# Spherical Mirrors: Focal Length, Ray Tracing, Mirror Equation and Magniﬁcation

## Spherical Mirrors

In This Chapter:

Focal Length

Ray Tracing

Mirror Equation

Magniﬁcation

Focal Length

Figure 18-1 shows how a concave mirror converges a parallel beam of light to a real focal point F, and Figure 18-2 shows how a convex mirror diverges a parallel beam of light so that the reﬂected rays appear to come from a virtual focal point F behind the mirror. In either case, if the radius of curvature of the mirror is R, the focal length f is R/2. For a concave mirror, f is positive, and for a convex mirror, f is negative. Thus

The axis of a mirror of either kind is the straight line that passes through

C and F.

Figure 18-1

Figure 18-2

Ray Tracing

The position and size of the image formed by a spherical mirror of an ob- ject in front of it can be found by constructing a scale drawing by tracing two different light rays from each point of interest in the object to where they (or their extensions, in the case of a virtual image) intersect after be- ing reﬂected by the mirror. Three rays especially useful for this purpose are shown in Figure 18-3; any two are sufﬁcient:

Figure 18-3

1. A ray that leaves the object parallel to the axis of the mirror. Af- ter reﬂection, this ray passes through the focal point of a con- cave mirror or seems to come from the focal point of a convex mirror.

2. A ray that passes through the focal point of a concave mirror or is directed toward the focal point of a convex mirror. After re- ﬂection, this ray travels parallel to the axis of the mirror.

3. A ray that leaves the object along a radius of the mirror. After reﬂection, this ray returns along the same radius.

###### Mirror Equation

When an object is a distance p from a mirror of focal length f, the image is located a distance q from the mirror, where

This equation holds for both concave and convex mirrors (see Figure 18-4). The mirror equation is readily solved for p, q, or f :

A positive value of p or q denotes a real object or image, and a negative value denotes a virtual object or image. A real object is in front of a mirror; a virtual object appears to be located behind the mirror and must itself be an image produced by another mirror or lens. A real image is formed by light rays that actually pass through the image, so a real image will appear on the screen placed at the position of the image. But a virtual image can be seen only by the eye since the light rays that appear to come from the image actually do not pass through it.

Remember

###### Real images are located in front of a mirror, virtual images behind it.

Figure 18.4

Magniﬁcation

The linear magniﬁcation m of any optical system is the ratio between the size (height or width or other transverse linear dimensions) of the image and the size of the object. In the case of a mirror,

A positive magniﬁcation signiﬁes an erect image, as in Figure 18-4(b); a negative one signiﬁes an inverted image, as in Figure 18-4(a). Table 18.1 is a summary of the sign conventions used in connection with spherical mirrors.

Table 18.1