Three glass cylinders of equal height H = 30 cm and same refractive index n = 1.5 are placed on a horizontal surface as 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 (R = 3 m). If $H_{1}, H_{2}, a n d H_{3}$ are the apparent depths of a point X on the bottom of the three cylinders, respectively, the correct statement(s) is/are :

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$0.8 cm < (H_{2} - H_{1}) < 0.9 cm$

$H_{2} > H_{1}$

$H_{2} > H_{3}$

$H_{3} > H_{1}$

answer is C, D.

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Detailed Solution

$H_{1} = \frac{H \times 2}{3} = \frac{2 \times 30}{3} = 20 cm = 0.2 m$

$\frac{1}{V} - \frac{1.5}{H} = \frac{1 - 1.5}{R} \frac{1}{V} = \frac{- 1}{2 \times 3} + \frac{1.5}{0.3} \frac{1}{V} = \frac{- 1}{6} + 5 \frac{1}{V} = \frac{29}{6} V = \frac{6}{29} = 0.2069 m$

$\frac{1}{V} - \frac{1.5}{H} = \frac{1 - 1.5}{- R} \frac{1}{V} = \frac{1}{2 \times 3} + \frac{1.5}{0.3} \frac{1}{V} = \frac{1}{6} + 5 \frac{1}{V} = \frac{31}{6} V = \frac{6}{31} = 0.1935 m$

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