Optics

A camera lens has a focal length of 180.0 mm and an aperture diameter of 16.36 mm. (a) What is the ƒ-number of the lens? (b) If the correct exposure of a certain scene is 1/30 s at f/11, what is the correct exposure at f/2.8?

Curvature of the Cornea: In a simplified model of the human eye, the aqueous and vitreous humors and the lens all have a refractive index of 1.40 , and all the bending occurs at the cornea, whose vertex is 2.60 cm from the retina. What should be the radius of curvature of the cornea such that the image of an object 40 cm from the cornea’s vertex is focused on the retina?

(a) Where is the near point of an eye for which a contact lens with a power of +2.75 diopters is prescribed? (b) Where is the far point of an eye for which a contact lens with a power of -1.30 diopters is prescribed for distant vision?

Contact Lenses. Contact lenses are placed right on the eyeball, so the distance from the eye to an object (or image) is the same as the distance from the lens to that object (or image). A certain person can see distant objects well, but his near point is 45.0 cm from his eyes instead of the usual 25.0 cm . (a) Is this person nearsighted or farsighted? (b) What type of lens (converging or diverging) is needed to correct his vision? (c) If the correcting lenses will be contact lenses, what focal length lens is needed and what is its power in diopters?

Ordinary Glasses: Ordinary glasses are worn in front of the eye and usually 2.0 cm in front of the eyeball. Suppose that the person in previous exercise prefers ordinary glasses to contact lenses. What focal length lenses are needed to correct his vision, and what is their power in diopters?

A person can see clearly up close but cannot focus on objects beyond 75.0 cm. She opts for contact lenses to correct her vision. (a) Is she nearsighted or farsighted? (b) What type of lens (converging or diverging) is needed to correct her vision? (c) What focal length contact lens is needed, and what is its power in diopters?

If the person in Exercise 34.55 chooses ordinary glasses over contact glasses, what power lens (in dioptres) does she need to correct her vision if the lenses are 2.0 cm in front of the eye?

A thin lens with a focal length of 6.00 cm is used as a simple magnifier. (a) What angular magnification is obtainable with the lens if the object is at the focal point? (b) When an object is examined through the lens, how close can it be brought to the lens? Assume that the image viewed by the eye is at the near point, 25.0 cm from the eye, and that the lens is very close to the eye.

The focal length of a simple magnifier is 8.0 cm. Assume the magnifier is a thin lens placed very close to the eye. (a) How far in front of the magnifier should an object be placed if the image is formed at the observers near point, 25.0 cm in front of the eye? (b) If the object is 1.0 mm high, what is the height of its image formed by the magnifier?

You want to view through a magnifier an insect that is 2.00 mm long. If the insect is to be at the focal point of the magnifier, what focal length will give the image of the insect an angular size of 0.032 radian?

The focal length of the eyepiece of a certain microscope is 18.0 mm. The focal length of the objective is 8.00 mm. The distance between objective and eyepiece is 19.7 cm. The final image formed by the eyepiece is at infinity. Treat all lenses as thin. (a) What is the distance from the objective to the object being viewed? (b) What is the magnitude of the linear magnification produced by the objective? (c) What is the overall angular magnification of the microscope?

Resolution of a Microscope. The image formed by a microscope objective with a focal length of 5.00 mm is 160 mm from its second focal point. The eyepiece has a focal length of 26.0 mm. (a) What is the angular magnification of the microscope? (b) The unaided eye can distinguish two points as its near point as separate if they are about 0.10 mm apart. What is the minimum separation between two points that can be observed (or resolved) through this microscope?

A telescope is constructed from two lenses with focal lengths of 95.0 cm and 15.0 cm, the 95.0 cm lens is being used as the objective. Both the object being viewed and the final image are at infinity. (a) Find the angular magnification for the telescope. (b) Find the height of the image formed by the objective of a building 60.0 m tall, 3.00 km away. (c) What is the angular size of the final image as viewed by an eye very close to the eyepiece?

