Lens Maker’s Formula: Definition, Derivation & Limitations

Lens Maker's Formula relates the focal length of a lens to the refractive index of the material of the lens and the radii of curvature of the two surfaces of the lens. It is used to make a lens of the required focal length from a glass of a given refractive index. Hence, its name. Its use varies from optical instruments such as magnifying glasses, telescopes, and microscopes to contact lenses which are used to correct eye disorders such as myopia, hyperopia, or hypermetropia. Before learning about the Lens maker’s formula, one must be well acquainted with the difference between a convex lens and a concave lens, the focal length, and the radius of curvature of a lens.

Key Terms: Lens Maker’s Formula, Focal Length, Refractive Index, Centre of Curvature, Radius of Curvature, Gaussian Lens Equation 

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What is Lens Maker’s Formula?

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The general equation for the lens maker’s formula is

\(\frac{1}{f}=(n-1)(\frac{1}{R1}-\frac{1}{R2})\)

Where, 

• f = focal length of the lens
• n = refractive index of the material of the lens with respect to the surrounding medium 
• R = Radius of curvature of the first surface of the lens
• R² = Radius of curvature of the second surface of the lens 

This equation is applicable to all kinds of lenses, ie- double convex or double concave, Plano- concave or plano-convex, and convexo-concave or concavo-convex lenses. This formula is used to make lenses of a particular desired focal length or power (reciprocal of focal length) from a piece of glass with a definite refractive index. Such lenses, (be it convex or concave) are used in telescopes, magnifying glasses, simple microscopes, or contact lenses.

Lens Maker’s Formula

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Lens Maker’s Formula Detailed Video Explanation:

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Derivation of Lens Maker’s Formula: Convex Lens

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In deriving this formula, we adopt the coordinate geometry sign convention and make the following assumptions:

1. The lens is taken to be so thin that the distances measured from the poles of its two surfaces can be taken as equal to the distances from its optical center.
2. The aperture of the lens is small.
3. An object is a point object placed on the principal axis of the lens.
4. The refracted and incident rays make small angles with the principal axis.
5. In order to derive the lens maker's formula, we must make sure that the refractive index of the material of the lens is always greater than the refractive index of the surrounding medium.

To derive the lens maker's formula for a convex lens let's take the example of a thin double convex lens of refractive index placed in a media of refractive index μ1, provided that μ2>μ1. Look at the figure given below.

Derivation for lens maker formula

• μ₁= refractive index of the surrounding medium
• μ₂= refractive index of the material of the lens
• C₁ and C₂= Centre of curvature of the first and second surface of the lens respectively
• R₁ and R₂ = Radius of curvature of the first and second surface of the lens respectively
• u = distance of the object O from the lens
• v₁ = distance of the image I₁
• v = distance of the actual image I formed by the lens

Step 1- Refraction at the first surface 

So, here an object O is placed on the principal axis in the surrounding medium of refractive index μ₁. In this step, we assume that the second surface of the lens that is, ADC is absent. A Ray of light from an object strikes the surface ABC.So, in the absence of the second surface, this ray would have refracted through the first surface of the lens and would have formed an image I₁ at a distance of v₁ from the lens. The equation for refraction at this surface is-

\(\frac{\mu2}{v}-\frac{\mu1}{u}=\frac{\mu2-\mu1}{R2}v = v1\)

\(\frac{\mu2}{v1}-\frac{\mu1}{u}=\frac{\mu2-\mu1}{R1}\)...(i)

Step 2- Refraction at the second surface of the lens 

The image I₁ serves as a virtual object for the second surface of the lens. Its image is formed at the point I behind the convex lens at a distance of v after undergoing refraction through the second surface of the lens. In this case, the equation of refraction takes the form of

\(\frac{\mu2}{v}-\frac{\mu1}{u}=\frac{\mu2-\mu1}{R2}v = v1\)

\(\frac{\mu2}{v}-\frac{\mu1}{v1}=\frac{\mu2-\mu1}{R2}\)...(ii)

Step 3- Adding equations (i) and (ii) 

\(\frac{\mu1}{v}-\frac{\mu2}{u}=({\mu2-\mu1})(\frac{1}{R1}-\frac{1}{R2})\)

