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 cm, while the planar surface (STU) acts as a mirror. [2024]

Question

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 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 O at a height of $h$ (OT in the figure) is formed onto itself, then, which of the following option(s) is(are) correct [2024]

Smart Achievers · Accepted Answer

(1, 2)

The image of a point object o at a height of $h$ in the figure is formed onto itself, i.e., object and image are at the same distance from the equivalent lens. Hence, $h = 2 f_{eq}$

$\Rightarrow - \frac{1}{f_{net}} = 2 (\frac{1}{f_{liq}}) + 2 (\frac{1}{f_{lens}}) + (\frac{- 1}{f_{mirror}})$ $\dots (i)$

$- \frac{1}{f_{net}} = 2 (\frac{n - 1}{- 9}) + 2 (\frac{0.6}{9}) + (\frac{- 1}{\infty})$ $\dots (i i)$

From (i) and (ii),

$\frac{1}{h / 2} = \frac{1.2}{9} + \frac{2 (1 - n)}{9}$ $\Rightarrow \frac{2}{h} = \frac{3.2 - 2 n}{9}$ or, $h = \frac{9}{1.6 - n} cm$

Therefore, $for (1) n = 1.42, h = 50 cm$

$(2) n = 1.35, h = 36 cm$

$(3) n = 1.45, h = 60 cm$

$(4) n = 1.48, h = 75 cm$

So, options (1) and (2) are correct.

Solution

1 For $n = 1.42, h = 50 cm$

2 For $n = 1.35, h = 36 cm$

3 For $n = 1.45, h = 65 cm$

4 For $n = 1.48, h = 85 cm$

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Solution

1 For n=1.42, h=50 cm

2 For n=1.35, h=36 cm

3 For n=1.45, h=65 cm

4 For n=1.48, h=85 cm

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1 For $n = 1.42, h = 50 cm$

2 For $n = 1.35, h = 36 cm$

3 For $n = 1.45, h = 65 cm$

4 For $n = 1.48, h = 85 cm$