Coherence Sunlight Refraction homework

[MednataMiza]

Homework Statement

On the wet road / asphalt / there is a oily stain,with refraction coeff 1.4 .Sunlight lights up the grease perpendicular .What colour will be a stain with thickness of d = 250 nm ? The water's refraction coeff is 1.33

Homework Equations

Since we are asked what colour will be the stain, we actually have a maximum and the formula will be 2nd=(m+1/2)λ
n - refraction coeff
d - thickness of the stain
m - integer showing the number of the maximum
λ - length of light

The Attempt at a Solution

I actually happen to have the answer : 470nm. Now I tried solving the equation using that m = 0 => 4nd = λ,but this gave so far away from the answer. I tried to reverse the equation,plugged in the expected wavelength and calculated that i need a refraction coeff of 0.47, which I don't know how to express ...

 

[MednataMiza]

Anyone ?

 

[gneill]

If you suppose that white sunlight contains a continuum of colors (wavelengths) then you want to find those that will deliver constructive interference for the given geometry of oil and water. There are an infinite number of such wavelengths, but only very few (probably only one!) will be in the visible spectrum.

That means that for the light passing through the oil and reflecting off of the water (no phase shift on reflection) you should look for a relationship between the overall phase shift and the fixed distance of travel, keeping in mind that the light's wavelength in the oil is not the same as the wavelength in air. The overall shift you're targeting is a half wavelength (180 degrees to match the light reflected off of the oil's surface).

Let's take your formula and rewrite is slightly:

(i + 1/2)*λ_oil = 2d, for i = 0,1,2,...

and λ_oil = λ_air/n_oil, with n_oil being the refractive index of oil.

Rearrange the formulas to solve for the wavelength (in air). By plugging in the values of i you will generate a list of wavelengths that will show constructive interference. Pick the one(s) that you might be able to see.

 

[ehild]

Homework Equations

Since we are asked what colour will be the stain, we actually have a maximum and the formula will be 2nd=(m+1/2)λ

You tried m=0, and you got too long wavelength. What about trying the next integer m?

ehild

 

[MednataMiza]

Ehild,gneill,
Thanks,both of you :)
I figured it out ... [: