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Interference for for N slits formula

  1. Nov 11, 2013 #1
    Interference for a lattice with N slits

    OK, so check out the first formula (the one for intensity) in the following pdf:

    http://folk.ntnu.no/magnud/OvLf/bolge/oving11.pdf

    It's the formula for Intensity as a function of angle from the interference for a lattice with N very thin slits.

    Mathematically, why is ##I_{max}## found only when BOTH the numerator and denominator are approaching zero?

    I agree that the intensity is largest when the ratio between the numerator and denominator is biggest, but why does this only happen when both approach zero?
     
    Last edited: Nov 11, 2013
  2. jcsd
  3. Nov 11, 2013 #2

    Philip Wood

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    Gold Member

    Replaces previous post, which was misleading.

    Consider when [itex]\theta[/itex] is close to [itex]n\pi[/itex]. Put [itex]\theta = n\pi + \epsilon[/itex].

    Then [itex]\frac{sin N\theta}{sin \theta} = \frac{sin (Nn\pi +N\epsilon)}{sin (n\pi + \epsilon)} = \frac{cos (Nn\pi) sin(N\epsilon)}{cos (n\pi) sin\epsilon} = ± \frac{sin N\epsilon}{sin \epsilon}[/itex]

    The limit of this as [itex]\epsilon[/itex] approaches zero is simply [itex]±\frac{N\epsilon}{\epsilon} = ±N[/itex].
     
    Last edited: Nov 11, 2013
  4. Nov 11, 2013 #3
    Oh I just figured it out. When the upper term approaches zero, then so must the lower turn because N is just a whole number.

    thanks
     
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