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Small Angle Approximation in Single Slit Interference

  1. May 16, 2015 #1
    1. The problem statement, all variables and given/known data
    A monochromatic light source is used with a double slit to create an interference pattern on a screen that is 2.00 meters away. If the 2nd bright spot is observed 8.73 mm above the central maximum, can the small angle approximation be used? Show and/or explain your reasoning

    2. Relevant equations
    ## d\sinθ=mλ ##

    3. The attempt at a solution
    I might be over-thinking this. I know you can have a small angle approximation if D >> d, but in this case I don't know d yet, so I first wanted to relate d to y without first assuming a small angle approximation (sinθ ≈ tanθ ≈ θ), but I get stuck:

    p6hLM2y.jpg

    I get the following relation:
    ## D=\frac{d}{2\tan(γ)} ##
    ## D= (y+\frac{d}{2}) * \frac{1}{ \tan(θ)} ##

    ## \frac{d}{2}\cot(γ) = y\cot(θ) + \frac{d}{2}\cot(θ) ##

    I'm not sure how to solve for d here so that I can show that D >> d.

    Thanks in advance!
     
  2. jcsd
  3. May 16, 2015 #2

    mfb

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    2016 Award

    Staff: Mentor

    D>>d/2 is necessary for other approximations (all your formulas wouldn't work without that condition).

    The small angle approximation that is relevant here is a small θ. You can calculate θ with D and y alone in a single step.
     
  4. May 16, 2015 #3
    mfb, that makes sense thank you!

    So in this case tan(θ) = y/D

    ## θ = \arctan(\frac{y}{D}) ##
    ## θ = \arctan(\frac{8.73*10e-3}{2}) ##

    θ = 0.00436 rad ~ 0.25 degrees
    That is a very small angle, so I can use a small angle approximation here!
     
  5. May 17, 2015 #4

    mfb

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    2016 Award

    Staff: Mentor

    Right.
     
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