Chapter 34

A candle 4.85 \(\mathrm{cm}\) tall is 39.2 \(\mathrm{cm}\) to the left of a plane mirror. Where is the image formed by the mirror, and what is the height of this image?

The image of a tree just covers the length of a plane mirror 4.00 \(\mathrm{cm}\) tall when the mirror is held 35.0 \(\mathrm{cm}\) from the eye. The tree is 28.0 \(\mathrm{m}\) from the mirror. What is its height?

A pencil that is 9.0 \(\mathrm{cm}\) long is held perpendicular to the surface of a plane mirror with the tip of the pencil lead 12.0 \(\mathrm{cm}\) from the mirror surface and the end of the eraser 21.0 \(\mathrm{cm}\) from the mirror surface. What is the length of the image of the pencil that is formed by the mirror? Which end of the image is closer to the mirror surface: the tip of the lead or the end of the eraser?

A concave mirror has a radius of curvature of 34.0 \(\mathrm{cm} .\) (a) What is its focal length? (b) If the mirror is immersed in water (refractive index 1.33), what is its focal length?

An object 0.600 \(\mathrm{cm}\) tall is placed 16.5 \(\mathrm{cm}\) to the left of the vertex of a concave spherical mirror having a radius of curvature of 22.0 \(\mathrm{cm} .\) (a) Draw a principal-ray diagram showing the formation of the image. (b) Determine the position, size, orientation, and nature (real or virtual) of the image.

An object is 24.0 \(\mathrm{cm}\) from the center of a silvered spherical glass Christmas tree ornament 6.00 \(\mathrm{cm}\) in diameter. What are the position and magnification of its image?

A coin is placed next to the convex side of a thin spherical glass shell having a radius of curvature of 18.0 \(\mathrm{cm} .\) Reflection from the surface of the shell forms an image of the \(1.5-\) cm-tall coin that is 6.00 \(\mathrm{cm}\) behind the glass shell. Where is the coin located? Determine the size, orientation, and nature (real or virtual) of the image.

You hold a spherical salad bowl 90 \(\mathrm{cm}\) in front of your face with the bottom of the bowl facing you. The salad bowl is made of polished metal with a \(35-\) cm radius of curvature. (a) Where is the image of your 2.0 -cm- tall nose located? (b) What are the image's size, orientation, and nature (real or virtual)?

Dental Mirror. A dentist uses a curved mirror to view teeth on the upper side of the mouth. Suppose she wants an erect image with a magnification of 2.00 when the mirror is 1.25 \(\mathrm{cm}\) from a tooth. (Treat this problem as though the object and image lie along a straight line.) (a) What kind of mirror (concave or convex) is needed? Use a ray diagram to decide, without performing any calculations. (b) What must be the focal length and radius of curvature of this mirror? (c) Draw a principal-ray diagram to check your answer in part (b).

A spherical, concave shaving mirror has a radius of curvature of 32.0 \(\mathrm{cm}\) . (a) What is the magnification of a person's face when it is 12.0 \(\mathrm{cm}\) to the left of the vertex of the mirror? (b) Where is the image? Is the image real or virtual? (c) Draw a principal-ray diagram showing the formation of the image.

A speck of dirt is embedded 3.50 \(\mathrm{cm}\) below the surface of a sheet of ice \((n=1.309) .\) What is its apparent depth when viewed at normal incidence?

A tank whose bottom is a mirror is filled with water to a depth of 20.0 \(\mathrm{cm} .\) A small fish floats motionless 7.0 \(\mathrm{cm}\) under the surface of the water. (a) What is the apparent depth of the fish when viewed at normal incidence? (b) What is the apparent depth of the image of the fish when viewed at normal incidence?

A person swimming 0.80 \(\mathrm{m}\) below the surface of the water in a swimming pool looks at the diving board that is directly overhead and sees the image of the board that is formed by refraction at the surface of the water. This image is a height of 5.20 \(\mathrm{m}\) above the swimmer. What is the actual height of the diving board above the surface of the water?

