(a) Show that if two thin lenses of focal lengths f1 and

Problem 2P (I) When you look at yourself in a 60-cm-tall plane mirror, you see the same amount of your body whether you are close to the mirror or far away. (Try it and see.) Use ray diagrams to show why this should be true.

Problem 2Q Archimedes is said to have burned the whole Roman fleet in the harbor of Syracuse, Italy, by focusing the rays of the Sun with a huge spherical mirror. Is this reasonable?

Problem 4Q If a concave mirror produces a real image, is the image necessarily inverted? Explain.

Problem 7Q What is the focal length of a plane mirror?What is the magnification of a plane mirror?

Problem 8Q When you look at the Moon's reflection from a ripply sea, it appears elongated (Fig. 23-47)

Problem 9Q What is the angle of refraction when a light ray is incident perpendicular to the boundary between two transparent materials?

Problem 10Q How might you determine the speed of light in a solid, rectangular, transparent object?

Problem 11Q When you look down into a swimming pool or a lake, are you likely to overestimate or underestimate its depth? Explain. How does the apparent depth vary with the viewing angle? (Use ray diagrams.)

Draw a ray diagram to show why a stick or straw looks bent when part of it is under water (Fig. 23-21).

Your eye looks into an aquarium and views a fish inside. One ray of light that emerges from the tank is shown in Fig. 23-45, as well as the apparent position of the fish. In the drawing, indicate the approximate position of the actual fish. Briefly justify your answer.

Problem 14Q Problem How can you "see" a round drop of water on a table even though the water is transparent and colorless?

Problem 15P (II) (a) Where should an object be placed in front of a concave mirror so that it produces an image at the same location as the object? (b) Is the image real or virtual? (c) Is the image inverted or upright? (d) What is the magnification of the image?

Problem 17P (II) Use two techniques, (a) a ray diagram, and (b) the mirror equation, to show that the magnitude of the magnification of a concave mirror is less than 1 if the object is beyond the center of curvature C (do > r), and is greater than 1 if the object is within C (do < r).

Problem 17Q Light rays from stars (including our Sun) always bend toward the vertical direction as they pass through the Earth's atmosphere. (a) Why does this make sense? (b)What can you conclude about the apparent positions of stars as viewed from Earth? Draw a circle for Earth, a dot for you, and 3 or 4 stars at different angles.

Problem 18P (III) Show, using a ray diagram, that the magnification m of a convex mirror is m =-di/do, just as for a concave mirror. [Hint Consider a ray from the top of the object that reflects at the center of the mirror.]

Problem 18Q Where must the film be placed if a camera lens is to make a sharp image of an object far away? Explain.

What type of mirror is shown in Fig. 23-46? Explain.

Problem 20Q A photographer moves closer to his subject and then refocuses. Does the camera lens move farther away from or closer to the camera film or sensor? Explain.

Problem 21Q Can a diverging lens form a real image under any circumstances? Explain.

Problem 23P (I) What is the speed of light in (a) ethyl alcohol, (b) lucite, (c) crown glass?

Problem 23Q Light rays are said to be "reversible." Is this consistent with the thin lens equation? Explain.

Problem 24Q Can real images be projected on a screen? Can virtual images? Can either be photographed? Discuss carefully.

Problem 25Q A thin converging lens is moved closer to a nearby object. Does the real image formed change (a) in position, (b) in size? If yes, describe how.

Problem 32Q The thicker a double convex lens is in the center as compared to its edges, the shorter its focal length for a given lens diameter. Explain.

(III) A beam of light enters the end of an optic fiber as shown in Fig. 23-52. Show that we can guarantee total internal reflection at the side surface of the material (at point a), if the index of refraction is greater than about 1.42. In other words, regardless of the angle , the light beam reflects back into the material at point a. Equation Transcription: Text Transcription: alpha beta gamma

Problem 43P A sharp image is located 78.0 mm behind a 65.0-mm-focal-length converging lens. Find the object distance (a) using a ray diagram, (b) by calculation.

Problem 1P Suppose that you want to take a photograph of yourself as you look at your image in a flat mirror 2.5 m away. For what distance should the camera lens be focused?

Problem 1Q What would be the appearance of the Moon if it had (a) a rough surface; (b) a polished mirrorlike surface?

