1. The problem statement, all variables and given/known data

I'm having some trouble understanding the following solved problem:

In Young’s experiment, narrow double slits 0.20 mm apart diffract monochromatic light onto a screen 1.5 m away. The distance between the 5th minima on either side of the zeroth-order maximum is measured to be 34.73 mm. Determine the wavelength of the light.

Solution:

I don't understand why they have used ##(m-\frac{1}{2}) \lambda## as the condition for destructive interference, instead of ##(m+\frac{1}{2}) \lambda##?

2. Relevant equations

For destructive interference my textbook uses:

##d \ sin \theta_{dark} = (m+\frac{1}{2}) \lambda##

(this is an approximation)

3. The attempt at a solution

If I use -1/2, then 5th minima turn out to be are 9λ apart, just like the model answer.

However when I use +1/2 I get a totally different solution:

##\therefore \ \lambda = \frac{\Delta y a}{(5.5-(-5.5)) L}= \frac{\Delta y a}{11 L} =421 \ nm##

So, why do they use -1/2? And which method is correct?

My textbook ("Optics" by Hecht) uses "m+1/2", while my other book ("Introduction to Optics" by Pedrotti) uses "m-1/2" for destructive interference. The two methods clearly produce different results!

Why in this particular situation, do we need "-1/2" and not "+1/2"?

##\Delta x## represents the distance between two consecutive nodal or antinodal fringes.

##x_n## represents the distance from a minimum to the right bisector (central maximum).

They are not talking about the distance between two consecutive node/antinodal fringes, rather the distance from the 5th minima on either side of the fringe pattern to the right bisector.

Also, because the fringe pattern is symmetric, I beleive it wouldn't matter whether you used ##(n - 1/2)## or ##(n + 1/2)##.

It depends on which direction the author likes to measure I believe.

You are using a different notation. I never defined ##\Delta y## as the distance between consecutive fringes! It is the distance between the position of 5th minima on one side, to the 5th minima on the other.

##y## itself is the linear positions of fringes measured along the screen from the bisector you mentioned.