A vessel ABCD of `10 cm` width has two small slits `S_(1)` and `S_(2)` sealed with idebtical glass plates of equal thickness. The distance between the slits is `0.8 mm`. POQ is the line perpendicular to the plane AB and passing through O, the middle point of `S_(1)` and `S_(2)`. A monochromatic light source is kept at `S, 40 cm` below `P` and `2 m` from the vessel, to illuminate the slits as shown in the figure. Calculate the position of the central bright fringe on the other wall CD with respect of the line `OQ`. Now, a liquid is poured into the vessel and filled up to `OQ`. The central bright fringe is fiund to be at Q. Calculate the refractive index of the liquid.

Answer

A, B, C

Explanation

(i) O is the middle point of two slits `S-1` and `S_2`.
`S_1S_2=d=0.8mm`
`tanalpha+(y_1)/(D_1)=(40)/(200)=(1)/(5)`
`:'` `sinalpha=(1)/(sqrt26)=(1)/(5.1)approx(1)/(1.5)approxtanalpha`
Path defference `triangleX_1=SS_1-SS_2`
or `triangleX_1=dsinalpha=(0.8mm)((1)/(5))=0.16`mm...(i)
Let R denotes the position of central bright fringe. Net path defference will be zero.
Now `triangleX_2=S_2R-S_1R` or `triangleX_2=dsintheta`...(ii)
For central bright fringe
`triangleX_2-triangleX_1=0` or `dsintheta-triangleX_1=0`
or `dsintheta=0.16` or ` sintheta=(0.16)/(0.8)=(1)/(5)` `:'` `tantheta=(1)/(sqrt24)=(1)/(4.9)approx(1)/(5)=sintheta` `:'``tantheta=(y_2)/(D_2)`
or `(1)/(5)=(y_2)/(D_2)` or `y_2=(D_2)/(5)=(10)/(5)=2 cm`
Hence position of central bright fringe is 2 cm above point Q on side CD.
(ii) `mu` of liquid poured if central fringe is at Q:
The liquid is poured into vessel upto OQ.
The central bright fringe is formed at Q.
For central bright fringe net path defference=0
`(mu-1)t=triangleX_1` or `(mu-1)(100)=0.16`
or `mu-1=0.0016` or ` mu=1.0016`