Chapter 24

\(\cdot\) 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?

What is the size of the smallest vertical plane mirror in which a 10 ft tall giraffe standing erect can see her full-length image? (Hint: Locate the image by drawing a number of rays from the giraffe's body that reflect off the mirror and go to her eye. Then eliminate that part of the mirror for which the reflected rays do not reach her eye.)

An object is placed between two plane mirrors arranged at right angles to each other at a distance \(d_{1}\) from the surface of one mirror and a distance \(d_{2}\) from the surface of the other. (a) How many images are formed? Show the location of the images in a diagram.

\(\bullet\) If you run away from a plane mirror at \(2.40 \mathrm{m} / \mathrm{s},\) at wha speed does your image move away from you?

A concave spherical mirror has a radius of curvature of 10.0 \(\mathrm{cm} .\) Calculate the location and size of the image formed of an 8.00 -mm-tall object whose distance from the mirror is (a) \(15.0 \mathrm{cm},(\mathrm{b}) 10.0 \mathrm{cm},(\mathrm{c}) 2.50 \mathrm{cm},\) and (d) 10.0 \(\mathrm{m}\)

\(\cdot\) A concave mirror has a radius of curvature of 34.0 \(\mathrm{cm}\) . (a) What is its focal length? (b) A ladybug 7.50 \(\mathrm{mm}\) tall is located 22.0 \(\mathrm{cm}\) from this mirror along the principal axis. Find the location and height of the image of the insect. (c) If the mirror is immersed in water (of refractive index \(1.33 ),\) what is its focal length?

Rearview mirror. A mirror on the passenger side of your car is convex and has a radius of curvature with magnitude 18.0 \(\mathrm{cm} .\) (a) Another car is seen in this side mirror and is 13.0 \(\mathrm{m}\) behind the mirror. If this car is 1.5 \(\mathrm{m}\) tall, what is the height of its image? (b) The mirror has a warning attached that objects viewed in it are closer than they appear. Why is this so?

\(\bullet\) Examining your image in a convex mirror whose radius of curvature is \(25.0 \mathrm{cm},\) you stand with the tip of your nose 10.0 \(\mathrm{cm}\) from the surface of the mirror. (a) Where is the image of your nose located? What is its magnification? (b) Your ear is 10.0 \(\mathrm{cm}\) behind the tip of your nose; where is the image of your ear located, and what is its magnification? Do your answers suggest reasons for your strange appearance in a convex mirror?

A \(\mathrm{A}\) coin is placed next to the convex side of a thin spherical glass shell having a radius of curvature of 18.0 \(\mathrm{cm} .\) An image of the 1.5 -cm-tall coin is formed 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.

\(\bullet\) (a) Show that when an object is outside the focal point of a concave mirror, its image is always inverted and real. Is there any limitation on the magnification? (b) Show that when an object is inside the focal point of a concave mirror, its image is always erect and virtual. Is there any limitation on the magnification?

A spherical, concave shaving mirror has a radius of curva- ture 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.

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) Calculate the position, size, orientation (erect or inverted), and nature (real or virtual) of the image.

\(\cdot\) The stainless steel rear end of a tanker truck is convex, shiny, and has a radius of curvature of 2.0 \(\mathrm{m} .\) You're tailgating the truck, with the front end of your car only 5.0 \(\mathrm{m}\) behind it. Making the not very realistic assumption that your car is on the axis of the mirror formed by the tank, (a) determine the position, orientation, magnification, and type (real or virtual) of the image of your car's front end that forms in this mirror; (b) draw a principal-ray diagram of the situation to check your answer.

\(\cdot\) 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) \(12.0 \mathrm{cm},\) and \((c) 2.00 \mathrm{cm} .\)

\(\cdot\) The left end of a long glass rod 8.00 \(\mathrm{cm}\) in diameter and 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 inci- dent on the convex surface. Is the image erect or inverted?

A large aquarium has portholes of thin transparent plastic with a radius of curvature of 1.75 \(\mathrm{m}\) and their convex sides fac- ing into the water. A shark hovers in front of a porthole, sizingup the dinner prospects outside the tank. (a) If one of the shark's teeth is exactly 45.0 \(\mathrm{cm}\) from the plastic, how far from the plastic does it appear to be to observers outside the tank?You can ignore refraction due to the plastic.) (b) Does the shark appear to be right side up or upside down? (c) If the tooth has an actual length of 5.00 \(\mathrm{cm}\) , how long does it appear to the observers?

