How Thick Is the Oil Film for Red and No Light Interference?

[quanticism]

Problem

A thin film of oil (n = 1.30) is located on a smooth, wet pavement. When viewed perpendicular to the pavement, the film reflects most strongly red light at 640 nm and reflects no light at 548 nm. How thick is the oil film?

Equations and Attempt

Refractive index:
air - 1
oil - 1.3
water - 1.33

Assumption: pavement doesn't reflect any light.

Light will undergo a 180 degree phase reversal at the air-oil and oil-water boundaries so the net phase difference in the reflected light is 0.

Thus, for constructive interference, path difference must be an integer number of wavelengths so:

2t = m (640 nm)/1.3, where m is an integer

Similarly, for destructive interference, path difference must be a 1/2 number of wavelength so:

2t = p (548nm)/(2*1.3), where p is an integer


So, I tried to find integers values for m and p for which I could get destructive and constructive interference simultaneously by finding the ratio of m and p:

m/p = 137/320

So minimum integer values for which that ratio is satisfied is m = 137 and p = 320

Subbing it back into the above for t, I obtain t = 33723nm...

A friend told me the answer should be in the domain of the visible spectrum ~(380nm-750nm). But I don't see where I made the mistake. Could someone please nudge me in the right direction?

Note: Alternately, for destructive interference, I could have said:
2t = [(m+1/2)](548nm)/1.3 = (2m+1)(548nm)/[2(1.3)] where m is the "same" integer/order as the one I used for constructive interference.

If I equate this to the eqn for constructive interference, I obtain m = 2.978. Using this, I'll get t = 733.04 nm. However, for this method to work, doesn't m need to be an integer? I guess 2.978 is close to 3 but wouldn't what I did above be more "valid"? (Assuming I didn't make some mistake)

 

[quanticism]

Well, I just tried m=3 and the answer was right. Still not happy about the reasoning behind it though so I'd still appreciate any insight.

 

[Doc Al]

Thus, for constructive interference, path difference must be an integer number of wavelengths so:

2t = m (640 nm)/1.3, where m is an integer

Good.

Similarly, for destructive interference, path difference must be a 1/2 number of wavelength so:

2t = p (548nm)/(2*1.3), where p is an integer

Not so good. Use 2t = (p + 1/2)(548nm/1.3). (You want an odd number of half wavelengths.)

Set those equal:
(p + 1/2)(548nm/1.3) = m (640 nm)/1.3
(p + 1/2)(548) = m (640)
(p + 1/2)(548/640) = m

Now just start plugging in values for p, starting with 0, to see which gives an integer value for m.

 

[quanticism]

Thanks for the reply :)

2t = (p + 1/2)(548nm/1.3, where p = 0,1,2,...
= (2p+1)(548nm)/(2*1.3)

Ah, I see. I should have said "2t = p (548nm)/(2*1.3), where p is an odd integer". But your expression is easier to deal with so I'll use it.

Now just start plugging in values for p, starting with 0, to see which gives an integer value for m.

I think equating the two expressions and finding the ratio m/(2p+1) is more efficient

m/(2p+1) = 137/320. But for 2p+1=320, p can't be an integer.

So I'm tempted to say that with the values given by the question, it's impossible to find a thickness that gives perfect constructive + destructive interference simultaneously.

The answer that the test accepted used m=3.

m/(2p+1) = 137/320 \approx 3/7.00729927... so m \approx 3, p \approx 3

Or using your method:

(p + 1/2)(548/640) = m

Now just start plugging in values for p, starting with 0, to see which gives an integer value for m.

When I plug p = 3 in, I get m = 2.996875. This method is probably better in this case because I honestly don't know how I could have gone from 137/320 to 3/7.00729927...

So this question probably gave "dodgy" values to me?

My friend got these values in his version:
Constructive interference: 640nm
Destructive interference: 512nm

m/(2p+1) = 512/(640*2) = 2/5. So m = 2, p=2. (Except he had m as the integer variable for both cases so he was able to "solve for m". Guess he was lucky that m = p in this case).

Or using Doc Al's expression.

(p + 1/2)(512/640) = m
m/(p+1/2) = 512/640= 4/5
m/(2p+1) = 2/5 (I converted p+1/2 to 2p+1 because this "lowest integer ratio" method requires the denominator and numerator to be integers)