Air wedge interference: why does change in m equal 1 for coinciding lights?

[chickens]

Hi there, I was doing some past year paper of my coming exam...and I got the answers for the question too, but one part I'm totally blur and can't understand that well... well here goes all the questions and solution:

http://server3.uploadit.org/files/chickens-phyQ.jpg

http://server3.uploadit.org/files/chickens-ans1.jpg

http://server2.uploadit.org/files/chickens-ans2.jpg

The part that I'm confused is with the part (b)...the answer given by both different book ... the solution seems...a little blur to me.

Unclear to me:

1. why "for the first time the two lights coincide, change in m = 1" ?
2. the long and madness working...i totally have no idea what is going on...

Anyone kind here could help me explain all the steps? I'm very blur... thanks in advance.

 

[chickens]

no one could help to explain ? :(

 

[Doc Al]

Do you understand the meaning of this equation?
$$2 t = (m + \frac{1}{2}) \lambda$$

There will be constructive interference (and thus a bright fringe) for values of m = 1, 2, 3, etc. You'll get a yellow fringe whenever a red and green fringe overlap. They can't overlap with both having the same m, so find the first point of overlap when $\Delta m = 1$.

 

[chickens]

yes that equation is for the condition to get constructive interference...but for the delta m...i know it can't have the same m, but can't the first point of overlap is when delta m = 2 or 3 or so on?

 

[Doc Al]

Sure, it could turn out that the first overlap happens at some higher difference. But the only way to find out is to try it and see. (You have to solve this problem semi-numerically.) Start with $\Delta m = 1$. If that provides a solution, you're done; if not, then increment to $\Delta m = 2$. And so on until you get a solution. Of course, this problem is a set up, so the solution appears with the first try!

 

[chickens]

ahhh, lol a set up q :P thanks for the explanation...

then what about the second solution? it seems simple but doesn't seem that right lol

 

[Doc Al]

error in second solution

I looked at the second solution. I believe it is incorrect. The mistake occurs when they say:

The distance between O and the thin film end is given by $$N_R x_R = N_G x_G$$

Not so: The first fringe occurs at half the separation distance.