Conceptual Physics

convention allows the use of a single equation to locate images and objects for both

types of mirrors.

Our study of mirrors will focus on paraxial rays. Paraxial rays are incident rays that are

relatively close to the principal axis. Rays that are far from the principal axis and do not

converge at a single image point are called nonparaxial rays. Nonparaxial rays cause

blurry images. This effect is called spherical aberration.

For our purposes, the base of an object is located on the principal axis. When an

object’s base is on the principal axis, then the base of its image will be too, though it

may be inverted, like the image shown in Concept 4. The height of such an object or

image is measured from the principal axis. Like most quantities associated with objects

and images in optics, height has a sign. A positive value means the image is upright. Its

top is still on top. A negative sign means an image is inverted, as you see in Concept 4.

Its base still lies on the principal axis, but its top is now below the axis. Focal point (F)

Image point of infinitely distant object

Focal length (f )

Distance from mirror to focal point

·Positive on object side

·Negative on far side

Height (h)

Image point and height h describe

image

31.9 - Spherical mirrors: focal length equation

In Concept 1, you see a diagram of a concave mirror with center of curvature C and

radius of curvature r. A distant object and its image are shown. (The object and image

distances as well as the relative heights are not drawn to scale.) Because the object is

far away, the image point is at the focal point. The focal length f is the distance from the

mirror to the focal point.

How does the focal length change as the mirror radius changes? Imagine you are using

a mirror to create an image of a distant object. A concave mirror with a very slight

curvature would correspond to a sphere with a large radius. If you consider the law of

reflection and how rays would reflect from this mirror, you will realize that the focal point

(and image) would be relatively far away from it. In contrast, a sharply curving mirror

would create an image quite close to the surface.

The equation shown in Equation 1 quantifies this general relationship. The magnitude of

the focal length f equals one-half the radius rof the sphere. It is positive for a concave

mirror, and negative for a convex mirror.

Focal length

Positive for concave mirror

Negative for convex mirror

(^580) Copyright 2007 Kinetic Books Co. Chapter 31