A person is lying on a diving board 3.00 m above the surface of the water in a swimming pool. The person looks at a penny that is on the bottom of the pool directly below her. The penny appears to the person to be a distance of 8.00 \(\mathrm{m}\) from her. What is the depth of the water at this point?

The left end of a long glass rod 6.00 \(\mathrm{cm}\) in diameter has a convex hemispherical surface 3.00 \(\mathrm{cm}\) in radius. The refractive index of the glass is 1.60. Determine the position of the image if an object is placed in air on the axis of the rod at the following distances to the left of the vertex of the curved end: (a) infinitely far, (b) \(12.0 \mathrm{cm} ;(c) 2.00 \mathrm{cm} .\)

The left end of a long glass rod 8.00 \(\mathrm{cm}\) in diameter, with an index of refraction of \(1.60,\) is ground and polished to a convex hemispherical surface with a radius of 4.00 \(\mathrm{cm} .\) An object in the form of an arrow 1.50 \(\mathrm{mm}\) tall, at right angles to the axis of the rod, is located on the axis 24.0 \(\mathrm{cm}\) to the left of the vertex of the convex surface. Find the position and height of the image of the arrow formed by paraxial rays incident on the convex surface. Is the image erect or inverted?

An insect 3.75 mm tall is placed 22.5 \(\mathrm{cm}\) to the left of a thin planoconvex lens. The left surface of this lens is flat, the right surface has a radius of curvature of magnitude \(13.0 \mathrm{cm},\) and the index of refraction of the lens material is 1.70. (a) Calculate the location and size of the image this lens forms of the insect. Is it real or virtual? Erect or inverted? (b) Repeat part (a) if the lens is reversed.

A lens forms an image of an object. The object is 16.0 \(\mathrm{cm}\) from the lens. The image is 12.0 \(\mathrm{cm}\) from the lens on the same side as the object. (a) What is the focal length of the lens? Is the lens converging or diverging? (b) If the object is 8.50 \(\mathrm{mm}\) tall, how tall is the image? Is it erect or inverted? (c) Draw a principal-ray diagram.

A converging lens with a focal length of 90.0 \(\mathrm{cm}\) forms an image of a 3.20 -cm-tall real object that is to the left of the lens. The image is 4.50 \(\mathrm{cm}\) tall and inverted. Where are the object and image located in relation to the lens? Is the image real or virtual?

A converging lens forms an image of an 8.00 -mm-tall real object. The image is 12.0 \(\mathrm{cm}\) to the left of the lens, 3.40 \(\mathrm{cm}\) tall, and erect. What is the focal length of the lens? Where is the object located?

A photographic slide is to the left of a lens. The lens projects an image of the slide onto a wall 6.00 \(\mathrm{m}\) to the right of the slide. The image is 80.0 times the size of the slide. (a) How far is the slide from the lens? (b) Is the image erect or inverted? (c) What is the focal length of the lens? (d) Is the lens converging or diverging?

A double-convex thin lens has surfaces with equal radii of curvature of magnitude 2.50 \(\mathrm{cm} .\) Looking through this lens, you observe that it forms an image of a very distant tree at a distance of 1.87 \(\mathrm{cm}\) from the lens. What is the index of refraction of the lens?

The Lens of the Eye. The crystalline lens of the human eye is a double-convex lens made of material having an index of refraction of 1.44 (although this varies). Its focal length in air is about 8.0 \(\mathrm{mm}\) , which also varies. We shall assume that the radii of curvature of its two surfaces have the same magnitude. (a) Find the radii of curvature of this lens. (b) If an object 16 \(\mathrm{cm}\) tall is placed 30.0 \(\mathrm{cm}\) from the eye lens, where would the lens focus it and how tall would the image be? Is this image real or virtual? Is it erect or inverted? (Note: The results obtained here are not strictly accurate because the lens is embedded in fluids having refractive indexes different from that of air.)

