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Diffraction - maxima

  1. Oct 28, 2010 #1
    1. The problem statement, all variables and given/known data
    Show that the angular locations of the first to fourth secondary maximas are [tex]\alpha[/tex] = a sin [tex]\Theta[/tex]/[tex]\Lambda[/tex] = 1.43030 2.45902 3.47089 4.46641 respectively.

    a is the slit width = 0.00016m
    [tex]\Lambda[/tex] wavelength of laser 650nm



    2. Relevant equations

    I([tex]\Theta[/tex]) = I0 [sin([tex]\Pi\alpha[/tex])/[tex]\Pi\alpha[/tex]]2

    where I0 is the intensity at the central peak

    3. The attempt at a solution

    I have no idea how to begin..
     
  2. jcsd
  3. Oct 29, 2010 #2
    You have an equation for the intensity of the diffraction pattern. The maxima and minima of the function will give the maxima and minima of the diffraction pattern. So, simply differentiate and set to zero. You'll get two factors. One of these gives the minima and the other gives the maxima, or, at least, another equation for the maxima. Solving the "maxima equation" will give the values at which the maxima occur.
     
  4. Oct 31, 2010 #3
    hi there
    thanks for your reply,

    after differentiating, i got this equation

    2I0 sin([tex]\pi[/tex][tex]\alpha[/tex])/[tex]\pi[/tex]^2[tex]\alpha[/tex]^3 ([tex]\pi \alpha[/tex] cos([tex]\pi\alpha[/tex]) - sin [tex]\pi\alpha[/tex] )

    i hope this is correct.. anyway i set this to zero and got 2 different sets of eqn.


    2I0 sin([tex]\pi[/tex][tex]\alpha[/tex])/[tex]\pi[/tex]^2[tex]\alpha[/tex]^3 = 0

    so sin([tex]\pi[/tex][tex]\alpha[/tex]) = 0
    [tex]\alpha[/tex] = 0,1,2,3,...

    and ([tex]\alpha[/tex]^2 cos([tex]\pi\alpha[/tex]) - sin [tex]\pi\alpha[/tex] ) = 0
    [tex]\pi\alpha[/tex] = tan ([tex]\pi\alpha[/tex])
    how do I solve this?
     
    Last edited: Nov 1, 2010
  5. Nov 1, 2010 #4
    sorry the correct maxima eqn should be this.

    [tex]\pi[/tex][tex]\alpha[/tex] = tan ([tex]\pi[/tex][tex]\alpha[/tex])

    how do i solve it?
     
  6. Nov 1, 2010 #5
    Let me rewrite this as
    b = tan(b)

    Plot y = b and y = tan(b) on the same plot. Where the two plots intersect are the solutions.
     
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