Shown from above in FIGURE P34.54 is one corner of a

What are light rays?

What is the law of reflection?

What is refraction?

How do lenses form images?

Why is optics important?

A long, thin lightbulb illuminates a vertical aperture. Which pattern of light do you see on a viewing screen behind the aperture? (a) (b) (c) (d)

A dressing mirror on a closet door is 1.50 m tall. The bottom is 0.50 m above the floor. A bare lightbulb hangs 1.00 m from the closet door, 2.50 m above the floor. How long is the streak of reflected light across the floor?

If your height is h, what is the shortest mirror on the wall in which you can see your full image? Where must the top of the mirror be hung?

Two plane mirrors form a right angle. How many images of the ball can you see in the mirrors? a. 1 b. 2 c. 3 d. 4

A laser beam is aimed at a 1.0-cm-thick sheet of glass at an angle 30 above the glass. a. What is the laser beams direction of travel in the glass? b. What is its direction in the air on the other side? c. By what distance is the laser beam displaced?

A light ray travels from medium 1 to medium 3 as shown. For these media, 30 20 10 n1 n2 n3 a. n3 7 n1 b. n3 = n1 c. n3 6 n1 d. We cant compare n1 to n3 without knowing n2

FIGURE 34.18 shows a laser beam deflected by a 30-60-90 prism. What is the prisms index of refraction?

A lens produces a sharply focused, inverted image on a screen. What will you see on the screen if the lens is removed? a. The image will be inverted and blurry. b. The image will be upright and sharp. c. The image will be upright and blurry. d. The image will be much dimmer but otherwise unchanged. e. There will be no image at all

A small lightbulb is set in the bottom of a 3.0-m-deep swimming pool. What is the diameter of the circle of light seen on the waters surface from above?

Which of these actions will move the real image point P farther from the boundary? More than one may work. a. Increase the radius of curvature R. b. Increase the index of refraction n. c. Increase the object distance s. d. Decrease the radius of curvature R. e. Decrease the index of refraction n. f. Decrease the object distance s.

A fish and a sailor look at each other through a 5.0-cm-thick glass porthole in a submarine. There happens to be an air bubble right in the center of the glass. How far behind the surface of the glass does the air bubble appear to the fish? To the sailor?

A lens forms a real image of a lightbulb, but the image of the bulb on a viewing screen is blurry because the screen is slightly in front of the image plane. To focus the image, should you move the lens toward the bulb or away from the bulb?

A 4.0-cm-diameter flower is 200 cm from the 50-cm-focal-length lens of a camera. How far should the light detector be placed behind the lens to record a well-focused image? What is the diameter of the image on the detector?

A concave mirror of focal length f forms an image of the moon. Where is the image located? a. At the mirrors surface b. Almost exactly a distance f behind the mirror c. Almost exactly a distance f in front of the mirror d. At a distance behind the mirror equal to the distance of the moon in front of the mirror

To see a flower better, a naturalist holds a 6.0-cm-focal-length magnifying glass 4.0 cm from the flower. What is the magnification?

A diverging lens with a focal length of 50 cm is placed 100 cm from a flower. Where is the image? What is its magnification?

One end of a 4.0-cm-diameter glass rod is shaped like a hemisphere. A small lightbulb is 6.0 cm from the end of the rod. Where is the bulbs image located?

A goldfish lives in a spherical fish bowl 50 cm in diameter. If the fish is 10 cm from the near edge of the bowl, where does the fish appear when viewed from the outside?

What is the focal length of the glass meniscus lens shown in FIGURE 34.42? Is this a converging or diverging lens?

The objective lens of a microscope uses a planoconvex glass lens with the flat side facing the specimen. A real image is formed 160 mm behind the lens when the lens is 8.0 mm from the specimen. What is the radius of the lenss curved surface?

A stamp collector uses a magnifying lens that sits 2.0 cm above the stamp. The magnification is 4.0. What is the focal length of the lens?

A 3.0-cm-high object is located 60 cm from a concave mirror. The mirrors focal length is 40 cm. Use ray tracing to find the position and height of the image.