Step 4- Dividing both sides by μ1, we get:

\(\frac{1}{v}-\frac{1}{u}=(\frac{\mu2}{\mu1}-1)(\frac{1}{R1}-\frac{1}{R2})\)

Step 5- Putting μ2/μ1 = n

\(\frac{1}{v}-\frac{1}{u}=(n-1)(\frac{1}{R1}-\frac{1}{R2})\)….(iii)

When the object is at infinity the image will be formed at the principle focus of the lens that is when u= ♾ then we V = f(the focal length of the lens) therefore from equation (iii) we have

\(\frac{1}{f}-1/\infty=(n-1)(\frac{1}{R1}-\frac{1}{R2})\)

This equation can also be written as:-

\(\frac{1}{f}=(n-1)(\frac{1}{R1}-\frac{1}{R2})\)

This is the lens maker's formula. The formula has been derived for a convex lens forming a real image, however, it is equally applicable to a convex lens leading to the formation of a virtual image and to a concave lens which forms only virtual images.

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Derivation of Lens Maker’s Formula: Concave Lens 

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To derive the lens maker's formula for a concave lens the same assumptions and conditions (as that for a convex lens) are used. However, the lens used is a concave lens instead of a convex lens. 

Now let’s consider a concave lens of refractive index μ₂ placed in a medium of refractive index μ₁ with a condition that μ₂<μ₁.

Step 1 - Refraction at the first surface

So, here an object O is placed on the principle axis of the concave lens of refractive index μ₂ which is in the surrounding medium of refractive index μ₁. In this step, we have assumed that the second surface of the lens, that is, ADC is absent. A Ray of light from object O strikes the surface ABC of the concave lens. So, in the absence of the second surface, this ray would have refracted and formed a virtual image I₁ in front of the concave lens. This is the equation for refraction at the first spherical surface of the lens:

v = v₁

\(\frac{\mu2}{v}-\frac{\mu1}{u}=\frac{\mu2-\mu1}{R2}\)

\(\frac{\mu2}{v1}-\frac{\mu1}{u}=\frac{\mu2-\mu1}{R1}\)..(i)

Step 2- Refraction at the second surface of the lens 

The image I₁ serves as a virtual object for the second surface of the lens. Its image is formed at the point I in front of the concave mirror after undergoing refraction through the second surface of the lens. In this step, the equation of refraction takes the form of

u = v₁

\(\frac{\mu2}{v}-\frac{\mu1}{u}=\frac{\mu2-\mu1}{R2}\)

\(\frac{\mu2}{v}-\frac{\mu1}{v1}=\frac{\mu2-\mu1}{R2}\)..(ii)

Step 3- Adding equations (i) and (ii) 

\(\frac{\mu1}{v}-\frac{\mu2}{u}=({\mu2-\mu1})(\frac{1}{R1}-\frac{1}{R2})\)

Step 4- Dividing both sides by μ1, we get:

\(\frac{1}{v}-\frac{1}{u}=(\frac{\mu2}{\mu1}-1)(\frac{1}{R1}-\frac{1}{R2})\)

Step 5- Putting μ2/μ1 = n

\(\frac{1}{v}-\frac{1}{u}=(n-1)(\frac{1}{R1}-\frac{1}{R2})\)….(iii)

When the object is at infinity the image will be formed at the principle focus of the lens that is when u= ♾ then we V = f (the focal length of the lens) therefore from equation (iii) we have

\(\frac{1}{f}-1/\infty=(n-1)(\frac{1}{R1}-\frac{1}{R2})\)

This equation can also be written as:-

\(\frac{1}{f}=(n-1)(\frac{1}{R1}-\frac{1}{R2})\)

This is the lens maker's formula. It has been derived from a concave lens forming a virtual image. 

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Gaussian Form of Lens Equation

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One comparison of the equations (iii) and (iv) we get that

\(\frac{1}{f}=\frac{1}{v}-\frac{1}{u}\)

This is the Gaussian form of the lens equation which can also be used to find out the focal distance of an object or the image formed by a convex or a concave lens of focal length f.

