Solved: Show that placing a 1-diopter lens in front of a 2-diopter lens gives the

How can you see a virtual image, when its not really there?

Under what circumstances will the image in a concave mirror be the same size as the object?

If youre handed a converging lens, what can you do to estimate its focal length quickly?

A diverging lens always makes a reduced image. Could you use such a lens to start a fire by focusing sunlight? Explain.

Is there any limit to the temperature you can achieve by focusing sunlight? (Hint: Think about the second law of thermodynamics.)

Can a concave mirror make a reduced real image? A reduced virtual image? An enlarged real image? An enlarged virtual image? Specify conditions for each possible image.

If you placed a screen at the location of a virtual image, would the image appear on the screen? Why or why not?

If you look into the bowl of a metal spoon, you see yourself upside down. Flip the spoon so youre looking at the back side, and now youre right-side up. Explain.

Is the image on a movie screen real or virtual? How do you know?

Does a fish in a spherical bowl appear larger or smaller than it actually is?

A block of ice contains a hollow, air-filled space in the shape of a double-convex lens. Describe the optical behavior of this space.

The refractive index of the human cornea is about 1.4. If you can see clearly in air, why cant you see clearly underwater? Why do goggles help?

Do you want a long or short focal length for a telescopes objective lens? What about a microscopes?

Give at least three reasons why reflecting telescopes are superior to refractors.

A shoe store uses small floor-level mirrors to let customers view prospective purchases. At what angle should such a mirror be inclined so that a person standing 50 cm from the mirror with eyes 140 cm off the floor can see her feet?

A candle is on the axis of a 15-cm-focal-length concave mirror, 36 cm from the mirror. (a) Where is its image? (b) How do the image and object sizes compare? (c) Is the image real or virtual?

An object is five focal lengths from a concave mirror. (a) How do the object and image heights compare? (b) Is the image upright or inverted?

A virtual image is located 40 cm behind a concave mirror with focal length 18 cm. (a) Where is the object? (b) By how much is the image magnified?

(a) Where on the axis of a concave mirror would you place an object to get a half-size image? (b) Where will the image be located? (c) Will the image be real or virtual?

A lightbulb is 56 cm from a convex lens. Its image appears on a screen 31 cm from the lens, on the other side. Find (a) the lenss focal length and (b) how much the image is enlarged or reduced.

By what factor is the image magnified for an object 1.5 focal lengths from a converging lens? Is the image upright or inverted?

A lens with 50-cm focal length produces a real image the same size as the object. How far from the lens are image and object?

By holding a magnifying glass 25 cm from your desk lamp, you can focus an image of the lamps bulb on a wall 1.6 m from the lamp. Whats the focal length of your magnifying glass?

A real image is four times as far from a lens as is the object. Whats the object distance, measured in focal lengths?

A magnifying glass enlarges print by 50% when its 9.0 cm from a page. Whats its focal length?

Youre writing specifications for a new line of magnifying glasses that have double-convex lenses with equal 32-cm curvature radii, made from glass with n = 1.52. What do you list for the focal length?

Youre standing in a wading pool and your feet appear to be cm below the surface. How deep is the pool?

The bottom of a swimming pool looks to be 1.5 m below the surface. Find the pools actual depth.

A tiny insect is trapped 1.0 mm from the center of a spherical dewdrop 4.0 mm in diameter. As you look straight into the drop, whats the insects apparent distance from the drops surface?

Youre underwater, looking through a spherical air bubble (Fig. 31.35). Whats its actual diameter if it appears, along your line of sight, to be 1.5 cm in diameter

You have to hold a book 55 cm from your eyes for the print to be in focus. What power lens would correct your farsightedness?

What focal length should you specify if you want a magnifying glass with angular magnification 3.2?

Youre an optometrist helping a nearsighted patient who claims he cant see clearly beyond 80 cm. Prescribe a lens that will put the images of distant objects at 80 cm, giving your patient clear vision at all distances beyond the normal near point.

A particular eye has a focal length of 2.0 cm instead of the 2.2 cm that would put a sharply focused image on the retina. (a) Is this eye nearsighted or farsighted? (b) What corrective lens is needed?