The Cornea As a Simple Lens. The cornea behaves as a thin lens of focal length approximately \(1.8 \mathrm{cm},\) although this varies a bit. The material of which it is made has an index of refraction of \(1.38,\) and its front surface is convex, with a radius of curvature of 5.0 \(\mathrm{mm} .\) (a) If this focal length is in air, what is the radius of curvature of the back side of the cornea? (b)The closest distance at which a typical person can focus on an object (called the near point) is about \(25 \mathrm{cm},\) although this varies considerably with age. Where would the cornea focus the image of an 8.0 -mm- tall object at the near point? (c) What is the height of the image in part (b)? Is this image real or virtual? Is it erect or inverted? (Note: The results obtained here are not strictly accurate because, on one side, the cornea has a fluid with a refractive index different from that of air.)

A converging lens with a focal length of 7.00 \(\mathrm{cm}\) forms an image of a \(4.00-\mathrm{mm}\)-tall real object that is to the left of the lens. The image is 1.30 \(\mathrm{cm}\) tall and erect. Where are the object and image located? Is the image real or virtual?

A converging lens with a focal length of 12.0 \(\mathrm{cm}\) forms a virtual image 8.00 \(\mathrm{mm}\) tall, 17.0 \(\mathrm{cm}\) to the right of the lens. Determine the position and size of the object. Is the image erect or inverted? Are the object and image on the same side or opposite sides of the lens? Draw a principal-ray diagram for this situation.

An object is 16.0 \(\mathrm{cm}\) to the left of a lens. The lens forms an image 36.0 \(\mathrm{cm}\) to the right of the lens. (a) What is the focal length of the lens? Is the lens converging or diverging? (b) If the object is 8.00 mm tall, how tall is the image? Is it erect or inverted? (c) Draw a principal-ray diagram.

Combination of Lenses I. A 1.20 -cm-tall object is 50.0 \(\mathrm{cm}\) to the left of a converging lens of focal length 40.0 \(\mathrm{cm} . \mathrm{A}\) second converging lens, this one having a focal length of \(60.0 \mathrm{cm},\) is located 300.0 \(\mathrm{cm}\) to the right of the first lens along the same optic axis. (a) Find the location and height of the image (call it \(I_{1} )\) formed by the lens with a focal length of 40.0 \(\mathrm{cm} .\) (b) \(I_{1}\) is now the object for the second lens. Find the location and height of the image produced by the second lens. This is the final image produced by the combination of lenses.

Combination of Lenses II. Two thin lenses with a focal length of magnitude \(12.0 \mathrm{cm},\) the first diverging and the second converging, are located 9.00 \(\mathrm{cm}\) apart. An object 2.50 \(\mathrm{mm}\) tall is placed 20.0 \(\mathrm{cm}\) to the left of the first (diverging) lens. (a) How far from this first lens is the final image formed? (b) Is the final image real or virtual? (c) What is the height of the final image? Is it erect or inverted? (Hint: See the preceding two problems.)

You wish to project the image of a slide on a screen 9.00 \(\mathrm{m}\) from the lens of a slide projector. (a) If the slide is placed 15.0 \(\mathrm{cm}\) from the lens, what focal length lens is required? (b) If the dimensions of the picture on a \(35-\mathrm{mm}\) color slide are 24 \(\mathrm{mm} \times 36 \mathrm{mm}\) , what is the minimum size of the projector screen required to accommodate the image?

When a camera is focused, the lens is moved away from or toward the film. If you take a picture of your friend, who is standing 3.90 m from the lens, using a camera with a lens with a 85 -mm focal length, how far from the film is the lens? Will the whole image of your friend, who is 175 \(\mathrm{cm}\) tall, fit on film that is \(24 \times 36 \mathrm{mm} ?\)

Figure 34.41 shows photographs of the same scene taken with the same camera with lenses of different focal length. If the object is 200 \(\mathrm{m}\) from the lens, what is the magnitude of the lateral magnification for a lens of focal length (a) \(28 \mathrm{mm} ;\) (b) 105 \(\mathrm{mm}\) ; (c) 300 \(\mathrm{mm}\) ?

