Chapter 34

A concave mirror is to form an image of the filament of a headlight lamp on a screen 8.00 \(\mathrm{m}\) from the mirror. The filament is 6.00 \(\mathrm{mm}\) tall, and the image is to be 24.0 \(\mathrm{cm}\) tall. (a) How far in front of the vertex of the mirror should the filament be placed? (b) What should be the radius of curvature of the mirror?

Rear-View 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 behind your car, 9.00 m from the mirror, and this car is viewed in the mirror by your passenger. If this car is 1.5 \(\mathrm{m}\) tall, what is the height of the image? (b) The mirror has a warning attached that objects viewed in it are closer than they appear. Why is this so?

A layer of benzene \((n=1.50) 4.20 \mathrm{cm}\) 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 layer when it is viewed at normal incidence?

cp calc You are in your car driving on a highway at 25 \(\mathrm{m} / \mathrm{s}\) when you glance in the passenger-side mirror (a convex mirror with radius of curvature 150 \(\mathrm{cm}\) ) and notice a truck approaching. If the image of the truck is approaching the vertex of the mirror at a speed of 1.9 \(\mathrm{m} / \mathrm{s}\) when the truck is 2.0 \(\mathrm{m}\) from the mirror, what is the speed of the truck relative to the highway?

Pinhole Camera. A pinhole camera is just a rectangular box with a tiny hole in one face. The film is on the face opposite this hole, and that is where the image is formed. The camera forms an image without a lens.(a) Make a clear ray diagram to show how a pinhole camera can form an image on the film without using a lens. (Hint: Put an object outside the hole, and then draw rays passing through the hole to the opposite side of the box.) (b) A certain pinhole camera is a box that is 25 \(\mathrm{cm}\) square and 20.0 \(\mathrm{cm}\) deep, with the hole in the middle of one of the 25 \(\mathrm{cm} \times 25 \mathrm{cm}\) faces. If this camera is used to photograph a fierce chicken that is 18 \(\mathrm{cm}\) high and 1.5 \(\mathrm{m}\) in front of the camera, how large is the image of this bird on the film? What is the magnification of this camera?

A Glass Rod. Both ends of a glass rod with index of refraction 1.60 are ground and polished to convex hemispherical surfaces. The radius of curvature at the left end is \(6.00 \mathrm{cm},\) and the radius of curvature at the right end is 12.0 \(\mathrm{cm} .\) The length of the rod between vertices is 40.0 \(\mathrm{cm} .\) The object for the surface at the left end is an arrow that lies 23.0 \(\mathrm{cm}\) to the left of the vertex of this surface. The arrow is 1.50 \(\mathrm{mm}\) tall and at right angles to the axis. (a) What constitutes the object for the surface at the right end of the rod? (b) What is the object distance for this surface? (c) Is the object for this surface real or virtual? (d) What is the position of the final image? (e) Is the final image real or virtual? Is it erect or inverted with respect to the original object? (f) What is the height of the final image?

A transparent rod 30.0 \(\mathrm{cm}\) long is cut flat at one end and rounded to a hemispherical surface of radius 10.0 \(\mathrm{cm}\) at the other end. A small object is embedded within the rod along its axis and halfway between its ends, 15.0 \(\mathrm{cm}\) from the flat end and 15.0 \(\mathrm{cm}\) from the vertex of the curved end. When viewed from the flat end of the rod, the apparent depth of the object is 9.50 \(\mathrm{cm}\) from the flat end. What is its apparent depth when viewed from the curved end?

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?

A transparent rod 50.0 \(\mathrm{cm}\) long and with a refractive index of 1.60 is cut flat at the right end and rounded to a hemispherical surface with a 15.0 -cm radius at the left end. An object is placed on the axis of the rod 12.0 \(\mathrm{cm}\) to the left of the vertex of the hemispherical end. (a) What is the position of the final image? (b) What is its magnification?

