Three glass cylinders of equal height H = 30cm and same refractive...

Three glass cylinders of equal height $H = 30cm$ and same refractive index $n = 1.5$ are placed on a horizontal surface shown in figure. Cylinder $I$ has a flat top, cylinder $II$ has a convex top and cylinder $III$ has a concave top. The radii of curvature of the two curved tops are same \left(R = 3m\right) . $If$ {H}_{1}, {H}_{2} and {H}_{3} $are the apparent depths of a point$ X on the bottom of the three cylinders, respectively, the correct statement(s) is/are $Three glass cylinders of equal height H = 30cm and same refractive index n = 1.5 are placed on a horizontal surface$

Options

A ${H}_{3} > {H}_{1}$
B $0.8cm < \left({H}_{2} - {H}_{1}\right) < 0.9cm$
C ${H}_{2} > {H}_{3}$
D ${H}_{2} > {H}_{1}$

Correct answer

C. ${H}_{2} > {H}_{3}$

Step-by-step solution

Case- $I:$
$Three glass cylinders of equal height H = 30cm and same refractive index n = 1.5 are placed on a horizontal surface$
Given, $H = 30cm$
$n = \dfrac{3}{2}$
Apparent depth $⇒ {H}_{1} = H / n ⇒ \dfrac{30 \times 2}{3} = 20cm$
Case- $II:$
$Three glass cylinders of equal height H = 30cm and same refractive index n = 1.5 are placed on a horizontal surface$
$R = 300cm$
$\dfrac{{n}_{2}}{v} - \dfrac{{n}_{1}}{u} = \dfrac{{n}_{2} - {n}_{1}}{R}$ (Refraction through curved surface)
$\dfrac{1}{- {H}_{2}} - \dfrac{3}{- 2 \times 30} = \dfrac{1 - \dfrac{3}{2}}{- 300}$
${⇒ H}_{2} = \dfrac{600}{29} = 20.684cm$
Case- $III:$
$Three glass cylinders of equal height H = 30cm and same refractive index n = 1.5 are placed on a horizontal surface$
$\dfrac{{n}_{2}}{v} - \dfrac{{n}_{1}}{u} = \dfrac{{n}_{2} - {n}_{1}}{R}; \dfrac{1}{- {H}_{3}} - \dfrac{3}{- 2 \times 30} = \dfrac{1 - \dfrac{3}{2}}{300}; {H}_{3} = \dfrac{600}{31} = 19.354cm$

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