# Lens Maker Formula Derivation With Diagram

In this article, we will be learning about the **Lens Maker Formula** which is also included in the syllabus of Physics in Class 12. We will first begin with the Definition, followed by some assumptions that are made to derive the lens maker formula. We will then go through the Sign Conventions used. Then we will derive Lens Maker Formula: **derive lens maker formula for convex lenses** and **lens maker formula for concave lenses** with diagrams.

The formula giving the relation between the focal length (f) of the lens, the refractive index of the material of the lens (n) and the radii of curvature of its surface (R_{1} & R_{2}) is known as **Lens maker formula**.

The manufacturers of the lenses use this relation to design the lens of a given focal length and hence it is called so.

## Assumptions made to derive Lens maker formula

(i) The lens is thin and all the distances are measured from the optical centre of the lens.

(ii) The aperture of the lens is small.

(iii) The object is a point object and lies on the principal axis.

(iv) The angle made by incident ray and refracted ray with the principal axis are small.

## New Cartesian Sign Conventions

(i) All the distances are measured from the optical centre of the lens.

(ii) Distances measured in the direction of the propagation of incident light are taken as positive while the distances measured in the direction opposite to the direction of propagation of incident light are taken as negative.

## Derivation of Lens Maker Formula for Convex lens

Consider a lens made of a material of absolute refractive index n_{2}. This lens is placed in a medium of absolute refractive index n_{1} (n_{1} < n_{2} ). This lens is bounded by two spherical refracting surfaces XP_{1}Y and XP_{2}Y. C_{1} and C_{2} be their centres of curvature and R_{1} and R_{2} be their radii of curvature respectively. C is the optical centre of the lens.

**STEP I. Refraction at Surface XP**_{1}**Y**

Let O be a point object lying in the rarer medium on the principal axis of the refracting surface XP_{1}Y. The incident ray OA after refraction at A bends towards the normal AC_{1} and meets the principal axis at I_{1} if the second surface XP_{2}Y were not present. So I_{1} is the real image of the object O.

Since object lies in the rarer medium, so we have

– n_{1}/u + n_{2}/v_{1} = n_{2} – n_{1}/R_{1} ……(i)

**STEP 2. Refraction at Surface XP**_{2}**Y**

In fact, the ray AB refracted by the ﬁrst surface XP_{1}Y is refracted at B by the second surface XP_{2}Y and it ﬁnally meets the principal axis at I. The point I_{1} acts as a virtual object placed in the denser medium for the spherical surface XP_{2}Y. Now it is the situation, when the object is placed in the denser medium and image is formed in the rarer medium.

– n_{2}/v_{1} + n_{1}/v= n_{1} – n_{2}/R_{2}

or – n_{2}/v_{1} + n_{1}/v= – (n_{2} – n_{1})/R_{2}

– n_{1}/u + n_{1}/v = (n_{2} – n_{1}) (1/R_{1} – 1/R_{2})

or – 1/u + 1/v = (n_{2}/n_{1} – 1) (1/R_{1} – 1/R_{2})

But n_{2}/n_{1} = n, relative refractive index of the lens w.r.t. the rarer medium.

– 1/u + 1/v = (n – 1) (1/R_{1} – 1/R_{2})

If the object is at infinity, the image is formed at the principal focus of the lens.

Hence eqn. (vii) becomes

– 1/∞ + 1/f = (n – 1) (1/R_{1} – 1/R_{2})

or 1/f = (n – 1) (1/R_{1} – 1/R_{2})

Eqn. (viii) represents Lens maker formula.

## Derivation of Lens Maker Formula for Concave Lens

Let a concave lens have two spherical surfaces X_{1}P_{1}Y_{1} and X_{2}P_{2}Y_{2} having radius of curvature as R_{1} and R_{2} respectively.

**STEP I. Refraction at X**_{1}**P**_{1}**Y**_{1}

Let O be a point object lying in the rarer medium on the principal axis of the refracting surface X_{1}P_{1}Y_{1}. The incident ray OA after refraction at A bends towards the normal NC_{1} and travel along AB. It appears to come from point I_{1} if the second surface X_{2}P_{2}Y_{2} were not present. So I_{1} is the virtual image of the object O.

So object lies in the rarer med1um, so we have

– n_{1}/u + n_{2}/v = n_{2} – n_{1}/R

Here v = v_{1} and R = R_{1}

∴ – n_{1}/u + n_{2}/v_{1} = n_{2} – n_{1}/R_{1} ……(i)

**STEP 2. Refraction at X**_{2}**P**_{2}**Y**_{2}

Ray AB inside the lens will stiffer another refraction at surface X_{2}P_{2}Y_{2}. After refracting from surface X_{2}P_{2}Y_{2}, the ray travels along BD and appears to come from the point I. So, I is the ﬁnal virtual image of the object O. Point I_{1} acts as the virtual object placed in the denser medium for the surface X_{2}P_{2}Y_{2}.

– n_{2}/u + n_{1}/v = n_{1} – n_{2}/R

Here u = v_{1} and R = R_{2}

∴ – n_{2}/v_{1} + n_{1}/v = n_{2} – n_{1}/- R_{2}

**STEP 3. Adding Eqs. (i) and (ii), we get**

– n_{1}/u + n_{1}/v = (n_{2} – n_{1}) (1/R_{1} – 1/R_{2})

or 1/v – 1/u = (n_{2}/n_{1} – 1) (1/R_{1} – 1/R_{2})

or 1/v – 1/u = (n – 1) (1/R_{1} – 1/R_{2}) (n2/n_{1} = n)

For u = infinity, the image is formed at principal focus so v = f (the focal length of the lens).

Then 1/f – 1/∞ = (n – 1) (1/R_{1} – 1/R_{2})

or 1/f = (n – 1) (1/R_{1} – 1/R_{2})