A 3.0-cm-high object is located 20 cm from a concave mirror. The mirrors radius of curvature is 80 cm. Determine the position, orientation, and height of the image

An endoscope is a thin bundle of optical fibers that can be inserted through a bodily opening or small incision to view the interior of the body. As FIGURE 34.52 shows, an objective lens forms a real image on the entrance face of the fiber bundle. Individual fibers, using total internal reflection, transport the light to the exit face, where it emerges. The doctor (or a TV camera) observes the object by viewing the exit face through an eyepiece lens.

Suppose you have two pinhole cameras. The first has a small round hole in the front. The second is identical except it has a square hole of the same area as the round hole in the first camera. Would the pictures taken by these two cameras, under the same conditions, be different in any obvious way? Explain.

You are looking at the image of a pencil in a mirror, as shown in FIGURE Q34.2. a. What happens to the image if the top half of the mirror, down to the midpoint, is covered with a piece of cardboard? Explain. b. What happens to the image if the bottom half of the mirror is covered with a piece of cardboard? Explain.

One problem with using optical fibers for communication is that a light ray passing directly down the center of the fiber takes less time to travel from one end to the other than a ray taking a longer, zig-zag path. Thus light rays starting at the same time but traveling in slightly different directions reach the end of the fiber at different times. This problem can be solved by making the refractive index of the glass change gradually from a higher value in the center to a lower value near the edges of the fiber. Explain how this reduces the difference in travel times.

A light beam passing from medium 2 to medium 1 is refracted as shown in FIGURE Q34.4. Is n1 larger than n2, is n1 smaller than n2, or is there not enough information to tell? Explain.

A fish in an aquarium with flat sides looks out at a hungry cat. To the fish, does the distance to the cat appear to be less than the actual distance, the same as the actual distance, or more than the actual distance? Explain

Consider one point on an object near a lens. a. What is the minimum number of rays needed to locate its image point? Explain. b. How many rays from this point actually strike the lens and refract to the image point?

The object and lens in FIGURE Q34.7 are positioned to form a well-focused, inverted image on a viewing screen. Then a piece of cardboard is lowered just in front of the lens to cover the top half of the lens. Describe what you see on the screen when the cardboard is in place.

A converging lens creates the image shown in FIGURE Q34.8. Is the object distance less than the focal length f, between f and 2f, or greater than 2f? Explain.

A concave mirror brings the suns rays to a focus in front of the mirror. Suppose the mirror is submerged in a swimming pool but still pointed up at the sun. Will the suns rays be focused nearer to, farther from, or at the same distance from the mirror? Explain

You see an upright, magnified image of your face when you look into a magnifying cosmetic mirror. Where is the image? Is it in front of the mirrors surface, on the mirrors surface, or behind the mirrors surface? Explain.

When you look at your reflection in the bowl of a spoon, it is upside down. Why?

A point source of light illuminates an aperture 2.0 m away. A 12.0-cm-wide bright patch of light appears on a screen 1.0 m behind the aperture. How wide is the aperture?

a. How long (in ns) does it take light to travel 1.0 m in vacuum? b. What distance does light travel in water, glass, and cubic zirconia during the time that it travels 1.0 m in vacuum?

A student has built a 15-cm-long pinhole camera for a science fair project. She wants to photograph her 180-cm-tall friend and have the image on the film be 5.0 cm high. How far should the front of the camera be from her friend?

A 5.0-cm-thick layer of oil is sandwiched between a 1.0-cmthick sheet of glass and a 2.0-cm-thick sheet of polystyrene plastic. How long (in ns) does it take light incident perpendicular to the glass to pass through this 8.0-cm-thick sandwich?

A light ray leaves point A in FIGURE EX34.5, reflects from the mirror, and reaches point B. How far below the top edge does the ray strike the mirror?

The mirror in FIGURE EX34.6 deflects a horizontal laser beam by 60. What is the angle f?

At what angle f should the laser beam in FIGURE EX34.7 be aimed at the mirrored ceiling in order to hit the midpoint of the far wall?

A laser beam is incident on the left mirror in FIGURE EX34.8. Its initial direction is parallel to a line that bisects the mirrors. What is the angle f of the reflected laser beam?

It is 165 cm from your eyes to your toes. Youre standing 200 cm in front of a tall mirror. How far is it from your eyes to the image of your toes?

A laser beam in air is incident on a liquid at an angle of 53 with respect to the normal. The laser beams angle in the liquid is 35. What is the liquids index of refraction?

