## The lens makers formula

The lens maker’s formula shows the relation between the focal length of a lens to the refractive index of its material.

## Lens

A lens has two surfaces, at least one of them should be curved. A thin lens has a very little gap between its surfaces. A lens having a positive focal length will be converging and a lens having a negative focal length will be diverging.

A convex lens shouldn’t necessarily be a converging lens and a concave lens should not necessarily be a diverging lens. The specific value of every lens can be computed using the lens makers formula.

## Lens Makers’ Formula

Real lenses have a definite thickness between their two surfaces. A thin lens having two surfaces with equal curvature has zero optical power. This lens would neither converge nor diverge light. A lens having a noticeable thickness is called a thick lens.

Lenses are of two types based on the curvature of the surfaces: convex and concave. The lens maker’s formula shows the relationship between the focal length of the lens to the refractive index of its material and the radii of the curvature of its two surfaces.

In the case of thin lenses, the distance measured between the poles of the two surfaces can be considered equal to the distance measured from the optical centre.

## Formula

The lens maker’s formula is used to study the lens of a specified focal length. The two curved surfaces of a lens are not exactly the same. With the help of the given refractive index and radius of the curvature of both the surfaces, the focal length of a lens can be determined using the lens maker’s formula.

1/f = (μ – 1) × (1/R₁) – (1/R₂)

‘f’ is the focal length of the lens

‘μ’ is the refractive index

‘R₁’ and ‘R₂’ are curvatures of both surfaces.

The medium on both sides of the lens should be the same. The refraction between the two refractive surfaces should also be small.

## Solved Example

**Question: The refractive index of a lens is 2. The radius of the curvature of the surfaces is 20 cm and (-35) cm respectively. Find out the focal length of the lens. **

**Answer: **

μ = 2

R₁ = 20

R₂ = (-35)

The lens makers formula: 1/f = (μ – 1) × (1/R₁) – (1/R₂)

1/f = (2-1) × (1/20) – {1 ÷ (-35) }

1/f = 1 × (0.05 + 0.028)

1/f = 0.078

f = 1/0.078

f = 12.82 cm