In the diagram given below, there are three lenses formed. Considering negligible thickness of each of them as compared to $[\mathrm{R}_{1}]$ and $[\mathrm{R}_{2}]$, i.e., the radii of curvature for upper and lower surfaces of the glass lens, the power of the combination is

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

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

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

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

Geometrical Optics — JEE Main · 2025 · 22 Jan 2025 · Shift 1 Question with Solution

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

Answer: (B)

Model the water–glass–water sandwich as three refracting surfaces in series and use the power of a single curved surface,

P = (μ - 1) (\frac{1}{v} - \frac{1}{u})

, applied at each surface.

Surface 1 – top curved face of the water layer, radius

R_{1}

(water,

μ = 4/3

P_{1} = (\frac{4}{3} - 1) (\frac{1}{\infty} - \frac{1}{- ∣ R _{1} ∣}) = \frac{1}{3∣ R _{1} ∣} .

Surface 2 – the glass lens (

μ = 3/2

) bounded by

R_{1}

above and

R_{2}

below:

P_{2} = (\frac{3}{2} - 1) (\frac{1}{- ∣ R _{1} ∣} - \frac{1}{- ∣ R _{2} ∣}) = \frac{1}{2} (\frac{1}{∣ R _{2} ∣} - \frac{1}{∣ R _{1} ∣}) .

Surface 3 – bottom curved face into the lower water layer, radius

R_{2}

(water,

μ = 4/3

P_{3} = (\frac{4}{3} - 1) (\frac{1}{- ∣ R _{2} ∣} - \frac{1}{\infty}) = - \frac{1}{3∣ R _{2} ∣} .

Adding the three:

P_{e q} = P_{1} + P_{2} + P_{3} = \frac{1}{3∣ R _{1} ∣} - \frac{1}{3∣ R _{2} ∣} + \frac{1}{2} (\frac{1}{∣ R _{2} ∣} - \frac{1}{∣ R _{1} ∣}) = (\frac{1}{3} - \frac{1}{2}) \frac{1}{∣ R _{1} ∣} + (\frac{1}{2} - \frac{1}{3}) \frac{1}{∣ R _{2} ∣} = - \frac{1}{6} (\frac{1}{∣ R _{1} ∣} - \frac{1}{∣ R _{2} ∣}) .

The power of the combination is

- \frac{1}{6} (\frac{1}{∣ R _{1} ∣} - \frac{1}{∣ R _{2} ∣})

JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Single correct option