(II) Two mirrors meet at a \(135^{\circ}\) angle, Fig. 23–47. If light rays strike one mirror at \(40^{\circ}\) as shown, at what angle \(\phi\) do they leave the second mirror? Equation Transcription: ???? ???? Text Transcription: 135^o 40^o 40^o phi phi

Although a plane mirror appears to reverse left and right, it doesn’t reverse up and down. Explain.

(II) A person whose eyes are 1.68 m above the floor stands 2.20 m in front of a vertical plane mirror whose bottom edge is 43 cm above the floor, Fig. 23–48. What is the horizontal distance x to the base of the wall supporting the mirror of the nearest point on the floor that can be seen reflected in the mirror?

Problem 5P Suppose you are 90 cm from a plane mirror. What area of the mirror is used to reflect the rays entering one eye from a point on the tip of your nose if your pupil diameter is 5.5 mm?

Problem 5Q An object is placed along the principal axis of a spherical mirror. The magnification of the object is ?3.0. Is the image real or virtual, inverted or upright? Is the mirror concave or convex? On which side of the mirror is the image located?

Using the rules for the three rays discussed with reference to Fig. 23–13, draw ray 2 for Fig. 23–17b.

A solar cooker, really a concave mirror pointed at the Sun, focuses the Sun’s rays 18.0 cm in front of the mirror. What is the radius of the spherical surface from which the mirror was made?

(I) How far from a concave mirror (radius 23.0 cm) must an object be placed if its image is to be at infinity?

Problem 9P If you look at yourself in a shiny Christmas tree ball with a diameter of 9.0 cm when your face is 30.0 cm away from it, where is your image? Is it real or virtual? Is it upright or inverted?

Problem 10P A mirror at an amusement park shows an upright image of any person who stands 1.4 m in front of it. If the image is three times the person’s height, what is the radius of curvature?

(II) A dentist wants a small mirror that, when 2.20 cm from a tooth, will produce a \(4.5 \times\) upright image. What kind of mirror must be used and what must its radius of curvature be?

Problem 12P Some rearview mirrors produce images of cars behind you that are smaller than they would be if the mirror were flat. Are the mirrors concave or convex? What is a mirror’s radius of curvature if cars 20.0 m away appear 0.33 × their normal size?

You are standing 3.0 m from a convex security mirror in a store. You estimate the height of your image to be half of your actual height. Estimate the radius of curvature of the mirror.

A luminous object 3.0 mm high is placed 20.0 cm from a convex mirror of radius of curvature 20.0 cm. (a) Show by ray tracing that the image is virtual, and estimate the image distance. (b) Show that the (negative) image distance can be computed from Eq. 23–2 using a focal length of ?10.0 cm. (c) Compute the image size, using Eq. 23–3.

When you look up at an object in air from beneath the surface in a swimming pool, does the object appear to be the same size as when you see it directly in air? Explain.

Problem 89GP When an object is placed 60.0 cm from a certain converging lens, it forms a real image. When the object is moved to 40.0 cm from the lens, the image moves 10.0 cm farther from the lens. Find the focal length of this lens.

Problem 16P The image of a distant tree is virtual and very small when viewed in a curved mirror. The image appears to be 18.0 cm behind the mirror. What kind of mirror is it, and what is its radius of curvature?

A small object is 25.0 cm from a diverging lens as shown in Fig. 23-59. A converging lens with a focal length of 12.0 cm is 30.0 cm to the right of the diverging lens. The two-lens system forms a real inverted image 17.0 cm to the right of the converging lens. What is the focal length of the diverging lens?

Problem 16Q How can a spherical mirror have a negative object distance?

Problem 94GP (a) Show that if two thin lenses of focal lengths f1 and 12 are placed in contact with each other, the focal length of the combination is given by fT = f1 f2/(f1 + f2). (b) Show that the power P of the combination of two lenses is the sum of their separate powers, P = P1 + P2.

(II) Use ray diagrams to show that the mirror equation, Eq. 23-2, is valid for a convex mirror as long as f is considered negative.

Problem 20P The magnification of a convex mirror is +0.65 × for objects 2.2 m from the mirror. What is the focal length of this mirror?

Problem 21P A 4.5 -cm-tall object is placed 28 cm in front of a spherical mirror. It is desired to produce a virtual image that is upright and 3.5 cm tall. (a) What type of mirror should be used? (b) Where is the image located? (c) What is the focal length of the mirror? (d) What is the radius of curvature of the mirror?

