A planoconvex lens has a thickness of . When placed on a horizontal surface ( table),

A planoconvex lens has a thickness of $=4 \mathrm{~cm}$ . When placed on a horizontal surface ( table), with the curved surface in contact with it, the apparent depth of the bottom most point of the lens is found to be $\mathrm{t_{1}=3 \mathrm{~cm}}$ . If the lens is inverted such that the plane face is in contact with the table, the apparent depth of the center of the plane surface is found to be $\mathrm{t_{2}=25 / 8 \mathrm{~cm}}$ .Find the focal length of the lens
Option: 1
$50 \mathrm{~cm}$

Option: 2
$60 \mathrm{~cm}$

Option: 3
$75 \mathrm{~cm}$

Option: 4
$90 \mathrm{~cm}$

Answers (1)

Refraction of flat surface:

Apparent depth $=($ Real depth $) /($ R.I $)$
$\Rightarrow \quad \mathrm{t}_{1}=\frac{\mathrm{t}}{\mathrm{n}}\quad \ldots(1)$

Refraction of curved surface
$\frac{\mathrm{n}_{2}}{\mathrm{v}}-\frac{\mathrm{n}_{1}}{\mathrm{u}}=\frac{\mathrm{n}_{2}-\mathrm{n}_{1}}{\mathrm{R}}$

$\text { Here } \mathrm{n}_{1}=\mathrm{n}, \mathrm{n}_{2}=1, \mathrm{u} \cong-\mathrm{t}, \mathrm{R} \text { is }-\mathrm{R}$
$\Rightarrow \quad \frac{1}{\mathrm{v}}-\frac{\mathrm{n}}{-\mathrm{t}}=\frac{1-\mathrm{n}}{-\mathrm{R}}$
$\Rightarrow \quad \frac{1}{\mathrm{v}}=\frac{\mathrm{n}-1}{\mathrm{R}}-\frac{\mathrm{n}}{\mathrm{t}}$
$\mathrm{Putting \: v=t_{2} we \: obtain, -\frac{1}{t_{2}}=\frac{n-1}{R}-\frac{n}{t}}\quad \ldots(2)$

$\frac{1}{f}=(n-1)\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}\right) \text {, Putting } R_{1}=R \text { and } R_{2}=\infty$
$\text { we obtain, } f=\frac{R}{n-1}\quad \ldots(3)$
$\text { Putting } \frac{n-1}{R}=\frac{1}{f} \text { from (3) in (2) we obtain, }-\frac{1}{t_{2}}=\frac{1}{f}-\frac{n}{t}$
$\Rightarrow \quad \frac{1}{f}=\frac{n}{t}-\frac{1}{t_{2}} \quad \Rightarrow \quad f=\frac{t_{2}}{n t_{2}-t} .$

Putting the value of $\mathrm{n}$ from equation (1)
$\Rightarrow \quad \mathrm{f}=\frac{\mathrm{t}_{1} \mathrm{t}_{2}}{\frac{\mathrm{t}}{\mathrm{t}_{1}} \mathrm{t}_{2}-\mathrm{t}}=\frac{\mathrm{t}_{1} \mathrm{t}_{2}}{\mathrm{tt}_{2}-\mathrm{t}_{1}}=75 \mathrm{~cm}$ .