Option 4 : Between the pole of the mirror and the principal focus
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Explanation:
- if an object is placed beyond the center of curvature then the image will be real, inverted and diminish
- if an object is placed at the center of curvature then the image will be real, inverted and of the same size
- if an object is placed at the principal focus no image will be formed i.e., the image will be formed at infinity and the image will be highly enlarged
- if an object is placed between principle focus and pole of the mirror then the image will be virtual, erect and magnifie
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Next: Image Formation by Convex
Up: Paraxial Optics
Previous: Spherical Mirrors
There are two alternative methods of locating the image
formed by a concave mirror. The first is purely graphical, and the
second uses simple algebraic analysis.
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:
- An incident ray which is parallel to the principal axis is
reflected through the focus of the mirror.
- An incident ray which passes through the focus
of the mirror is reflected parallel to the principal axis.
- An incident ray which passes through the centre of
curvature of the mirror is reflected back along its own
path (since it is normally incident on the mirror).
- An incident ray which strikes the mirror at its
vertex is reflected such that its angle of incidence with respect to
the principal axis is equal to its angle of reflection.
The validity of these rules in the
paraxial approximation is fairly self-evident.
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 71:
Formation of a real image by a concave mirror.
|
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.
Figure 72:
Formation of a virtual image by a concave mirror.
|
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.
Figure 73:
Image formation by a concave mirror.
|
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
Thus, the magnification
of the image with respect
to the object is given by
By convention,
is negative if the image is inverted with
respect to the object, and
positive if the image is upright. It is clear that the
magnification of the image is just determined by the
ratio of the image and object distances from the vertex.
From triangles
and
, we have
and
, respectively.
These expressions yield
Equations (352) and (353) can be combined to give
which easily reduces to
This expression relates the object distance, the image distance,
and the radius of curvature of the mirror.
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 yields
In other words, in the paraxial approximation, the focal length
of a concave spherical mirror is half of its radius of
curvature. Equations (355) and (357) can be combined to give
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.
Table 5:
Rules for image formation by concave mirrors.
Position of object |
Position of image |
Character of image |
At |
At |
Real, zero size |
Between and |
Between and |
Real, inverted, diminished |
At |
At |
Real, inverted, same size |
Between and |
Between and |
Real, inverted, magnified |
At |
At |
|
Between and |
From to |
Virtual,
upright, magnified |
At |
At |
Virtual, upright, same size |
|
Next: Image Formation by Convex
Up: Paraxial Optics
Previous: Spherical Mirrors
Richard Fitzpatrick
2007-07-14