Question

Three glass cylinders of equal height $H=30cm$ 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 $H_1,H_2$ 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

• A. $H_2>H_1$
• B. $H_3>H_1$
• C. $0.8cm<(H_2-H_1)<0.9cm$
• D. $H_2>H_3$

Answer 1

Answer: (A), (D)

$H_2>H_3$

Case - I (Refraction at plane surface, denser to rarer medium)
$H=30cm$
$n=3/2$
$H_1=H/n\Rightarrow \frac{30\times 2}{3}=20cm$
Case - II (Refraction at curved surface)
$R=300cm$
$\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$
$\frac{1}{-H_2}-\frac{3}{-2\times 30}=\frac{1-\frac{3}{2}}{-300}$
$H_2=\frac{600}{29}=20.684cm$
Case - III (Refraction at curved surface)
$\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R};\frac{1}{H_3}-\frac{3}{-2\times 30}=\frac{1-\frac{3}{2}}{300}$
$H_3=\frac{600}{31}=19.354cm$
Hence $H_2>H_3$ and $H_2>H_1$