A 1.0-cm-thick layer of water stands on a horizontal slab of glass. A light ray in the air is incident on the water 60 from the normal. What is the rays direction of travel in the glass?

A costume jewelry pendant made of cubic zirconia is submerged in oil. A light ray strikes one face of the zirconia crystal at an angle of incidence of 25. Once inside, what is the rays angle with respect to the face of the crystal?

An underwater diver sees the sun 50 above horizontal. How high is the sun above the horizon to a fisherman in a boat above the diver?

The glass core of an optical fiber has an index of refraction 1.60. The index of refraction of the cladding is 1.48. What is the maximum angle a light ray can make with the wall of the core if it is to remain inside the fiber?

A thin glass rod is submerged in oil. What is the critical angle for light traveling inside the rod?

FIGURE EX34.16 shows a transparent hemisphere with radius R and index of refraction n. What is the maximum distance d for which a light ray parallel to the axis refracts out through the curved surface?

A fish in a flat-sided aquarium sees a can of fish food on the counter. To the fishs eye, the can looks to be 30 cm outside the aquarium. What is the actual distance between the can and the aquarium? (You can ignore the thin glass wall of the aquarium.)

A biologist keeps a specimen of his favorite beetle embedded in a cube of polystyrene plastic. The hapless bug appears to be 2.0 cm within the plastic. What is the beetles actual distance beneath the surface?

A 150-cm-tall diver is standing completely submerged on the bottom of a swimming pool full of water. You are sitting on the end of the diving board, almost directly over her. How tall does the diver appear to be?

To a fish in an aquarium, the 4.00-mm-thick walls appear to be only 3.50 mm thick. What is the index of refraction of the walls?

An object is 20 cm in front of a converging lens with a focal length of 10 cm. Use ray tracing to determine the location of the image. Is the image upright or inverted?

An object is 30 cm in front of a converging lens with a focal length of 5 cm. Use ray tracing to determine the location of the image. Is the image upright or inverted?

An object is 6 cm in front of a converging lens with a focal length of 10 cm. Use ray tracing to determine the location of the image. Is the image upright or inverted?

An object is 15 cm in front of a diverging lens with a focal length of -15 cm. Use ray tracing to determine the location of the image. Is the image upright or inverted?

Find the focal length of the glass lens in FIGURE EX34.25.

Find the focal length of the planoconvex polystyrene plastic lens in FIGURE EX34.26.

Find the focal length of the glass lens in FIGURE EX34.27

Find the focal length of the meniscus polystyrene plastic lens in FIGURE EX34

An air bubble inside an 8.0-cm-diameter plastic ball is 2.0 cm from the surface. As you look at the ball with the bubble turned toward you, how far beneath the surface does the bubble appear to be?

A goldfish lives in a 50-cm-diameter spherical fish bowl. The fish sees a cat watching it. If the cats face is 20 cm from the edge of the bowl, how far from the edge does the fish see it as being? (You can ignore the thin glass wall of the bowl.)

A 1.0-cm-tall candle flame is 60 cm from a lens with a focal length of 20 cm. What are the image distance and the height of the flames image?

A 1.0-cm-tall object is 10 cm in front of a converging lens that has a 30 cm focal length. a. Use ray tracing to find the position and height of the image. To do this accurately, use a ruler or paper with a grid. Determine the image distance and image height by making measurements on your diagram. b. Calculate the image position and height. Compare with your ray-tracing answers in part a

A 2.0-cm-tall object is 40 cm in front of a converging lens that has a 20 cm focal length. a. Use ray tracing to find the position and height of the image. To do this accurately, use a ruler or paper with a grid. Determine the image distance and image height by making measurements on your diagram. b. Calculate the image position and height. Compare with your ray-tracing answers in part a

A 1.0-cm-tall object is 75 cm in front of a converging lens that has a 30 cm focal length.a. Use ray tracing to find the position and height of the image. To do this accurately, use a ruler or paper with a grid. Determine the image distance and image height by making measurements on your diagram. b. Calculate the image position and height. Compare with your ray-tracing answers in part a

A 2.0-cm-tall object is 15 cm in front of a converging lens that has a 20 cm focal length. a. Use ray tracing to find the position and height of the image. To do this accurately, use a ruler or paper with a grid. Determine the image distance and image height by making measurements on your diagram. b. Calculate the image position and height. Compare with your ray-tracing answers in part a

A 1.0-cm-tall object is 60 cm in front of a diverging lens that has a -30 cm focal length. a. Use ray tracing to find the position and height of the image. To do this accurately, use a ruler or paper with a grid. Determine the image distance and image height by making measurements on your diagram. b. Calculate the image position and height. Compare with your ray-tracing answers in part a.