The eyepiece of a refracting telescope (see Fig. 34.53) has a focal length of 9.00 cm. The distance between objective and eyepiece is 1.20 m, and the final image is at infinity. What is the angular magnification of the telescope?

Question: A reflecting telescope (Fig. E34.63) is to be made by using a spherical mirror with a radius of curvature of 1.30 m and an eyepiece with a focal length of 1.10 cm. The final image is at infinity. (a) What should the distance between the eyepiece and the mirror vertex be if the object is taken to be at infinity? (b) What will the angular magnification be?

If you run away from a plane mirror at 3.60m/s, at what speed does your image move away from you?

Where must you place an object in front of a concave mirror with radius R so that the image is erect and  times the size of the object? Where is the image?

Question: A concave mirror is to form an image of the filament of a headlight lamp on a screen 8.00m from the mirror. The filament is 6.00 mm tall, and the image is to be 24.0cm tall. (a) How far in front of the vertex of the mirror should the filament be placed? (b) What should be the radius of curvature of the mirror?

A light bulb is 3.00 m from a wall. You are to use a concave mirror to project an image of the bulb on the wall, with the image 3.50 times the size of the object. How far should the mirror be from the wall? What should its radius of curvature be?

You are in your car driving on a highway at 25m/s when you glance in the passenger-side mirror (a convex mirror with radius of curvature 150 cm) and notice a truck approaching. If the image of the truck is approaching the vertex of the mirror at a speed of 1.9 m/s when the truck is 2.0 m/s from the mirror, what is the speed of the truck relative to the highway?

Question: A layer of benzene (n = 1.50) that is 4.20 cm deep floats on water (n = 1.33) that is 5.70 cm deep. What is the apparent distance from the upper benzene surface to the bottom of the water when you view these layers at normal incidence?

(a) Prove that when two thin lenses with focal lengths ${\mathit{f}}_{\mathbf{1}}$ and ${\mathit{f}}_{\mathbf{2}}$ are placed in contact, the focal length ƒ of the combination is given by the relationship  $\frac{\mathbf{1}}{\mathbf{f}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{f}}_{\mathbf{1}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{f}}_{\mathbf{2}}}$ (b) A converging meniscus lens (see Fig. 34.32a) has an index of refraction of 1.55 and radii of curvature for its surfaces of magnitudes 4.50 cm and 9.00 cm. The concave surface is placed upward and filled with carbon tetrachloride $\left(CC{I}_4\right)$ , which has n = 1.46. What is the focal length of the $\mathit{C}\mathit{C}{\mathit{I}}_{\mathbf{4}}$ -glass combination?

When an object is placed at the proper distance to the left of a converging lens, the image is focused on a screen 30.0 cm to the right of the lens. A diverging lens is now placed 15.0 cm to the right of the converging lens, and it is found that the screen must be moved 19.2 cm farther to the right to obtain a sharp image. What is the focal length of the diverging lens?

(a) Repeat the derivation of Eq. (34.19) for the case in which the lens is totally immersed in a liquid of refractive index ${\mathit{n}}_{\mathbf{l}\mathbf{i}\mathbf{q}}$ . (b) A lens is made of glass that has refractive index 1.60. In air, the lens has focal length +18.0 cm. What is the focal length of this lens if it is totally immersed in a liquid that has refractive index 1.42?

A convex spherical mirror with a focal length of magnitude 24.0 cm is placed 20.0 cm to the left of a plane mirror. An object 0.250 cm tall is placed midway between the surface of the plane mirror and the vertex of the spherical mirror. The spherical mirror forms multiple images of the object. Where are the two images of the object formed by the spherical mirror that are closest to the spherical mirror, and how tall is each image?

A camera with a 90-mm-focal-length lens is focused on an object 1.30 m from the lens. To refocus on an object 6.50 m from the lens, by how much must the distance between the lens and the sensor be changed? To refocus on the more distant object, is the lens moved toward or away from the sensor?

