JEE Advanced · Physics · 24. Ray Optics

A glass beaker has a solid, plano-convex base of refractive index 1.60, as shown in the figure. The radius of curvature of the convex surface (SPU) is $9 \mathrm{~cm}$, while the planar surface (STU) acts as a mirror. This beaker is filled with a liquid of refractive index $n$ up to the level QPR. If the image of a point object $\mathrm{O}$ at a height of $h$ (OT in the figure) is formed onto itself, then, which of the following option(s) is(are) correct?

A For $n=1.42, h=50 \mathrm{~cm}$.
B For $n=1.35, h=36 \mathrm{~cm}$.
C For $n=1.45, h=65 \mathrm{~cm}$.
D For $n=1.48, h=85 \mathrm{~cm}$.

Verified Solution

Answer & Solution

Correct Answer

(B) For $n=1.35, h=36 \mathrm{~cm}$.

Step-by-step Solution

Detailed explanation

Since STU is a plane mirror, we can take mirror image of the whole situation about it and final image can be assumed to be at a distance $\mathrm{h}$ below the base.

Since object and image are at same distance from equivalent lens, hence $\mathrm{h}=2 \mathrm{~F}_{\mathrm{eq}}$
$\begin{aligned} & \frac{1}{\mathrm{~F}_{\mathrm{eq}}}=\left(\frac{1.6-1}{1}\right)\left(\frac{2}{9}\right)+\frac{(\mathrm{n}-1)}{1}\left(\frac{-2}{9}\right) \\ & \frac{1}{\frac{\mathrm{h}}{2}}=\frac{1.2}{9}+\frac{2(1-\mathrm{n})}{9} \\ & \frac{2}{\mathrm{~h}}=\frac{3.2-2 \mathrm{n}}{9} \\ & \mathrm{~h}=\frac{9}{1.6-\mathrm{n}} \mathrm{cm}\end{aligned}$
(1) for $\mathrm{n}=1.42, \mathrm{~h}=50 \mathrm{~cm}$
(2) for $\mathrm{n}=1.35, \mathrm{~h}=36 \mathrm{~cm}$
(3) for $\mathrm{n}=1.45, \mathrm{~h}=60 \mathrm{~cm}$
(4) for $\mathrm{n}=1.48, \mathrm{~h}=75 \mathrm{~cm}$

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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