Question

Two equi-convex lenses each of focal lengths 20 cm and refractive index 1.5 are placed in contact and space between them is filled with water of refractive index 4/3. The combination works as :

• A. converging lens of focal length 50 cm.
• B. diverging lens of focal length 15 cm
• C. converging lens of focal length 15 cm
• D. diverging lens of focal length 50 cm

Answer 1

Answer: (C)

converging lens of focal length 15 cm

$\begin{array}{c}\text{Ths is a combination of convex, concave and convex}\\ \text{lens. Number them}1,2,3\\ \begin{array}{cc}\text{given}{f}_1& =f_3=20cm\\ {\mu }_1& ={\mu }_3=1.5\\ {\mu }_2& =4/3\end{array}\end{array}$

$\begin{array}{c}{\mu }_2=4/3\\ \text{Now densmaker formula}\frac{1}{f}=({\mu }_r-1)(\frac{1}{R_1}-\frac{1}{R_2})\\ \text{for (1)}\frac{1}{f_1}=(1.5-1)(\frac{1}{R_1}-(-\frac{1}{R}))\end{array}$$\text{(equi convex lenS}{R}_1=R_2)$

$\begin{array}{c}\frac{1}{20}=\frac{1}{2}\times \frac{2}{R}\\ R=20cm\\ \text{for (2)}\\ \begin{array}{c}\frac{1}{f_2}=(\frac{4}{3}-1)(\frac{1}{R_1}-\frac{1}{R_2})\\ \frac{1}{f_2}=\frac{1}{3}\left(\frac{-2}{R}\right)\\ f_2=-30cm\end{array}\end{array}$

$\begin{array}{c}\text{Now we know that}\\ \begin{array}{cc}\text{ieffective}& =P_1+P_2+P_3\\ \frac{1}{f_{eq}}& =\frac{1}{20}-\frac{1}{30}+\frac{1}{20}\\ f_{\text{eq}}=15cm& \end{array}\\ \text{So it is a converging lens of focal length}15cm.\end{array}$