Question

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

• A. $1.56$
• B. $1.78$
• C. $2.46$
• D. $3.84$

Answer 1

The correct option is B $1.78$

Let the object be placed at $A$ and apparent depth be $BD$.


Let $BD=x$.
According to the figure, $\frac{x}{3}=\cot r\dots \dots \left(1\right)$
Also, $\frac{\sin i}{\sin r}=\frac{1}{1.33}=\frac{3}{4}$
$\Rightarrow \sin r=\frac{4}{3}\sin i=\frac{4}{3}\times \frac{3}{5}=\frac{4}{5}(\text{because}\sin i=\frac{BC}{AC}=\frac{3}{5})$
$\Rightarrow \cot r=\frac{3}{4}\dots \dots \left(2\right)$
From (1) and (2) $\Rightarrow \frac{x}{3}=\frac{3}{4}$
$\Rightarrow x=\frac{9}{4}=2.25\text{cm}$
$\therefore$ Ratio of real and apparent depth $=\frac{4}{2.25}=1.78$

Why this question? Caution: The formula for apparent depth: $\frac{\text{Real depth}}{\text{Apparent depth}}=\mu$ can only be used for small angles of incidence.