Problem 22P A shaving/makeup mirror is designed to magnify your face by a factor of 1.33 when your face is placed 20.0 cm in front of it. (a) What type of mirror is it? (b) Describe the type of image that it makes of your face. (c) Calculate the required radius of curvature for the mirror.

Problem 22Q Use ray diagrams to show that a real image formed by a thin lens is always inverted, whereas a virtual image is always upright if the object is real.

(I) The speed of light in ice is \(2.29 \times 10^8 \mathrm{~m} / \mathrm{s}\). What is the index of refraction of ice?

Problem 25P The speed of light in a certain substance is 89% of its value in water. What is the index of refraction of this substance?

Problem 26P A flashlight beam strikes the surface of a pane of glass (n = 1.58) at a 63° angle to the normal. What is the angle of refraction?

(I) A diver shines a flashlight upward from beneath the water at a \(42.5^\circ\) angle to the vertical. At what angle does the light leave the water?

Problem 26Q A lens is made of a material with an index of refraction n = 1.30. In air, it is a converging lens. Will it still be a converging lens if placed in water? Explain, using a ray diagram.

Problem 27Q A dog with its tail in the air stands facing a converging lens. If the nose and the tail are each focused on a screen in turn, which will have the greater magnification?

Problem 28P A light beam coming from an underwater spotlight exits the water at an angle of 66.0° to the vertical. At what angle of incidence does it hit the air–water interface from below the surface?

Problem 28Q A cat with its tail in the air stands facing a converging lens. Under what circumstances (if any) would the image of the nose be virtual and the image of the tail be real? Where would the image of the rest of the cat be?

Problem 29P Rays of the Sun are seen to make a 31.0° angle to the vertical beneath the water. At what angle above the horizon is the Sun?

Why, in Example 23–13, must the converging lens have a shorter focal length than the diverging lens if the latter’s focal length is to be determined by combining them?

(II) An aquarium filled with water has flat glass sides whose index of refraction is 1.52. A beam of light from outside the aquarium strikes the glass at a \(43.5^{\circ}\) angle to the perpendicular (Fig. 23–49). What is the angle of this light ray when it enters (a) the glass, and then (b) the water? (c) What would be the refracted angle if the ray entered the water directly? ________________ Equation Transcription: Text Transcription: 43.5^o 43.5^o

Problem 30Q Explain how you could have a virtual object.

In searching the bottom of a pool at night, a watchman shines a narrow beam of light from his flashlight, 1.3 m above the water level, onto the surface of the water at a point 2.7 m from the edge of the pool (Fig. 23–50). Where does the spot of light hit the bottom of the pool, measured from the wall beneath his foot, if the pool is 2.1m deep?

Problem 31Q An unsymmetrical lens (say, planoconvex) forms an image of a nearby object. Does the image point change if the lens is turned around?

(II) Light is incident on an equilateral glass prism at a \(45.0^{\circ}\) angle to one face, Fig. 23–51. Calculate the angle at which light emerges from the opposite face. Assume that \(n=1.58\). Equation Transcription: Text Transcription: 45.0^o n=1.58 45.0^o theta_3 theta_2 ?=theta_4

Problem 33P A beam of light in air strikes a slab of glass (n = 1.52) and is partially reflected and partially refracted. Find the angle of incidence if the angle of reflection is twice the angle of refraction.

Problem 33Q Consider two converging lenses separated by some distance. An object is placed so that the image from the first lens lies exactly at the focal point of the second lens. Will this combination produce an image? If so, where? If not, why not?

(III) Prove in general that for a light beam incident on a uniform layer of transparent material, as in Fig. 23–22, the direction of the emerging beam is parallel to the incident beam, independent of the incident angle \(\theta\). Assume air on both sides of the glass. Equation Transcription: Text Transcription: theta

(III) A light ray is incident on a flat piece of glass with index of refraction n as in Fig. 23–22. Show that if the incident angle \(\theta\) is small, the emerging ray is displaced a distance \(d=t \theta(n-1)/n\) from the incident ray, where t is the thickness of the glass and \(\theta\) is in radians. [Hint: for small \(\theta\), \(\sin \theta \approx \tan \theta \approx \theta\) in radians.]

(I) What is the critical angle for the interface between water and Lucite? To be totally internally reflected, the light must start in which material?