A compound microscope has objective and eyepiece focal lengths of 6.1 mm and 1.7 cm, respectively. If the lenses are 8.3 cm apart, what is the instruments magnification?

(a) Find the focal length of a concave mirror if an object placed 38.4 cm in front of the mirror has a real image 55.7 cm from the mirror. (b) Where and what type will the image be if the object is moved to a point 16.0 cm from the mirror?

A 12-mm-high object is 10 cm from a concave mirror with focal length 17 cm. (a) Where is the image, (b) how high is it, and (c) what type is it?

Repeat Problem 37 for a convex mirror, assuming all numbers stay the same.

An objects image in a 27-cm-focal-length concave mirror is upright and magnified by a factor of 3. Where is the object?

Youre asked to design a concave mirror that will produce a virtual image, enlarged 1.8 times, of an object 22 cm from the mirror. What do you specify for the mirrors curvature radius?

Viewed from Earth, the Moon subtends an angle of 0.52 in the sky. What will be the physical size of the Moons image formed by either of the twin Keck telescopes, with 10-m-diameter mirrors and 17.5-m focal length? At what two distances could you place an object from a

At what two distances could you place an object from a cm-focal-length concave mirror to get an image 1.5 times the objects size?

LCD projectors commonly used for computer and video projection create an image on a small LCD display (see Application on page 369). The display is mounted before a lens and illuminated from behind. In a projector using a 7.50-cm-focal-length convex lens, where should the LCD display be located so the projected image is focused on a screen 6.30 m from the lens?

An object 15 cm from a concave mirror has a virtual image magnified 2.5 times. Whats the mirrors focal length?

How far from a page should you hold a lens with 32-cm focal length in order to see the print magnified 1.6 times?

A converging lens has focal length 4.0 cm. A 1.0-cm-high arrow is located 7.0 cm from the lens with its lowest point 5.0 mm above the lens axis. Make a full-scale ray-tracing diagram to locate both ends of the image. Confirm using the lens equation.

A lens has focal length f = 35 cm. Find the type and height of the image produced when a 2.2-cm-high object is placed at distances (a) f + 10 cm and (b) f - 10 cm.

How far apart are the object and image produced by a converging lens with 35-cm focal length when the object is (a) 40 cm and (b) 30 cm from the lens?

A candle and a screen are 70 cm apart. Find two points between candle and screen where you could put a convex lens with 17-cm focal length to give a sharp image of the candle on the screen

The cornea of the human eye has refractive index 1.38, while the eyes lens has a graduated index in the range 1.38 to 1.40; use 1.39 for this problem. For the aqueous humor between cornea and lens, n = 1.34. Find the angle through which light is deflected at the first surface of (a) the cornea and (b) the lens, if its incident at 20 to the normal at each surface. Your result shows that the cornea is the dominant refractive element in the eye

How far from a 25-cm-focal-length lens should you place an object to get an upright image magnified 1.8 times?

An object and its lens-produced real image are 2.4 m apart. If the lens has 55-cm focal length, what are the possible values for the object distance and magnification?

An object is 68 cm from a plano-convex lens whose curved side has curvature radius 26 cm. The refractive index of the lens is 1.62. Where is the image, and what type is it?

Use Equation 31.6 to show that an object at the center of a glass sphere will appear to be its actual distanceone radiusfrom the edge. Draw a ray diagram showing why this makes sense.

Rework Example 31.4 for a fish 15.0 cm from the far wall of the tank.

Consider the inverse of Example 31.4: Youre inside a 70.0-cmdiameter hollow tube containing air, and the tip of your nose is 15.0 cm from the tubes wall. The tube is immersed in water, and a fish looks in. To the fish, whats the apparent distance from your nose to the tube wall?

Two specks of dirt are trapped in a crystal ball, one at the center and the other halfway to the surface. If you peer into the ball on a line joining the two specks, the outer one appears to be only onethird of the way to the other. Find the refractive index of the ball

A contact lens is in the shape of a convex meniscus (see Fig. 31.25). The inner surface is curved to fit the eye, with curvature radius 7.80 mm. The lens is made from plastic with refractive index n = 1.56. If it has a 44.4-cm focal length, whats the curvature radius of its outer surface?