In one form of cataract surgery the person’s natural lens, which has become cloudy, is replaced by an artificial lens. The refracting properties of the replacement lens can be chosen so that the person’s eye focuses on distant objects. But there is no accommodation, and glasses or contact lenses are needed for close vision. What is the power, in diopters, of the corrective contact lenses that will enable a person who has had such surgery to focus on the page of a book at a distance of 24 cm?

A certain very nearsighted person cannot focus on anything farther than 36.0 cm from the eye. Consider the simplified model of the eye described in Exercise 34.50. If the radius of curvature of the cornea is 0.75 cm when the eye is focusing on an object 36.0 cm from the cornea vertex and the indexes of refraction are as described in Exercise 34.50, what is the distance from the cornea vertex to the retina? What does this tell you about the shape of the nearsighted eye?

A person with a near point of 85 cm, but excellent distant vision, normally wears corrective glasses. But he loses them while traveling. Fortunately, he has his old pair as a spare. (a) If the lenses of the old pair have a power of +2.25 diopters, what is his near point (measured from his eye) when he is wearing the old glasses if they rest 2.0 cm in front of his eye? (b) What would his near point be if his old glasses were contact lenses instead?

The Galilean Telescope. Figure P34.100 is a diagram of a Galilean telescope, or opera glass, with both the object and its final image at infinity. The image serves as a virtual object for the eyepiece. The final image is virtual and erect. (a) Prove that the angular magnification is $\mathbf{M}\mathbf{=}\mathbf{-}{\mathbf{f}}_{\mathbf{1}}\mathbf{/}{\mathbf{f}}_{\mathbf{2}}$. (b) A Galilean telescope is to be constructed with the same objective lens as in Exercise 34.61. What focal length should the eyepiece have if this telescope is to have the same magnitude of angular magnification as the one in Exercise 34.61? (c) Compare the lengths of the telescopes. Figure P34.100  

Focal Length of a Zoom Lens. Figure P34.101 shows a simple version of a zoom lens. The converging lens has focal length ${\mathbf{f}}_{\mathbf{2}}\mathbf{=}\mathbf{-}\left|f_2\right|$ and the diverging lens has focal length . The two lenses are separated by a variable distance  that is always less than  Also, the magnitude of the focal length of the diverging lens satisfies the inequality $\mathbf{\left|}{\mathbf{f}}_{\mathbf{2}}\mathbf{\right|}\mathbf{>}\left(f_1-d\right)$. To determine the effective focal length of the combination lens, consider a bundle of parallel rays of radius ${\mathit{r}}_{\mathbf{0}}$ entering the converging lens. (a) Show that the radius of the ray bundle decreases ${\mathbf{r}}_{\mathbf{0}}^{\mathbf{\prime }}\mathbf{=}{\mathbf{r}}_{\mathbf{0}}\left(f_1-d\right)\mathbf{/}{\mathbf{f}}_{\mathbf{1}}$ at the point that it enters the diverging lens. (b) Show that the final image  is formed a distance ${\mathbf{s}}_{\mathbf{2}}^{\mathbf{\prime }}\mathbf{=}\left|f_2\right|\left(f_1-d\right)\mathbf{/}\left(\left|f_2\right|-f_1+d\right)$ to the right of the diverging lens. (c) If the rays that emerge from the diverging lens and reach the final image point are extended  backward to the left of the diverging lens, they will eventually expand to the original radius  at some point . The distance from the final image I′ to the point  is the effective focal length  of the lens combination; if the combination were replaced by a single lens of focal length f placed at , parallel rays would still be brought to a focus at . Show that the effective focal length is given by $\mathit{f}={\mathit{f}}_1\left|{\mathit{f}}_2\right|I\left(\left|{\mathit{f}}_2\right|-{\mathbf{f}}_1+\mathbf{d}\right)$ . (d) If  and the separation  is adjustable between  and  find the maximum and minimum focal lengths of the combination. What value d of  gives f = 30.0 cm ?

