400 Week 12: Lenses and Mirrors
However, this is not necessarily easier to use for the purposes of computation, as one still
(ultimately) has to do the same algebra to actually computesand/ors′.
At this point we have derived a simple equation relatings,s′andf. The only rule
we have used so far in deriving that equation (which you can easily seeholds for plane
mirrors as well) is the law of reflection. We have deduced as atheoremof this the rule
that parallel paraxial rays are diverted by a converging mirror to an image at the focal
distance from the mirror. We now need to take these two rules (anda third that is a
restatement of the second) and use them to constructray diagramsthat permit us to
visualize how a convergingordiverging mirror forms an image out of rays diverging from
an object. Constructing such diagrams, and answering a more or less standard set of
questions, will constitute most of theproblemsassociated with this chapter.12.3: Ray Diagrams for Ideal Mirrors
To construct our ray diagrams, we need to begin by idealizing spherical mirrors in a way
that “hides” things like the fact that many rays we might wish to imagewith arenot
paraxial. Later in this chapter we’ll deal with many of the aberrations that are features
of real lenses and mirrors as deviations from ideal behavior in the focussing elements
themselves or the light that goes through them, but these will be “corrections” that
should not cloud our perception of how things basically work.
First, when drawing rays in a ray diagram, one always assumes thatall deflection by the
lens or mirror occurs in a single plane.This is an idealization, to be sure – the reason
mirrors and lenses focus light is because they arecurved, not planar. But paraxial rays
by definition strike close enough to the center that the deviation from planar can be
ignored, and we idealize this to the entire plane.
Given this, the following three rays have rules that can be used to locate images and
compute magnification for any mirror (and eventually, lens):a)The Parallel Ray:A ray from the object that is parallel to the axis of the mirror
is reflected by the mirrorthrough the focal point.
b) The Focal Ray:A ray from the object that strikes the mirror eitherthroughthe
focal point or along a line thatcomes fromthe focal point is reflectedparallel to the
axis of the mirror.
c)The Central Ray:A ray from the object that strikes the mirror in the center is
reflected by the mirrorwith angle of incidence equal to the angle of reflection
which means that the reflected ray is symmetric across the axis from the incident
one.Now consider the following ray diagrams for various positions of our archetypical arrow
object for converging (+) and diverging (-) ideal mirrors.
In this figure,f= 10 cm,s= 25 cm. Therefore:1
25+
1
s′=
1
10
1
s′=
1
10
−
1
25
1
s′=
1. 5
25
s′ =25
1. 5
= 16.7 cm (979)