Question

What is the refractive index of the material of Plano-convex lens, if the radius of curvature of the convex is $10cm$ and focal length of the lens is $30cm?$
(A) $\dfrac{1}{3}$
(B) $1$
(C) $\dfrac{4}{3}$
(D) $\dfrac{2}{3}$

Answer 1

Hint In the question they have provided the radius of curvature of the convex and the focal length of the lens. The given lens is a Plano convex lens. We have to find the refractive index of the material of the Plano-convex lens.
Hint:

Complete step by step answer
A lens is an optical device that focuses or disperses a light beam by refraction. Lens are of two types a simple lens and a compound lens. A simple lens has a single piece of transparent material that can be a glass or plastic, while a compound lens consists of several simple lenses, arranged along a common axis. A lens usually focuses light by refraction to form an image.
Plano convex lens is a lens that has one plane side and one convex side.
The relation between the radius of curvature of the lens and the focal length the lens is given by:
\[\dfrac{1}{f} = (n - 1)\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)\]
f is the focal length of the lens
n is the refractive index of the lens
${R_1}$ is the radius of curvature of the convex surface
${R_2}$ is the radius of curvature of the plane-surface
Given,
Focal length of the lens is $30cm$
The radius of curvature of the convex is $10cm$
The radius of the curvature of the plane-surface is infinite for a of Plano-convex lens
So, the radius of the curvature of the plane-surface is $\infty $
We have seen that the relation between the radius of curvature of the lens and the focal length the lens is given by:
\[ \Rightarrow \dfrac{1}{f} = (n - 1)\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)\]
Substitute the given values
\[ \Rightarrow \dfrac{1}{{30}} = (n - 1)\left( {\dfrac{1}{{10}} - \dfrac{1}{\infty }} \right)\]
\[ \Rightarrow \dfrac{1}{{30}} = (n - 1)\left( {\dfrac{1}{{10}} - 0} \right)\]
\[ \Rightarrow \dfrac{1}{{30}} = (n - 1)\dfrac{1}{{10}}\]
\[ \Rightarrow 10 = 30(n - 1)\]
\[ \Rightarrow 10 = 30n - 30\]
\[ \Rightarrow 10 + 30 = 30n\]
\[ \Rightarrow 40 = 30n\]
\[ \therefore n = \dfrac{4}{3}\]
The refractive index of the Plano convex lens is \[\dfrac{4}{3}\]

Hence the correct answer is option (C) \[\dfrac{4}{3}\]

Note We have seen that the relation between the radius of curvature of the lens and the focal length the lens is:
\[ \Rightarrow \dfrac{1}{f} = (n - 1)\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)\]
This relation is known as the lens maker’s formula because it helps to find the curvature needed to make a lens of desired focal length. This formula is applicable to find curvature of concave lenses also.