A photographer takes a photograph of a Boeing 747 airliner (length 70.7 \(\mathrm{m} )\) when it is flying directly overhead at an altitude of 9.50 \(\mathrm{km} .\) The lens has a focal length of 5.00 \(\mathrm{m} .\) How long is the image of the airliner on the film?

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

Recall that the intensity of light reaching film in a camera is proportional to the effective area of the lens. Camera A has a lens with an aperture diameter of 8.00 \(\mathrm{mm} .\) It photographs an object using the correct exposure time of \(\frac{1}{30} \mathrm{s}\) . What exposure time should be used with camera \(\mathrm{B}\) in photographing the same object with the same film if this camera has a lens with an aperture diameter of 23.1 \(\mathrm{mm}\) ?

Photography. A 35 -mm camera has a standard lens with focal length 50 \(\mathrm{mm}\) and can focus on objects between 45 \(\mathrm{cm}\) and infinity. (a) Is the lens for such a camera a concave or a convex lens? (b) The camera is focused by rotating the lens, which moves it on the camera body and changes its distance from the film. In what range of distances between the lens and the film plane must the lens move to focus properly over the 45 \(\mathrm{cm}\) to infinity range?

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 refraction occurs at the cornea, whose vertex is 2.60 \(\mathrm{cm}\) from the retina. What should be the radius of curvature of the cornea such that the image of an object 40.0 \(\mathrm{cm}\) 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 \(\mathrm{cm}\) from his eyes instead of the usual 25.0 \(\mathrm{cm} .(\mathrm{a})\) 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?

A person can see clearly up close but cannot focus on objects beyond 75.0 \(\mathrm{cm} .\) 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?

A thin lens with a focal length of 6.00 \(\mathrm{cm}\) 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 \(\mathrm{cm}\) from the eye, and that the lens is very close to the eye.

The focal length of a simple magnifier is 8.00 \(\mathrm{cm} .\) 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 observer's near point, 25.0 \(\mathrm{cm}\) in front of her eye? (b) If the object is 1.00 mm high, what is the height of its image formed by the magnifier?

You want to view an insect 2.00 \(\mathrm{mm}\) in length through a magnifier. 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 0.025 radian?

A certain microscope is provided with objectives that have focal lengths of \(16 \mathrm{mm}, 4 \mathrm{mm},\) and 1.9 \(\mathrm{mm}\) and with eye-pieces that have angular magnifications of \(5 \times\) and \(10 \times .\) Each objective forms an image 120 \(\mathrm{mm}\) beyond its second focal point. Determine (a) the largest overall angular magnification obtainable and (b) the smallest overall angular magnification obtainable.

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

The focal length of the eyepiece of a certain microscope is 18.0 \(\mathrm{mm}\) . The focal length of the objective is 8.00 \(\mathrm{mm}\) . The distance between objective and eyepiece is 19.7 \(\mathrm{cm} .\) 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?

A telescope is constructed from two lenses with focal lengths of 95.0 \(\mathrm{cm}\) and \(15.0 \mathrm{cm},\) the 95.0 -cm lens 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 \(\mathrm{m}\) tall, 3.00 \(\mathrm{km}\) away. (c) What is the angular size of the final image as viewed by an eye very close to the eyepiece?

Where must you place an object in front of a concave mirror with radius \(R\) so that the image is erect and 2\(\frac{1}{2}\) times the size of the object? Where is the image?

If you run away from a plane mirror at \(3.60 \mathrm{m} / \mathrm{s},\) at what speed does your image move away from you?

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 2.25 times the size of the object. How far should the mirror be from the wall? What should its radius of curvature be?