In setting up an experiment for a high school biology lab, you use a concave spherical mirror to produce real images of a 4.00 - mm -tall firefly. The  firefly is to the right of the mirror, on the mirror’s optic axis, and serves as a real object for the mirror. You want to determine how far the object must be from the mirror’s vertex (that is, object distance ) to produce an image of a specified height. First, you place a square of white cardboard to the right of the object and find what its distance from the vertex needs to be so that the image is sharply focused on it. Next you measure the height of the sharply focused images for five values of . For each value, you then calculate the lateral magnification . You find that if you graph your data with on the vertical axis and on the horizontal axis, then your measured points fall close to a straight line. (a) Explain why the data plotted this way should fall close to a straight line. (b) Use the graph in Fig. P34.102 to calculate the focal length of the mirror. (c) How far from the mirror’s vertex should you place the object in order for the image to be real 8.00 mm, tall, and inverted? (d) According to Fig. P34.102, starting from the position that you calculated in part (c), should you move the object closer to the mirror or farther from it to increase the height of the inverted, real image? What distance should you move the object in order to increase the image height from to 12.00 mm? (e) Explain why approaches zero as approaches . Can you produce a sharp image on the cardboard when s = 25 cm ? (f) Explain why you can’t see sharp images on the cardboard when s < 25 cm (and is positive).

DATA It is your first day at work as a summer intern at an optics company. Your supervisor hands you a diverging lens and asks you to measure its focal length. You know that with a converging lens, you can measure the focal length by placing an object a distance to the left of the lens, far enough from the lens for the image to be real, and viewing the image on a screen that is to the right of the lens. By adjusting the position of the screen until the image is in sharp focus, you can determine the image distance and then use Eq. (34.16) to calculate the focal length f of the lens. But this procedure won’t work with a diverging lens—by itself, a diverging lens produces only virtual images, which can’t be projected onto a screen. Therefore, to determine the focal length of a diverging lens, you do the following: First you take a converging lens and measure that, for an object 20.0 cm to the left of the lens, the image is 29.7 cm to the right of the lens. You then place a diverging lens to the right of the converging lens and measure the final image to be 42.8 cm to the right of the converging lens. Suspecting some inaccuracy in measurement, you repeat the lens-combination measurement with the same object distance for the converging lens but with the diverging lens 20.0 cm to the right of the converging lens. You measure the final image to be 31.6 cm to the right of the converging lens. (a) Use both lens-combination measurements to calculate the focal length of the diverging lens. Take as your best experimental value for the focal length the average of the two values. (b) Which position of the diverging lens, 20.0 cm to the right or 25.0 cm to the right of the converging lens, gives the tallest image?  Answer

The science museum where you work is constructing a new display. You are given a glass rod that is surrounded by air and was ground on its left-hand end to form a hemispherical surface there. You must determine the radius of curvature of that surface and the index of refraction of the glass. Remembering the optics portion of your physics course, you place a small object to the left of the rod, on the rod’s optic axis, at a distance s from the vertex of the hemispherical surface. You measure the distance   of the image from the vertex of the surface, with the image being to the right of the vertex. Your measurements are as follows:

Recalling that the object–image relationships for thin lenses and spherical mirrors include reciprocals of distances, you plot your data as 1/s' versus 1/s . (a) Explain why your data points plotted this way lie close to a straight line. (b) Use the slope and y - intercept of the best-fit straight line to your data to calculate the index of refraction of the glass and the radius of curvature of the hemispherical surface of the rod. (c) Where is the image if the object distance is  ? 

CALC (a) For a lens with focal length $\mathit{f}$, find the smallest distance possible between the object and its real image. (b) Graph the distance between the object and the real image as a function of the distance of the object from the lens. Does your graph agree with the result you found in part (a)?