For what refractive index would the focal length of a planoconvex lens be equal to the curvature radius of its one curved surface?

An object is 28 cm from a double-convex lens with n = 1.5 and curvature radii 35 cm and 55 cm. Where is the image, and what type is it?

Youre an optician whos been asked to design a new replacement lens for cataract patients. The lens must be 5.5 mm in diameter, with focal length 17 mm, and it cant be thicker than 0.8 mm. For the lens material, you have a choice of plastic with refractive index 1.49 or more expensive silicone with n = 1.58. Which material do you choose, and why?

A double-convex lens with equal 28.5-cm curvature radii is made from glass with refractive indices nred = 1.512 and nviolet = 1.547. If a point source of white light is located on the lens axis at 75.0 cm from the lens, over what distance will its visible image be smeared?

An object placed 17.5 cm from a convex lens of glass with n = 1.524 forms a virtual image twice the objects size. If the lens is replaced with an identically shaped one made of diamond, (a) what type of image will appear and (b) what will be its magnification?

Youre taking a photography class, working with a camera whose zoom lens covers the focal-length range 38 mm110 mm. Your instructor asks you to compare the sizes of the images of a distant object when photographed at the two zoom extremes. Your answer?

A camera can normally focus as close as 60 cm, but it has provisions for mounting additional lenses just in front of the main lens to provide close-up capability. What type and power of auxiliary lens will allow the camera to focus as close as 20 cm?

A 300-power compound microscope has a 4.5-mm-focal-length objective lens. If the distance from objective to eyepiece is 10 cm, what should be the focal length of the eyepiece?

To the unaided eye, Jupiter has an angular diameter of 50 arcseconds. What will its angular size be when viewed through a 1-m-focal-length refracting telescope with a 40-mmfocal-length eyepiece?

A Cassegrain telescope like that shown in Fig. 31.34b has 1.0-m focal length, and the convex secondary mirror is located 0.85 m from the primary. What should be the focal length of the secondary in order to put the final image 0.12 m behind the front surface of the primary mirror?

You stand with your nose 6.0 cm from the surface of a reflecting ball, and your noses image appears three-quarters full size. Whats the balls diameter?

A contact lens prescription calls for +2.25-diopter lenses with inner curvature radius 8.6 mm to fit the patients cornea. (a) If the lenses are plastic with n = 1.56, what should be the outer curvature radius? (b) With these lenses, the patient comfortably reads a newspaper 30 cm from her eyes. Wheres the image as viewed through the lenses?

Show that placing a 1-diopter lens in front of a 2-diopter lens gives the equivalent of a single 3-diopter lens (i.e., the powers of closely spaced lenses add).

Derive an expression for the thickness t of a plano-convex lens with diameter d, focal length f, and refractive index n

Show that identical objects placed equal distances on either side of the focal point of a concave mirror or converging lens produce images of equal size. Are the images of the same type?

Generalize the derivation of the lensmakers formula (Equation 31.7) to show that a lens of refractive index nlens in an external medium with index next has focal length given by 1 f = a nlens next - 1ba 1 R1 - 1 R2 b

Draw a diagram like Fig. 31.10, but showing a ray from the arrowhead through the center of curvature. Using the fact that this ray reflects back on itself, draw similar triangles with object and image as their vertical sides, and show that the center of curvature is twice as far from the mirror as the focal pointthat is, R = 2f, with R the curvature radius

Galileos first telescope used the arrangement shown in Fig. 31.36, with a double-concave eyepiece slightly before the focus of the objective lens. Use ray tracing to show that this design gives an upright image, which makes the Galilean telescope useful in terrestrial observing.

The maximum magnification of a simple magnifier occurs with the image at the 25-cm near point. Show that the angular magnification is m = 1 + 125 cm/f2, where f is the focal length.

Chromatic aberration results from variation of the refractive index with wavelength. Starting with the lensmakers formula, find an expression for the fractional change df/f in the focal length of a thin lens in terms of the change dn in refractive index.