Problem 37P The critical angle for a certain liquid–air surface is 47.7°. What is the index of refraction of the liquid?

Problem 38P A beam of light is emitted in a pool of water from a depth of 62.0 cm. Where must it strike the air–water interface, relative to the spot directly above it, in order that the light does not exit the water?

Problem 39P A beam of light is emitted 8.0 cm beneath the surface of a liquid and strikes the surface 7.0 cm from the point directly above the source. If total internal reflection occurs, what can you say about the index of refraction of the liquid?

(III) Suppose a ray strikes the left face of the prism in Fig. 23–51 at \(45.0^{\circ}\) as shown, but is totally internally reflected at the opposite side. If the prism apex angle (at the top) is \(\phi=75.0^{\circ}\), what can you say about the index of refraction of the prism? Equation Transcription: ???? Text Transcription: 45.0^o phi=75.0^o 45.0^o theta_2 theta_3 ?=theta_4

(III) (a) What is the minimum index of refraction for a glass or plastic prism to be used in binoculars (Fig. 23–26) so that total internal reflection occurs at \(45^{\circ}\)? (b) Will binoculars work if its prisms (assume \(n=1.50\)) are immersed in water? (c) What minimum n is needed if the prisms are immersed in water? Equation Transcription: Text Transcription: 45^o n=1.50

Problem 44P Sunlight is observed to focus at a point 18.5 cm behind a lens. (a) What kind of lens is it? (b) What is its power in diopters?

Problem 45P A certain lens focuses light from an object 2.75 m away as an image 48.3 cm on the other side of the lens. What type of lens is it and what is its focal length? Is the image real or virtual?

Problem 46P (a) What is the power of a 20.5-cm-focal-length lens? ________________ (b) What is the focal length of a ?6.25-diopter lens? ________________ (c) Are these lenses converging or diverging?

(II) A stamp collector uses a converging lens with focal length 24 cm to view a stamp 18 cm in front of the lens. (a) Where is the image located? (b) What is the magnification?

Problem 48P A ?5.5-D lens is held 14.0 cm from an object 4.0 mm high. What are the position, type, and height of the image?

Problem 49P An 80-mm-focal-length lens is used to focus an image on the film of a camera. The maximum distance allowed between the lens and the film plane is 120 mm. (a) How far ahead of the film should the lens be if the object to be photographed is 10.0 m away? (b) 3.0 m away? (c) 1.0 m away? (d) What is the closest object this lens could photograph sharply?

Problem 50P It is desired to magnify reading material by a factor of 2.5× when a book is placed 8.0 cm behind a lens. (a) Draw a ray diagram and describe the type of image this would be. (b) What type of lens is needed? (c) What is the power of the lens in diopters?

(II) An object is located \(1.5 \mathrm{~m}\) from an 8.0-D lens. By how much does the image move if the object is moved (a) \(1.0 \mathrm{~m}\) closer to the lens, and (b) \(1.0 \mathrm{~m}\) farther from the lens?

(II) How far from a converging lens with a focal length of \(25 \mathrm{~cm}\) should an object be placed to produce a real image which is the same size as the object?

Problem 53P (a) How far from a 50.0-mm-focal-length lens must an object be placed if its image is to be magnified 2.00× and be real? (b) What if the image is to be virtual and magnified 2.00×?

Problem 54P Repeat Problem 53 for a ?50.0-mm-focal-length lens. [Hint: consider objects real or virtual (formed by some other piece of optics).] Problem 53 (a) How far from a 50.0-mm-focal-length lens must an object be placed if its image is to be magnified 2.00× and be real? (b) What if the image is to be virtual and magnified 2.00×?

Problem 55P (a) A 2.00-cm-high insect is 1.20 m from a 135-mm-focal-length lens. Where is the image, how high is it, and what type is it? (b) What if f = ?135 mm?

(III) How far apart are an object and an image formed by a \(75-\mathrm{cm}\)-focal-length converging lens if the image is \(2.5 \times\) larger than the object and is real?

Problem 57P A bright object and a viewing screen are separated by a distance of 66.0 cm. At what location(s) between the object and the screen should a lens of focal length 12.5 cm be placed in order to produce a crisp image on the screen? [Hint: first draw a diagram.]

Problem 59P A diverging lens with f = ?31.5 cm is placed 14.0 cm behind a converging lens with f = 20.0 cm. Where will an object at infinity be focused?

