derivation of lens maker's formula from ab-initio
It is a relation that connects focal length of a lens to radii of curvature of the two surfaces of the lens and refractive index of the material of the lens.
Consider a convex lens (or concave lens) of absolute refractive index m 2 to be placed in a rarer medium of absolute refractive index m 1.
Considering the refraction of a point object on the surface XP1Y, the image is formed at I1 who is at a distance of V1.
CI1= P1I1 = V1 (as the lens is thin)
CC1 = P1C1 = R1
CO = P1O = u
It follows from the refraction due to convex spherical surface XP1Y
The refracted ray from A suffers a second refraction on the surface XP2Y and emerges along BI. Therefore I is the final real image of O.
Here the object distance is
(Note P1P2 is very small)
(Final image distance)
Let R2 be radius of curvature of second surface of the lens.
It follows from refraction due to concave spherical surface from denser to rarer medium that
Adding (1) & (2)