25.5

The Formation of Images by Spherical Mirrors

CONCEPTS AT A GLANCE As we have seen, some of the light rays emitted from an object in front of a mirror strike the mirror, reflect from it, and form an image. We can analyze the image produced by either concave or convex mirrors by using a graphical method called ray tracing. As the Concepts-at-a-Glance chart in Figure 25-17 shows, this method is based on the law of reflection and the notion that a spherical mirror has a center of curvature C and a focal point F. Ray tracing enables us to find the location of the image, as well as its size, by taking advantage of the following fact: paraxial rays leave from a point on the object and intersect at a corresponding point on the image after reflection.

Figure 25-17 CONCEPTS AT A GLANCE The method of ray tracing is used to predict the location and size of an image produced by a spherical mirror. Ray tracing is based on the law of reflection and two points associated with the mirror, its center of curvature C and its focal point F. The red, green, and blue light beams in the photograph pass through the focal point after reflecting from the mirror. (Ken Kay/Fundamental Photographs)

| Concave Mirrors

Three specific paraxial rays are especially convenient to use in the ray-tracing method. Figure 25-18 shows an object in front of a concave mirror, and these three rays leave from a point on the top of the object. The rays are labeled 1, 2, and 3, and when tracing their paths, we use the following reasoning strategy.

Figure 25-18 The rays labeled 1, 2, and 3 are useful in locating the image of an object placed in front of a concave spherical mirror. The object is represented as a vertical arrow.

| Reasoning Strategy

Ray Tracing for a Concave Mirror
Ray 1. | This ray is initially parallel to the principal axis and, therefore, passes through the focal point F after reflection from the mirror.
Ray 2. | This ray initially passes through the focal point F and is reflected parallel to the principal axis. Ray 2 is analogous to ray 1 except that the reflected, rather than the incident, ray is parallel to the principal axis.
Ray 3. | This ray travels along a line that passes through the center of curvature C and follows a radius of the spherical mirror; as a result, the ray strikes the mirror perpendicularly and reflects back on itself.

If rays 1, 2, and 3 are superimposed on a scale drawing, they converge at a point on the top of the image, as can be seen in Figure 25-19a.* Although three rays have been used here to locate the image, only two are really needed; the third ray is usually drawn to serve as a check. In a similar fashion, rays from all other points on the object locate corresponding points on the image, and the mirror forms a complete image of the object. If you were to place your eye as shown in the drawing, you would see an image that is larger and inverted relative to the object. The image is real because the light rays actually pass through the image.

Figure 25-19 (a) When an object is placed between the focal point F and the center of curvature C of a concave mirror, a real image is formed. The image is enlarged and inverted relative to the object. (b) When the object is located beyond the center of curvature C, a real image is created that is reduced in size and inverted relative to the object.

If the locations of the object and image in Figure 25-19a are interchanged, the situation in part b of the drawing results. The three rays in part b are the same as those in part a, except the directions are reversed. These drawings illustrate the principle of reversibility, which states that if the direction of a light ray is reversed, the light retraces its original path (Problem solving insight). This principle is quite general and is not restricted to reflection from mirrors. The image is real, and it is smaller and inverted relative to the object.
When the object is placed between the focal point F and a concave mirror, as in Figure 25-20a, three rays can again be drawn to find the image. But now ray 2 does not go through the focal point on its way to the mirror, since the object is inside the focal point. However, when projected backward, ray 2 appears to come from the focal point. Therefore, after reflection, ray 2 is directed parallel to the principal axis. In this case the three reflected rays diverge from each other and do not converge to a common point. However, when projected behind the mirror, the three rays appear to come from a common point; thus, a virtual image is formed. This virtual image is larger than the object and upright. Makeup and shaving mirrors are concave mirrors. When you place your face between the mirror and its focal point, you see an enlarged virtual image of yourself, as part b of the drawing shows.

Figure 25-20 (a) When an object is located between a concave mirror and its focal point F, an enlarged, upright, and virtual image is produced. (b) A makeup mirror (or shaving mirror) is a concave mirror that can form an enlarged virtual image, as this photograph suggests. (Stockdisc/Getty Images, Inc.)