An Object at an Angle. A 16.0cm  long pencil is placed at a 45.0° angle, with its center 15.0cm above the optic axis and 45.0 cm from a lens with a 20.0cm  focal length as shown in Fig. P34.106. (Note that the figure is not drawn to scale.) Assume that the diameter of the lens is large enough for the paraxial approximation to be valid. (a) Where is the image of the pencil? (Give the location of the images of the points and on the object, which are located at the eraser, point, and center of the pencil, respectively.) (b) What is the length of the image (that is, the distance between the images of points and )? (c) Show the orientation of the image in a sketch.

People with normal vision cannot focus their eyes underwater if they aren’t wearing a face mask or goggles and there is water in contact with their eyes (see Discussion Question Q34.23). (a) Why not? (b) With the simplified model of the eye described in Exercise 34.50, what corrective lens (specified by focal length as measured in air) would be needed to enable a person underwater to focus an infinitely distant object? (Be careful—the focal length of a lens underwater is not the same as in air! See Problem 34.92. Assume that the corrective lens has a refractive index of and that the lens is used in eyeglasses, not goggles, so there is water on both sides of the lens. Assume that the eyeglasses are 2.00cm in front of the eye.)

The eyes of amphibians such as frogs have a much flatter cornea but a more strongly curved (almost spherical) lens than do the eyes of air-dwelling mammals. In mammalian eyes, the shape (and therefore the focal length) of the lens changes to enable the eye to focus at different distances. In amphibian eyes, the shape of the lens doesn’t change. Amphibians focus on objects at different distances by using specialized muscles to move the lens closer to or farther from the retina, like the focusing mechanism of a camera. In air, most frogs are near-sighted; correcting the distance vision of a typical frog in air would require contact lenses with a power of about -6.0 D .A frog can see an insect clearly at a distance of 10cm. At that point the effective distance from the lens to the retina is 8 mm. If the insect moves farther from the frog, by how much and in which direction does the lens of the frog’s eye have to move to keep the insect in focus? (a) 0.02 cm toward the retina; (b)0.02 cm, away from the retina; (c)0.06 cm, toward the retina; (d)0.06 cm, away from the retina.

The eyes of amphibians such as frogs have a much flatter cornea but a more strongly curved (almost spherical) lens than do the eyes of air-dwelling mammals. In mammalian eyes, the shape (and therefore the focal length) of the lens changes to enable the eye to focus at different distances. In amphibian eyes, the shape of the lens doesn’t change. Amphibians focus on objects at different distances by using specialized muscles to move the lens closer to or farther from the retina, like the focusing mechanism of a camera. In air, most frogs are near-sighted; correcting the distance vision of a typical frog in air would require contact lenses with a power of about -6.0 D .What is the farthest distance at which a typical “near-sighted” frog can see clearly in air? (a) 12 m; (b) 6.0 m; (c) 80 cm (d) 17 cm.

The eyes of amphibians such as frogs have a much flatter cornea but a more strongly curved (almost spherical) lens than do the eyes of air-dwelling mammals. In mammalian eyes, the shape (and therefore the focal length) of the lens changes to enable the eye to focus at different distances. In amphibian eyes, the shape of the lens doesn’t change. Amphibians focus on objects at different distances by using specialized muscles to move the lens closer to or farther from the retina, like the focusing mechanism of a camera. In air, most frogs are near-sighted; correcting the distance vision of a typical frog in air would require contact lenses with a power of about  -6.0 D .Given that frogs are nearsighted in air, which statement is most likely to be true about their vision in water? (a) They are even more nearsighted; because water has a higher index of refraction than air, a frog’s ability to focus light increases in water. (b) They are less nearsighted, because the cornea is less effective at refracting light in water than in air. (c) Their vision is no different, because only structures that are internal to the eye can affect the eye’s ability to focus. (d) The images projected on the retina are no longer inverted, because the eye in water functions as a diverging lens rather than a converging lens.  

