Hi, this is about thin film interference in ray optics. My book presents two similar problems, but somehow uses different paths for the same variable. I'll list out the given variables:
--Problem A--
n1 < nf > n2, delta phase is therefore lambda over two.
Film thickness is 200 nm (L = 200 nm)
With nf = 1.40, find the dark spot in the visible range.
--Problem B--
n1 > nf < n2, delta phase is ditto
Film thickness is 380 nm (L = 380 nm)
with nf = 1.34, find the dark spot in the visible range.
--
The equation I am using is this,
Lambda = 2Lnf/m
Since we are trying to find a dark spot when the waves are already destructive, I thought there was no reason to use (m-1/2). In my head, using (m-1/2) will shift the phase another pi, creating constructive interference (and bright spots). So therefore, finding a dark spot in the visible range should use m = 1.
This method gave me the book's answer for problem A: 560 nm.
However, I got twice the amount for problem B. Now I hope I haven't lost my algebras yet, but even using (m-1/2) would actually yield quadruple the book's answer. The book says problem B's wavelength should be 509 nm, but I got 1,018 nm.
Am I missing something, or is the book wrong? This is very puzzling.
To solve thin film interference problems, the equation you provided is correct:
λ = 2Lnf/m
where:
- λ is the wavelength of light in the film
- L is the thickness of the film
- nf is the refractive index of the film
- m is the order of the interference maximum/dark spot
In your case, since you are looking for a dark spot, you correctly chose m = 1.
Now let's analyze the problems individually:
Problem A: n1 < nf > n2
Here, the refractive index of the film is higher than the neighboring mediums, indicating a dark spot in the visible range. You are given nf = 1.40 and L = 200 nm. By substituting these values into the equation, you correctly found the book's answer of 560 nm.
Problem B: n1 > nf < n2
In this case, the refractive index of the film is lower than the surrounding mediums, implying another dark spot in the visible range. You are given nf = 1.34 and L = 380 nm. By plugging these values into the equation, you obtained 1,018 nm. However, this does not match the book's answer of 509 nm.
It seems that there may be an error in the book, or there might be some additional information missing from your question. Based on the information provided, your calculations appear to be correct.
To further investigate, you could:
1. Double-check the given data in the book for problem B.
2. Check if there are any additional constraints or information provided in the book that might affect the calculation.
3. Consult with your teacher or instructor to clarify the problem and determine if there is an error.
It's important to note that discrepancies between your calculated answer and the book's answer can arise due to different rounding approaches, different assumptions, or errors in either your calculations or the book's answer. Double-checking and seeking clarification will help you determine the correct solution.