Lens makers formula:

It is a relation between the focal length of a lens to the refractive index of its material and the radii of curvature of its two surfaces. It is so called because it is used by lens manufacturers to make lenses of particular power from the glass of given refractive index.

Assumptions:

1. The lens is thin so that the distance measured from the poles of the two surfaces of the lens can be taken to be equal to the distances measured from the optical centre. b) The object is a point object which is situated on the principal axis. c) The aperture of the lens is small. d) The incident as well as refracted ray makes small angle with the principal axis.

Derivation:

Consider a thin convex lens made of a material of absolute refractive index n₂ , placed in a rarer medium of absolute refractive index n₁.

Also, n = n₂ / n₁ be the refractive index of the material of the lens with respect to the medium surrounding it.

R1 and R2 are the radii of curvature of surfaces XP₁Y and XP₂Y respectively.

For refraction at surface XP₁Y : O is the object and I¹ is its real image.

Using the formula, n₂ / v – n₁ / u = n₂ – n₁ / R₁

we get,  n₂ / v¹ – n₁ / u = n₂ – n₁ / R₁ -------------------Eqn (1)

For refraction at surface XP₂Y : I¹ is the virtual object and I is its real image which is the final image.

Using the formula, n₁ / v – n₂ / u = n₁ – n₂ / R₁

we get, n₁ / v – n₂ / v¹ = n₁ – n₂ / R₂ -------------------Eqn (2)

Adding equations (1) and (2), we get n₁ (1/v – 1/u) = (n₂ – n₁)(1/R₁ – 1/R₂)

Divide both sides by n₁ we get, (1/v – 1/u) = (n₂ – n₁)/n₁ (1/R₁ – 1/R₂)

1/f = (1/v – 1/u) = (n₂/n₁ – 1) (1/R₁ – 1/R₂)

1/f = (1/v – 1/u) = (n – 1) (1/R₁ – 1/R₂)

This is called lens maker s formula.