26.9

Lenses In Combination

Many optical instruments, such as microscopes and telescopes, use a number of lenses together to produce an image. Among other things, a multiple-lens system can produce an image that is magnified more than is possible with a single lens. For instance, Figure 26-33a shows a two-lens system used in a microscope. The first lens, the lens closest to the object, is referred to as the objective. The second lens is known as the eyepiece (or ocular). The object is placed just outside the focal point Fo of the objective. The image formed by the objective—called the “first image” in the drawing—is real, inverted, and enlarged compared to the object. This first image then serves as the object for the eyepiece. Since the first image falls between the eyepiece and its focal point Fe, the eyepiece forms an enlarged, virtual, final image, which is what the observer sees.

Figure 26-33 (a) This two-lens system can be used as a compound microscope to produce a virtual, enlarged, and inverted final image. (b) The objective forms the first image and (c) the eyepiece forms the final image.

The location of the final image in a multiple-lens system can be determined by applying the thin-lens equation to each lens separately. The key point to remember in such situations is that the image produced by one lens serves as the object for the next lens, as the next example illustrates.

Problem solving insight

| Example 11 | | | A Microscope—Two Lenses in Combination

The objective and eyepiece of the compound microscope in Figure 26-33 are both converging lenses and have focal lengths of fo = 15.0 mm and fe = 25.5 mm. A distance of 61.0 mm separates the lenses. The microscope is being used to examine an object placed do1 = 24.1 mm in front of the objective. Find the final image distance.
Reasoning The thin-lens equation can be used to locate the final image produced by the eyepiece. We know the focal length of the eyepiece, but to determine the final image distance from the thin-lens equation we also need to know the object distance, which is not given. To obtain this distance, we recall that the image produced by one lens (the objective) is the object for the next lens (the eyepiece). We can use the thin-lens equation to locate the image produced by the objective, since the focal length and the object distance for this lens are given. The location of this image relative to the eyepiece will tell us the object distance for the eyepiece.
Solution The final image distance relative to the eyepiece is di2, and we can determine it by using the thin-lens equation:

The focal length fe of the eyepiece is known, but to obtain a value for the object distance do2 we must locate the first image produced by the objective. The first image distance di1 (see Figure 26-33b) can be determined using the thin-lens equation with do1 = 24.1 mm and fo = 15.0 mm.

The first image now becomes the object for the eyepiece (see part c of the drawing). Since the distance between the lenses is 61.0 mm, the object distance for the eyepiece is . Noting that the focal length of the eyepiece is fe = 25.5 mm, the final image distance can be found the thin-lens equation:

The fact that di2 is negative indicates that the final image is virtual. It lies to the left of the eyepiece, as the drawing shows.

Need more practice?

Interactive LearningWare26.2
A diverging lens (f = −14.0 cm) is 30.0 cm to the left of a converging lens (f = 12.0 cm). An object is placed 16.0 cm to the left of the diverging lens. Find the image distance of the first image, measured from the diverging lens, and the image distance of the final image, measured from the converging lens.
Related Homework: Problem 62

The overall magnification m of a two-lens system is the product of the magnifications m1 and m2 of the individual lenses, or . Suppose, for example, that the image of lens 1 is magnified by a factor of 5 relative to the original object. As we know, the image of lens 1 serves as the object for lens 2. Suppose, in addition, that lens 2 magnifies this object further by a factor of 8. The final image of the two-lens system, then, would be times as large as the original object. In other words, .

Problem solving insight

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