can anyone derive lens maker formula for a concave lens please ????

Dasvidaniya

Lens Maker's Formula

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.

The following assumptions are made for the derivation:

The lens is thin, so that distances measured from the poles of its surfaces can be taken as equal to the distances from the optical centre of the lens.

The aperture of the lens is small.

Point object is considered.

Incident and refracted rays make small angles.

Consider a convex lens (or concave lens) of absolute refractive index m₂ to be placed in a rarer medium of absolute refractive index m₁.

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)

source; -google search ....