Optics

The focal length of a simple lens depends on the color (wavelength) of light passing through it. Why? Is it possible for a lens to have a positive focal length for some colors and negative for others? Explain.

When a converging lens is immersed in water, does itsfocal length increase or decrease in comparison with the value inair? Explain.

A spherical air bubble in water can function as a lens. Is it a converging or diverging lens? How is its focal length related to its radius?

Can an image formed by one reflecting or refractingsurface serve as an object for a second reflection or refraction?Does it matter whether the first image is real or virtual? Explain.

According to the discussion in Section 34.2, light rays are reversible. Are the formulas in the table in this chapter’s Summary still valid if object and image are interchanged? What does reversibility imply with respect to the forms of the various formulas?

You can’t see clearly underwater with the naked eye, but you can if you wear a face mask or goggles (with air between your eyes and the mask or goggles). Why is there a difference? Could you instead wear eyeglasses (with water between your eyes and the eyeglasses) in order to see underwater? If so, should the lenses be converging or diverging? Explain.

You take a lens and mask it so that light can pass through only the bottom half of the lens. How does the image formed by the masked lens compare to the image formed before masking?

You take a lens and mask it so that light can pass through only the bottom half of the lens. How does the image formed by the masked lens compare to the image formed before masking?

An object 0.6 cm tall is placed 16.5 cm to the left of the vertex of a concave spherical mirror having a radius of curvature of 22 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 0.6cm tall is placed 16.5cm to the left of the vertex of a concave spherical mirror having a radius of curvature of 22cm. (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.

The diameter of Mars is 6794km, and its minimum distance from the earth is $\mathbf{5}\mathbf{.}\mathbf{58}\mathbf{\times }{\mathbf{10}}^{\mathbf{7}}\mathit{k}\mathit{m}$. When Mars is at this distance, find the diameter of the image of Mars formed by a spherical, concave telescope mirror with a focal length of 1.75m.

An object is 18cm from the centre of a spherical silvered-glass Christmas tree ornament 6cm 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 18cm. Reflection from the surface of the shell forms an image of the 1.5cm tall coin that is 6cm behind the glass shell. Where is the coin located? Determine the size, orientation, and nature (real or virtual) of the image.

A spherical, concave shaving mirror has a radius of curvature of 32.0 cm. (a) What is the magnification of a person’s face when it is 12.0 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.

34.12 For a concave spherical mirror that has focal length f=+18 cm, what is the distance of an object from the mirror’s vertex if the image is real and has the same height as the object?

34.13 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 1.25cm 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).

34.14 For a convex spherical mirror that has focal length $\mathbf{f}\mathbf{=}\mathbf{-}\mathbf{12}\mathbf{cm}$, what is the distance of an object from the mirror’s vertex if the height of the image is half the height of the object?
 

34.15 The thin glass shell shown in Fig. E34.15 has a spherical shape with a radius of curvature of 12cm, and both of its surfaces can act as mirrors. A seed  high is placed 15.0cm from the center of the mirror along the optic axis, as shown in the figure. (a) Calculate the location and height of the image of this seed. (b) Suppose now that the shell is reversed. Find the location and height of the seed’s image.

34.16 A tank whose bottom is a mirror is filled with water to a depth of 20.0cm. A small fish floats motionless 7.0cm 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?

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

34.18 A transparent liquid fills a cylindrical tank to a depth of 3.60m. There is air above the liquid. You look at normal incidence at a small pebble at the bottom of the tank. The apparent depth of the pebble below the liquid’s surface is 2.45m. What is the refractive index of this liquid?

34.19 A person swimming 0.80cm 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 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. She looks at a penny that is on the bottom of the pool directly below her. To her, the penny appears to be a distance of 7.00 m from her. What is the depth of the water at this point?

A Spherical Fish Bowl. A small tropical fish is at the centre of a water-filled, spherical fish bowl 28.0 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?

The left end of a long glass rod 6.00 cm in diameter has a convex hemispherical surface 3.00 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 cm; (c) 2.00 cm.

The glass rod of Exercise 34.22 is immersed in oil (n = 1.452). An object placed to the left of the rod on the rod’s axis is to be imaged 1.20 m inside the rod. How far from the left end of the rod must the object be located to form the image?

