Do I understand these two formulas on interference?

[grandpa2390]

Homework Statement

This isn't really a question about a specific problem, just about 2 formulas. I am not sure what the difference is when to use the two formulas. I have a hypothesis, but I need it confirmed or corrected. thankyou :)

Homework Equations

If you have the Halliday and Resnick, Fundamental of Physics 9th edition book, the two problems that have me questioning is chapter 35 1&2. Not important to state them verbatim.

Basically we have 2 beams of light that bounce of a mirror, one beam travels twice the distance of the other beam.
in problem 1 the two beams start out exactly in phase, and we want to find the small distance between the mirrors (L) that the light must travel in order to become exactly out of phase.

in problem 2 we have the same setup, but reversed. the light beams start out exactly out of phase, and we want to find the distance L that would put them exactly in phase.

so basically beam 1 travel the distance L 4 times. Beam 2 travels the distance L 2 times.

anyways. my question is about solving this. It was my understanding that if you want to calculate in phase, you must use the formula dL=m(lambda) and if you want to calculate out of phase, you do dL =( 2m+1)/2 * (lambda)
only for both problems we use the second formula. and that kind of makes sense that we would use the same formula for both since, effectively, they are asking us to do the same thing. even though one starts out in phase or out of phase, we can pretend that it is the opposite since we are trying to change it exactly a half of a wave backwards. it doesn't matter whether we are trying to change it half a wave backwards into phase or out of phase, it is the same formula. that makes sense.

but what about the dL=m(lambda) formula? what kind of problem would we use it for? Is it only used to find bright fringes if the rays start out in phase and remain in phase?
the dL= (2m+1)/2 (lambda) formula is used to find dark fringes when the rays start in phase yet here it is used to change the phase of our wave.

The Attempt at a Solution

I am thinking the formula dL = m*(lambda) is only useful for rays that are already in phase and we want to find all of the length L where they are still in phase. L= some multiple of lambda. or if they are exactly out of phase, and we want to find the distance L where they remain exactly out of phase.

and I am thinking that dL = (2m+1)/2 (lambda) is useful in any other situation where they start out in phase and we want to find the distance L that they are out of phase
or they start out out of phase and we want to find the Distance L that they become in phase.

is this correct?

 

[NascentOxygen]

You seem to understand the formulae okay. Exactly "out of phase" just means one path length effectively differs from the other by λ/2. Any number of whole λs, plus half a λ, in any situation.

Don't forget that a reflection itself can introduce a λ/2 difference, because of the likely phase reversal.