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## Homework Statement

A thin film of oil (n = 1.50) floats on water (n = 1.33). If the film appears yellow-green ([tex]\lambda[/tex] = 550 nm) when viewed at normal incidence, how thick is it?

Given:

[tex]n_{f} = 1.50[/tex]

[tex]n_{water} = 1.33[/tex]

[tex]\lambda_{0} = 550 nm[/tex]

[tex]\theta_{i} = 0[/tex]

Required:

Thickness (d)

## Homework Equations

[tex]\Delta\phi_{net} = 2m\pi[/tex] (Since we are looking for constructive interference)

## The Attempt at a Solution

[tex]\Delta\phi_{1} = \pi[/tex] since the first beam is reflected from a material with a higher index of refraction

[tex]\Delta\phi_{2} = 2dk_{f}[/tex] due to path difference

[tex]k_{f} = \frac{2\pi}{\lambda_{f}}[/tex]

[tex]\lambda_{f} = \frac{\lambda_{0}}{n_{f}}[/tex]

Substituting everything in, we get:

[tex]\Delta\phi_{2} = \frac{4dn_{f}\pi}{\lambda_{0}}[/tex]

[tex]\Delta\phi_{net} = \Delta\phi_{2} - \Delta\phi_{1}[/tex]

[tex]\Delta\phi_{net} = (\frac{4dn_{f}}{\lambda_{0}}-1)\pi[/tex]

So, from the equation given for constructive interference, we're left with:

[tex](\frac{4dn_{f}}{\lambda_{0}}-1)\pi = 2m\pi[/tex]

Solving for d, I get:

[tex]d = \frac{(2m+1)\lambda_{0}}{4n_{f}}[/tex]

Assuming I did everything else right (a big assumption), I'm still left with one problem. What do I use for my integer (m = 0, 1, 2, 3, ...)? The question does not specifically state that I'm looking for the thinnest value for the film. All the similar examples in my book solve for the wavelength of light that you see due to a film of a given thickness. For those problems, you choose the value of m such that the resulting wavelength is in the visible range. In this case, however, I am not sure what constraints I should apply such that I get one thickness out of the problem.