For visible wavelengths, the refractive index of the polycarbonate plastic widely used in eyeglasses is given approximately by n(l) = b + c/l2 , where b = 1.55 and c = 11,500 nm2 . (a) Find an expression for the change in refractive index dn corresponding to a small wavelength change dl. (b) Use the results of part (a) and of Problem 78 to determine the variation df in focal length for a +2.25-diopter polycarbonate lens, over a wavelength range of 10.0 nm centered on 589 nm.

The table below shows measurements of magnification versus object distance for a lens. Determine a quantity that, when you plot object distance against it, should give a straight line. Make your plot, establish a best-fit line, and use your line to find the focal length of the lens.

The speed of a camera lens measures its ability to photograph in dim light. Speed is characterized by f-ratio, also called the f-number, defined as the ratio of focal length f to lens diameter d. Thus an f/2.8 lens, for example, has diameter d = f/2.8. The actual amount of light a lens admits depends on its area A, but the inverse-square law shows that the light intensity at the cameras imaging sensor is proportional to A/f 2. Most cameras have an adjustable iris that obscures part of the lens to change the f-ratio in response to available light. Pointand-shoot cameras adjust the f-ratio automatically, but serious photographers use their cameras manual f-ratio adjustment (Fig. 31.37). Stopping down is the photographers term for reducing the lens area using the adjustable iris. Zooming your cameras lens for telephoto shots increases the focal length. With no change in the lens area, this will a. increase the f-ratio and increase the lens speed. b. decrease the f-ratio and decrease the lens speed. c. increase the f-ratio and decrease the lens speed. d. not change the f-ratio or the lens speed.

The speed of a camera lens measures its ability to photograph in dim light. Speed is characterized by f-ratio, also called the f-number, defined as the ratio of focal length f to lens diameter d. Thus an f/2.8 lens, for example, has diameter d = f/2.8. The actual amount of light a lens admits depends on its area A, but the inverse-square law shows that the light intensity at the cameras imaging sensor is proportional to A/f 2. Most cameras have an adjustable iris that obscures part of the lens to change the f-ratio in response to available light. Pointand-shoot cameras adjust the f-ratio automatically, but serious photographers use their cameras manual f-ratio adjustment (Fig. 31.37). Stopping down is the photographers term for reducing the lens area using the adjustable iris. Increasing the f-ratio from 2.8 to 5.6 a. decreases the light admitted by a factor of 2. b. decreases the light admitted by a factor of 4. c. increases the light admitted by a factor of 2. d. increases the light admitted by a factor of 4.

The speed of a camera lens measures its ability to photograph in dim light. Speed is characterized by f-ratio, also called the f-number, defined as the ratio of focal length f to lens diameter d. Thus an f/2.8 lens, for example, has diameter d = f/2.8. The actual amount of light a lens admits depends on its area A, but the inverse-square law shows that the light intensity at the cameras imaging sensor is proportional to A/f 2. Most cameras have an adjustable iris that obscures part of the lens to change the f-ratio in response to available light. Pointand-shoot cameras adjust the f-ratio automatically, but serious photographers use their cameras manual f-ratio adjustment (Fig. 31.37). Stopping down is the photographers term for reducing the lens area using the adjustable iris. Youre given two lenses with different diameters. Knowing nothing else, you can conclude that a. the larger lens is faster. b. the smaller lens has the shorter focal length. c. the smaller lens suffers less spherical aberration. d. none of the above

The speed of a camera lens measures its ability to photograph in dim light. Speed is characterized by f-ratio, also called the f-number, defined as the ratio of focal length f to lens diameter d. Thus an f/2.8 lens, for example, has diameter d = f/2.8. The actual amount of light a lens admits depends on its area A, but the inverse-square law shows that the light intensity at the cameras imaging sensor is proportional to A/f 2. Most cameras have an adjustable iris that obscures part of the lens to change the f-ratio in response to available light. Pointand-shoot cameras adjust the f-ratio automatically, but serious photographers use their cameras manual f-ratio adjustment (Fig. 31.37). Stopping down is the photographers term for reducing the lens area using the adjustable iris. If a lens suffers from spherical aberration, stopping down will a. worsen the focus. b. improve the focus. c. not affect the focus.