Derivation Of Lens Maker Formula: Assumptions, Limitations and Solved Examples

Lens maker formula gives the relationship between the focal length (f) of the lens, the refractive index of the material of the lens (μ), and the radii of the curvature of its surface (R₁ and R₂).

• A lens is a piece of transparent material bounded by two refracting surfaces out of which at least one is curved.
• There are two types of lenses: 1) Concave lens and 2) Convex lens
• If the central portion of a lens is thinner than its edges, it behaves as a divergent lens known as a concave lens.
• If the central portion of a lens is thicker than its edges, it behaves as a convergent lens known as a convex lens.

Key Terms: Lens maker formula, Concave lens, Convex lens, Focal length, Radius of curvature, Center of curvature, Convergent lens, Divergent lens

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Lens Maker Formula

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The lens maker formula is given by

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

Where

• f is the focal length of the lens
• µ is the relative refractive index of the lens with respect to the rarer medium or air
• R₁ and R₂ are the radii of curvature of surfaces of the lens

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Assumptions Used in the Derivation of Lens Maker Formula

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The following are the assumptions used in the derivation of lens maker formula

• Consider a lens made of material of absolute refractive index µ₂. 
• This lens is placed in a medium of absolute refractive index µ₁, such that µ₁ < µ₂. 
• The lens is bounded by two spherical refracting surfaces. 
• C₁ and C₂ be centers of curvature, and R₁ and R₂ be their radii of curvature respectively. 
• C is the optical center of the lens.

Read More:

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Derivation of Lens Maker Formula For a Convex Lens

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For the first refracting surface XP₁Y, I₁ is the real image of the object O. Therefore

– \(\frac{\mu_1}{u} + \frac{\mu_2}{v}= \frac{\mu_2- \mu_1}{R_1}\) ……(i)

Convex Lens

For the second refracting surface XP₂Y, the point I₁ acts as a virtual object placed in the denser medium. Therefore

\(-\frac{\mu_2}{v_1} + \frac{\mu_1}{v} = \frac{\mu_1-\mu_2}{R_2}\) ……(ii)

Adding equations (i) and (ii), we get

\(-\frac{\mu_1}{u} + \frac{\mu_1}{v} = (\mu_2- \mu_1)[\frac{1}{R_1}-\frac{1}{R_2}]\)

⇒ \(-\frac{1}{u} + \frac{1}{v} = [\frac{\mu_2}{\mu_1}-1][\frac{1}{R_1} - \frac{1}{R_2}]\)

But µ₂ / µ₁ = µ (relative refractive index of the lens with respect to rarer medium or air).

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

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

if u = - ∞ then v = f

⇒ \(-\frac{1}{\infty} + \frac{1}{v} = [\mu -1][\frac{1}{R_1} - \frac{1}{R_2}]\)

Since 1/∞ = 0, therefore we can write

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

This is lens maker formula derivation.

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Derivation of Lens Maker Formula For a Concave Lens

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Consider a concave lens of two refracting surfaces X₁P₁Y₁ and X₂P₂Y₂.

Concave Lens

For the first refracting surface X₁P₁Y₁ object lying in the rarer medium and I₁ is the virtual image of the object O. Therefore

\(-\frac{\mu_1}{u} - \frac{\mu_2}{v_1} = \frac{\mu_2-\mu_1}{R_2}\) ……(i)

For the second refracting surface X₂P₂Y₂ the point I₁ acts as the virtual object and point I act as the virtual image of object O. Therefore

\(-\frac{\mu_2}{v_1} - \frac{\mu_1}{v}= \frac{\mu_1- \mu_2}{-R_1}\) ……(ii)

Adding equations (i) and (ii), we get

\(-\frac{\mu_1}{u} - \frac{\mu_1}{v} = (\mu_2- \mu_1)[\frac{1}{R_1}-\frac{1}{R_2}]\)

⇒ \(-\frac{1}{u} - \frac{1}{v} = [\frac{\mu_2}{\mu_1}-1][\frac{1}{R_1} - \frac{1}{R_2}]\)

But µ2 / µ1 = µ (relative refractive index of the lens with respect to rarer medium or air).

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

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

if u = - ∞ then v = - f

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

Since 1/∞ = 0, therefore we can write

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

Read More: Formation of Convex and Concave Lenses

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Limitations of The Lens Maker Formula

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The followings are the limitations of the lens maker formula

• Ideally, the lens should be thin so that there is a minimal gap between the two refracting surfaces.
• There should always be the same medium used on both sides of the lens.

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Solved Examples

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Ques. The radius of curvature of the faces of a double convex lens is 10 cm and 15 cm. If the focal length of the lens is 12 cm, find the refractive index of the material of the lens.

Ans. Given

• The radius of curvature of the first surface, R₁ = 10 cm
• The radius of curvature of the second surface, R₂ = - 15 cm
• Focal length of the lens, f = 12 cm

Using the lens maker formula

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

⇒ \(\frac{1}{12} = (\mu -1)[ \frac{1}{10}+\frac{1}{15}] = \frac{\mu - 1}{6}\)

⇒ \(\frac{1}{12}= \frac{\mu -1}{6}\)

⇒ μ = \(\frac{3}{2}\) = 1.5

Ques. The focal length of an equi-convex lens is equal to the radius of curvature of either face. What is the refractive index of the material of the lens?

Ans. Let R be the radius of curvature of the faces of the lens, then

• Focal length of the lens, f = R
• The radius of curvature of the first face, R₁ = +R
• The radius of curvature of the second face, R₂ = -R

Using the lens maker formula

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

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

⇒ 2μ – 2 = 1

⇒ μ = \(\frac{3}{2}\) = 1.5

Also Read:

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Things to Remember

• Lens maker formula gives the relationship between the focal length, the refractive index, and the radii of the curvature of the lens.
• A piece of transparent material bounded by two refracting surfaces out of which at least one is curved is called a Lens.
• Concave and Convex are the two types of lenses.
• Concave lens is also known as Divergent lens
• Convex lens is also known as Convergent lens
• The lens maker formula is applicable for any type of lens.
• The lens maker formula is given by \(\frac{1}{f} = (\mu -1)[ \frac{1}{R_1}-\frac{1}{R_2}]\)

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