Question

The focal length of a convex lens of refractive index $1.5$ is $f$ when it is placed in the air. When it is immersed in a liquid it behaves as a converging lens its focal length becomes$xf\left( {x > 1} \right)$ . The refractive index of the liquid
A. $> 3/2$
B. $< \left( {3/2} \right)$ and $> 1$
C. $< 3/2$
D. All of these

Answer 1

Hint:- In this question, we can use the Lens Maker’s formula to find two different equations. In the first equation, the refractive index of lens and air is used. In the second equation, the refractive index of lens and liquid is used.

Complete step-by-step solution :
According to the question, we have a convex lens whose radii of curvatures are ${R_1}$ and ${R_2}$. Now, when a lens is placed in the air, then the focal length is $f$. So, using lens maker, we have-
$\dfrac{1}{f} = (\dfrac{{{n_2}}}{{{n_1}}} - 1)\left( {\dfrac{1}{{{R_1}}} + \dfrac{1}{{{R_2}}}} \right)$
Where ${n_2}$ and ${n_1}$ are the refractive indexes.
$\Rightarrow \dfrac{1}{f} = \left( {\dfrac{{1.5}}{1} - 1} \right)\left( {\dfrac{1}{{{R_1}}} + \dfrac{1}{{{R_2}}}} \right)$ (i)
Now, when the lens is immersed in the liquid, the focal length becomes $xf$. Let the refractive index of the liquid is $y$ , then by using lens makers formula, we have-
$\Rightarrow \dfrac{1}{{xf}} = \left( {\dfrac{{1.5}}{y} - 1} \right)\left( {\dfrac{1}{{{R_1}}} + \dfrac{1}{{{R_2}}}} \right)$ (ii)
Now, dividing equation (i) and (ii), we get
$\dfrac{{xf}}{f} = \dfrac{{\dfrac{{1.5}}{1} - 1}}{{\dfrac{{1.5}}{y} - 1}}$
$\Rightarrow x = \dfrac{{0.5y}}{{1.5 - y}}$
$\Rightarrow 1.5x - xy = 0.5y$
$\Rightarrow 1.5x = 0.5y + xy$
$\Rightarrow \left( {0.5 + x} \right)y = 1.5x$

$$\Rightarrow y = \dfrac{{1.5x}}{{0.5 + x}} \\\ \Rightarrow y = \dfrac{{\dfrac{3}{2}x}}{{\dfrac{1}{2} + x}} \\\$$

$\Rightarrow y = \dfrac{3}{{2 + \dfrac{1}{x}}}$
Thus, the refractive index of the liquid is $\dfrac{3}{{2 + \dfrac{1}{x}}}$ .
Hence, option B is correct.

Additional Information:- A convex lens may have one or two radii of curvature. Each and every medium has a different refractive index. For example- the refractive index of air is $1$, the refractive index of the ice is $1.31$, the refractive index of the salt is $1.52$.

Note:- In this question, we use the Lens Maker’s formula. In the lens maker’s formula, there are two refractive indexes. We have three mediums in the question. So, we have to keep in mind that there will be two equations by using the lens maker’s formula. In both the equations, the refractive index of the lens is the same. We also need to remember that the converging lens means a convex lens.