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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 = 3m ). 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 :

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Solution

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Case - I

$$H = 30 \, cm$$

$$n = 3/2$$

$$H_1 = H/n \Rightarrow \dfrac{30 \times 2}{3} = 20 \, cm$$

Case - II

$$R = 300 cm$$

$$\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$$

$$\dfrac{1}{-H_2} - \dfrac{3}{-2 \times 30} = \dfrac{1 - \dfrac{3}{2}}{-300}$$

$$H_2 = \dfrac{600}{29} = 20.684 cm$$

Case - III :

$$\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R} ; \dfrac{1}{H_3} - \dfrac{3}{-2 \times 30} = \dfrac{1 - \dfrac{3}{2}}{300}$$

$$; H_3 = \dfrac{600}{31} = 19.354 cm$$

$$H = 30 \, cm$$

$$n = 3/2$$

$$H_1 = H/n \Rightarrow \dfrac{30 \times 2}{3} = 20 \, cm$$

Case - II

$$R = 300 cm$$

$$\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$$

$$\dfrac{1}{-H_2} - \dfrac{3}{-2 \times 30} = \dfrac{1 - \dfrac{3}{2}}{-300}$$

$$H_2 = \dfrac{600}{29} = 20.684 cm$$

Case - III :

$$\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R} ; \dfrac{1}{H_3} - \dfrac{3}{-2 \times 30} = \dfrac{1 - \dfrac{3}{2}}{300}$$

$$; H_3 = \dfrac{600}{31} = 19.354 cm$$

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