The physics of makeup and shaving mirrors.

Concave mirrors are also used in one method for displaying the speed of a car. The method presents a digital readout (e.g., “55 mph”) that the driver sees when looking directly through the windshield, as in Figure 25-21a. The advantage of the method, which is called a head-up display (HUD), is that the driver does not need to take his or her eyes off the road to monitor the speed. Figure 25-21b shows how a HUD works. Located below the windshield is a readout device that displays the speed in digital form. This device is located in front of a concave mirror but within its focal point. The arrangement is similar to that in Figure 25-20a and produces a virtual, upright, and enlarged image of the speed readout (see virtual image 1 in Figure 25-21b). Light rays that appear to come from this image strike the windshield at a place where a so-called combiner is located. The purpose of the combiner is to combine the digital readout information with the field of view that the driver sees through the windshield. The combiner is virtually undetectable by the driver because it allows all colors except one to pass through it unaffected. The one exception is the color produced by the digital readout device. For this color, the combiner behaves as a plane mirror and reflects the light that appears to originate from image 1. Thus, the combiner produces image 2, which is what the driver sees. The location of image 2 is out above the front bumper. The driver can then read the speed with his eyes focused just as they are to see the road.

Figure 25-21 (a) A head-up display (HUD) presents the driver with a digital readout of the car’s speed in the field of view seen through the windshield. (Copyright GM Corporation. All rights reserved.) (b) One version of a HUD uses a concave mirror. (See text for explanation.)

The physics of a head-up display for automobiles.

| Convex Mirrors

The ray-tracing procedure for determining the location and size of an image in a convex mirror is similar to that for a concave mirror. The same three rays are used. However, the focal point and center of curvature of a convex mirror lie behind the mirror, not in front of it. Figure 25-22a shows the rays. When tracing their paths, we use the following reasoning strategy, which takes into account these locations of the focal point and center of curvature.

Figure 25-22 (a) An object placed in front of a convex mirror always produces a virtual image behind the mirror. The virtual image is reduced in size and upright. (b) The sun shield on this pilot’s helmet acts as a convex mirror and reflects an image of his plane. (Chad Slattery/Stone/Getty Images)

| Reasoning Strategy

Ray Tracing for a Convex Mirror
Ray 1. | This ray is initially parallel to the principal axis and, therefore, appears to originate from the focal point F after reflection from the mirror.
Ray 2. | This ray heads toward F, emerging parallel to the principal axis after reflection. Ray 2 is analogous to ray 1, except that the reflected, rather than the incident, ray is parallel to the principal axis.
Ray 3. | This ray travels toward the center of curvature C; as a result, the ray strikes the mirror perpendicularly and reflects back on itself.

The three rays in Figure 25-22a appear to come from a point on a virtual image that is behind the mirror. The virtual image is diminished in size and upright, relative to the object. A convex mirror always forms a virtual image of the object, no matter where in front of the mirror the object is placed. Figure 25-22b shows an example of such an image.
Because of its shape, a convex mirror gives a wider field of view than do other types of mirrors. Therefore, they are often used in stores for security purposes. A mirror with a wide field of view is also needed to give a driver a good rear view. Thus, the outside mirror on the passenger side is often a convex mirror. Printed on such a mirror is usually the warning “VEHICLES IN MIRROR ARE CLOSER THAN THEY APPEAR.” The reason for the warning is that the virtual image in Figure 25-22a is reduced in size and therefore looks smaller, just as a distant object would appear in a plane mirror. An unwary driver, thinking that the side-view mirror is a plane mirror, might incorrectly deduce from the small size of the image that the car behind is far enough away to ignore.

The physics of passenger-side automobile mirrors.

Check Your Understanding 3

An object is placed between the focal point and the center of curvature of a concave mirror. The object is then moved closer to the mirror, but still remains between the focal point and the center of curvature. What happens to the magnitudes of the image distance and the image height? Larger or smaller? (The answers are given at the end of the book.)
Background: The location and height of an image can be found by constructing a ray diagram and using the three rays discussed in this section.
For similar questions (including calculational counterparts), consult Self-Assessment Test 25.2, which is described at the end of Section 25.6.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.