Lens Maker’s Formula

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Lens Maker’s Formula

Have you ever wondered how opticians determine the exact curvature of glass needed to fix someone's vision? Or how camera manufacturers design lenses to capture perfectly focused images? They rely on a fundamental equation in optics called the Lens Maker’s Formula.

For Class 10 students, this topic serves as a brilliant bridge between geometry and physics, helping you understand how a lens's physical shape directly influences its ability to focus light.

1.0What is the Lens Maker’s Formula?

The Lens Maker’s Formula is a mathematical equation that relates the focal length (f) of a lens to the refractive index of its material (n) and the radii of curvature (R₁ and R₂) of its two spherical surfaces.

The Universal Equation:

$\frac{1}{f} = (\frac{n _{lens}}{n _{medium}} - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})$

If the lens is placed in air, the refractive index of the medium $(n_{medium})$ is equal to 1. The formula simplifies to:

$\frac{1}{f} = (n - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})$

f = Focal length of the lens
n = Refractive index of the lens material (usually glass)
R₁ = Radius of curvature of the first spherical surface (where light enters)
R₂ = Radius of curvature of the second spherical surface (where light exits)

Assumptions

The following assumptions are taken for the derivation of lens maker formula.

Let us consider the thin lens shown in the image above with 2 refracting surfaces having the radii of curvatures R₁ and R₂, respectively.
Let the refractive indices of the surrounding medium and the lens material be n₁ and n₂, respectively.

2.0Derivation of Lens Maker’s Formula

Spherical Refraction from 1st surface,

$\frac{μ _{2}}{v} - \frac{μ _{1}}{u} = \frac{μ _{2} - μ _{1}}{R} μ_{2} = μ_{l}, μ_{1} = μ_{s}, u = u, v = v_{1}, R = R_{1} \frac{μ _{l}}{v _{1}} - \frac{μ _{s}}{u} = \frac{μ _{l} - μ _{s}}{R _{1}} \dots\dots\dots .. (1)$

Spherical Refraction from 2nd surface,

$\frac{μ _{2}}{v} - \frac{μ _{1}}{u} = \frac{μ _{2} - μ _{1}}{R} μ_{2} = μ_{s}, μ_{1} = μ_{l}, u = v_{1}, v = v, R = R_{2} \frac{μ _{s}}{v} - \frac{μ _{l}}{v _{1}} = \frac{μ _{s} - μ _{l}}{R _{2}} \dots\dots\dots (2)$

Add equations (1) and (2)

$\frac{μ _{s}}{v} - \frac{μ _{s}}{u} = \frac{μ _{l} - μ _{s}}{R _{1}} + \frac{μ _{s} - μ _{l}}{R _{2}} \frac{μ _{s}}{v} - \frac{μ _{s}}{u} = (μ_{l} - μ_{s}) (\frac{1}{R _{1}} - \frac{1}{R _{2}})$

$\frac{1}{v} - \frac{1}{u} = (\frac{μ _{l}}{μ _{s}} - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})$

If an object is placed at infinity, i.e. u = −∞ , then v = f

$\frac{1}{f} - \frac{1}{- \infty} = (\frac{μ _{l}}{μ _{s}} - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}}) \frac{1}{f} = (\frac{μ _{l}}{μ _{s}} - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}}) \Rightarrow Lens Maker’s Formula$

3.0Sign Convention for Convex Lens

For a convex lens:

Focal length is positive
Radius of curvature of first surface is positive
Radius of curvature of second surface is negative

Convex lenses are converging lenses.

4.0Sign Convention for Concave Lens

For a concave lens:

Focal length is negative
Radius of curvature follows sign convention

Concave lenses are diverging lenses.

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(Session 2026 - 27)

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I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Lens Maker’s Formula

Have you ever wondered how opticians determine the exact curvature of glass needed to fix someone's vision? Or how camera manufacturers design lenses to capture perfectly focused images? They rely on a fundamental equation in optics called the Lens Maker’s Formula.

For Class 10 students, this topic serves as a brilliant bridge between geometry and physics, helping you understand how a lens's physical shape directly influences its ability to focus light.

1.0What is the Lens Maker’s Formula?

