The graphical method of locating the image produced by a concave mirror consists of drawing light-rays emanating from key points on the object, and finding where these rays are brought to a focus by the mirror. This task can be accomplished using just four simple rules:
Consider an object which is placed a distance from a concave spherical mirror, as shown in Fig. 71. For the sake of definiteness, let us suppose that the object distance is greater than the focal length of the mirror. Each point on the object is assumed to radiate light-rays in all directions. Consider four light-rays emanating from the tip of the object which strike the mirror, as shown in the figure. The reflected rays are constructed using rules 1-4 above, and the rays are labelled accordingly. It can be seen that the reflected rays all come together at some point . Thus, is the image of (i.e., if we were to place a small projection screen at then we would see an image of the tip on the screen). As is easily demonstrated, rays emanating from other parts of the object are brought into focus in the vicinity of such that a complete image of the object is produced between and (obviously, point is the image of point ). This image could be viewed by projecting it onto a screen placed between points and . Such an image is termed a real image. Note that the image would also be directly visible to an observer looking straight at the mirror from a distance greater than the image distance (since the observer's eyes could not tell that the light-rays diverging from the image were in anyway different from those which would emanate from a real object). According to the figure, the image is inverted with respect to the object, and is also magnified.
Figure 72 shows what happens when the object distance is less than the focal length . In this case, the image appears to an observer looking straight at the mirror to be located behind the mirror. For instance, rays emanating from the tip of the object appear, after reflection from the mirror, to come from a point which is behind the mirror. Note that only two rays are used to locate , for the sake of clarity. In fact, two is the minimum number of rays needed to locate a point image. Of course, the image behind the mirror cannot be viewed by projecting it onto a screen, because there are no real light-rays behind the mirror. This type of image is termed a virtual image. The characteristic difference between a real image and a virtual image is that, immediately after reflection from the mirror, light-rays emitted by the object converge on a real image, but diverge from a virtual image. According to Fig. 72, the image is upright with respect to the object, and is also magnified.
The graphical method described above is fine for developing an intuitive understanding of image formation by concave mirrors, or for checking a calculation, but is a bit too cumbersome for everyday use. The analytic method described below is far more flexible.
Consider an object placed a distance in front of a concave mirror of radius of curvature . In order to find the image produced by the mirror, we draw two rays from to the mirror--see Fig. 73. The first, labelled 1, travels from to the vertex and is reflected such that its angle of incidence equals its angle of reflection. The second ray, labelled 2, passes through the centre of curvature of the mirror, strikes the mirror at point , and is reflected back along its own path. The two rays meet at point . Thus, is the image of , since point must lie on the principal axis.
In the triangle , we have
, and in the
triangle we have
, where is
the object distance, and is the image distance. Here,
is the height of the object, and is the height of
the image. By convention, is a negative number, since
the image is inverted (if the image were upright then
would be a positive number). It follows that
From triangles and , we have
These expressions yield
For an object which is very far away from
the mirror (i.e.,
so that light-rays from the object are parallel to the principal
axis, we expect the image to form at the focal point
of the mirror. Thus, in this case, , where is
the focal length of the mirror, and Eq. (355) reduces to
The above expression was derived for the case of a real
image. However, as is easily demonstrated, it also applies
to virtual images provided that the following sign convention
is adopted. For real images, which always form in front
of the mirror, the image distance is positive. For
virtual images, which always form behind the mirror,
the image distance is negative. It immediately follows,
from Eq. (352), that real images are always inverted, and
virtual images are always upright. Table 5
shows how the location and character of the image formed
in a concave spherical mirror depend on the location of
the object, according to Eqs. (352) and (358). It is
clear that the modus operandi of a shaving mirror,
or a makeup mirror, is to place the object (i.e., a
face) between the mirror and the focus of the mirror. The image
is upright, (apparently) located behind the mirror, and magnified.