Final answer:
To derive the mirror formula for a concave mirror, we use ray tracing and the small-angle approximation, arriving at 1/f = 1/do + 1/di, with R = 2f, where f is the focal length and R is the radius of curvature of the mirror.
Step-by-step explanation:
The derivation of the mirror formula for a concave mirror involves geometrical optics and the small-angle approximation. By using ray tracing, we find that when an object is placed in front of a concave mirror, rays reflecting from the mirror converge at the focal point if they are parallel to the principal axis. The mirror formula is given by 1/f = 1/do + 1/di, where f is the focal length, do is the distance of the object from the mirror, and di is the distance of the image from the mirror. This formula implies that the focal length (f) of a concave mirror is positive, and it is half of its radius of curvature (R), which can be represented as R = 2f. The signs in this formula are important: a negative value for di indicates a virtual image, a positive di a real image; a negative h indicates an inverted image, while a positive h indicates an erect image.