Question

A small object is placed at the centre of the bottom of a cylindrical vessel of radius 3 cm and height 4 cm filled completely with water. Consider the ray leaving the vessel through a corner. Suppose this ray and the ray along the axis of the vessel are used to trace the image. Find the apparent depth of the image and the ratio of real depth to the apparent depth under the assumptions taken. Refractive index of water = 1.33.

Answer 1

Given,
Refractive index of water μ = 1.33
Radius of the cylindrical vessel = 3 cm
Height of the cylindrical vessel = 4 cm

Let x be the length of BD
According to the diagram,
$\frac{x}{3}=cotr....\left(\mathrm{i}\right)$

Using Snell's law,
$$\begin{gathered} \frac{\sin i}{\sin r}=\frac{1}{1.33}=\frac{3}{4} \\ \Rightarrow \sin r=\frac{4}{3}\sin i=\frac{4}{3}\times \frac{3}{5}=\frac{4}{5} \\ \mathrm{As}\sin i=\frac{BC}{AC}=\frac{3}{5} \\ \Rightarrow \cot r=\frac{3}{4}.....\left(\mathrm{ii}\right) \end{gathered}$$
From (i) and (ii),

$$\begin{gathered} \frac{x}{3}=\frac{3}{4} \\ \Rightarrow x=\frac{9}{4}=2.25\mathrm{cm} \end{gathered}$$

Hence, the ratio of real and apparent depth of the image will be $4:2.25=1.78$