25.4

Spherical Mirrors

The most common type of curved mirror is a spherical mirror. As Figure 25-10 shows, a spherical mirror has the shape of a section from the surface of a sphere. If the inside surface of the mirror is polished, it is a concave mirror. If the outside surface is polished, it is a convex mirror. The drawing shows both types of mirrors, with a light ray reflecting from the polished surface. The law of reflection applies, just as it does for a plane mirror. For either type of spherical mirror, the normal is drawn perpendicular to the mirror at the point of incidence. For each type, the center of curvature is located at point C, and the radius of curvature is R. The principal axis of the mirror is a straight line drawn through C and the midpoint of the mirror.

Figure 25-10 A spherical mirror has the shape of a segment of a spherical surface. The center of curvature is point C and the radius is R. For a concave mirror, the reflecting surface is the inner one; for a convex mirror it is the outer one.

Figure 25-11 shows a tree in front of a concave mirror. A point on this tree lies on the principal axis of the mirror and is beyond the center of curvature C. Light rays emanate from this point and reflect from the mirror, consistent with the law of reflection. If the rays are near the principal axis, they cross it at a common point after reflection. This point is called the image point. The rays continue to diverge from the image point as if there were an object there. Since light rays actually come from the image point, the image is a real image.

Figure 25-11 A point on the tree lies on the principal axis of the concave mirror. Rays from this point that are near the principal axis are reflected from the mirror and cross the axis at the image point.

If the tree in Figure 25-11 is infinitely far from the mirror, the rays are parallel to each other and to the principal axis as they approach the mirror. Figure 25-12 shows rays near and parallel to the principal axis, as they reflect from the mirror and pass through an image point. In this special case the image point is referred to as the focal point F of the mirror. Therefore, an object infinitely far away on the principal axis gives rise to an image at the focal point of the mirror. The distance between the focal point and the middle of the mirror is the focal length f of the mirror.

Figure 25-12 Light rays near and parallel to the principal axis are reflected from a concave mirror and converge at the focal point F. The focal length f is the distance between F and the mirror.

We can show that the focal point F lies halfway between the center of curvature C and the middle of a concave mirror. In Figure 25-13, a light ray parallel to the principal axis strikes the mirror at point A. The line CA is the radius of the mirror and, therefore, is the normal to the spherical surface at the point of incidence. The ray reflects from the mirror such that the angle of reflection equals the angle of incidence. Furthermore, the angle ACF is also because the radial line CA is a transversal of two parallel lines. Since two of its angles are equal, the colored triangle CAF is an isosceles triangle; thus, sides CF and FA are equal. But when the incoming ray lies close to the principal axis, the angle of incidence is small, and the distance FA does not differ appreciably from the distance FB. Therefore, in the limit that is small, , and so the focal point F lies halfway between the center of curvature and the mirror. In other words, the focal length f is one-half of the radius R:

| (25.1)

Figure 25-13 This drawing is used to show that the focal point F of a concave mirror is halfway between the center of curvature C and the mirror at point B.

Rays that lie close to the principal axis are known as paraxial rays,* and Equation 25.1 is valid only for such rays. Rays that are far from the principal axis do not converge to a single point after reflection from the mirror, as Figure 25-14 shows. The result is a blurred image. The fact that a spherical mirror does not bring all rays parallel to the principal axis to a single image point is known as spherical aberration. Spherical aberration can be minimized by using a mirror whose height is small compared to the radius of curvature.

Figure 25-14 Rays that are farthest from the principal axis have the greatest angle of incidence and miss the focal point F after reflection from the mirror.

The physics of capturing solar energy with mirrors; automobile headlights. A sharp image point can be obtained with a large mirror, if the mirror is parabolic in shape instead of spherical. The shape of a parabolic mirror is such that all light rays parallel to the principal axis, regardless of their distance from the axis, are reflected through a single image point. However, parabolic mirrors are costly to manufacture and are used where the sharpest images are required, as in research-quality telescopes. Parabolic mirrors are also used in one method of collecting solar energy for commercial purposes. Figure 25-15 shows a long row of concave parabolic mirrors that reflect the sun’s rays to the focal point. Located at the focal point and running the length of the row is an oil-filled pipe. The focused rays of the sun heat the oil. In a solar-thermal electric plant, the heat from many such rows is used to generate steam. The steam, in turn, drives a turbine connected to an electric generator. Another application of parabolic mirrors is in automobile headlights. Here, however, the situation is reversed from the operation of a solar collector. In a headlight, a high-intensity light bulb is placed at the focal point of the mirror, and light emerges parallel to the principal axis.

Figure 25-15 This long row of parabolic mirrors focuses the sun’s rays to heat an oil-filled pipe located at the focal point of each mirror. It is one of many that are used by a solar-thermal electric plant in the Mojave Desert. (Mastrorillo/Corbis Stock Market)

A convex mirror also has a focal point, and Figure 25-16 illustrates its meaning. In this picture, parallel rays are incident on a convex mirror. Clearly, the rays diverge after being reflected. If the incident parallel rays are paraxial, the reflected rays seem to come from a single point F behind the mirror. This point is the focal point of the convex mirror, and its distance from the midpoint of the mirror is the focal length f. The focal length of a convex mirror is also one-half of the radius of curvature, just as it is for a concave mirror. However, we assign the focal length of a convex mirror a negative value because it will be convenient later on:

| (25.2)

Figure 25-16 When paraxial light rays that are parallel to the principal axis strike a convex mirror, the reflected rays appear to originate from the focal point F. The radius of curvature is R and the focal length is f.

Spherical aberration is a problem with convex mirrors, just as it is with concave mirrors. Rays that emanate from a single point on an object but are far from the principal axis do not appear to originate from a single image point after reflection from the mirror. As with a concave mirror, the result is a blurred image.

Check Your Understanding 2

A section of a sphere has a radius of curvature of 0.60 m, and both the inside and outside surfaces have a mirror-like polish. What are the focal lengths of the inside and outside surfaces? (The answers are given at the end of the book.) Background: The two types of spherical mirrors, concave and convex, are made from a section of a sphere. The radius of curvature determines the focal lengths of these types of mirrors.
For similar questions (including calculational counterparts), consult Self-Assessment Test 25.2, which is described at the end of Section 25.6.

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