The characteristics and location of an image formed by a spherical mirror can be determined from an equation which is called spherical mirror formula.
Concave mirror formula
To derive concave mirror formula consider fig. (14.4) where an object OA, is placed in front of a concave mirror. A ray of light starting from the end point A of the object and moving parallel to the principal axis strikes the mirror at the point E. it is reflected at E and passes through the principal focus F. A second ray also starting from A falls on the mirror at pole P. it is reflected by making an angle of reflection equal to the angle of incidence and intersects the first reflected ray at point B i. e., . Thus, point B is the real image of point A.
Generally, the distance of the object from the mirror is denoted by P and that of image as q. focal length of the mirror is denoted by f.
Therefore, the above equation can be written in the following manner:
Convex Mirror Formula
Consider an object OA placed in front of a convex mirror (fig. 14.5). A ray of light starts from the end point A of the object. It moves parallel to the principal axis. It strikes the mirror at the point E and reflected in the direction EM. If this ray is produced backwards (in dotted lines), it meets the principal axis at the principal focus F. this ray appears to be diverged from F. another ray starting from end point ‘A’ falls on the pole P of the mirror and is reflected by making an angle of reflection equal to the angle of incidence.
If this ray is produced backwards (in dotted lines), it intersects the first ray at the point B. thus, point B is the virtual image of ‘A’. if this process is repeated for other points of the object OA then the image IB of the object OA is obtained. This image is virtual, erect and diminished.
Using Fig. 14.5, we can prove that the relationship between the object distance p, from the pole, the image distance q from the pole and the focal length of convex mirror f is the same as given by Eq. 14.3 i.e.,
This equation is known as a spherical mirror formula. Since in case of convex mirror, image is always virtual and according to sign conventions, distance of virtual image and focal length of convex mirror is taken as negative.