Chapter 25

Two diverging light rays, originating from the same point, have an angle of \(10^{\circ}\) between them. After the rays reflect from a plane mirror, what is the angle between them? Construct one possible ray diagram that supports your answer.

The drawing shows a laser beam shining on a plane mirror that is perpendicular to the floor. The beam's angle of incidence is \(33.0^{\circ} .\) The beam emerges from the laser at a point that is \(1.10 \mathrm{~m}\) from the mirror and \(1.80 \mathrm{~m}\) above the floor. After reflection, how far from the base of the mirror does the beam strike the floor?

The drawing shows a top view of a square room. One wall is missing, and the other three are each mirrors. From point \(P\) in the center of the open side, a laser is fired, with the intent of hitting a small target located at the center of one wall. Identify six directions in which the laser can be fired and score a hit, assuming that the light does not strike any mirror more than once. Draw the rays to confirm your choices.

Two plane mirrors are facing each other. They are parallel, \(3.00 \mathrm{~cm}\) apart, and \(17.0 \mathrm{~cm}\) in length, as the drawing indicates. A laser beam is directed at the top mirror from the left edge of the bottom mirror. What is the smallest angle of incidence with respect to the top mirror, such that the laser beam (a) hits only one of the mirrors and (b) hits each mirror only once?

Two plane mirrors are facing each other. They are parallel, \(3.00 \mathrm{~cm}\) apart, and \(17.0 \mathrm{~cm}\) in length, as the drawing indicates. A laser beam is directed at the top mirror from the left edge of the bottom mirror. What is the smallest angle of incidence with respect to the top mirror, such that the laser beam (a) hits only one of the mirrors and (b) hits each mirror only once?

At illustrates the concepts pertinent to this problem. A \(2.0-\mathrm{cm}-\) high object is situated \(15.0 \mathrm{~cm}\) in front of a concave mirror that has a radius of curvature of \(10.0 \mathrm{~cm}\). Using a ray diagram drawn to scale, measure (a) the location and (b) the height of the image. The mirror must be drawn to scale.

A concave mirror has a focal length of \(20.0 \mathrm{~cm} .\) A \(2.0-\mathrm{cm}\) -high object is located \(12.0 \mathrm{~cm}\) in front of this mirror. Using a ray diagram drawn to scale, measure (a) the location and (b) the height of the image. The mirror must be drawn to scale.

At illustrates the concepts pertinent to this problem. A convex mirror has a focal length of \(-40.0 \mathrm{~cm}\). A \(12.0\) -cm-tall object is located \(40.0 \mathrm{~cm}\) in front of this mirror. Using a ray diagram drawn to scale, determine the (a) location and (b) size of the image. Note that the mirror must be drawn to scale.

A plane mirror and a concave mirror \((f=8.0 \mathrm{~cm})\) are facing each other and are separated by a distance of \(20.0 \mathrm{~cm}\). An object is placed \(10.0 \mathrm{~cm}\) in front of the plane mirror. Consider the light from the object that reflects first from the plane mirror and then from the concave mirror. Using a ray diagram drawn to scale, find the location of the image that this light produces in the concave mirror. Specify this distance relative to the concave mirror.

A mirror produces an image that is located \(34.0 \mathrm{~cm}\) behind the mirror when the object is located \(7.50 \mathrm{~cm}\) in front of the mirror. What is the focal length of the mirror, and is the mirror concave or convex?

The image behind a convex mirror (radius of curvature \(=68 \mathrm{~cm}\) ) is located \(22 \mathrm{~cm}\) from the mirror. (a) Where is the object located and (b) what is the magnification of the mirror? Determine whether the image is (c) upright or inverted and (d) larger or smaller than the object.

A concave mirror has a focal length of \(12 \mathrm{~cm}\). This mirror forms an image located \(36 \mathrm{~cm}\) in front of the mirror. What is the magnification of the mirror?

The outside mirror on the passenger side of a car is convex and has a focal length of \(-7.0 \mathrm{~m}\). Relative to this mirror, a truck traveling in the rear has an object distance of \(11 \mathrm{~m}\). Find (a) the image distance of the truck and (b) the magnification of the mirror.

A small postage stamp is placed in front of a concave mirror (radius \(=R\) ), such that the image distance equals the object distance. (a) In terms of \(R\), what is the object distance? (b) What is the magnification of the mirror? (c) State whether the image is upright or inverted relative to the object. Draw a ray diagram to guide your thinking.

