Question

In a vessel, as shown in fig, point P is just visible when no liquid is filled in vessel through a telescope in the air. When liquid is filled in the vessel completely, point Q is visible without moving the vessel or telescope. Find the refractive index of the liquid.

• A. $\frac{\sqrt{14}}{3}$
• B. $\frac{\sqrt{85}}{5}$
• C. $\sqrt{2}$
• D. $\sqrt{3}$

Answer 1

Answer: (B)

$\frac{\sqrt{85}}{5}$

When liquid is not filled in the vessel, point P can just be seen.

Thus the triangles, ABP and BCD are similar.

Thus $\frac{PA}{AB}=\frac{BC}{CD}$

Hence, $CD=4R$

Now after filling in a liquid of refractive index $\mu$, point Q can be seen. Hence the angle of refraction at D is $r$ as shown in the figure.
Applying snell's law to the refraction at point D,

$\sin i=\mu \sin r$

$\frac{1}{\sqrt{5}}=\mu \frac{1}{\sqrt{17}}$

Hence, $\mu =\frac{\sqrt{85}}{5}$