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Lens Maker’s Formula for a Convex and Concave Lens

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Dependence upon Radii of Curvature:-

The basic equation of the lens maker’s formula holds true for both, the convex and the concave lens with changes in the signs of the values of R₁ and R₂ according to the coordinate geometry sign convention. In other words, we can say that the focal length of a lens is dependent upon the radii of curvature of the two surfaces of the lens and it changes according to the coordinate geometry sign convention. 

For a double convex lens, R₁ is positive and R₂ is negative. Hence from the lens maker’s equation, we have

\(\frac{1}{f}=(n-1)(\frac{1}{+R1}-\frac{1}{-R2})\)

For a double concave lens, R₁ is negative and R₂ is positive. Hence, 

\(\frac{1}{f}=(n-1)(\frac{1}{-R1}-\frac{1}{+R2})\)

It is clear from these formulae that the focal length of a lens of large radii of curvature is large and that of a lens of small radii of curvature is small. In other words, the focal length of a thin lens is large and that of a thick lens is small.

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Limitations of Lens Maker’s Formula 

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• Lens maker's formula can be applied to lenses that have big apertures and are very thin. This formula cannot be used for a thick lens.
• The lens maker’s formula is applicable when the medium on both sides of the lens has the same refractive index. If there are two different media on both sides of the lens, then the equation will have to be derived accordingly.

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Things to Remember

• The lens maker’s formula shows the relation between the focal length of a lens, the refractive index of the surrounding medium, and the radii of curvatures of the lens.
• The lens maker's formula can be applied to any double convex or double concave lens. 
• The greater the radii of curvature of a lens the greater the focal length of the lens and vice versa.
• For a spherical drop of any liquid, it is assumed to be the double convex spherical lens and the lens maker's formula is equally applicable to it.
• Another form of the lens maker's formula is known as the Gaussian length form of the lens equation. It states the relationship between the focal length of the length and the distance of the object and the distance of the image from the optical center of the lens.
• The lens maker’s formula is used in the making of the lens which is used in microscopes, telescopes, magnifying glasses, and contact lenses, etc.

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Previous Year Questions

1. Light rays from a luminous (bright point) object in a denser medium of… [COMEDK UGET 2010]
2. The limit of resolution of a telescope is 4.88 x 10… [COMEDK UGET 2005]
3. The twinkling effect of starlight is due to… [JCECE 2009]
4. The lateral shift produced by a glass slab is X. When the slab is immersed in a liquid… [COMEDK UGET 2006]
5. If A is the angle of the prism, r is the angle of refraction, then the condition for minimum… [COMEDK UGET 2014]
6. A prism with a refracting angle of 30 degrees, made of material of refractive index 1.732… [COMEDK UGET 2011]
7. All lights are switched off, except for a bright point-light source kept at the… [COMEDK UGET 2015]
8. A plano-convex lens (f=20cm) is silvered at the plane surface… [COMEDK UGET 2010]
9. A person inside water nw=4/3 sees the 'setting sun' at about… [COMEDK UGET 2008]
10. A concave mirror gives an image three times as large as its object placed at a… [COMEDK UGET 2015]
11. An equiangular glass prism of refractive index 1.6 is kept fully immersed in water… [COMEDK UGET 2015]
12. A glass hemisphere of radius 0.1 cm and refractive index 1.5 is placed over… [COMEDK UGET 2012]
13. A concave mirror forms an enlarged, erect, virtual image of an object, only… [COMEDK UGET 2014]
14. A telescope has an objective of focal length 100cm and an eye-piece of focal… [BCECE 2009]
15. A bulb is kept at a depth h inside the water of refractive index n… [COMEDK UGET 2007]
16. A critical angle for a medium is 60∘. Then the refractive index of the medium… [GUJCET 2006]
17. The limit of resolution of an optical instrument arises on account of… [GUJCET 2007]
18. An object is placed at the focus of a convex mirror. If its focal length is… [BCECE 2009]
19. A person sees clearly at a distance of 100 cm, then the power of the lens used… [JCECE 2006]
20. A plano-convex lens of the refractive index of 1.5 and radius of curvature of 30cm… [BCECE 2004]

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