Show with the aid of a ray diagram that the radius of curvature of a concave mirror is twice the focal length of the mirror.

Show with the aid of a ray diagram that the radius of curvature of a concave mirror is twice the focal length of the mirror.

Consider the reflection of a paraxial ray, AX, parallel to the principal axis of a concave mirror as shown.

Taking FP = focal length and C = center of curvature, then, CX is the normal to the mirror surface and CP is the radius of curvatures.

since i and 2i are small angles;

tan i≈ i = XP/ CP  …………………..(i)

tan 2i ≈ 2i = XP/CP ……………… (ii)

Combining (i) and (ii)

2(XP/CP) =XP/FP

Dividing buy XP on both sides

2/CP  = 1/ FP  or  CP =2FP

But CP = r and FP = f

therefore, r = 2f

Thus, the radius of curvature is twice the focal length