A concave mirror \((f=45 \mathrm{~cm})\) produces an image whose distance from the mirror is onethird the object distance. Determine (a) the object distance and (b) the (positive) image distance.

When viewed in a spherical mirror, the image of a setting sun is a virtual image. The image lies \(12.0 \mathrm{~cm}\) behind the mirror. (a) Is the mirror concave or convex? Why? (b) What is the radius of curvature of the mirror?

Consult Interactive Solution \(\underline{25.25}\) at for insight into this problem. An object is placed in front of a convex mirror, and the size of the image is one-fourth that of the object. What is the ratio \(d_{0} / f\) of the object distance to the focal length of the mirror?

The same object is located at the same distance from two spherical mirrors, \(\mathrm{A}\) and \(\mathrm{B}\). The magnifications produced by the mirrors are \(m_{\mathrm{A}}=4.0\) and \(m_{\mathrm{B}}=2.0 .\) Find the ratio \(f_{\mathrm{A}} / f_{\mathrm{B}}\) of the focal lengths of the mirrors.

An image formed by a convex mirror \((f=-24.0 \mathrm{~cm})\) has a magnification of 0.150. Which way and by how much should the object be moved to double the size of the image?

A concave mirror has a focal length of \(30.0 \mathrm{~cm} .\) The distance between an object and its image is \(45.0 \mathrm{~cm} .\) Find the object and image distances, assuming that (a) the object lies beyond the center of curvature and (b) the object lies within the focal point.

Using the mirror equation and the magnification equation, show that for a convex mirror the image is always (a) virtual (i.e., \(d_{\mathrm{i}}\) is always negative) and (b) upright and smaller, relative to the object (i.e., \(m\) is positive and less than one).

An object is placed in front of a convex mirror. Draw the convex mirror (radius of curvature \(=15 \mathrm{~cm}\) ) to scale, and place the object \(25 \mathrm{~cm}\) in front of it. Make the object height \(4 \mathrm{~cm}\). Using a ray diagram, locate the image and measure its height. Now move the object closer to the mirror, so the object distance is \(5 \mathrm{~cm}\). Again, locate its image using a ray diagram. As the object moves closer to the mirror, (a) does the magnitude of the image distance become larger or smaller, and (b) does the magnitude of the image height become larger or smaller? (c) What is the ratio of the image height when the object distance is \(5 \mathrm{~cm}\) to its height when the object distance is \(25 \mathrm{~cm} ?\) Give your answer to one significant figure.

The image produced by a concave mirror is located \(26 \mathrm{~cm}\) in front of the mirror. The focal length of the mirror is \(12 \mathrm{~cm}\). How far in front of the mirror is the object located?

A clown is using a concave makeup mirror to get ready for a show and is \(27 \mathrm{~cm}\) in front of the mirror. The image is \(65 \mathrm{~cm}\) behind the mirror. Find (a) the focal length of the mirror and (b) the magnification.

An object that is \(25 \mathrm{~cm}\) in front of a convex mirror has an image located \(17 \mathrm{~cm}\) behind the mirror. How far behind the mirror is the image located when the object is \(19 \mathrm{~cm}\) in front of the mirror?

A concave mirror has a focal length of \(42 \mathrm{~cm}\). The image formed by this mirror is \(97 \mathrm{~cm}\) in front of the mirror. What is the object distance?

A ray of light strikes a plane mirror at a \(45^{\circ}\) angle of incidence. The mirror is then rotated by \(15^{\circ}\) into the position shown in red in the drawing, while the incident ray is kept fixed. (a) Through what angle \(\phi\) does the reflected ray rotate? (b) What is the answer to part (a) if the angle of incidence is \(60^{\circ}\) instead of \(45^{\circ} ?\)

A candle is placed \(15.0 \mathrm{~cm}\) in front of a convex mirror. When the convex mirror is replaced with a plane mirror, the image moves \(7.0 \mathrm{~cm}\) farther away from the mirror. Find the focal length of the convex mirror.

An object is located \(14.0 \mathrm{~cm}\) in front of a convex mirror, the image being \(7.00 \mathrm{~cm}\) behind the mirror. A second object, twice as tall as the first one, is placed in front of the mirror, but at a different location. The image of this second object has the same height as the other image. How far in front of the mirror is the second object located?

A spherical mirror is polished on both sides. When the convex side is used as a mirror, the magnification is \(+1 / 4\). What is the magnification when the concave side is used as a mirror, the object remaining the same distance from the mirror?