A glass rod with a refractive index of 1.55 is ground and polished at both ends to hemispherical surfaces with radii of 6.00 \(\mathrm{cm} .\) When an object is placed on the axis of the rod, 25.0 \(\mathrm{cm}\) to the left of the left-hand end, the final image is formed 65.0 \(\mathrm{cm}\) to the right of the right-hand end. What is the length of the rod measured between the vertices of the two hemispherical surfaces?

The radii of curvature of the surfaces of a thin converging meniscus lens are \(R_{1}=+12.0 \mathrm{cm}\) and \(R_{2}=+28.0 \mathrm{cm} .\) The index of refraction is 1.60 . (a) Compute the position and size of the image of an object in the form of an arrow 5.00 \(\mathrm{mm}\) tall, perpendicular to the lens axis, 45.0 \(\mathrm{cm}\) to the left of the lens. (b) A second converging lens with the same focal length is placed 3.15 \(\mathrm{m}\) to the right of the first. Find the position and size of the final image. Is the final image erect or inverted with respect to the original object? (c) Repeat part (b) except with the second lens 45.0 \(\mathrm{cm}\) to the right of the first.

An object to the left of a lens is imaged by the lens on a screen 30.0 \(\mathrm{cm}\) to the right of the lens. When the lens is moved 4.00 \(\mathrm{cm}\) to the right, the screen must be moved 4.00 \(\mathrm{cm}\) to the left to refocus the image. Determine the focal length of the lens.

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 -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?

One end of a long glass rod is ground to a convex hemispherical shape. This glass has an index of refraction of 1.55 . When a small leaf is placed 20.0 \(\mathrm{cm}\) in front of the center of the hemisphere along the optic axis, an image is formed inside the glass 9.12 \(\mathrm{cm}\) from the spherical surface. Where would the image be formed if the glass were now immersed in water (refractive index 1.33 ) but nothing else were changed?

When an object is placed at the proper distance to the left of a converging lens, the image is focused on a screen 30.0 \(\mathrm{cm}\) to the right of the lens. A diverging lens is now placed 15.0 \(\mathrm{cm}\) to the right of the converging lens, and it is found that the screen must be moved 19.2 \(\mathrm{cm}\) farther to the right to obtain a sharp image. What is the focal length of the diverging lens?

A convex spherical mirror with a focal length of magnitude 24.0 \(\mathrm{cm}\) is placed 20.0 \(\mathrm{cm}\) to the left of a plane mirror. An object 0.250 \(\mathrm{cm}\) tall is placed midway between the surface of the plane mirror and the vertex of the spherical mirror. The spherical mirror forms multiple images of the object. Where are the two images of the object formed by the spherical mirror that are closest to the spherical mirror, and how tall is each image?

A glass plate 3.50 \(\mathrm{cm}\) thick, with an index of refraction of 1.55 and plane parallel faces, is held with its faces horizontal and its lower face 6.00 \(\mathrm{cm}\) above a printed page. Find the position of the image of the page formed by rays making a small angle with the normal to the plate.

You have a camera with a 35.0 - \(\mathrm{mm}\) -focal-length lens and 36.0 -mm- wide film. You wish to take a picture of a \(12.0\)-m-long sailboat but find that the image of the boat fills only \(\frac{1}{4}\) of the width of the film. (a) How fare you from the boat? (b) How much closer must the boat be to you for its image to fill the width of the film?

What Is the Smallest Thing We Can See? The smallest object we can resolve with our eye is limited by the size of the light receptor cells in the retina. In order for us to distinguish any detail in an object, its image cannot be any smaller than a single retinal cell. Although the size depends on the type of cell (rod or cone), a diameter of a few microns \((\mu \mathrm{m})\) is typical near the center of the eye. We shall model the eye as a sphere 2.50 \(\mathrm{cm}\) in diameter with a single thin lens at the front and the retina at the rear, with light receptor cells 5.0\(\mu \mathrm{m}\) in diameter. (a) What is the smallest object you can resolve at a near point of 25 \(\mathrm{cm}\) ? (b) What angle is subtended by this object at the eye? Express your answer in units of minutes \(\left(1^{\circ}=60 \mathrm{min}\right),\) and compare it with the typical experimental value of about 1.0 min. (Note: There are other limitations, such as the bending of light as it passes through the pupil, but we shall ignore them here.)

