Will Reflected Light Produce a Bright Spot with Thin Film Interference?

[NATURE.M]

Homework Statement

Light strikes two plane sheets of glass with a thin air space between them as shown. If the light has a wavelength of 580 nm and the air space between the glass has a thickness of 870 nm, predict whether the reflected light as demonstrated by rays A and B will cancel or produce a bright spot.

Homework Equations

2t=mλ, m=1,2,3...

The Attempt at a Solution

So using the above equation, and substituting the values in the question, I obtain, m=3. So the path difference is 3λ (extra distance ray B travels)
Then since the second ray (ray B) is reflected from a denser medium (glass), it is in phase reversal or λ/2 out of phase. So, then the total path difference is 3.5 λ. Since its a half-integral multiple it implies the rays will cancel-produce a dark spot.
Is this logic correct?

I've been told this is the correct answer, but initially I would have said that the path difference is 3λ, since the rays are already out of phase-and destructive interference would occur-producing a dark spot. Then the whole integral number of wavelengths (3λ in this case), wouldn't change the status of the system, so the rays will cancel. But this is supposedly wrong, with the first answer being correct.
Can someone please clarify why this is true.

 

[ehild]

Both arguments are correct, integer number of wavelengths in the path difference do not count.

ehild

 

[NATURE.M]

Both arguments are correct, integer number of wavelengths in the path difference do not count.

ehild

Would you advise its better to use the first argument, or does it really not matter?

 

[ehild]

I think the firs argument is better to use. But I prefer argument referring to phase difference. If the phase diference between two waves is odd number of pi, the interference is destructive, if it is even number of pi, it is constructive.

Travelling across λ distance changes the phase by 2pi. Reflecting from a denser medium changes the phase by pi.
One ray reflects from the front surface of the air gap with no phase change. The other ray traverses the air gap (1.5 λ), changing the phase by 3 pi, reflects from a denser medium, that changes the phase by pi, then traverses the gap again, so the net phase difference between the waves is 7pi. Odd number of pi, so the waves interfere destructively.


ehild