The left end of a long glass rod 8.00 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 cm. An object in the form of an arrow 1.50 mm tall, at right angles to the axis of the rod, is located on the axis 24.0 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?

Repeat Exercise 34.24 for the case in which the end of the rod is ground to a concave hemispherical surface with a radius of 4.00 cm.

The glass rod of Exercise 34.25 is immersed in a liquid. An object 14.0 cm from the vertex of the left end of the rod and on its axis is imaged at a point 9.00 cm from the vertex inside the liquid. What is the index of refraction of the liquid?

An insect 3.75 mm tall is placed 22.5 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 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 cm from the lens. The image is 12.0 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 mm tall, how tall is the image? Is it erect or inverted? 

(c) Draw a principal-ray diagram.

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

A converging lens with a focal length of 70.0 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 cm tall and inverted. Where are the object and image located in relation to the lens? Is the image real or virtual?

A photographic slide is to the left of a lens. The lens projectsan image of the slide onto a wall 6.00 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 2.50cm  . Using this lens, you observe that it forms an image of a very distant tree at a distance of  1.87cm from the lens. What is the index of refraction of the lens?

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

The cornea behaves as a thin lens of focal length approximately1.8 cm, although this varies a bit. The material of which it is made has an index of refraction of 1.38 cm , and its front surface is convex, with a radius of curvature of 5.0 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 25cm, although this varies considerably with age. Where would the cornea focus the image of an 8.00 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 lensmaker wants to make a magnifying glass from glass that has an index of refraction n = 1.55 and a focal length of 20.0 cm. If the two surfaces of the lens are to have equal radii, what should that radius be?

For each thin lens shown in Fig. E34.37, calculate the location of the image of an object that is 1.80 cm to the left of the lens. The lens material has a refractive index of 1.50, and the radii of curvature shown are only the magnitudes.

A converging lens with a focal length of 12.0 cm forms a virtual image 8.00 mm tall, 17.0 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.

Repeat Exercise 34.38 for the case in which the lens is diverging, with a focal length of -48.0 cm .

An object is 16.0 cm to the left of a lens. The lens forms an image 36.0 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 8.0 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 cm to the left of a converging lens of focal length 40.0 cm. A second converging lens, this one having a focal length of 60.0 cm, is located 300.0 cm to the right of the first lens along the same optic axis. (a) Find the location and height of the image (call it I₁) formed by the lens with a focal length of 40.0 cm. (b) I₁ 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: Repeat previous question using the same lenses except for the following changes: (a) The second lens is a diverging lens having a focal length of magnitude 60.0 cm. (b) The first lens is a diverging lens having a focal length of magnitude 40.0 cm. (c) Both lenses are diverging lenses having focal lengths of the same magnitudes as before.

Combination of Lenses III: Two thin lenses with a focal length of magnitude 12.0 cm, the first diverging and the second converging, are located 9.00 cm apart. An object 2.50 mm tall is placed 20.0 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.)

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 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 cm tall is placed 30.0 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.)

A camera lens has a focal length of 200mm . How far from the lens should the subject for the photo be if the lens is 20.4cm from the sensor?

You wish to project the image of a slide on a screen 9.00m from the lens of a slide projector. (a) If the slide is placed 15.0cm from the lens, what focal length lens is required? (b) If the dimensions of the picture on a 35 mm color slide are 24mm x 36mm, 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 digital image sensor. If you take a picture of your friend, who is standing 3.90 m from the lens, using a camera with a lens with an 85-mm focal length, how far from the sensor is the lens? Will the whole image of your friend, who is 175 cm tall, fit on a sensor that is 24 mm * 36 mm?

Zoom Lens: Consider the simple model of the zoom lens as shown in (A). The converging lens has focal length f₁ = 12 cm, and the diverging lens has focal length f₂ = -12 cm. The lenses are separated by 4 cm. (a) For a distant object, where is the image of the converging lens? (b) The image of the converging lens serves as the object for the diverging lens. What is the object distance for the diverging lens? (c) Where is the final image? (d) Repeat parts (a), (b), and (c) for the situation shown in (B), in which the lenses are separated by 8 cm.