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carnot cycle
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Homework Statement
I'm having difficulty understanding why this equation makes sense for the following m values.
Homework Equations
The following is stated in my textbook.
For dark fringes,
sinθ = (mλ)/a
where λ is the wavelength of light in nanometers and a is the width of the single slit.
m is ±1, ±2, ±3, ±4, etc..
The Attempt at a Solution
I understand where the equation arises from, however, what I am not understanding is why all values of m are used in this equation and not just the odd ones. Because if I go backwards, here is what happens:
The path length difference is (a/2)sinθ, and for destructive interference, the following must hold:
path length difference = (m/2)λ, where m is an odd integer because the waves have to be out of phase to destroy each other through superposition.
So this is true then:
(a/2)sinθ = (m/2)λ but m has to be odd.
My book canceled the twos in the denominator and divided by a on both sides, but made it so all values of m could work. I find this very confusing. Why isn't it only odd values of m?