Light of wave length$\lambda$is incident on two thin films that are in contact and surrounded by air. The top layer is $\lambda$/4 thick and has an index of refraction of 1.33. The bottom layer is $\lambda$/2 thick and has an index of refraction of 1.50. At normal incidence, the reflected rays that are in phase with each other are
A) 1 and 2
B) 1 and 3
C) 2 and 3
D) 1, 2, and 3
E) None of these is correct.

Question

Light of wave length$\lambda$is incident on two thin films that are in contact and surrounded by air. The top layer is $\lambda$/4 thick and has an index of refraction of 1.33. The bottom layer is $\lambda$/2 thick and has an index of refraction of 1.50. At normal incidence, the reflected rays that are in phase with each other are
A) 1 and 2 
B) 1 and 3 
C) 2 and 3 
D) 1, 2, and 3 
E) None of these is correct.

Quizplus · Accepted Answer

For two thin films in contact, the condition for constructive interference is given by:

2nt = mλ

where n is the refractive index of the film, t is the thickness of the film, m is an integer, and λ is the wavelength of light.

At normal incidence, the phase change upon reflection is π. Therefore, the reflected rays will be in phase with each other if the path difference between them is an integer multiple of the wavelength.

Let's consider the path differences for the three rays in the figure:

- Path difference between Ray 1 and Ray 2: (1/4)λ + (1/2)λ = 3/4 λ
- Path difference between Ray 1 and Ray 3: (1/4)λ + λ = 5/4 λ
- Path difference between Ray 2 and Ray 3: (1/2)λ + λ = 3/2 λ

For constructive interference, the path difference must be an integer multiple of the wavelength. Therefore, only Ray 1 and Ray 3 are in phase with each other, and the correct answer is B) 1 and 3.

Light of Wave Length λ\lambdaλ Is Incident on Two Thin Films That Are in Contact

Light of Wave Length $\lambda$ Is Incident on Two Thin Films That Are in Contact