A lamp is twice as far in front of a plane mirror as a person is. Light from the lamp reaches the person via two paths, reflected and direct. It strikes the mirror at a \(30.0^{\circ}\) angle of incidence and reflects from it before reaching the person. The total time for the light to travel this path includes the time to travel to the mirror and the time to travel from the mirror to the person. The light also travels directly to the person without reflecting. Find the ratio of the total travel time along the reflected path to the travel time along the direct path.

Questions A small mirror is attached to a vertical wall, and it hangs a distance \(y\) above the floor. A ray of sunlight strikes the mirror, and the reflected ray forms a spot on the floor. (a) From a knowledge of \(y\) and the horizontal distance \(x\) from the base of the wall to the spot, describe how one can determine the angle of incidence of the ray striking the mirror. If it is morning and the mirror is facing due east, would (b) the angle of incidence and (c) the distance \(x\) increase or decrease in time? Why? Suppose the mirror is \(1.80 \mathrm{~m}\) above the floor. The reflected ray of sunlight strikes the floor at a distance of \(3.86 \mathrm{~m}\) from the base of the wall. Later in the morning, the ray is observed to strike the floor at a distance of \(1.26 \mathrm{~m}\) from the wall. The earth rotates at a rate of \(15.0^{\circ}\) per hour. How much time (in hours) has elapsed between the two observations?

A small mirror is attached to a vertical wall, and it hangs a distance \(y\) above the floor. A ray of sunlight strikes the mirror, and the reflected ray forms a spot on the floor. (a) From a knowledge of \(y\) and the horizontal distance \(x\) from the base of the wall to the spot, describe how one can determine the angle of incidence of the ray striking the mirror. If it is morning and the mirror is facing due east, would (b) the angle of incidence and (c) the distance \(x\) increase or decrease in time? Why? Problem Suppose the mirror is \(1.80 \mathrm{~m}\) above the floor. The reflected ray of sunlight strikes the floor at a distance of \(3.86 \mathrm{~m}\) from the base of the wall. Later in the morning, the ray is observed to strike the floor at a distance of \(1.26 \mathrm{~m}\) from the wall. The earth rotates at a rate of \(15.0^{\circ}\) per hour. How much time (in hours) has elapsed between the two observations?

For an inverted image that is in front of a mirror, is the image distance positive or negative and is the image height positive or negative? Explain. (b) Given the image distance, what additional information is needed to determine the focal length? Explain. (c) Given the object and image heights and a statement as to whether the image is upright or inverted, what additional information is needed to determine the object distance? A small statue has a height of \(3.5 \mathrm{~cm}\) and is placed in front of a concave mirror. The image of the statue is inverted, \(1.5 \mathrm{~cm}\) tall, and is located \(13 \mathrm{~cm}\) in front of the mirror. Find the focal length of the mirror.

(a) For an inverted image that is in front of a mirror, is the image distance positive or negative and is the image height positive or negative? Explain. (b) Given the image distance, what additional information is needed to determine the focal length? Explain. (c) Given the object and image heights and a statement as to whether the image is upright or inverted, what additional information is needed to determine the object distance? Problem A small statue has a height of \(3.5 \mathrm{~cm}\) and is placed in front of a concave mirror. The image of the statue is inverted, \(1.5 \mathrm{~cm}\) tall, and is located \(13 \mathrm{~cm}\) in front of the mirror. Find the focal length of the mirror.

A tall tree is growing across a river from you. You would like to know the distance between yourself and the tree, as well as its height, but are unable to make the measurements directly. However, by using a mirror to form an image of the tree, and then measuring the image distance and the image height, you can calculate the distance to the tree, as well as its height. (a) What kind of mirror, concave or convex, must you use? Why? (b) You will need to know the focal length of the mirror. The sun is shining. You aim the mirror at the sun and form an image of it. How is the image distance of the sun related to the focal length of the mirror? (c) Having measured the image distance \(d_{\mathrm{i}}\) and the image height \(h_{\mathrm{i}}\) of the tree, as well as the image distance of the sun, describe how you would use these numbers to determine the distance and height of the tree. Problem A mirror produces an image of the sun, and the image is located \(0.9000 \mathrm{~m}\) from the mirror. The same mirror is then used to produce an image of the tree. The image of the tree is \(0.9100 \mathrm{~m}\) from the mirror. (a) How far away is the tree? (b) The image height of the tree has a magnitude of \(0.12 \mathrm{~m}\). How tall is the tree?