Lens Maker’s Formula

A lens is a transparent medium surrounded by two surfaces, at least one of which must be curved. If the gap between the two surfaces is very small, the lens is said to be very thin. The lens converges when the focal length is positive and diverges when the focal length is negative. Therefore, we can conclude from this that the convex lens is not necessarily converging, and the concave lens is not necessarily diverging.

The relation between the focal length of the lens, the refractive index of its material, and the radii of curvature of its two surfaces is known as the lens maker's formula. In this article, we will understand about lens maker’s formula, lens maker equation, Focal Length and Radius of Curvature. So without any further ado, let’s begin by learning about Lens maker’s formula:

The lens is a refracting device consisting of a transparent material. It can have two curved surfaces or one curved and one plane surface. The lenses can be classified as converging (convex) and diverging lenses (concave). Lens maker’s formula relates the focal length, radii of curvature of the curved surfaces, and the refractive index of the transparent material. The formula is used to construct lenses with desired focal lengths. The formula applies to both types of lenses. The sign convention should be followed in the application of the lens maker’s equation. 

Focal Length and Radius of Curvature Definition

When parallel light rays are incident on a lens, the refracted rays converge to a point (for a converging lens) or appear to diverge from a point (for a diverging lens). This point is known as the focus of the lens. The distance between the optical centre and the focus is known as the focal length.

The curved surfaces of a lens belong to two spheres. The radii of these spheres are known as the radii of curvature of the lens. Depending on the shape of the lens, the radii change.

Lens Maker Equation

The lens maker formula for a lens of thickness d and refractive index μ is given by,

\[\frac{1}{f}=(\mu -1)[\frac{1}{R_1}-\frac{1}{R_2}+(1-\frac{1}{\mu })\frac{d}{R_1R_2}]\]

Here, R1 and  R2 are the radii of curvature of the two surfaces.

For a thin lens, the thickness d is taken to be zero. The lens maker equation for a thin lens is given by,

\[\frac{1}{f}=(\mu -1)(\frac{1}{R_1}-\frac{1}{R_2})\]

General Equation of a Convex Lens

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A convex lens of negligible thickness is considered in the above figure. The refractive index of the lens is mu2, and it is kept in a medium of refractive index mu1. A point object O is kept in front of the lens. An image I' is formed due to refraction at the first surface with a radius of curvature R1. This intermediate image serves as an object for the second surface with a radius of curvature R2. The final image I is formed due to refraction at the second surface. The object and the images lie on the principal axis. The principal axis intersects the surfaces at points C and D. The optical centre of the lens is at P. Since the lens is thin, the points C, D and P are considered to be overlapping. 

According to the sign convention for refraction at a curved surface, 

• Distances, measured along the direction of the incident light, are positive. 

• Any distance, opposite to the direction of the incident light, is negative.  

If an object (at a medium with refractive index mu1) is placed at a distance u, in front of a spherical surface of refractive index mu2, having a radius of curvature R, an image is formed at a distance v from that surface such that,

\[\frac{\mu _2}{v}-\frac{\mu _1}{u}=\frac{\mu _2\mu _1}{R}\]

For refraction at the first surface, object distance OC ≈ OP = - u and image distance is I'CI'P=v'. The radius of curvature is R1. Then the object-image relation due to refraction at the first surface gives,

\[\frac{\mu _2}{v}-\frac{\mu _1}{u}=\frac{\mu _2\mu _1}{R_1}\]

The intermediate image serves as the object for the second surface. Therefore, the object distance is I'DI'P=v' and the final image distance is IDIP=v. The radius of curvature is negative, i.e. -R2. Applying the object-image relation due to refraction at the second surface,

\[\frac{\mu _2}{v}-\frac{\mu _1}{v’}=\frac{\mu _2\mu _1}{R_2}\]

Adding the two equations,

\[\frac{1}{v}-\frac{1}{u}=(\frac{\mu _1}{\mu _2}-1)(\frac{1}{R_1}-\frac{1}{R_2})\]

If the object is at infinity, the image is formed at the focus, i.e. u=∞ and v=f. This gives,

\[\frac{1}{f}=(\frac{\mu _1}{\mu _2}-1)(\frac{1}{R_1}-\frac{1}{R_2})\]

If the ambient medium is taken to be air, i.e.

μ1≈1  and μ2=μis considered, the lens maker formula can be given in the usual form.