A 2.0-cm-tall object is 15 cm in front of a diverging lens that has a -20 cm focal length. a. Use ray tracing to find the position and height of the image. To do this accurately, use a ruler or paper with a grid. Determine the image distance and image height by making measurements on your diagram. b. Calculate the image position and height. Compare with your ray-tracing answers in part a.

An object is 40 cm in front of a concave mirror with a focal length of 20 cm. Use ray tracing to locate the image. Is the image upright or inverted?

An object is 12 cm in front of a concave mirror with a focal length of 20 cm. Use ray tracing to locate the image. Is the image upright or inverted?

An object is 30 cm in front of a convex mirror with a focal length of -20 cm. Use ray tracing to locate the image. Is the image upright or inverted?

A 1.0-cm-tall object is 20 cm in front of a concave mirror that has a 60 cm focal length. Calculate the position and height of the image. State whether the image is in front of or behind the mirror, and whether the image is upright or inverted.

A 1.0-cm-tall object is 20 cm in front of a convex mirror that has a -60 cm focal length. Calculate the position and height of the image. State whether the image is in front of or behind the mirror, and whether the image is upright or inverted.

An advanced computer sends information to its various parts via infrared light pulses traveling through silicon fibers. To acquire data from memory, the central processing unit sends a light-pulse request to the memory unit. The memory unit processes the request, then sends a data pulse back to the central processing unit. The memory unit takes 0.5 ns to process a request. If the information has to be obtained from memory in 2.0 ns, what is the maximum distance the memory unit can be from the central processing unit?

A red ball is placed at point A in FIGURE P34.44. a. How many images are seen by an observer at point O? b. What are the 1x, y2 coordinates of each image?

The place you get your hair cut has two nearly parallel mirrors 5.0 m apart. As you sit in the chair, your head is 2.0 m from the nearer mirror. Looking toward this mirror, you first see your face and then, farther away, the back of your head. (The mirrors need to be slightly nonparallel for you to be able to see the back of your head, but you can treat them as parallel in this problem.) How far away does the back of your head appear to be? Neglect the thickness of your head.

A microscope is focused on a black dot. When a 1.00-cm-thick piece of plastic is placed over the dot, the microscope objective has to be raised 0.40 cm to bring the dot back into focus. What is the index of refraction of the plastic?

A light ray in air is incident on a transparent material whose index of refraction is n. a. Find an expression for the (non-zero) angle of incidence whose angle of refraction is half the angle of incidence. b. Evaluate your expression for light incident on glass

The meter stick in FIGURE P34.48 lies on the bottom of a 100-cmlong tank with its zero mark against the left edge. You look into the tank at a 30 angle, with your line of sight just grazing the upper left edge of the tank. What mark do you see on the meter stick if the tank is (a) empty, (b) half full of water, and (c) completely full of water?

The 80-cm-tall, 65-cm-wide tank shown in FIGURE P34.49 is completely filled with water. The tank has marks every 10 cm along one wall, and the 0 cm mark is barely submerged. As you stand beside the opposite wall, your eye is level with the top of the water. a. Can you see the marks from the top of the tank (the 0 cm mark) going down, or from the bottom of the tank (the 80 cm mark) coming up? Explain. b. Which is the lowest or highest mark, depending on your answer to part a, that you can see?

A horizontal meter stick is centered at the bottom of a 3.0-m-deep, 3.0-m-wide pool of water. How long does the meter stick appear to be as you look at it from the edge of the pool?

A 4.0-m-wide swimming pool is filled to the top. The bottom of the pool becomes completely shaded in the afternoon when the sun is 20 above the horizon. How deep is the pool?

Its nighttime, and youve dropped your goggles into a 3.0-m-deep swimming pool. If you hold a laser pointer 1.0 m above the edge of the pool, you can illuminate the goggles if the laser beam enters the water 2.0 m from the edge. How far are the goggles from the edge of the pool?

