26.7

The Formation of Images by Lenses

| Ray Diagrams and Ray Tracing

Each point on an object emits light rays in all directions, and when some of these rays pass through a lens, they form an image. As with mirrors, ray diagrams can be drawn to determine the location and size of the image. Lenses differ from mirrors, however, in that light can pass through a lens from left to right or from right to left. Therefore, when constructing ray diagrams, begin by locating a focal point F on each side of the lens; each point lies on the principal axis at the same distance f from the lens. The lens is assumed to be thin, in that its thickness is small compared with the focal length and the distances of the object and the image from the lens. For convenience, it is also assumed that the object is located to the left of the lens and is oriented perpendicular to the principal axis. There are three paraxial rays that leave a point on the top of the object and are especially helpful in drawing ray diagrams. They are labeled 1, 2, and 3 in Figure 26-25. When tracing their paths, we use the following reasoning strategy.

Figure 26-25 The rays shown here are useful in determining the nature of the images formed by converging and diverging lenses

BOX 26.1 | Reasoning Strategy

Ray Tracing for Converging and Diverging Lenses

Converging Lens | Diverging Lens
Ray 1
This ray initially travels parallel to the principal axis. In passing through a converging lens, the ray is refracted toward the axis and travels through the focal point on the right side of the lens, as Figure 26-25a shows. | This ray initially travels parallel to the principal axis. In passing through a diverging lens, the ray is refracted away from the axis, and appears to have originated from the focal point on the left of the lens. The dashed line in Figure 26-25d represents the apparent path of the ray.
Ray 2
This ray first passes through the focal point on the left and then is refracted by the lens in such a way that it leaves traveling parallel to the axis, as in Figure 26-25b. | This ray leaves the object and moves toward the focal point on the right of the lens. Before reaching the focal point, however, the ray is refracted by the lens so as to exit parallel to the axis. See Figure 26-25e, where the dashed line indicates the ray’s path in the absence of the lens.
Ray 3*
This ray travels directly through the center of the thin lens without any appreciable bending, as in Figure 26-25c. | This ray travels directly through the center of the thin lens without any appreciable bending, as in Figure 26-25f.

*) |  | Ray 3 does not bend as it proceeds through the lens because the left and right surfaces of each type of lens are nearly parallel at the center. Thus, in either case, the lens behaves as a transparent slab. As Figure 26-26 shows, the rays incident on and exiting from a slab travel in the same direction with only a lateral displacement. If the lens is sufficiently thin, the displacement is negligibly small.

| Image Formation by a Converging Lens

Figure 26-26a illustrates the formation of a real image by a converging lens. Here the object is located at a distance from the lens that is greater than twice the focal length (beyond the point labeled 2F). To locate the image, any two of the three special rays can be drawn from the tip of the object, although all three are shown in the drawing. The point on the right side of the lens where these rays intersect locates the tip of the image. The ray diagram indicates that the image is real, inverted, and smaller than the object. This optical arrangement is similar to that used in a camera, where a piece of film records the image (see part b of the drawing).

Figure 26-26 (a) When the object is placed to the left of the point labeled 2F, a real, inverted, and smaller image is formed. (b) The arrangement in part a is like that used in a camera.

The physics of a camera.

The physics of a slide or film projector.

When the object is placed between 2F and F, as in Figure 26-27a, the image is still real and inverted; however, the image is now larger than the object. This optical system is used in a slide or film projector in which a small piece of film is the object and the enlarged image falls on a screen. However, to obtain an image that is right-side up, the film must be placed in the projector upside down.

Figure 26-27 (a) When the object is placed between 2F and F, the image is real, inverted, and larger than the object. (b) This arrangement is found in projectors.

When the object is located between the focal point and the lens, as in Figure 26-28, the rays diverge after leaving the lens. To a person viewing the diverging rays, they appear to come from an image behind (to the left of) the lens. Because none of the rays actually come from the image, it is a virtual image. The ray diagram shows that the virtual image is upright and enlarged. A magnifying glass uses this arrangement, as can be seen in part b of the drawing.

Figure 26-28 (a) When an object is placed within the focal point F of a converging lens, an upright, enlarged, and virtual image is created. (b) Such an image is seen when looking through a magnifying glass.

| Image Formation by a Diverging Lens

Light rays diverge upon leaving a diverging lens, as Figure 26-29 shows, and the ray diagram indicates that a virtual image is formed on the left side of the lens. In fact, regardless of the position of a real object, a diverging lens always forms a virtual image that is upright and smaller relative to the object.

Figure 26-29 (a) A diverging lens always forms a virtual image of a real object. The image is upright and smaller relative to the object. (b) The image seen through a diverging lens.

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