A spherical fishbowl. A small tropical fish is at the center of a water-filled spherical fishbowl \(28.0 \mathrm{~cm}\) in diameter. (a) Find the apparent position and magnification of the fish to an observer outside the bowl. The effect of the thin walls of the bowl may be ignored. (b) A friend advised the owner of the bowl to keep it out of direct sunlight to avoid blinding the fish, which might swim into the focal point of the parallel rays from the sun. Is the focal point actually within the bowl?

Focus of the eye. The cornea of the eye has a radius of curvature of approximately \(0.50 \mathrm{cm},\) and the aqueous humor bbehind it has an index of refraction of \(1.35 .\) The thickness of the cornea itself is small enough that we shall neglect it. The depth of a typical human eye is around 25 5 \(\mathrm{mm}\) . (a) What would have to be the radius of curvature of the cornea so that it alone would focus the image of a distant mountain on the retina, which is at the back of the eye opposite the cornea? (b) If the cornea focused the mountain correctly on the retina as described in part \((a),\) would it also focus the text from a computer screen on the retina if that screen were 25 \(\mathrm{cm}\) in front of the eye? If not, where would it focus that text, in front of or behind the retina? (c) Given that the cornea has a radius of curvature of about \(5.0 \mathrm{mm},\) where does it actually focus the mountain? Is this in front of or behind the retina? Does this help you see why the eye needs help from a lens to complete the task of focusing?

A speck of dirt is embedded 3.50 \(\mathrm{cm}\) below the surface of a sheet of ice having a refractive index of \(1.309 .\) What is the apparent depth of the speck, when viewed from directly above?

\(\cdot\) A skin diver is 2.0 \(\mathrm{m}\) below the surface of a lake. A bird flies overhead 7.0 \(\mathrm{m}\) above the surface of the lake. When the bird is directly overhead, how far above the diver does it appear to be?

\(\bullet \mathrm{A}\) zoo aquarium has transparent walls, so that spectators on both sides of it can watch the fish. The aquarium is 5.50 \(\mathrm{m}\) across, and the spectators on both sides of it are standing 1.20 \(\mathrm{m}\) from the wall. How far away do spectators on one side of the aquarium appear to those on the other side? (Ignore any refraction in the walls of the aquarium.)

\(\cdot\) To a person swimming 0.80 \(\mathrm{m}\) beneath the surface of the water in a swimming pool, the diving board directly overhead appears to be 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?

\(\cdot\) A converging lens with a focal length of 7.00 \(\mathrm{cm}\) forms an image of a 4.00 -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 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?

\(\cdot\) You are standing in front of a lens that projects an image of you onto a wall 1.80 \(\mathrm{m}\) on the other side of the lens. This image is three times your height. (a) How far are you from the lens? (b) Is your image erect or inverted? (c) What is the focal length of the lens? Is the lens converging or diverging?

\(\cdot\) The two surfaces of a plastic converging lens have equal radii of curvature of \(22.0 \mathrm{cm},\) and the lens has a focal length of 20.0 \(\mathrm{cm} .\) Calculate the index of refraction of the plastic.

\(\cdot\) The front, convex, surface of a lens made for eyeglasses has a radius of curvature of \(11.8 \mathrm{cm},\) and the back, concave, surface has a radius of curvature of 6.80 \(\mathrm{cm} .\) The index of refraction of the plastic lens material is 1.67 . Calculate the local length of the lens.

\(\cdot\) 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 thisimage 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.)

\(\cdot\) An insect 3.75 \(\mathrm{mm}\) 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) Calcu- late 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 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 refrac- tion of the lens?

(. A converging meniscus lens (see Fig. 24.31\()\) with a refrac- tive index of 1.52 has spherical surfaces whose radii are 7.00 \(\mathrm{cm}\) and 4.00 \(\mathrm{cm} .\) What is the position of the image if an object is placed 24.0 \(\mathrm{cm}\) to the left of the lens? What is the magnification?

\(\bullet\) A converging lens with a focal length of 12.0 \(\mathrm{cm}\) forms 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 o opposite sides of the lens?

