25.7

Concepts & Calculations

Relative to the object in front of a spherical mirror, the image can differ in a number of respects. The image can be real (in front of the mirror) or virtual (behind the mirror). It can be larger or smaller than the object, and it can be upright or inverted. As you solve problems dealing with spherical mirrors, keep these image characteristics in mind. They can help guide you to the correct answer, as Examples 8 and 9 illustrate.

| Concepts & Calculations Example 8 | | | Finding the Focal Length

An object is located 7.0 cm in front of a mirror. The virtual image is located 4.5 cm away from the mirror and is smaller than the object. Find the focal length of the mirror. Concept Questions and Answers Based solely on the fact that the image is virtual, is the mirror concave or convex, or is either type possible?

Answer Either type is possible. A concave mirror can form a virtual image if the object is within the focal point of the mirror, as Figure 25-20a illustrates. A convex mirror always forms a virtual image, as Figure 25-22a shows.

The image is smaller than the object, as well as virtual. Do these characteristics together indicate a concave or convex mirror, or do they indicate either type?

Answer They indicate a convex mirror. A concave mirror can produce an image that is smaller than the object if the object is located beyond the center of curvature of the mirror, as in Figure 25-19b. However, the image in Figure 25-19b is real, not virtual. A convex mirror, in contrast, always produces an image that is virtual and smaller than the object, as Figure 25-22a illustrates.

Is the focal length positive or negative?

Answer The focal length is negative because the mirror is convex. A concave mirror has a positive focal length.

Solution The virtual image is located behind the mirror and, therefore, has a negative image distance, . Using this value together with the object distance of , we can apply the mirror equation to find the focal length:

As expected, the focal length is negative.

| Concepts & Calculations Example 9 | | | Two Choices

The radius of curvature of a mirror is 24 cm. A diamond ring is placed in front of this mirror. The image is twice the size of the ring. Find the object distance of the ring. Concept Questions and Answers Is the mirror concave or convex, or is either type possible?

Answer A convex mirror always forms an image that is smaller than the object, as Figure 25-22a shows. Therefore, the mirror must be concave.

How many places are there in front of a concave mirror where the ring can be placed and produce an image that is twice the size of the object?

Answer There are two places. Figure 25-19a illustrates that one of the places is between the center of curvature and the focal point. The enlarged image is real and inverted. Figure 25-20a shows that another possibility is between the focal point and the mirror, in which case the enlarged image is virtual and upright.

What are the possible values for the magnification of the image of the ring?

Answer Since the image is inverted in Figure 25-19a, the magnification for this possibility is . In Figure 25-20a, however, the image is upright, so the magnification is for this option. In either case, the image is twice the size of the ring.

Solution According to the mirror equation and the magnification equation, we have

We can solve the magnification equation for the image distance and obtain . Substituting this expression for di into the mirror equation gives

Applying this result with the two magnifications (and noting that ), we obtain

At the end of the problem set for this chapter, you will find homework problems that contain both conceptual and quantitative parts. These problems are grouped under the heading Concepts & Calculations.

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