Thin films: White light is incident normally on a thin soap film having an index of refraction of 1.34. It reflects with an interference maximum at 684 nm and an interference minimum at 570 nm with no minima between those two values. The film has air on both sides of it. What is the thickness of the soap film?
A) 766 nm
B) 627 nm
C) 638 nm
D) 894 nm
E) 510 nm

Question

Quizplus · Accepted Answer

For interference maxima, the path difference between the two reflected waves is an integer multiple of the wavelength. For interference minima, the path difference is an odd multiple of half the wavelength.

Let the thickness of the soap film be "t".

For the interference maximum at 684 nm: 
2t = mλ
where m is an integer and λ is the wavelength of light (684 nm in this case).

For the interference minimum at 570 nm: 
2t + λ/2 = (2m + 1)(λ/2)
where m is an integer and λ is the wavelength of light (570 nm in this case).

Solving these two equations simultaneously, we get: 
t = [(2m + 1)λ/4] - [λ/4] 
which simplifies to: 
t = [(2m + 1)λ/4] - [λ/4]

Substituting the given values, we get: 
t = [(2m + 1)(684 nm)/4] - [684 nm/4] = [(2m + 1)(684 nm) - 684 nm]/4 = (2m)(684 nm)/4 = 171m nm

So, the thickness of the soap film is a multiple of 171 nm. From the given information, we know that there is no interference minimum between 570 nm and 684 nm. This means that the path difference at 627 nm (the average of 570 nm and 684 nm) is not an odd multiple of half the wavelength.

Therefore, the path difference at 627 nm must be the minimum path difference for which there is no interference minimum between 570 nm and 684 nm.

Using the equation for path difference: 
2t + λ/2 = (2m + 1)(λ/2) 
we can solve for "t" at 627 nm: 
2t + 627 nm/2 = (2m + 1)(627 nm/2) 
2t + 313.5 nm = (2m + 1)(313.5 nm) 
2t = (2m + 1)(313.5 nm) - 313.5 nm 
2t = (2m)(313.5 nm) + 313.5 nm 
t = (2m)(313.5 nm)/2 + 313.5 nm/2 
t = 313.5m + 156.75 nm

We need to find a value of "m" such that "t" is closest to the given options.

Option A: t = 766 nm = 313.5m + 156.75 nm
Option B: t = 627 nm = 313.5m + 156.75 nm
Option C: t = 638 nm = 313.5m + 156.75 nm
Option D: t = 894 nm = 313.5m + 156.75 nm
Option E: t = 510 nm = 313.5m + 156.75 nm

Option C is the closest to the solution we obtained for "t". Therefore, the answer is: 
Answer: C

Thin Films: White Light Is Incident Normally on a Thin