Topic
21. Spherical Mirrors, Part 2: Image formation.
The
rules of the preceding topic (see Topic
20. Spherical Mirrors, Part 1.) can be used to calculate the position and
size of the image produced by any spherical mirror.
1.
The image produced by a convex mirror. The image is formed behind the mirror
and is therefore virtual. The image is erect and is smaller than the object.
The image is formed at a point behind the mirror that is almost always less
than the distance from the front of the mirror to the object. The image
distance varies in the same direction as the object distance – an increase in
one produces a corresponding increase in the other. However, the maximum image
distance is the focal distance when the object is very, very far away, and the
minimum image distance is 0 when the object is right at the mirror.
The
magnification is the ratio of the size of the image to the size of the object.
Since the object and image vectors form the bases of two similar triangles, the
magnification is the ratio of the distance of the image to the focal point
divided by the distance of the object to the focal point, and this ratio is
almost always less than 1. It approaches 1 as the object comes closer and
closer to the mirror surface.
If a point source of light is very far away from a convex mirror, then, although the wavefronts are spheres, they have very large radii and so look like plane waves. The rays that strike the mirror are essentially parallel to each other in this case.
If the rays strike the mirror parallel to its axis, then all of them are reflected so that the reflected rays appear to come from the focal point. The reflected rays therefore look like a single point source of light located at the focal point.
If the rays strike the mirror at an angle to the axis, then image point can be found using the usual 3 rays: one that is aimed at the center of curvature and is reflected back on itself, one that is aimed at the focal point and is reflected parallel to the axis, and one that is directed at the center of the mirror and is reflected symmetrically below the axis. The image point is on the focal plane, but not at the focal point. That is, it is located directly above (or below) the focal point.+
2. The image produced by a concave mirror when the object is outside of the focal point. The image is inverted and is real. The rays intersect at the image point and really do appear to diverge again from that point.
When the object is further away from the mirror than the center of curvature (as shown in the figure), the image will be formed somewhere between the center of curvature and the focal length. The image will always be inverted and will be smaller than the object.
3.
As the object in the diagram above moves in towards the mirror, the image moves
out to meet it. The following diagram shows the object and image when the
object has moved in far enough so that it is located at the center of curvature
of the mirror. No ray from the top of the object can pass through the center of
curvature in this configuration, but we can find the position of the image
using other standard rays: a ray parallel to the axis and a ray that strikes
the mirror exactly at its center.
The
diagram below is symmetric between the object and the image: both are located
at the center of curvature and are equal in size. The image is inverted, but it
has now grown to be the same size as the object.
4.
This process continues as the object is moved closer and closer to the mirror.
The image continues to increase in size, but it remains inverted and it continues
to move further and further away from the mirror. Nothing special happens as
long as the object does not actually get to the focal point. The image is
always inverted, and it is always larger than the object.
5.
The image produced by a concave mirror when the object is at the focal point.
The rays from the bottom of the arrow all emerge from the focal point and
therefore are reflected parallel to the axis. The rays do not converge to an
image point at any finite distance. We can think of this as if the image is
formed at infinity, where all of the parallel lines meet again.
Rays
from the top of the arrow cannot pass through the center of curvature or the
focal point. A ray that left the top of the arrow parallel to the axis would be
reflected through the focal point, and a ray from the top of the arrow that was
headed for the center of the mirror would be reflected downward by the same
angle. These two rays are divergent and do not converge to a single point at
any finite distance. This confirms the statement that there is no image of the
entire object at any finite distance – only at infinity.
The
same figure shows the image produced when the object is very far away. These
two cases are symmetrical – rays entering parallel to the axis from infinity
converge at the focal point, and rays emerging from the focal point in any
direction are reflected parallel to the axis.
If
the rays are parallel (from an object at infinity, for example), but strike the
mirror at an angle to the axis, then the same thing happens here as happened
above with a convex mirror. The image is formed in the focal plane, but not at
the focal point. The position of the image is found using the same technique as
was used above.
6.
When the object moves inside of the focal point, then the image changes in a
fundamental way. It is no longer possible for a ray from the object to actually
pass through either the center of curvature or the focal point, since the
object is inside of both of these distances.
However, there are still rays which leave the object along these same
directions so that they might have seemed to emerge from the focal point or the
center of curvature, and these rays can be used to construct the image in this
case. Likewise, a ray which leaves the object headed for the center of the
mirror is reflected symmetrically about the axis of the mirror as usual.
These
rays do not converge to any finite point, and therefore there is no real image
formed. However, the rays appear to be coming from a point on the right hand
side of the mirror, and therefore there is a virtual image formed on the right
hand side. Note that this image is not inverted as the real images above were,
but is erect. Also, it is larger than the object.