Three thin lenses, each with a focal length of \(40.0 \mathrm{cm},\) are aligned on a common axis; adjacent lenses are separated by 52.0 \(\mathrm{cm} .\) Find the position of the image of a small object on the axis, 80.0 \(\mathrm{cm}\) to the left of the first lens.

A camera with a 90 -mm-focal-length lens is focused on an object 1.30 \(\mathrm{m}\) from the lens. To refocus on an object 6.50 \(\mathrm{m}\) from the lens, by how much must the distance between the lens and the film be changed? To refocus on the more distant object, is the lens moved toward or away from the film?

A person with a near point of \(85 \mathrm{cm},\) but excellent distant vision, normally wears corrective glasses. But he loses them while traveling. Fortunately, he has his old pair as a spare. (a) If the lenses of the old pair have a power of \(+2.25\) diopters, what is his near point (measured from his eye) when he is wearing the old glasses if they rest 2.0 \(\mathrm{cm}\) in front of his eye? (b) What would his near point be if his old glasses were contact lenses instead?

Focal Length of a Zoom Lens. Figure P34.113 shows a simple version of a zoom lens. The converging lens has focal length \(f_{1},\) and the diverging lens has focal length \(f_{2}=-\left|f_{2}\right|\) The two lenses are separated by a variable distance \(d\) that is always less than \(f_{1} .\) Also, the magnitude of the focal length of the diverging lens satisfies the inequality \(\left|f_{2}\right|>\left(f_{1}-d\right) .\) To determine the effective focal length of the combination lens, consider a bundle of parallel rays of radius \(r_{0}\) entering the converging lens. (a) Show that the radius of the ray bundle decreases to \(r_{0}^{\prime}=r_{0}\left(f_{1}-d\right) / f_{1}\) at the point that it enters the diverging lens. (b) Show that the final image \(I^{\prime}\) is formed a distance \(s_{2}^{\prime}=\left|f_{2}\right|\left(f_{1}-d\right) /\left(\left|f_{2}\right|-f_{1}+d\right)\) to the right of the diverging lens. (c) If the rays that emerge from the diverging lens and reach the final image point are extended backward to the left of the diverging lens, they will eventually expand to the original radius \(r_{0}\) at some point \(Q .\) The distance from the final image \(I^{\prime}\) to the point \(Q\) is the effective focal length \(f\) of the lens combination; if the combination were replaced by a single lens of focal length \(f\) placed at \(Q,\) parallel rays would still be brought to a focus at \(I^{\prime}\) . Show that the effective focal length is given by \(f=f_{1}\left|f_{2}\right| /\left(\left|f_{2}\right|-f_{1}+d\right) .\) (d) If \(f_{1}=12.0 \mathrm{cm}\) \(f_{2}=-18.0 \mathrm{cm},\) and the separation \(d\) is adjustable between 0 and \(4.0 \mathrm{cm},\) find the maximum and minimum focal lengths of the combination. What value of \(d\) gives \(f=30.0 \mathrm{cm} ?\)

A microscope with an objective of focal length 8.00 \(\mathrm{mm}\) and an eyepiece of focal length 7.50 \(\mathrm{cm}\) is used to project an image on a screen 2.00 \(\mathrm{m}\) from the eyepiece. Let the image distance of the objective be 18.0 \(\mathrm{cm} .\) (a) What is the lateral magnification of the image? (b) What is the distance between the objective and the eyepiece?

(a) For a lens with focal length \(f,\) find the smallest distance possible between the object and its real image. (b) Graph the distance between the object and the real image as a function of the distance of the object from the lens. Does your graph agree with the result you found in part (a)?