Derivation Of Lens Maker Formula

What is a Lens Maker Formula?

The Lens Maker’s Formula is used to determine the focal length of a lens based on the refractive index of the material and the radii of curvature of its two surfaces.

$\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

• $f=$ focal length of the lens
• $\mu=$ refractive index of lens material
• $R_1=$ radius of curvature of first surface
• $R_2=$ radius of curvature of second surface

Recommended Topic Video

NCERT Physics Notes :

• NCERT Notes class 11 physics
• NCERT Notes class 12 physics
• NCERT Notes for all subject

Recommended Topic Video

Derivation of Lens Maker’s Formula

It is derived by applying the formula for refraction at two spherical surfaces of a thin lens and then combining the two equations to relate focal length with radii of curvature and refractive index.

The lens maker formula is derived using the assumptions listed below

Consider the thin lens in the picture above, which has two refracting surfaces with curvature radii R₁ and R₂, respectively. Assume that the surrounding medium and the lens material have refractive indices of n₁ and n₂, respectively. The whole derivation of the lens maker formula is provided further below. We can say that, using the formula for refraction at a single spherical surface,

For refraction at the first spherical surface (from air, refractive index $n_1$, to lens, refractive index $n_2$ ):
$\frac{n_2}{v_1}-\frac{n_1}{u}=\frac{n_2-n_1}{R_1}$. ---------(1)

For refraction at the second spherical surface (from lens $\left(n_2\right)$ to air $\left(n_1\right)$ ):
$\frac{n_1}{v}-\frac{n_2}{v_1}=\frac{n_1-n_2}{R_2}$---------(2)

Adding equations (1) and (2), we get
$
\frac{n_1}{v}-\frac{n_1}{u}=\left(n_2-n_1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)
$

Dividing both sides by $n_1$ :
$
\frac{1}{v}-\frac{1}{u}=\left(\frac{n_2}{n_1}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)
$

If the object is at infinity, $u=\infty$ and the image is formed at the focal length $f$ of the lens, so $v=f$. Then
$
\frac{1}{f}=\left(\frac{n_2}{n_1}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)
$

For a lens in air, $n_1=1$ and $n_2=\mu$. Therefore, the above equation becomes
$
\frac{1}{f}=(\mu-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)
$

Where $\mu$ is the material's refractive index.

This is the derivation of the lens maker formula. Examine the constraints of the lens maker's formula to have a better understanding of the lens maker's formula derivation.

Also, check-

• NCERT Exemplar Class 11th Physics Solutions
• NCERT Exemplar Class 12th Physics Solutions
• NCERT Exemplar Solutions for All Subjects

Limitations of the Lens Maker’s Formula

• The lens should not be thick so that the space between the 2 refracting surfaces can be small.
• The medium used on both sides of the lens should always be the same.

Uses of Lens Maker’s Formula

1. The formula calculates the focal length of a lens through its refractive index and radii of curvature measurements.
2. The formula enables the creation of optical instruments which range from microscopes to telescopes and cameras to spectacles.
3. The formula enables the selection of appropriate lens materials based on their refractive index which matches the required focal length.
4. The formula shows how varying lens surface curvatures affect the lens's ability to converge or diverge light.
5. The formula enables lens power calculation which researchers use to study how lenses create images.
6. Researchers study geometric optics principles through their application in real-world scenarios.

Importance of Lens Maker Formula Derivation Class 12

• It helps us understand how a lens works and how it bends light.
• It shows how focal length depends on the shape of the lens and material.
• It makes it easier to solve numerical questions in exams (Class 12, JEE, NEET).
• It helps in understanding image formation clearly.
• It is useful in making things like spectacles, cameras, and microscopes.
• It connects what we study in theory with real-life uses of lenses.

Solved Examples on Lens Maker’s Formula

Example 1: A double convex lens is made of glass of refractive index $\mu=1.5$. Its two spherical surfaces have radii $R_1=+20 \mathrm{~cm}$ and $R_2=-20 \mathrm{~cm}$. Find the focal length of the lens in air.
Solution:
$
\begin{gathered}
\frac{1}{f}=(1.5-1)\left(\frac{1}{20}-\frac{1}{-20}\right) \\
\frac{1}{f}=0.5\left(\frac{1}{20}+\frac{1}{20}\right) \\
\frac{1}{f}=0.5 \times \frac{2}{20}=\frac{1}{20} \\
f=20 \mathrm{~cm}
\end{gathered}
$
Example 2 : A plano-convex lens has refractive index $\mu=1.5$.
The radius of curvature of its curved surface is $R=30 \mathrm{~cm}$.
Find the focal length.
Solution:
For a plane surface: $R=\infty$
$
\begin{gathered}
\frac{1}{f}=(1.5-1)\left(\frac{1}{\infty}-\frac{1}{-30}\right) \\
\frac{1}{f}=0.5\left(0+\frac{1}{30}\right) \\
\frac{1}{f}=\frac{1}{60} \\
f=60 \mathrm{~cm}
\end{gathered}
$
Example 3: A double concave lens is made of refractive index $\mu=1.5$. Its radii of curvature are $R_1=-20 \mathrm{~cm}$ and $R_2=+30 \mathrm{~cm}$.
Find the focal length.
Solution:
$
\begin{gathered}
\frac{1}{f}=(1.5-1)\left(\frac{1}{-20}-\frac{1}{30}\right) \\
\frac{1}{f}=0.5\left(-\frac{1}{20}-\frac{1}{30}\right) \\
-\frac{1}{20}-\frac{1}{30}=-\frac{3}{60}-\frac{2}{60}=-\frac{5}{60} \\
\frac{1}{f}=0.5 \times\left(-\frac{5}{60}\right)=-\frac{5}{120}=-\frac{1}{24} \\
f=-24 \mathrm{~cm}
\end{gathered}
$