Question

The focal length of a lens of refractive index $\dfrac{3}{2}$ is 10cm in the air. The focal length of that lens in a medium of refractive index $\dfrac{7}{5}$ is:
a) -70 cm
b) $\dfrac{10}{7}$ cm
c) 70 cm
d) None of these

Answer 1

Let us assume that the focal length of a lens in air is ${{f}_{1}}$ and the focal length of the lens in a medium is ${{f}_{2}}$. Now apply the maker's formula for each focal length in terms of radius, then divide both the equations to cancel out the terms and get the value of ${{f}_{2}}$.

Formula used:
The lens maker’s formula for thin lens is used: $\dfrac{1}{f}=\left( \mu -1 \right)\left( \dfrac{1}{{{R}_{1}}}-\dfrac{1}{{{R}_{2}}} \right)$
Here, ${{R}_{1}}$ and ${{R}_{2}}$ are the radius of curvature of the lens.
$f$ is the focal length of the lens.
${\mu}$ is the refractive index of the lens.

Complete step by step answer:
As we have the following data for air as a medium:
$\begin{aligned} &\Rightarrow {{f}_{1}}=10 \\\ &\Rightarrow {{\mu }_{1}}=\dfrac{3}{2} \\\ \end{aligned}$
So, by applying maker’s formula, we get:
$\Rightarrow\dfrac{1}{10}=\left( \dfrac{3}{2}-1 \right)\left( \dfrac{1}{{{R}_{1}}}-\dfrac{1}{{{R}_{2}}} \right)......(1)$
Also, we have the refractive index of lens in another medium as:
$\Rightarrow{{\mu }_{2}}=\dfrac{7}{5}$
So, by applying maker’s formula, we get:
$\Rightarrow\dfrac{1}{{{f}_{2}}}=\left( \dfrac{7}{5}-1 \right)\left( \dfrac{1}{{{R}_{1}}}-\dfrac{1}{{{R}_{2}}} \right)......(2)$
Now, divide equation (1) by equation (2), we get:
$\begin{aligned} &\Rightarrow \dfrac{\dfrac{1}{10}}{\dfrac{1}{{{f}_{2}}}}=\dfrac{\left( \dfrac{3}{2}-1 \right)\left( \dfrac{1}{{{R}_{1}}}-\dfrac{1}{{{R}_{2}}} \right)}{\left( \dfrac{7}{5}-1 \right)\left( \dfrac{1}{{{R}_{1}}}-\dfrac{1}{{{R}_{2}}} \right)} \\\ &\Rightarrow \dfrac{1}{10}{{f}_{2}}=\dfrac{1}{2}\times \dfrac{5}{2} \\\ &\Rightarrow {{f}_{2}}=\dfrac{5}{4}\times \dfrac{1}{10} \\\ &\Rightarrow {{f}_{2}}=12.5cm \\\ \end{aligned}$
Hence, option (d) is the correct answer.
Additional Information:
The limitations for using the lens maker’s formula are:
-The lens must be thin because the separation between the two refracting surfaces will also be small.
-The medium on both sides of the lens needs to be the same.

Note: In this type of questions,major mistakes occurs while taking the sign convention so you should always remember that all the distances which are measured along the direction of incident rays are taken as positive and vice-versa.Also, you should remember that the distance measured upon the principal axis are taken as positive while the distances measured below the it is taken as negative.