Focus of the eye. The cornea of the eye has a radius of curvature of approximately \(0.50 \mathrm{~cm},\) and the aqueous humor behind it has an index of refraction of \(1.35 .\) The thickness of the cornea itself is small enough that we shall neglect it. The depth of a typical human eye is around \(25 \mathrm{~mm}\). (a) What would have to be the radius of curvature of the cornea so that it alone would focus the image of a distant mountain on the retina, which is at the back of the eye opposite the cornea? (b) If the cornea focused the mountain correctly on the retina as described in part (a), would it also focus the text from a computer screen on the retina if that screen were \(25 \mathrm{~cm}\) in front of the eye? If not, where would it focus that text, in front of or behind the retina? (c) Given that the cornea has a radius of curvature of about \(5.0 \mathrm{~mm},\) where does it actually focus the mountain? Is this in front of or behind the retina? Does this help you see why the eye needs help from a lens to complete the task of focusing?

\(\cdot(\) a) You want to use a lens with a focal length of 35.0 \(\mathrm{cm}\) to produce a real image of an object, with the image twice as long as the object itself. What kind of lens do you need, and where should the object be placed? (b) Suppose you want a virtual image of the same object, with the same magnification - what kind of lens do you need, and where should the object be placed?

A converging lens has a focal length of 14.0 \(\mathrm{cm} .\) For each of two objects located to the left of the lens, one at a distance of 18.0 \(\mathrm{cm}\) and the other at a distance of \(7.00 \mathrm{cm},\) determine (a) the image position, (b) the magnification, (c) whether the image is real or virtual, and (d) whether the image is erect or inverted. Draw a principal-ray diagram in each case.

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. (a) What is the focal length of the lens? (b) Where is the object located? (c) Draw a principal-ray dia- gram for this situation.

A diverging lens with a focal length of \(-48.0 \mathrm{cm}\) forms a virtual image 8.00 \(\mathrm{mm}\) tall, 17.0 \(\mathrm{cm}\) to the right of the lens. (a) 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? (b) Draw a principal-ray diagram for this situation.

\(\cdot\) When an object is 16.0 \(\mathrm{cm}\) from a lens, an image is formed 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.

\bullet A layer of benzene \((n=1.50) 2.60 \mathrm{cm}\) deep floats on water \((n=1.33)\) that is 6.50 \(\mathrm{cm}\) deep. What is the apparent distance from the upper benzene surface to the bottom of the water laver when it is viewed at normal incidence?

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

A luminous object is 4.00 \(\mathrm{m}\) from a wall. You are to use a concave mirror to project an image of the object 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?

A concave mirror is to form an image of the filament of a headlight lamp on a screen 8.00 m from the mirror. The filament is 6.00 mm tall, and the image is to be 36.0 \(\mathrm{cm}\) tall. (a) How far in front of the vertex of the mirror should the filament be placed? (b) To what radius of curvature should you grind the mirror?

\(\cdot\) A 3.80 -mm-tall object is 24.0 \(\mathrm{cm}\) from the center of a sil- vered spherical glass Christmas tree ornament 6.00 \(\mathrm{cm}\) in diameter, What are the position and height of its image?

A A lensmaker wants to make a magnifying glass from glass with \(n=1.55\) and with a focal length of 20.0 \(\mathrm{cm} .\) If the two surfaces of the lens are to have equal radii, what should that radius be?

An object is placed 18.0 \(\mathrm{cm}\) from a screen. (a) At what two points between object and screen may a converging lens with a 3.00 \(\mathrm{cm}\) focal length be placed to obtain an image on the screen? (b) What is the magnification of the image for each position of the lens?

If you place a concave glass lens into a tank of a liquid that has an index of refraction that is greater than that of the lens, what will happen? A. The lens will no longer be able to create any images. B. The focal length of the lens will become longer. C. The focal length of the lens will become shorter. D. The lens will become a converoing lens

If you place a concave mirror with a focal length of 1 \(\mathrm{m}\) into a liquid that has an index of refraction of \(3,\) what will happen? A. The mirror will no longer be able to focus light. B. The focal length of the mirror will decrease. C. The focal length of the mirror will increase. D. Nothing will happen.