18.3 Images Formed by Curved Mirrors

18.3 Images Formed by Curved Mirrors

In the previous section we have seen that an object at infinity has an image at the focal point after reflection in a curved mirror; light leaves the mirror as if it had originated there. But what happens if the object is not at infinity? In understanding image formation, ray diagrams will prove very useful. There are three rays of light whose reflections are especially easy to handle or to understand. We call these the principal rays and will make frequent use of them. They are shown in Figure 18.7. We have already seen that a ray of light that comes in parallel to the optic axis will strike a concave mirror and be reflected toward the focal point-that is the definition of the focal point. From a convex mirror it will be reflected as if it had come from the focal point. Both of these are illustrated in the figure by ray 1. And a ray is reversible. A ray of light which passes through the focal point on its way to a concave mirror leaves parallel to the optic axis. A ray of light that is headed toward the focal point of a convex mirror leaves parallel to the optic axis. Both of these are illustrated in the figure by ray 2. The third principal ray is ray 3 that strikes a mirror at the optic axis. The optic axis is normal to the mirror so that ray leaves with an angle of reflection equal to the angle of incidence as indicated in the figure.

Figure 18.7 Principal rays will be of great use in understanding image formation.

Figure 18.8 shows an object placed some distance from a concave mirror. We have used an arrow for convenience. The tail of the arrow is on the optic axis. Simply from symmetry, we know that the image of this tail must also lie somewhere on the optic axis; whatever argument you can think of for the image's being above the axis is just as valid for its being below the axis. Therefore, all we need to do is to locate the image of the tip of the arrow. We can then draw in the rest of it just by dropping a line to the axis. While an infinite number of rays leave the tip of the arrow (or any other point on the arrow), we will concentrate on just the three principal rays shown in the first part of the figure. Having an image means that all of the rays that leave the object and are reflected from the mirror will pass through a single point that locates the image. If we construct a ray diagram using just two rays and find that they intersect we have determined the location of the image. However, it is prudent to confirm this with a third ray. If you draw three rays and they do not intersect at a single, common point then you know an error has occurred and you can track it down or begin again. Notice that the real image is inverted.

Figure 18.8 Ray diagrams locate a real image formed by a concave mirror. All the rays that leave a point on the object and are reflected from the mirror will pass through a common point if a real image is formed. For efficiency we concentrate on the three principal rays.

Figure 18.8 shows a real image produced. For a real image, the reflected light actually passes through the image. If a card or screen is placed at the location of the image an image will be projected on the card or screen. But a mirror can also produce a virtual image. Figure 18.9 shows additional examples of real images being produced by a concave mirror when the object is placed at various distances from the mirror. A far distant object produces a small, inverted, real image when reflected in a concave or converging mirror. Bringing the image in closer to the mirror enlarges the size of the image. When the object is at a distance of twice the focal length from the mirror, the image is the same size as the object (and still inverted). Moving the object in even closer makes the image larger than the object.

Figure 18.9 A far distant object produces a small, inverted, real image when reflected in a concave or converging mirror. Bringing the image in closer to the mirror enlarges the size of the image. When the object is at a distance of twice the focal length from the mirror, the image is the same size as the object (and still inverted). Moving the object in even closer makes the image larger than the object.

The magnification of a situation is the ratio of the image height to the object height. If the image is inverted, we will consider the image height negative so the magnification will be negative. If Figure 18.9, when the object is far away, the image is smaller in size and is upside down so the magnification is small and negative, like M = - 0.75 or M = - 0.50. When the object distance is two times the focal length, the image is the same size as the object and is upside down so the magnification is M = - 1.00. As the object moves closer than this, the image increases in size but remains upside down so the absolute value of the magnification continues to increase although the magnification is still negative.

Figure 18.10 shows two examples of producing a virtual image by a mirror. One is by a concave or converging mirror; the other, a convex or diverging mirror. Concave mirrors can produce either real or virtual images from a real object depending upon where the object is. If the object is beyond the focal point the mirror will produce a real image; if inside the focal point (between the mirror and the focal point), a virtual image. Convex mirrors produce only virtual images from real objects. A virtual image is very "real" in that you can see it quite clearly. But it can not be projected. If you place a card behind the mirror at the location of the image, there will be nothing projected upon it. Think again of your own virtual image in the bathroom mirror this morning. If you had held a card behind the mirror you would not have found your image projected upon it. While a virtual image can easily be seen, the light does not actually pass through the location of the image. That is precisely what is meant by a virtual image.

Figure 18.10 Ray diagrams locate a virtual image. All the rays that leave a point on the object and are reflected from the mirror leave as if they came from a common point when a virtual image is formed. We concentrate on the three principal rays because they are easy to handle.

Virtual images due to a reflection in a mirror will be right side up so the magnification will be positive in these cases. In Figure 18.10, the magnification is greater than one (M > 1.00) for the enlarged virtual image due to the concave mirror. The magnification is less than one (M < 1.00) for the reduced virtual image due to the convex mirror.

Ray diagrams are essential in understanding image formation. If they are carefully constructed all the dimensions can be accurately measured.

Figure 18.F A converging mirror can produce an upright and enlarged virtual image.

Figure 18.G A diverging mirror, such as the right outside rearview mirror on your car, produces an upright and reduced virtual image.

Q: What kind of images can a concave mirror produce?

A: Depending upon the distance the object is from the mirror, a concave mirror can produce a real or virtual image and the image can be enlarged or reduced in size. A shaving mirror or a make-up mirror is a good example of a concave mirror. The inside of a shiney spoon is another example of a concave mirror.

Q: What kind of images can a convex mirror produce?

A: For a real object, a convex mirror will always produce a virtual image that is reduced in size. The passenger-side rearview mirror on a car is a good example of a convex mirror. The outside of a shiney spoon is another example of a convex mirror.