air1.n = 1[a]The figure shows a soap film of thicknessdį in air. Light impinges on the film withan angle of incidence 01 and is reflectedand refracted at the interfaces. Rays 1and 2 represents reflected light, whilerays 3 and 4 represent light that has beentransmitted into the glass.soap\[c]n, = 1.33|d[b]airn= 134The bracketed letters (a], [b], and [c] indicate reflections at the air-soap, and soap-airinterfaces. For which of these reflections does a phase change of n occur? Check allthat apply.[a][b]O (c) What values of the phase difference (Aþ) between rays 1 an 2 lead to constructiveinterference in the reflected light? Aộ =Light with a wavelength in air of A = 557 nm illuminates the soap film with normalincidence (01 = 0). Calculate the wave number k, of the light when it travels in thesoap film. Please write the value (to 3 decimals) in the first space and the units in thesecond.Calculate the minimum thickness of the soap film that leads to constructiveinterference in the reflected light. Please write the numerical value in the first spaceand the units in the second.

Wave length of the light in the air is λ=557 nm=5.57×10-7 m Refractive index of Air is n=1…

Answered: What values of the phase difference…

Physics for Scientists and Engineers: Foundations and Connections

1st Edition

ISBN:9781133939146

Author:Katz, Debora M.

Publisher:Katz, Debora M.

Chapter38: Refraction And Images Formed By Refraction

Section: Chapter Questions

Problem 92PQ

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Question

air
1.
n = 1
[a]
The figure shows a soap film of thickness
dį in air. Light impinges on the film with
an angle of incidence 01 and is reflected
and refracted at the interfaces. Rays 1
and 2 represents reflected light, while
rays 3 and 4 represent light that has been
transmitted into the glass.
soap
\[c]
n, = 1.33
|d
[b]
air
n= 1
3
4
The bracketed letters (a], [b], and [c] indicate reflections at the air-soap, and soap-air
interfaces. For which of these reflections does a phase change of n occur? Check all
that apply.
[a]
[b]
O (c)

What values of the phase difference (Aþ) between rays 1 an 2 lead to constructive
interference in the reflected light? Aộ =
Light with a wavelength in air of A = 557 nm illuminates the soap film with normal
incidence (01 = 0). Calculate the wave number k, of the light when it travels in the
soap film. Please write the value (to 3 decimals) in the first space and the units in the
second.
Calculate the minimum thickness of the soap film that leads to constructive
interference in the reflected light. Please write the numerical value in the first space
and the units in the second.

Expert Solution

Step 1

Wave length of the light in the air is $λ = 557 nm = 5.57 \times 10^{- 7} m$

Refractive index of Air is n=1

Refractive index of soap is n_s=1.33

Step 2

By Snell's law,

$n_{1} \times {sinθ}_{1} = n_{2} \times {sinθ}_{2} . . . . . . (1) Here, n_{1} is the refractive index of air n_{2} is the refractive index of soap θ_{1} is the angle of inclination of light at air - soap interface θ_{2} is the angle of inclination of refracted light at soap - air interface$

Again applying Snell's law,

$n_{2} \times {sinθ}_{2} = n_{1} \times {sinθ}_{4} . . . . . . (2) Here, n_{1} is the refractive index of air n_{2} is the refractive index of soap θ_{1} is the angle of inclination of light at air - soap interface θ_{4} is the angle of inclination of refracted light at soap - air interface$

On comparing expression 1 and 2, we can say that

${sinθ}_{1} = {sinθ}_{4} or θ_{1} = θ_{4}$

From the above we can say that ray 1 and ray 4 are parallel and therefore, for the constructive interference, the phase difference should be even multiple of $π$ .

i.e $ϕ = 2 nπ$

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