Vision
Before we start analysing mirrors, it is useful
to first understand a little bit about how
human eyes and the human brain provides
us with depth perception. To understand
this, and many other issues in geometrical
optics, we need to first understand
how we perceive the distance to a point
source. The light from a point
source reaches the retina of our two eyes at
different positions. The brain notices this ``disparity''
and uses it to estimate the distance to the source.
To a lesser extent this can works for
one eye however the resolution is much worse.
Even the resolution of distance to
a point source using the disparity
in two eyes does not give high resolution.
Our eyes provide excellent depth resolution
when the landscape has many features. The
brain uses all of these features as
landmarks in forming an overall perception
of the scene. Depth resolution is much
worse when those features are absent, for
example a point source of light on
a dark night. Disparity is due
to diverging rays of light. By
tracing these rays back to their
source we perceive depth. Subconsciously our brain
carries out this process all the time.
Now we have to do it consciously. The
focus of our analysis is thus to
find points from which light appears to
emanate. These points are
perceived by our brains as an ``object''. If they
are not a true object or source of the original
light, we call them an image.
The bottom line: If
light rays diverge from a common point,
our eyes locate the image at that
common ``source''.
All of our analysis in geometrical optics thus
focuses on identifying points
at which light rays converge and then diverge. These points are
identified as images. We will be interested in
controlling the location and size of these images.
Two main results for plane mirrors
1. If a point object is placed a distance d
in front of a plane mirror, it produces a virtual
image a distance d behind the mirror.
2. The image is reversed.
Note that our eyes cannot distinguish
between the virtual image and
the real object (with no mirror!)
at that same location. Thus
if we now place a second mirror
in front of the first, the rays
from the virtual
image are reflected as if the
virtual image were a real object.
It is thus evident that we
can construct ray diagrams
recursively, treating one mirror
at a time. This is very helpful in
analysing complex geometrical systems.
Spherical mirrors
Concave mirror : A section of the interior surface of a sphere.
Principle rays used to identify images in mirrors
1. A ray parallel to the axis reflects through the focal point.
2. A ray through the center of curvature of the spherical mirror reflects directly back to the point of origin.
3. A ray reflecting from the point where the central axis meets the mirror reflects symmetrically.
4. A ray through the focal point reflects parallel to the central axis.
From the ray construction of the image, it is possible to
find the key relations,
 |
(1) |
 |
(2) |
- f is the focal length
- s is the object distance
- i is the image distance
- M is the magnification
Sign conventions for mirrors
1. A real image is on the same side of the mirror as the source. A virtual image is behind the mirror.
2. Real images have positive image distance, virtual images have negative image distance.
3. Inverted images have negative magnification. Upright images have positive magnification.
4. Diverging (convex) mirrors have negative focal length.
Convex mirror: The exterior of a section
of a sphere.
Summary of results for spherical mirrors
You need to know have to use ray diagrams to illustrate each of these
cases.
Concave mirrors
- If s>R, the image is real, inverted and reduced
in size.
- If f<s<R, the image is real inverted and magnified.
- If s<f, the image is virtual, upright and magnified.
Convex mirrors
- For all object(source) distances, the image is
virtual, upright and reduced in size.