An astronaut is exploring an unknown planet when she accidentally drops an oxygen canister into a 1.50-m-deep pool filled with an unknown liquid. Although she dropped the canister 21 cm from the edge, it appears to be 31 cm away when she peers in from the edge. What is the liquids index of refraction? Assume that the planets atmosphere is similar to earths.

Shown from above in FIGURE P34.54 is one corner of a rectangular box filled with water. A laser beam starts 10 cm from side A of the container and enters the water at position x. You can ignore the thin walls of the container. a. If x = 15 cm, does the laser beam refract back into the air through side B or reflect from side B back into the water? Determine the angle of refraction or reflection. b. Repeat part a for x = 25 cm. c. Find the minimum value of x for which the laser beam passes through side B and emerges into the air.

A light beam can use reflections to form a closed, N-sided polygon inside a solid, transparent cylinder if N is sufficiently large. What is the minimum possible value of N for light inside a cylinder of (a) water, (b) polystyrene plastic, and (c) cubic zirconia? Assume the cylinder is surrounded by air

Optical engineers need to know the cone of acceptance of an optical fiber. This is the maximum angle that an entering light ray can make with the axis of the fiber if it is to be guided down the fiber. What is the cone of acceptance of an optical fiber for which the index of refraction of the core is 1.55 while that of the cladding is 1.45? You can model the fiber as a cylinder with a flat entrance face

One of the contests at the school carnival is to throw a spear at an underwater target lying flat on the bottom of a pool. The water is 1.0 m deep. Youre standing on a small stool that places your eyes 3.0 m above the bottom of the pool. As you look at the target, your gaze is 30 below horizontal. At what angle below horizontal should you throw the spear in order to hit the target? Your raised arm brings the spear point to the level of your eyes as you throw it, and over this short distance you can assume that the spear travels in a straight line rather than a parabolic trajectory

Theres one angle of incidence b onto a prism for which the light inside an isosceles prism travels parallel to the base and emerges at angle b. a. Find an expression for b in terms of the prisms apex angle a and index of refraction n. b. A laboratory measurement finds that b = 52.2 for a prism shaped like an equilateral triangle. What is the prisms index of refraction?

Youre visiting the shark tank at the aquarium when you see a 2.5-m-long shark that appears to be swimming straight toward you at 2.0 m/s. What is the sharks actual speed through the water? You can ignore the glass wall of the tank.

Paraxial light rays approach a transparent sphere parallel to an optical axis passing through the center of the sphere. The rays come to a focus on the far surface of the sphere. What is the spheres index of refraction?

To determine the focal length of a lens, you place the lens in front of a small lightbulb and then adjust a viewing screen to get a sharply focused image. Varying the lens position produces the following data: Bulb to lens (cm) Lens to screen (cm) 20 61 22 47 24 39 26 37 28 32 Use the best-fit line of an appropriate graph to determine the focal length of the lens

The illumination lights in an operating room use a concave mirror to focus an image of a bright lamp onto the surgical site. One such light uses a mirror with a 30 cm radius of curvature. If the mirror is 1.2 m from the patient, how far should the lamp be from the mirror?

A dentist uses a curved mirror to view the back side of teeth in the upper jaw. Suppose she wants an upright image with a magnification of 1.5 when the mirror is 1.2 cm from a tooth. Should she use a convex or a concave mirror? What focal length should it have?

A keratometer is an optical device used to measure the radius of curvature of the eyes corneaits entrance surface. This measurement is especially important when fitting contact lenses, which must match the corneas curvature. Most light incident on the eye is transmitted into the eye, but some light reflects from the cornea, which, due to its curvature, acts like a convex mirror. The keratometer places a small, illuminated ring of known diameter 7.5 cm in front of the eye. The optometrist, using an eyepiece, looks through the center of this ring and sees a small virtual image of the ring that appears to be behind the cornea. The optometrist uses a scale inside the eyepiece to measure the diameter of the image and calculate its magnification. Suppose the optometrist finds that the magnification for one patient is 0.049. What is the absolute value of the radius of curvature of her cornea?

The mirror in FIGURE P34.65 is covered with a piece of glass. A point source of light is outside the glass. How far from the mirror is the image of this source?

A 2.0-cm-tall candle flame is 2.0 m from a wall. You happen to have a lens with a focal length of 32 cm. How many places can you put the lens to form a well-focused image of the candle flame on the wall? For each location, what are the height and orientation of the image?

