
In a vessel, as shown in figure, point P is just visible when no liquid is filled in the 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. $\dfrac{\sqrt{14}}{3} \\ $
B. $\dfrac{\sqrt{85}}{5} \\ $
C. $\sqrt{2}$
D. $\sqrt{3}$
Answer
162.9k+ views
Hint:In this question, a figure is given in which a point P is visible when no liquid is filled in the vessel. When we fill the liquid, a point Q is visible and we have to find the refractive index of that point. We find the refractive index with the help of Snell's law. By putting the values in the snell’s law, we find out the value of the refractive index.
Formula Used:
We solve this question with the help of snell’s law:-
$\mu \sin i=\sin r$
Here, $\mu$ is the refractive index, $i$ is the angle of incidence and $r$ is the angle of refraction.
Complete step by step solution:
When the vessel is not filled with the liquid, Point P can just be seen.

Then the triangles ABP and BCD are similar.
Thus $\dfrac{PA}{AB}=\dfrac{BC}{CD}$
Hence CD = 4 R
Now after filling in a liquid of refractive index $\mu $, point Q can be seen. Hence the angle of refraction at D is r can be seen in the figure.
Now we apply snell’s law to the refraction at point D,
$\mu \sin i=\sin r$
$\Rightarrow \mu \sin (\angle CDQ)=1\times \sin (\angle CDB)$
Hence $\mu \times \dfrac{1}{\sqrt{5}}=1\times \dfrac{2R}{R\sqrt{20}}$
Solving further, we get
$\mu =\dfrac{2\sqrt{17}}{\sqrt{20}}$
Hence $\mu =\dfrac{\sqrt{85}}{5}$
Hence, option B is the correct answer.
Note: Remember that in the snell’s law, when the ray of light is incident perpendicularly, the speed changes, but the direction remains unaltered. When light passes from a rare medium to a denser medium, it is inclined closer towards the normal. When light rays pass from a dense medium to a rare medium, it is inclined away from the normal.
Formula Used:
We solve this question with the help of snell’s law:-
$\mu \sin i=\sin r$
Here, $\mu$ is the refractive index, $i$ is the angle of incidence and $r$ is the angle of refraction.
Complete step by step solution:
When the vessel is not filled with the liquid, Point P can just be seen.

Then the triangles ABP and BCD are similar.
Thus $\dfrac{PA}{AB}=\dfrac{BC}{CD}$
Hence CD = 4 R
Now after filling in a liquid of refractive index $\mu $, point Q can be seen. Hence the angle of refraction at D is r can be seen in the figure.
Now we apply snell’s law to the refraction at point D,
$\mu \sin i=\sin r$
$\Rightarrow \mu \sin (\angle CDQ)=1\times \sin (\angle CDB)$
Hence $\mu \times \dfrac{1}{\sqrt{5}}=1\times \dfrac{2R}{R\sqrt{20}}$
Solving further, we get
$\mu =\dfrac{2\sqrt{17}}{\sqrt{20}}$
Hence $\mu =\dfrac{\sqrt{85}}{5}$
Hence, option B is the correct answer.
Note: Remember that in the snell’s law, when the ray of light is incident perpendicularly, the speed changes, but the direction remains unaltered. When light passes from a rare medium to a denser medium, it is inclined closer towards the normal. When light rays pass from a dense medium to a rare medium, it is inclined away from the normal.
Recently Updated Pages
JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Chemical Properties of Hydrogen - Important Concepts for JEE Exam Preparation

JEE Energetics Important Concepts and Tips for Exam Preparation

JEE Isolation, Preparation and Properties of Non-metals Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Electric field due to uniformly charged sphere class 12 physics JEE_Main

Degree of Dissociation and Its Formula With Solved Example for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

Charging and Discharging of Capacitor

Wheatstone Bridge for JEE Main Physics 2025

Formula for number of images formed by two plane mirrors class 12 physics JEE_Main

In which of the following forms the energy is stored class 12 physics JEE_Main