The Lens Maker’s Formula is a mathematical equation that relates the focal length (f) of a lens to the refractive index of its material (n) and the radii of curvature (R₁ and R₂) of its two spherical surfaces.

The Universal Equation:

$\frac{1}{f} = (\frac{n _{lens}}{n _{medium}} - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})$

If the lens is placed in air, the refractive index of the medium $(n_{medium})$ is equal to 1. The formula simplifies to:

$\frac{1}{f} = (n - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})$

f = Focal length of the lens
n = Refractive index of the lens material (usually glass)
R₁ = Radius of curvature of the first spherical surface (where light enters)
R₂ = Radius of curvature of the second spherical surface (where light exits)

Assumptions

The following assumptions are taken for the derivation of lens maker formula.

Let us consider the thin lens shown in the image above with 2 refracting surfaces having the radii of curvatures R₁ and R₂, respectively.
Let the refractive indices of the surrounding medium and the lens material be n₁ and n₂, respectively.

2.0Derivation of Lens Maker’s Formula

Spherical Refraction from 1st surface,

$\frac{μ _{2}}{v} - \frac{μ _{1}}{u} = \frac{μ _{2} - μ _{1}}{R} μ_{2} = μ_{l}, μ_{1} = μ_{s}, u = u, v = v_{1}, R = R_{1} \frac{μ _{l}}{v _{1}} - \frac{μ _{s}}{u} = \frac{μ _{l} - μ _{s}}{R _{1}} \dots\dots\dots .. (1)$

Spherical Refraction from 2nd surface,

$\frac{μ _{2}}{v} - \frac{μ _{1}}{u} = \frac{μ _{2} - μ _{1}}{R} μ_{2} = μ_{s}, μ_{1} = μ_{l}, u = v_{1}, v = v, R = R_{2} \frac{μ _{s}}{v} - \frac{μ _{l}}{v _{1}} = \frac{μ _{s} - μ _{l}}{R _{2}} \dots\dots\dots (2)$

Add equations (1) and (2)

$\frac{μ _{s}}{v} - \frac{μ _{s}}{u} = \frac{μ _{l} - μ _{s}}{R _{1}} + \frac{μ _{s} - μ _{l}}{R _{2}} \frac{μ _{s}}{v} - \frac{μ _{s}}{u} = (μ_{l} - μ_{s}) (\frac{1}{R _{1}} - \frac{1}{R _{2}})$

$\frac{1}{v} - \frac{1}{u} = (\frac{μ _{l}}{μ _{s}} - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})$

If an object is placed at infinity, i.e. u = −∞ , then v = f

$\frac{1}{f} - \frac{1}{- \infty} = (\frac{μ _{l}}{μ _{s}} - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}}) \frac{1}{f} = (\frac{μ _{l}}{μ _{s}} - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}}) \Rightarrow Lens Maker’s Formula$

3.0Sign Convention for Convex Lens

For a convex lens:

Focal length is positive
Radius of curvature of first surface is positive
Radius of curvature of second surface is negative

Convex lenses are converging lenses.

4.0Sign Convention for Concave Lens

For a concave lens:

Focal length is negative
Radius of curvature follows sign convention

Concave lenses are diverging lenses.

Frequently Asked Questions

What is Lens Maker Formula?

What is the SI unit of optical power?

Why is focal length positive for convex lens?

What happens if refractive index of lens and medium are equal?

Which lens has negative optical power?

Frequently Asked Questions

What is Lens Maker Formula?

What is the SI unit of optical power?

Why is focal length positive for convex lens?

What happens if refractive index of lens and medium are equal?

Which lens has negative optical power?

Join ALLEN!

Lens Maker’s Formula

1.0What is the Lens Maker’s Formula?

The Universal Equation:

2.0Derivation of Lens Maker’s Formula

3.0Sign Convention for Convex Lens

4.0Sign Convention for Concave Lens

On this page

Join ALLEN!

Lens Maker’s Formula

1.0What is the Lens Maker’s Formula?

The Universal Equation:

2.0Derivation of Lens Maker’s Formula

3.0Sign Convention for Convex Lens

4.0Sign Convention for Concave Lens

On this page