The minimum distance between an object and its real image formed by a concave mirror of focal length f is

The minimum distance between an object and its real image formed by a concave mirror can be found using the mirror formula:

1/f = 1/v + 1/u

Where: f is the focal length of the mirror, v is the image distance, and u is the object distance.

For a real image formed by a concave mirror, the object distance (u) and the image distance (v) are both negative (since they are measured against the direction of incident light).

The minimum distance between the object and the image occurs when the object is at the center of curvature (C) of the mirror. At this point, the object distance u = -2f (since the radius of curvature is twice the focal length for a spherical mirror), and the image is also formed at the center of curvature, so v = -2f.

Substituting these values into the mirror formula gives:

1/f = 1/(-2f) + 1/(-2f) 1/f = -1/2f - 1/2f 1/f = -1/f

This equation is only satisfied when f = f, confirming that the minimum distance between the object and its real image formed by a concave mirror is indeed 2f, or twice the focal length of the mirror.