In the first section of Lesson 4, we learned that light
is reflected by convex mirrors in a manner that a virtual image is formed. We
also learned that there are two simple rules of reflection for convex mirrors.These rules represent slight revisions of the
two rules given for concave mirrors. The revised rules can be stated as follows:
· Any incident
ray traveling parallel to the
principal axis on the way to a
convex mirror will reflect in such a manner that its extension will pass
through the focal point.
· Any incident
ray traveling towards a convex mirror such that its extension passes through
the focal point will reflect and travel parallel to the principal axis.
These two rules will be used to construct ray diagrams. A ray
diagram is a tool that is used to determine the location, size, orientation,
and type of image formed by a mirror. Ray diagrams for concave mirrors were drawn in
Lesson 3. In this lesson, we will see a similar method for constructing ray
diagrams for convex mirrors.
The method of drawing ray diagrams for convex mirrors is
described below.
1. Pick a point on the top of the object and draw two incident rays
traveling towards the mirror.
Using
a straight edge, accurately draw one ray so that it travels towards the focal
point on the opposite side of the mirror; this ray will strike the mirror
before reaching the focal point; stop the ray at the point of incidence with
the mirror. Draw the second ray such that it travels exactly parallel to the
principal axis. Place arrowheads upon the rays to indicate their direction of
travel.
2. Once these incident rays strike the mirror, reflect them according to
the two rules of reflection for convex mirrors.
The
ray that travels towards the focal point will reflect and travel parallel to
the principal axis. Use a straight edge to accurately draw its path. The ray
that traveled parallel to the principal
axis on the way to the mirror will reflect and travel in a direction such that
its extension passes through the focal point. Align a straight edge with the
point of incidence and the focal point, and draw the second reflected ray.
Place arrowheads upon the rays to indicate their direction of travel. The two
rays should be diverging upon reflection.
3. Locate and mark the image of the top of the object.
The
image point of the top of the object is the point where the two reflected rays
intersect. Since the two reflected rays are diverging, they must be extended
behind the mirror in order to intersect. Using a straight edge, extend each of
the rays using dashed lines. Draw the extensions until they intersect. The
point of intersection is the image point of the top of the object. Both
reflected rays would appear to diverge from this point. If your were to draw a
third pair of incident and reflected rays, then the extensions of the third
reflected ray would also pass through this point. This is merely the point
where all light from the top of the object would appear to diverge from upon
reflecting off the mirror. Of course, the rest of the object has an image as
well and it can be found by applying the same three steps for another chosen
point. See note below.
4. Repeat the process for the bottom of the object.
The
goal of a ray diagram is to determine the location, size, orientation, and type
of image that is formed by the convex mirror. Typically, this requires
determining where the image of the upper and lower extreme of the object is
located and then tracing the entire image. After completing the first three
steps, only the image location of the top extreme of the object has been found.
Thus, the process must be repeated for the point on the bottom of the object.
If the bottom of the object lies upon the principal axis (as it does in this
example), then the image of this point will also lie upon the principal axis
and be the same distance from the mirror as the image of the top of the object.
At this point the complete image can be filled in.
Some students have difficulty understanding how the entire
image of an object can be deduced once a single point on the image has been
determined. If the object is merely a vertical object (such as the arrow object
used in the example below), then the process is easy. The image is merely a
vertical line. This is illustrated in the diagram below. In theory, it would be
necessary to pick each point on the object and draw a separate ray diagram to
determine the location of the image of that point. That would require a lot of
ray diagrams as illustrated in the diagram below.
Fortunately, a shortcut exists. If the object is a vertical
line, then the image is also a vertical line. For our purposes, we will only
deal with the simpler situations in which the object is a vertical line that
has its bottom located upon the principal axis. For such simplified situations,
the image is a vertical line with the lower extremity located upon the
principal axis.
The ray diagram above illustrates that the image of an object
in front of a convex mirror will be located at a position behind the convex
mirror. Furthermore, the image will be upright, reduced in size (smaller than
the object), and virtual. This is the type of information that we wish to
obtain from a ray diagram. The characteristics of this image will be discussed
in more detail in the next section of Lesson 4.
Once the method of drawing ray diagrams is practiced a couple
of times, it becomes as natural as breathing. Each diagram yields specific
information about the image. It is suggested that you take a few moments to
practice a few ray diagrams on your own and to describe the characteristics of
the resulting image. The diagrams below provide the setup; you must merely draw
the rays and identify the image. If necessary, refer to the method described
above.