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.
Indeed I did post this again. I'd rather have this cleared up then dwell on it.
It seems like you are experiencing some confusion regarding the use of the equation for thin film interference and your results for two similar problems. Let's break down the problem and try to understand what might be causing the discrepancy.
The equation you are using is λ = 2Lnf/m, where λ is the wavelength of light in the film, L is the film thickness, nf is the refractive index of the film, m is the order of interference, and n1 and n2 are the refractive indices of the medium before and after the film, respectively.
In both Problem A and Problem B, you correctly identify the delta phase (Δφ) as λ/2, as the wave is half a wavelength out of phase upon reflection.
For Problem A, with n1 < nf > n2, you have L = 200 nm and nf = 1.40. In order to find the dark spot, you correctly choose m = 1 (since further integers would correspond to bright spots). Plugging the values into the equation, you get λ = 2 * 200 nm * 1.40 / 1 = 560 nm, which matches the book's answer.
For Problem B, with n1 > nf < n2, you have L = 380 nm and nf = 1.34. Again, you choose m = 1 since you are looking for a dark spot. Plugging the values into the equation, you obtain λ = 2 * 380 nm * 1.34 / 1 = 1,018 nm, which is different from the book's answer of 509 nm.
It seems like you may have made an error in your calculation for Problem B. When dealing with interference, it's essential to use the correct values for the variables to obtain accurate results.
To double-check your calculations, try redoing the calculation for Problem B using the correct values of L = 380 nm and nf = 1.34. Make sure to choose m = 1 for the dark spot, as you did before. This should help clarify whether the discrepancy is due to a calculation error or another issue.
If you still obtain a different result from the book, it's possible that there may be an error in the book. In that case, it would be beneficial to consult additional resources or seek clarification from your teacher or peers to ensure accurate understanding.
Remember, the equation you are using is a simplification of the more general equation for thin film interference. In this simplified version, the terms involving (m-1/2) account for the phase shift and constructive interference for bright spots. However, for dark spots, m = 1 should be used.
Keep exploring and analyzing the problem carefully, and don't hesitate to ask for further assistance if needed.