Thin Film Problem: Find Film Thickness from Reflected Light Wavelengths

[JSGandora]

Homework Statement

A thin film of alcohol (n=1.36) lies on a flat glass plate (n=1.51). When monochromatic light whose wavelength can be changed, is incident normally, the reflected light is minimum for λ=512 nm and a maximum for λ=640 nm. What is the thickness of the film?

Homework Equations

λ_n=λ/n

The Attempt at a Solution

Let the thickness be t, then we have
t/(512/1.36)=m+1/2 where m is an integer.
t/(640/1.36)=n where n is an integer.
Solving for t and setting the expressions equal, we get 376m+188=471n (using significant figures). I feel like I'm doing something wrong because of the resulting equation. Can someone help me? Thanks.

 

[anathema]

Hello there,

Your approach is on the right track, but there are a few things to consider. First, when using the equation λ_n=λ/n, we need to make sure that both sides of the equation are in the same units. In this case, since the thickness (t) is in units of length, we need to convert the wavelengths into the same units as well. We can do this by multiplying the wavelengths by the speed of light (c=3x10^8 m/s). This gives us:
λ_n=c/λ
Next, we need to consider the phase shift that occurs when light is reflected off a medium with a higher refractive index. This phase shift is equal to π when the light is reflected at normal incidence. So, for the minimum reflection at 512 nm, we have:
t/(512/1.36)=m+1/2+π
And for the maximum reflection at 640 nm, we have:
t/(640/1.36)=n+π
Solving for t and setting these two expressions equal, we get:
(512/1.36)(n+π)=(640/1.36)(m+1/2+π)
Simplifying, we get:
376m+188=471n+471π
Since m and n are integers, we can ignore the decimals and simplify further to:
188=471n+471π
Now, we don't know the exact value of π, but we do know that it is approximately equal to 3.14. So we can rewrite the equation as:
188=471n+1475.94
Solving for n, we get:
n=-2.24
Since n must be an integer, we can round this to n=-2. This means that m must be equal to -1, and we can plug these values into our original equations to solve for t:
t/(512/1.36)=-1+1/2+π
t/(640/1.36)=-2+π
Solving for t, we get:
t=276.8 nm
So, the thickness of the film is approximately 277 nm. I hope this helps!