Question

Focal length of a convex lens is \[0.1\,{\text{m}}\]. A liquid lens is formed between a plane surface and one face of this lens if radius of curvature \[0.12\,{\text{m}}\]. The converging combination formed is found to have a focal length \[0.18\,{\text{m}}\]. Calculate the refractive index of the liquid.

Answer 1

Hint: In this question, you are asked to calculate the refractive index of the liquid. You will need to use this lens maker formula so recall the lens maker formula, find the required quantities which will be needed in the formula, put the values of these quantities and calculate the value of the refractive index of the liquid.

Complete step by step answer:
Given, focal length of a convex lens, \[{f_1} = 0.1\,{\text{m}}\].Radius of curvature, \[{R_1} = 0.12\,{\text{m}}\].Focal length of the combination, \[f = 0.18\,{\text{m}}\].The combination formed will act as a plano-convex lens as the liquid will act as plane surface.Let the radius of curvature of the plane surface be \[{R_2}\] .The radius of a plane surface will be infinite so, \[{R_2} = \infty \].In this question, we will use lens maker formula. Lens maker formula is used for combination of lenses. The formula is,
\[\dfrac{1}{f} = \left( {\mu - 1} \right)\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)\]..(i)
where \[f\] is the focal length of the combination, \[\mu \] is the refractive index of the material and \[{R_1}\] and \[{R_2}\] are the radii of the two surfaces.
Now, putting the values of \[f\], \[{R_1}\] and \[{R_2}\] in equation (i) we get,
\[\dfrac{1}{{0.18}} = \left( {\mu - 1} \right)\left( {\dfrac{1}{{0.12}} - \dfrac{1}{\infty }} \right)\]
\[ \Rightarrow \dfrac{1}{{0.18}} = \left( {\mu - 1} \right)\left( {\dfrac{1}{{0.12}}} \right)\]
\[ \Rightarrow \left( {\mu - 1} \right) = \dfrac{{0.12}}{{0.18}}\]
\[ \Rightarrow \left( {\mu - 1} \right) = 0.66\]
\[ \Rightarrow \mu = 0.66 + 1\]
\[ \therefore \mu = 1.66\]

Therefore, the refractive index of the liquid is \[1.66\].

Note: Refractive index can be defined as the ratio of the speed of light in vacuum to the speed of light in the given medium. From the refractive index we can know about the material used or how the material will affect the speed of light passing through it. Also, remember refractive index does not have any unit or we can say it is dimensionless.