Lens Maker Formula for Concave Lens and Convex Lens

For a concave lens, R1 is negative and R2 is positive. The lens maker formula for the concave lens is given by,

\[\frac{1}{f}=-(\frac{\mu _1}{\mu _2}-1)(\frac{1}{R_1}+\frac{1}{R_2})\]

For a convex lens, R1 is positive and R2 is negative. The lens maker formula takes the following form,

\[\frac{1}{f}=-(\frac{\mu _1}{\mu _2}-1)(\frac{1}{R_1}+\frac{1}{R_2})\]

Limitations of Lens Maker’s Formula

The limitations of the lens maker’s formula are as follows:

• The lens needs to be thin. This is because the distance between the two refractive surfaces is also very small. 

• The medium on both sides of the lens needs to be the same. 

If the limitation is violated, refraction at the surface formula is used for both surfaces.

Solved examples

1.The focal length of a lens, made up of glass, is 5 cm in air. What would be the focal length of the same lens in water? The refractive indices of glass and water are 1.51 and 1.33 respectively.

Solution: The radii of curvature of the lens are R1 and R2. The focal length of the lens in the air is fa= 5 cm. The refractive index of glass is μg = 1.51 Applying lens maker equation for air,

\[\frac{1}{fa}=(\mu g-1)(\frac{1}{R_1}-\frac{1}{R_2})\]

\[\frac{1}{5cm}=(1.51-1)(\frac{1}{R_1}-\frac{1}{R_2})\]

\[(\frac{1}{R_1}-\frac{1}{R_2})=\frac{1}{2.55cm}\]

If the focal length is fw in water, the lens maker equation gives,

\[\frac{1}{fw}=(\frac{\mu g}{\mu w}-1)(\frac{1}{R_1}-\frac{1}{R_2})\]

Here,μw=1.33 is the refractive index of water. Substituting the value of(1/R1−1/R2),  

\[\frac{1}{fw}=(\frac{1.51}{1.33}-1)\frac{1}{2.55cm}\]

\[fw=18.84cm\]

The focal length in water is 18.84 cm.

2.An equiconvex lens is made up of material with a refractive index of 1.5, and it has a radius of curvature of 20 cm. What is the focal length?

Solution: The radii of curvature of the two surfaces are equal, i.e. 

R1= -  R2 = R.

Refractive index =1.5 and R=20 cm. If the focal length is f, applying lens maker formula, 

\[\frac{1}{f}=(\mu -1)\frac{2}{R}\]

\[\frac{1}{f}=(1.5-1)\frac{2}{20cm}\]

f=20 cm

The focal length is 20 cm.

3. Find out the focal length of the lens whose refractive index is 2. The radius of curvatures of is 20 cm and -35 cm of each surface respectively.

Solution: Given parameters are:

μ=2

R1 = 20 cm and R2= – 35 cm

Lens maker’s formula is:

\[\frac{1}{f}=(\mu -1)\times (\frac{1}{R_1}-\frac{1}{R_2})\]

\[\frac{1}{f}=(2-1)\times (\frac{1}{20}-\frac{1}{-35})\]

\[\frac{1}{f}=1\times (0.05+0.028)\]

\[\frac{1}{f}=0.078\]

\[f=\frac{1}{0.078}\]

\[f=12.82cm\]

4. Given that the focal length of a bi-convex (convex on both sides) lens made with glass (refractive index 1.5) having equal radii of curvature is 20 cm, find the radii of curvature of the lens.   

Solution: We have the following information given

R2=R1

Let R1=R. Then R2= −R 

μ=1.5

f=20cm

Since the lens is surrounded by air (uniform medium) and has the same radii of curvature on both sides, the system is symmetrical.

Substituting the above data in the lens maker’s formula: 

\[\frac{1}{20}=(1.5-1)\times (\frac{1}{R}-\frac{1}{-R})\] 

\[\frac{1}{20}=\frac{0.5\times 2}{R}\]

\[\frac{1}{20}=\frac{1}{R}\]

Therefore,

R=20cm 

Did you know?

• If a convex lens, having focal length f, is cut along the principal axis, both the resulting pieces will have the same focal length f. 

• Water droplets can be considered as convex lenses and the lens maker formula is applicable.

• The reciprocal of the focal length is called the refractive power that has units dioptre (inverse meter).