Young's Experiment: Fringe Shift Due to Refractive Index

Problem Statement

There is a Young's double-slit interference setup. Let $d$d be the separation between the two slits $S_1$S1​ and $S_2$S2​, and $L$L be the distance from the slits to the screen. When monochromatic light of wavelength $\lambda$λ is irradiated perpendicularly to the slits, interference fringes are observed on the screen.

A transparent thin film of thickness $t$t and refractive index $n$n is placed just behind slit $S_1$S1​, causing the central bright fringe (the point of zero path difference) to shift on the screen. Let the position of the original central bright fringe (before placing the film) be the origin $x = 0$x=0. Find the new position $x_1$x1​ of the central bright fringe after the film is placed. Assume the position $x$x on the screen satisfies $x \ll L$x≪L.

Constraints

• Slit separation: $d = 0.50 \text{ mm} = 5.0 \times 10^{-4} \text{ m}$d=0.50 mm=5.0×10−4 m
• Distance to screen: $L = 1.5 \text{ m}$L=1.5 m
• Film thickness: $t = 2.0 \times 10^{-5} \text{ m}$t=2.0×10−5 m ($20\ \mu$20 μm)
• Film refractive index: $n = 1.4$n=1.4
• The refractive index of air may be approximated as 1.0.

Input Format

Find the value of $x_1$x1​ in mm and give the integer part of the answer.