerkes refracting telescope has a $1.00$ -m-diameter objective lens of focal length $20.0 \mathrm{~m}$. Assume it is used with an eyepiece of focal length $2.50 \mathrm{~cm}$. (a) Determine the magnification of the planet Mars as seen through the telescope. (b) Are the observed Martian polar caps right side up or upside down?

Eduard Sanchez

Numerade Educator

Problem 54

Estimate the minimum angle subtended at the eye of a hawk flying at an altitude of $50 \mathrm{~m}$ necessary to recognize a mouse on the ground.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 55

The wavelengths of the sodium spectrum are $\lambda_{1}=$ $589.00 \mathrm{~nm}$ and $\lambda_{2}=589.59 \mathrm{~nm} .$ Determine the minimum number of lines in a grating that will allow resolution of the sodium spectrum in (a) the first order and (b) the third order.

Keshav Singh

Numerade Educator

Problem 56

A person with a nearsighted eye has near and far points of $16 \mathrm{~cm}$ and $25 \mathrm{~cm}$, respectively. (a) Assuming a lens is placed $2.0 \mathrm{~cm}$ from the eye, what power must the lens have to correct this condition? (b) Suppose contact lenses placed directly on the cornea are used to correct the person's eyesight. What is the power of the lens required in this case, and what is the new near point? Hint: The contact lens and the eyeglass lens require slightly different

Eduard Sanchez

Numerade Educator

Problem 57

The near point of an eye is $75.0 \mathrm{~cm} .$ (a) What should be the power of a corrective lens prescribed to enable the eye to see an object clearly at $25.0 \mathrm{~cm} ?$ (b) If, using the corrective lens, the person can see an object clearly at $26.0 \mathrm{~cm}$ but not at $25.0 \mathrm{~cm}$, by how many diopters did the lens grinder miss the prescription?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 58

If a typical eyeball is $2.00 \mathrm{~cm}$ long and has a pupil opening that can range from about $2.00 \mathrm{~mm}$ to $6.00 \mathrm{~mm}$, what are (a) the focal length of the eye when it is focused on objects $1.00 \mathrm{~m}$ away, (b) the smallest $f$ -number of the eye when it is focused on objects $1.00 \mathrm{~m}$ away, and (c) the largest $f$ -number of the eye when it is focused on objects $1.00 \mathrm{~m}$ away?

Eduard Sanchez

Numerade Educator

Problem 59

A cataract-impaired lens in an eye may be surgically removed and replaced by a manufactured lens. The focal length required for the new lens is determined by the lens-to-retina distance, which is measured by a sonarlike device, and by the requirement that the implant provide for correct distance vision. (a) If the distance from lens to retina is $22.4 \mathrm{~mm}$, calculate the power of the implanted lens in diopters. (b) Since there is no accommodation and the implant allows for correct distance vision, a corrective lens for close work or reading must be used. Assume a reading distance of $33.0 \mathrm{~cm}$, and calculate the power of the lens in the reading glasses.

Eduard Sanchez

Numerade Educator

Problem 60

If the aqueous humor of the eye has an index of refraction of $1.34$ and the distance from the vertex of the cornea to the retina is $2.00 \mathrm{~cm}$, what is the radius of curvature of the cornea for which distant objects will be focused on the retina? (For simplicity, assume all refraction occurs in the aqueous humor.)

Netra Sharma

University of Wisconsin - Milwaukee

Problem 61

A Boy Scout starts a fire by using a lens from his eyeglasses to focus sunlight on kindling $5.0 \mathrm{~cm}$ from the lens. The Boy Scout has a near point of $15 \mathrm{~cm} .$ When the lens is used as a simple magnifier, (a) what is the maximum magnification that can be achieved and (b) what is the magnification when the eye is relaxed? Caution: The equations derived in the text for a simple magnifier assume a "normal" eye.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 62

A laboratory (astronomical) telescope is used to view a scale that is $300 \mathrm{~cm}$ from the objective, which has a focal length of $20.0 \mathrm{~cm}$; the eyepiece has a focal length of $2.00$ $\mathrm{cm}$. Calculate the angular magnification when the telescope is adjusted for minimum eyestrain. Note: The object is not at infinity, so the simple expression $m=f_{o} / f_{e}$ is not sufficiently accurate for this problem. Also, assume small angles, so that $\tan \theta \approx \theta$.

Adnan Gill

Numerade Educator