Three experiments involving a thin film (in air) are shown. If t denotes the film thickness and $\lambda$ denotes the wavelength of the light in the film, which experiments will produce constructive interference as seen by the observer?
A) I only
B) II only
C) III only
D) I and III only
E) II and III only

Question

Three experiments involving a thin film (in air) are shown. If t denotes the film thickness and $\lambda$ denotes the wavelength of the light in the film, which experiments will produce constructive interference as seen by the observer? 
A) I only 
B) II only 
C) III only 
D) I and III only 
E) II and III only

Quizplus · Accepted Answer

For constructive interference to occur, the path difference between the reflected waves from the top and bottom surfaces of the thin film should be equal to an integer multiple of the wavelength λ in the film.

In experiment I, the path difference is 2t, so for constructive interference, we need:

2t = mλ

where m is an integer. This equation can be rearranged to:

λ/2t = 1/m

Since λ and t are both positive, the value of λ/2t is always less than or equal to 1. Therefore, the only integer value for m that satisfies the equation is 1. So constructive interference can only occur for m = 1, which means:

λ/2t = 1/1

λ = 2t

Therefore, only one wavelength λ (namely 2t) can produce constructive interference in experiment I.

In experiment II, the light passes through the thin film and then reflects off a mirror before returning to the film. This means that the path difference between the reflected waves is 4t (twice the thickness of the film). So for constructive interference, we need:

4t = mλ

where m is an integer. This equation can be rearranged to:

λ/4t = 1/m

Again, since λ and t are both positive, the value of λ/4t is always less than or equal to 1, which means that only one integer value for m can satisfy the equation (namely m = 1). Therefore, there is only one wavelength (namely 4t) that can produce constructive interference in experiment II.

In experiment III, the path difference is again 2t, so the same equation as in experiment I applies:

λ/2t = 1/m

However, in this case, there are multiple integer values for m that can satisfy the equation. For example, if we let m = 1, we get λ = 2t, as in experiment I. But if we let m = 2, we get:

λ/2t = 1/2

λ = t

which means that there is another wavelength (namely t) that can produce constructive interference in experiment III.

Therefore, the only experiments that can produce constructive interference are experiments I and III.

Three Experiments Involving a Thin Film (In Air) Are Shown λ\lambdaλ

Three Experiments Involving a Thin Film (In Air) Are Shown $\lambda$