Two 28.0-cm-focal-length converging lenses are placed 16.5 cm apart. An object is placed 36.0 cm in front of one lens. Where will the final image formed by the second lens be located? What is the total magnification?

Problem 60P A 31.0-cm-focal-length converging lens is 21.0 cm behind a diverging lens. Parallel light strikes the diverging lens. After passing through the converging lens, the light is again parallel. What is the focal length of the diverging lens? [Hint: first draw a ray diagram.]

(II) The two converging lenses of Example 23–12 are now placed only 20.0 cm apart. The object is still 60.0 cm in front of the first lens as in Fig. 23–41. In this case, determine (a) the position of the final image, and (b) the overall magnification. (c) Sketch the ray diagram for this system.

Problem 62P Two converging lenses are placed 30.0 cm apart. The focal length of the lens on the right is 20.0 cm, and the focal length of the lens on the left is 15.0 cm. An object is placed to the left of the 15.0-cm-focal-length lens. A final image from both lenses is inverted and located halfway between the two lenses. How far to the left of the 15.0-cm-focal-length lens is the original object?

(II) A diverging lens with a focal length of \(-14 \mathrm{~cm}\) is placed \(12 \mathrm{~cm}\) to the right of a converging lens with a focal length of \(18 \mathrm{~cm}\). An object is placed \(33 \mathrm{~cm}\) to the left of the converging lens (a) Where will the final image be located? (b) Where will the image be if the diverging lens is \(38 \mathrm{~cm}\) from the converging lens?

Problem 64P Two lenses, one converging with focal length 20.0 cm and one diverging with focal length ?10.0 cm, are placed 25.0 cm apart. An object is placed 60.0 cm in front of the converging lens. Determine (a) the position and (b) the magnification of the final image formed. (c) Sketch a ray diagram for this system.

(II) A diverging lens is placed next to a converging lens of focal length \(f_{\mathrm{C}}\), as in Fig. 23–42. If \(f_{\mathrm{T}}\) represents the focal length of the combination, show that the focal length of the diverging lens, \(f_{\mathrm{D}}\), is given by \(\frac{1}{f_{\mathrm{D}}}=\frac{1}{f_{\mathrm{T}}}-\frac{1}{f_{\mathrm{C}}}\) Equation Transcription: Text Transcription: f_C f_T f_D {1 over f_D} ={1 over f_T}={1 over f_C}

Problem 66P A double concave lens has surface radii of 34.2 cm and 23.8 cm. What is the focal length if n = 1.52?

Problem 67P Both surfaces of a double convex lens have radii of 31.0 cm. If the focal length is 28.9 cm, what is the index of refraction of the lens material?

Problem 68P A planoconcave lens (n = 1.50) has a focal length of ?23.4 cm. What is the radius of the concave surface?

(II) A Lucite planoconcave lens (see Fig. 23–29b) has one flat surface and the other has \(R=-18.4 \mathrm{~cm}\). What is the focal length? Equation Transcription: Text Transcription: R=-18.4 cm

A symmetric double convex lens with a focal length of \(25.0 \mathrm{~cm}\) is to be made from glass with an index of refraction of 1.52. What should be the radius of curvature for each surface?

Problem 71P A prescription for a corrective lens calls for +1.50D. The lensmaker grinds the lens from a “blank” with n = 1.56 and a preformed convex front surface of radius of curvature of 40.0 cm. What should be the radius of curvature of the other surface?

Two plane mirrors face each other 2.0 m apart as in Fig. 23–53. You stand 1.5 m away from one of these mirrors and look into it. You will see multiple images of yourself. (a) How far away from you are the first three images in the mirror in front of you? (b) Are these first three images facing toward you or away from you?

We wish to determine the depth of a swimming pool filled with water. We measure the width \((x=5.50 \mathrm{~m})\) and then note that the bottom edge of the pool is just visible at an angle of \(14.0^{\circ}\) above the horizontal as shown in Fig. 23–54. Calculate the depth of the pool. Equation Transcription: Text Transcription: (x=5.50 m) 14.0^o 14.0^o

Problem 74GP The critical angle of a certain piece of plastic in air is ?C = 37.3°. What is the critical angle of the same plastic if it is immersed in water?

