Question

In the diagram given below, there are three lenses formed. Considering negligible thickness of each of them as compared to \([R_1]\) and \([R_2]\), i.e., the radii of curvature for upper and lower surfaces of the glass lens, the power of the combination is

(1) \(-\frac{1}{6}\left(\frac{1}{\left|\mathrm{R}_{1}\right|}+\frac{1}{\left|\mathrm{R}_{2}\right|}\right)\)

(2) \(-\frac{1}{6}\left(\frac{1}{\left|\mathrm{R}_{1}\right|}-\frac{1}{\left|\mathrm{R}_{2}\right|}\right)\)

(3) \(\frac{1}{6}\left(\frac{1}{\left|\mathrm{R}_{1}\right|}+\frac{1}{\left|\mathrm{R}_{2}\right|}\right)\)

(4) \(\frac{1}{6}\left(\frac{1}{\left|\mathrm{R}_{1}\right|}-\frac{1}{\left|\mathrm{R}_{2}\right|}\right)\)

Answer 1

Correct option is:  (2) \(-\frac{1}{6}\left(\frac{1}{\left|\mathrm{R}_{1}\right|}-\frac{1}{\left|\mathrm{R}_{2}\right|}\right)\)

\(\Rightarrow\mathrm{p}_{\mathrm{eq}} = p_1 + p_2 + p_3\)

\(\Rightarrow \mathrm{p}_{1}=\left(\frac{4}{3}-1\right)\left(\frac{1}{\infty}-\frac{1}{-\left|\mathrm{R}_{1}\right|}\right) \\\)

\(\Rightarrow \mathrm{p}_{1}=\left(\frac{1}{3\left|\mathrm{R}_{1}\right|}\right) \\\)

\(\Rightarrow \mathrm{p}_{2}=\left(\frac{1}{2}\right)\left(\frac{1}{-\left|\mathrm{R}_{1}\right|}-\frac{1}{-\left|\mathrm{R}_{2}\right|}\right) \\\)

\(\Rightarrow \mathrm{p}_{2}=\frac{1}{2}\left(\frac{1}{\left|\mathrm{R}_{2}\right|}-\frac{1}{\left|\mathrm{R}_{1}\right|}\right) \\\)

\(\Rightarrow \mathrm{p}_{3}=\left(\frac{1}{3}\right)\left(\frac{1}{-\left|\mathrm{R}_{2}\right|}-\frac{1}{\infty}\right)=-\frac{1}{3\left|\mathrm{R}_{2}\right|} \\\)

\(\Rightarrow \mathrm{p}_{\mathrm{eq}}=\frac{1}{3}\left(\frac{1}{\left|\mathrm{R}_{1}\right|}-\frac{1}{\left|\mathrm{R}_{2}\right|}\right)-\frac{1}{2}\left(\frac{1}{\left|\mathrm{R}_{1}\right|}-\frac{1}{\left|\mathrm{R}_{2}\right|}\right) \\\)

\( =-\frac{1}{6}\left(\frac{1}{\left|\mathrm{R}_{1}\right|}-\frac{1}{\left|\mathrm{R}_{2}\right|}\right)\)