Thinnest film that will give constructive interference for red light

In summary, the desired equation for finding the thickness of a thin soap film is (2nt/λ) + 1/2 = m, where n is the index of refraction and λ is the wavelength of light. By setting the phase change to 2π for the first constructive interference, we can solve for t to find the thickness of the film. In this case, the thickness is 141 nm.
  • #1
prettykitty
8
0

Homework Statement



When a thin soap film is very thin, we see it as black. How thick (in nm) is the film
in the region where we see the first red band? Take the wavelength of red light to be 752
nm and the index of refraction of the soap to be n = 1.33.

Homework Equations



[2nt/λ] - 1/2 = m


The Attempt at a Solution



Solving for t:
First with m = 1 I moved the 1/2 over to get 3/2, Then I multiplied 3/2 by the wavelength 752e-9 and divided this result by 2n, n being 1.33.
The answer I attained after solving for t was: 424 nm.
The correct answer is 141 nm.

I still don't know what I'm doing wrong here. I have noticed that if I divide my answer by 3 I get the correct answer, though this may just be coincidence.
Also if anyone is using Walker edition 4, on page 988 the practice problem seems to be this exact same scenario and I am having the exact same problem.
Thanks!
 
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  • #3
So then the desired equation is 2ndcosθ=(m-.5)λ?
How would I figure out the angle? I thought it was the right equation.

Here is my instructors answer:
If the thickness of the film is t, the difference in phase between a ray reflecting off the
top surface of the film and one reflecting off the bottom surface of the film is (2πn/λ)2t + π
the factor π coming from the phase change on reflection of the ray in air reflecting off
the film. Set this phase change to 2π for the first constructive interference to get that
t = λ/4n = 752/(4 × 1.33) = 141nm.

I don't really understand how he got his answer either even though it is spelled out quite succinctly.
 
  • #4
prettykitty said:
So then the desired equation is 2ndcosθ=(m-.5)λ?
How would I figure out the angle? I thought it was the right equation.
It's the angle of incidence, so for the OP it's 0.
the difference in phase between a ray reflecting off the
top surface of the film and one reflecting off the bottom surface of the film is (2πn/λ)2t + π
the factor π coming from the phase change on reflection of the ray in air reflecting off
the film. Set this phase change to 2π for the first constructive interference to get that
t = λ/4n = 752/(4 × 1.33) = 141nm.
That's the same equation as I referenced. For the mth constructive interference phase, (2πn/λ)2t + π = 2πm. Rearranging: (2nt/λ) + 1/2 = m. You just had the sign wrong on the 1/2.
 
  • #5



Hello,

The main issue with your solution is that you are using the wrong equation. The equation you are using, [2nt/λ] – 1/2 = m, is for the general case of constructive interference in a thin film. However, in this problem, we are specifically looking for the thinnest film that will give constructive interference for red light.

In this case, we can use the equation t = (m + 1/2)λ/2n, where m is the order of the interference (in this case, m = 1), λ is the wavelength of the light, and n is the index of refraction of the soap. Plugging in the given values, we get:

t = (1 + 1/2)(752e-9)/2(1.33) = 141 nm

So, the thinnest film that will give constructive interference for red light is 141 nm, which is the correct answer. I hope this helps clarify things for you.
 

What is the concept of constructive interference in relation to thin films and red light?

Constructive interference occurs when two waves of the same frequency and wavelength overlap and their amplitudes combine to create a larger wave. In the context of thin films and red light, this means that when the thickness of the film is such that the reflected red light waves overlap and add together, the resulting wave will be brighter and more intense.

Why is red light specifically mentioned in relation to the thinnest film for constructive interference?

Red light has a longer wavelength than other visible light colors, which makes it easier to achieve constructive interference for this specific wavelength. This is because the spacing between the layers of the film needs to be precise in order for the reflected waves to overlap and add constructively. With red light having a longer wavelength, the required spacing between the layers is larger, making it easier to achieve constructive interference.

What factors determine the thinnest film that will produce constructive interference for red light?

The thinnest film that will produce constructive interference for red light is determined by the wavelength of the light, the refractive index of the film material, and the angle of incidence at which the light hits the film. These factors all affect the spacing between the layers of the film, which is crucial for achieving constructive interference.

How does the color of light affect the thinnest film that will give constructive interference?

The color of light, or its wavelength, plays a significant role in determining the thinnest film that will give constructive interference. As mentioned before, longer wavelengths, such as red light, require a greater spacing between the layers of the film for constructive interference to occur. On the other hand, shorter wavelengths, like blue light, require a smaller spacing, making it more challenging to achieve constructive interference.

What are some real-life applications of understanding the thinnest film for constructive interference for red light?

Understanding the thinnest film for constructive interference is crucial in fields such as optics, nanotechnology, and electronics. In optics, this knowledge is used to design and create thin film interference filters for various applications, such as anti-reflective coatings for lenses. In nanotechnology, thin films are used to create various structures and devices, such as solar cells. In electronics, thin film interference is utilized in the construction of LCD screens for electronic devices.

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