Question

A fish, looking up through the water sees the outside world contained in a circular horizon. If the refractive index of water is $4/3$ and the fish is $12cm$ below the surface of water, the radius of the circle in centimetre is:

• A. $\frac{12\times 3}{\sqrt{5}}$
• B. $12\times 3\times \sqrt{5}$
• C. $\frac{12\times 3}{\sqrt{7}}$
• D. $12\times 3\times \sqrt{7}$

Answer 1

Answer: (C)

$\frac{12\times 3}{\sqrt{7}}$

Given : Refractive index of water $\mu =\frac{4}{3}$ and $h=12$ cm

The ray after the refraction grazes along the surface of water.

Using $sin{\theta }_c=\frac{1}{\mu }$

$\therefore$ $sin{\theta }_c=\frac{3}{4}=0.75$

$⟹$ $cos{\theta }_c=\sqrt{1-si{n}^2{\theta }_c}=\sqrt{1-\frac{9}{16}}=\frac{\sqrt{7}}{4}$

Let the radius of the circle be $r$

$\therefore$ $r=h$ $tan{\theta }_c=h\frac{sin{\theta }_c}{cos{\theta }_c}=12\times \frac{3/4}{\sqrt{7}/4}$

$⟹$ $r=\frac{12\times 3}{\sqrt{7}}$ cm