The eyes of amphibians such as frogs have a much flatter cornea but a more strongly curved (almost spherical) lens than do the eyes of air-dwelling mammals. In mammalian eyes, the shape (and therefore the focal length) of the lens changes to enable the eye to focus at different distances. In amphibian eyes, the shape of the lens doesn’t change. Amphibians focus on objects at different distances by using specialized muscles to move the lens closer to or farther from the retina, like the focusing mechanism of a camera. In air, most frogs are near-sighted; correcting the distance vision of a typical frog in air would require contact lenses with a power of about -6.0 D .To determine whether a frog can judge distance by means of the amount its lens must move to focus on an object, researchers covered one eye with an opaque material. An insect was placed in front of the frog, and the distance that the frog snapped its tongue out to catch the insect was measured with  high-speed video. The experiment was repeated with a contact lens over the eye to determine whether the frog could correctly judge the distance under these conditions. If such an experiment is performed twice, once with a lens of power -9 D and once with a lens of power -15 D, in which case does the frog have to focus at a shorter distance, and why? (a) With the -9-D lens; because the lenses are diverging, the lens with the longer focal length creates an image that is closer to the frog. (b) With the -15-D lens; because the lenses are diverging, the lens with the shorter focal length creates an image that is closer to the frog. (c) With the -9-D lens; because the lenses are converging, the lens with the longer focal length creates a larger real image. (d) With the -15-D lens; because the lenses are converging, the lens with the shorter focal length creates a larger real image.

A two-slit interference experiment is set up, and the fringes are displayed on a screen. Then the whole apparatus is immersed in the nearest swimming pool. How does the fringe pattern change? 

 Could an experiment similar to Young’s two-slit experiment be performed with sound? How might this be carried out? Does it matter that sound waves are longitudinal and electromagnetic waves are transverse? Explain.

Monochromatic coherent light passing through two thin slits is viewed on a distant screen. Are the bright fringes equally spaced on the screen? If so, why? If not, which ones are closest to being equally spaced?

In a two-slit interference pattern on a distant screen, are the bright fringes midway between the dark fringes? Is this ever a good approximation?

Two small stereo speakers A and B that are 1.40 m apart are sending out sound of wavelength 34 cm in all directions and all in phase. A person at point P starts out equidistant from both speakers and walks so that he is always 1.50 m from speaker B (Fig. E35.1). For what values of x will the sound this person hears be (a) maximally reinforced, (b) cancelled? Limit your solution to the cases where x $\le$ 1.50 m.

Two speakers that are 15.0 m apart produce in-phase sound waves of frequency 250.0 Hz in a room where the speed of sound is 340.0 m/s. A woman starts out at the midpoint between the two speakers. The room’s walls and ceiling are covered with absorbers to eliminate reflections, and she listens with only one ear for best precision. (a) What does she hear: constructive or

destructive interference? Why? (b) She now walks slowly toward one of the speakers. How far from the centre must she walk before she first hears the sound reach a minimum intensity? (c) How far from the centre must she walk before she first hears the sound maximally enhanced?

A radio transmitting station operating at a frequency of 120 MHz has two identical antennas that radiate in phase. Antenna B is 9.00 m to the right of antenna A. Consider point P between the antennas and along the line connecting them, a horizontal distance x to the right of antenna A. For what values of x will constructive interference occur at point P?

 Two radio antennas A and B radiate in phase. Antenna B is 120 m to the right of antenna A. Consider point Q along the extension of the line connecting the antennas, a horizontal distance of 40 m to the right of antenna B. The frequency, and hence the wavelength, of the emitted waves can be varied. (a) What is the longest wavelength for which there will be destructive interference at point Q? (b) What is the longest wavelength for which there will be constructive interference at point Q?

Would the headlights of a distant car form a two-source interference pattern? If so, how might it be observed? If not, why not?