Problem 75GP (a) A plane mirror can be considered a limiting case of a spherical mirror. Specify what this limit is. (b) Determine an equation that relates the image and object distances in this limit of a plane mirror. (c) Determine the magnification of a plane mirror in this same limit. (d) Are your results in parts (b) and (c) consistent with the discussion of Section 23–2 on plane mirrors?

Stand up two plane mirrors so they form a \(90^{\circ}\) angle as in Fig. 23–55. When you look into this double mirror, you see yourself as others see you, instead of reversed as in a single mirror. Make a careful ray diagram to show how this occurs. Equation Transcription: Text Transcription: 90^o 90^o

Problem 77GP Show analytically that a diverging lens can never form a real image of a real object. Can you describe a situation in which a diverging lens can form a real image?

Problem 78GP Each student in a physics lab is assigned to find the location where a bright object may be placed in order that a concave mirror with radius of curvature r = 40 cm will produce an image three times the size of the object. Two students complete the assignment at different times using identical equipment, but when they compare notes later, they discover that their answers for the object distance are not the same. Explain why they do not necessarily need to repeat the lab, and justify your response with a calculation.

If the apex angle of a prism is \(\phi=72^{\circ}\) (see Fig. 23–56), what is the minimum incident angle for a ray if it is to emerge from the opposite side (i.e., not be totally internally reflected), given \(n=1.50\)? Equation Transcription: ???? ???? Text Transcription: phi=72^o n=1.50 phi

Problem 80GP The end faces of a cylindrical glass rod (n = 1.54) are perpendicular to the sides. Show that a light ray entering an end face at any angle will be totally internally reflected inside the rod when the ray strikes the sides. Assume the rod is in air. What if it were in water?

A lighted candle is placed 33 cm in front of a converging lens of focal length \(f_1=15\mathrm{~cm}\), which in turn is 55 cm in front of another converging lens of focal length \(f_2=12\mathrm{~cm}\) (see Fig. 23–57). (a) Draw a ray diagram and estimate the location and the relative size of the final image. (b) Calculate the position and relative size of the final image. Equation Transcription: Text Transcription: f_1=15 cm f_2=12 cm f_1=15 cm f_2=12 cm

In a slide or movie projector, the film acts as the object whose image is projected on a screen (Fig. 23–58). If a 105-mm-focal-length lens is to project an image on a screen 8.00 m away, how far from the lens should the slide be? If the slide is 36 mm wide, how wide will the picture be on the screen?

Problem 84GP A 35-mm slide (picture size is actually 24 by 36 mm) is to be projected on a screen 1.80 m by 2.70 m placed 7.50 m from the projector. What focal-length lens should be used if the image is to cover the screen?

Problem 86GP A movie star catches a reporter shooting pictures of her at home. She claims the reporter was trespassing. To prove her point, she gives as evidence the film she seized. Her 1.75-m height is 8.25 mm high on the film, and the focal length of the camera lens was 210 mm. How far away from the subject was the reporter standing?

Problem 87GP How large is the image of the Sun on film used in a camera with (a) a 28-mm-focal-length lens, (b) a 50-mm-focal-length lens, and (c) a 135-mm-focal-length lens? (d) If the 50-mm lens is considered normal for this camera, what relative magnification does each of the other two lenses provide? The Sun has diameter 1.4 × 106 km, and it is 1.5 × 108 km away.

Problem 88GP (a) An object 34.5 cm in front of a certain lens is imaged 8.20 cm in front of that lens (on the same side as the object). What type of lens is this, and what is its focal length? Is the image real or virtual? (b) If the image were located, instead, 41.5 cm in front of the lens, what type of lens would it be and what focal length would it have?

An object is placed \(15 \mathrm{~cm}\) from a certain mirror. The image is half the size of the object, inverted, and real. How far is the image from the mirror, and what is the radius of curvature of the mirror?

(a) Show that the lens equation can be written in the Newtonian form \(x x^{\prime}=f^{2}\), where x is the distance of the object from the focal point on the front side of the lens, and x? is the distance of the image to the focal point on the other side of the lens. Calculate the location of an image if the object is placed 45.0 cm in front of a convex lens with a focal length f of 32.0 cm using (b) the standard form of the thin lens equation, and (c) the Newtonian form, stated above.

Problem 93GP A converging lens with focal length of 10.0 cm is placed in contact with a diverging lens with a focal length of ?20.0 cm. What is the focal length of the combination, and is the combination converging or diverging?