25.3

The Formation of Images by a Plane Mirror

When you look into a plane (flat) mirror, you see an image of yourself that has three properties:

1. | The image is upright.
2. | The image is the same size as you are.
3. | The image is located as far behind the mirror as you are in front of it.

As Figure 25-5a illustrates, the image of yourself in the mirror is also reversed right to left and left to right. If you wave your right hand, it is the left hand of the image that waves back. Similarly, letters and words held up to a mirror are reversed. Ambulances and other emergency vehicles are often lettered in reverse, as in Figure 25-5b, so that the letters will appear normal when seen in the rearview mirror of a car.

Figure 25-5 (a) The person’s right hand becomes the image’s left hand when viewed in a plane mirror. (b) Many emergency vehicles are reverse-lettered so the lettering appears normal when viewed in the rearview mirror of a car. (Zoran Milich/Masterfile)

To illustrate why an image appears to originate from behind a plane mirror, Figure 25-6a shows a light ray leaving the top of an object. This ray reflects from the mirror (angle of reflection equals angle of incidence) and enters the eye. To the eye, it appears that the ray originates from behind the mirror, somewhere back along the dashed line. Actually, rays going in all directions leave each point on the object, but only a small bundle of such rays is intercepted by the eye. Part b of the figure shows a bundle of two rays leaving the top of the object. All the rays that leave a given point on the object, no matter what angle they have when they strike the mirror, appear to originate from a corresponding point on the image behind the mirror (see the dashed lines in part b). For each point on the object, there is a single corresponding point on the image, and it is this fact that makes the image in a plane mirror a sharp and undistorted one.

Figure 25-6 (a) A ray of light from the top of the chess piece reflects from the mirror. To the eye, the ray seems to come from behind the mirror. (b) The bundle of rays from the top of the object appears to originate from the image behind the mirror.

Although rays of light seem to come from the image, it is evident from Figure 25-6b that they do not originate from behind the plane mirror where the image appears to be. Because none of the light rays actually emanate from the image, it is called a virtual image. In this text the parts of the light rays that appear to come from a virtual image are represented by dashed lines. Curved mirrors, on the other hand, can produce images from which all the light rays actually do emanate. Such images are known as real images and are discussed later.
With the aid of the law of reflection, it is possible to show that the image is located as far behind a plane mirror as the object is in front of it. In Figure 25-7 the object distance is do and the image distance is di. A ray of light leaves the base of the object, strikes the mirror at an angle of incidence , and is reflected at the same angle. To the eye, this ray appears to come from the base of the image. For the angles and in the drawing it follows that and . But the angle is equal to the angle of reflection , since the two are opposite angles formed by intersecting lines. Therefore, . As a result, triangles ABC and DBC are identical (congruent) because they share a common side BC and have equal angles () at the top and equal angles (90°) at the base. Thus, the magnitude of the object distance do equals the magnitude of the image distance di.

Figure 25-7 This drawing illustrates the geometry used with a plane mirror to show that the image distance di equals the object distance do.

By starting with a light ray from the top of the object, rather than the bottom, we can extend the line of reasoning associated with Figure 25-7 to show also that the height of the image equals the height of the object.
Conceptual Examples 1 and 2 discuss some interesting features of plane mirrors.

| Conceptual Example 1 | | | Full-Length Versus Half-Length Mirrors

In Figure 25-8 a woman is standing in front of a plane mirror. What is the minimum mirror height necessary for her to see her full image?

Figure 25-8 For the woman to see her full-sized image, only a half-sized mirror is needed.

Reasoning and Solution The mirror is labeled ABCD in the drawing and is the same height as the woman. Light emanating from her body is reflected by the mirror, and some of this light enters her eyes. Consider a ray of light from her foot F. This ray strikes the mirror at B and enters her eyes at E. According to the law of reflection, the angles of incidence and reflection are both . Any light from her foot that strikes the mirror below B is reflected toward a point on her body that is below her eyes. Since light striking the mirror below B does not enter her eyes, the part of the mirror between B and A may be removed. The section BC of the mirror that produces the image is one-half the woman’s height between F and E. This follows because the right triangles FBM and EBM are identical. They are identical because they share a common side BM and have two angles, and 90°, that are the same.
The blowup in Figure 25-8 illustrates a similar line of reasoning, starting with a ray from the woman’s head at H. This ray is reflected from the mirror at P and enters her eyes. The top mirror section PD can be removed without disturbing this reflection. The necessary section CP is one-half the woman’s height between her head at H and her eyes at E. We find, then, that only the sections BC and CP are needed for the woman to see her full height. The height of section BC plus section CP is exactly one-half the woman’s height. The conclusions here are valid regardless of how far the person stands from the mirror. Thus, to view one’s full length in a mirror, only a half-length mirror is needed.
Related Homework: Problem 2 and 6

| Conceptual Example 2 | | | Multiple Reflections

A person is sitting in front of two mirrors that intersect at a 90° angle. As Figure 25-9a illustrates, the person sees three images of herself. (The person herself is not completely visible.) Why are there three images rather than two?

Figure 25-9 (a) These two perpendicular plane mirrors produce three images of the person (not completely visible) sitting in front of them. (Andy Washnik) (b) A “double” reflection, one from each mirror, gives rise to image 3.

Reasoning and Solution Figure 25-9b shows a top view of the person in front of the mirrors. It is a straightforward matter to understand two of the images that she sees. These are the images that are normally seen when one sits in front of a mirror. Sitting in front of mirror 1, she sees image 1, which is located as far behind that mirror as she is in front of it. She also sees image 2 behind mirror 2, at a distance that matches her distance in front of that mirror. Each of these images arises from light emanating from her body and reflecting from a single mirror. However, it is also possible for light to undergo two reflections in sequence, first from one mirror and then from the other. When a double reflection occurs, an additional image becomes possible. Figure 25-9b shows two rays of light that strike mirror 1. Each one, according to the law of reflection, has an angle of reflection that equals the angle of incidence. The rays then strike mirror 2, where they again are reflected according to the law of reflection. When the outgoing rays are extended backward (see the dashed lines in the drawing), they intersect and appear to originate from image 3.
Related Homework: Problems 3

Check Your Understanding 1

The drawing shows a light ray undergoing multiple reflections from a mirrored corridor. The walls of the corridor are either parallel or perpendicular to one another. If the initial angle of incidence is 35°, what is the angle of reflection when the ray makes its last reflection? (The answer is given at the end of the book.)
Background: Understanding the law of reflection is the key to answering this question. To use this law, one must know how the angles of incidence and reflection are measured.
For similar questions (including calculational counterparts), consult Self-Assessment Test 25.1, which is described next.

Test your understanding of the material in Sections 25.1, 25.2 and 25.3 :

· | Wave Fronts and Rays
· | The Reflection of Light
· | The Formation of Images by a Plane Mirror

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