For paraxial rays, show that the focal length of a spherical mirror is one-half of its radius of curvature.
Consider a ray of light AB, parallel to the principal axis and incident on a spherical mirror at point B. The normal to the surface at point B is CB and CP = CB = R is the radius of curvature. The ray AB, after reflection from the mirror, will pass through F (concave mirror) or will appear to diverge from F (convex mirror) and obeys the law of reflection i.e. i = r.
From the geometry of the figure,
InCFB,∠BCP =∠ABC = i (Alternate angles)
∠CBF = r
BF = FC (because i = r)
If the aperture of the mirror is small, B lies close to P, and therefore BF = PF
Or FC = FP = PF
Or PC = PF + FC = PF + PF
Or R = 2 PF = 2f
Or `"f" = "R"/2`
A similar relation holds for convex mirror also. In deriving this relation, we have assumed that the aperture of the mirror is small.