A 25 g rubber ball is dropped from a height of 3.0 m above the center of a horizontal, concave mirror. The ball and its image coincide 0.65 s after the ball is released. What is the mirrors radius of curvature?

In recent years, physicists have learned to create metamaterials engineered materials not found in naturewith negative indices of refraction. Its not yet possible to form a lens from a material with a negative index of refraction, but researchers are optimistic. Suppose you had a planoconvex lens (flat on one side, a 15 cm radius of curvature on the other) that is made from a metamaterial with n = -1.25. If you place an object 12 cm from this lens, (a) what type of image will be formed and (b) where will the image be located?

A lightbulb is 3.0 m from a wall. What are the focal length and the position (measured from the bulb) of a lens that will form an image on the wall that is twice the size of the lightbulb?

An old-fashioned slide projector needs to create a 98-cm-high image of a 2.0-cm-tall slide. The screen is 300 cm from the slide. a. What focal length does the lens need? Assume that it is a thin lens. b. How far should you place the lens from the slide?

Some electro-optic materials can change their index of refraction in response to an applied voltage. Suppose a planoconvex lens (flat on one side, a 15.0 cm radius of curvature on the other), made from a material whose normal index of refraction is 1.500, is creating an image of an object that is 50.0 cm from the lens. By how much would the index of refraction need to be increased to move the image 5.0 cm closer to the lens?

A point source of light is distance d from the surface of a 6.0-cm-diameter glass sphere. For what value of d is there an image at the same distance d on the opposite side of the sphere?

A lens placed 10 cm in front of an object creates an upright image twice the height of the object. The lens is then moved along the optical axis until it creates an inverted image twice the height of the object. How far did the lens move?

An object is 60 cm from a screen. What are the radii of a symmetric converging plastic lens (i.e., two equally curved surfaces) that will form an image on the screen twice the height of the object?

A wildlife photographer with a 200-mm-focal-length telephoto lens on his camera is taking a picture of a rhinoceros that is 100 m away. Suddenly, the rhino starts charging straight toward the photographer at a speed of 5.0 m/s. What is the speed, in mm/s, of the image of the rhinoceros? Is the image moving toward or away from the lens?

A concave mirror has a 40 cm radius of curvature. How far from the mirror must an object be placed to create an upright image three times the height of the object?

A 2.0-cm-tall object is placed in front of a mirror. A 1.0-cmtall upright image is formed behind the mirror, 150 cm from the object. What is the focal length of the mirror?

Consider a lens having index of refraction n2 and surfaces with radii R1 and R2. The lens is immersed in a fluid that has index of refraction n1. a. Derive a generalized lens makers equation to replace Equation 34.26 when the lens is surrounded by a medium other than air. That is, when n1 1. b. A symmetric converging glass lens (i.e., two equally curved surfaces) has two surfaces with radii of 40 cm. Find the focal length of this lens in air and the focal length of this lens in water

FIGURE CP34.80 shows a light ray that travels from point A to point B. The ray crosses the boundary at position x, making angles u1 and u2 in the two media. Suppose that you did not know Snells law. a. Write an expression for the time t it takes the light ray to travel from A to B. Your expression should be in terms of the distances a, b, and w; the variable x; and the indices of refraction n1 a b. The time depends on x. Theres one value of x for which the light travels from A to B in the shortest possible time. Well call it xmin. Write an expression (but dont try to solve it!) from which xmin could be found. c. Now, by using the geometry of the figure, derive Snells law from your answer to part b. Youve proven that Snells law is equivalent to the statement that light traveling between two points follows the path that requires the shortest time. This interesting way of thinking about refraction is called Fermats pri

A fortune tellers crystal ball (actually just glass) is 10 cm in diameter. Her secret ring is placed 6.0 cm from the edge of the ball. a. An image of the ring appears on the opposite side of the crystal ball. How far is the image from the center of the ball? b. Draw a ray diagram showing the formation of the image. c. The crystal ball is removed and a thin lens is placed where the center of the ball had been. If the image is still in the same position, what is the focal length of the lens?

Consider an object of thickness ds (parallel to the axis) in front of a lens or mirror. The image of the object has thickness ds. Define the longitudinal magnification as M = ds/ds. Prove that M = -m2 , where m is the lateral magnification.