18.2 Reflection from a Curved Mirror

Curved mirrors can produce all sorts of images. We will restrict our attention to spherical mirrors. Mirrors that reflect on the inside of the spherical surface are called concave mirrors; they will cause parallel light to converge on a point. Mirrors that reflect on the outside of the spherical surface are called convex mirrors; they will cause parallel light to diverge as if coming from a common point. Figure 18.5 shows a cross sectional view of both a convex mirror and a concave mirror. The axis of symmetry is known as the optic axis; the axis of symmetry will pass through the center of curvature of the mirror. The optic axis will be a useful reference line throughout our study of image formation.

Figure 18.D Curved "fun house" mirrors produce strange and unusual images. Cylindrical mirrors can even "decode" strange pictures and turn them into recognizable figures.

Figure 18.5 A line perpendicular to a spherical mirror is called the optic axis. The optic axis passes through the center of curvature of the mirror and the focal point. The optic axis is an axis of symmetry.

Rays of light from an object that is infinitely far away are parallel by the time we see them. Such parallel rays, after reflecting from a concave (or converging) spherical mirror, are bent so they converge on a single point. They pass through that point and then diverge from that point. After reflecting from a convex (or diverging) spherical mirror, such parallel rays are bent so they diverge as if they had come from a single point. If our eyes intercept these rays after their reflection they will look exactly as if they had originated from this point. For both mirrors, this point from which the light seems to have originated is called the focal point and is labeled by a capital letter F. The distance from the mirror to the focal point is the focal length and is labeled with a small letter f. We will adopt the convention that the focal length is positive for a concave mirror (f > 0) and is negative for a convex mirror (f < 0). These ideas are illustrated in Figure 18.6.

Figure 18.6 Rays of light parallel to the optic axis are focused at a single point by a spherical mirror. This point is called the focal point of the mirror. The distance from the focal point to the mirror is the focal length.

One note of caution; this description is only a first approximation. All that we have said is true as long as the size of the mirror is small compared to its radius of curvature. Another way of saying this is to limit ourselves to rays of light that lie close to the optic axis.

The focal length of a spherical mirror is one half the radius of curvature of the mirror,

f = R / 2

This equation also holds for convex mirrors as well as concave mirrors. By convention, the radius R is considered positive for concave or converging mirrors and is considered negative for convex or diverging mirrors. This means the focal length f will also be positive for concave or converging mirrors and negative for convex or diverging mirrors.

Light from an object infinitely far away, after reflection from a spherical mirror, behaves as if it had originated from this point. We call this point the focal point of the mirror. And we can say that an infinitely distant object has an image formed at the focal point of the mirror. For a concave or converging mirror, the rays actually pass through this point so we say a real image is formed. For a convex or diverging mirror, the rays do not actually pass through this point-this point is behind the mirror-so we say a virtual image is formed.

Figure 18.E The focal length of a spherical mirror is one-half its radius. Triangle CFM is an isosceles triangle and, for rays near the optic axis, distances CF, FM, and FV are equal so the focal length f = FV = R / 2.

Q: How are you able to see a virtual image?

A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image.

Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach?

A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.