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Young's double-slit interference

In a Young's experiment with λ = 550 nm, the two slits are d = 0.40 mm apart, the screen is at L = 2.0 m.\nCalculate: (a) the fringe spacing, (b) the position of the 3rd maximum, (c) the position of the 2nd minimum.
Given data
λ = 550×10⁻⁹ md = 0.40×10⁻³ mL = 2.0 m
Review the theory: Ottica
Steps
0 / 3
  1. Calculate the fringe spacing Δy (distance between adjacent fringes, in mm).
  2. Calculate the position y₃ of the 3rd maximum (m=3) on the screen (in mm).
  3. Calculate the position of the 2nd minimum on the screen (in mm).
Full worked solution
  1. Calculate the fringe spacing Δy (distance between adjacent fringes, in mm).
    Δy=λLd=550×10−9⋅2.00.40×10−3\Delta y = \dfrac{\lambda L}{d} = \dfrac{550\times10^{-9} \cdot 2.0}{0.40\times10^{-3}}
    Δy=λLd=550×10−9×2.00.40×10−3=1100×10−94×10−4=2.75×10−3 m=2.75 mm\Delta y = \frac{\lambda L}{d} = \frac{550\times10^{-9} \times 2.0}{0.40\times10^{-3}} = \frac{1100\times10^{-9}}{4\times10^{-4}} = \mathbf{2.75\times10^{-3}\,m} = \mathbf{2.75\,mm}. The fringe spacing is proportional to wavelength and screen distance, inversely proportional to slit separation.
  2. Calculate the position y₃ of the 3rd maximum (m=3) on the screen (in mm).
    y3=mλLd=3⋅550×10−9⋅2.00.40×10−3y_3 = \dfrac{m\lambda L}{d} = 3 \cdot \dfrac{550\times10^{-9} \cdot 2.0}{0.40\times10^{-3}}
    y3=m Δy=3×2.75=8.25 mmy_3 = m\,\Delta y = 3 \times 2.75 = \mathbf{8.25\,mm}. In general, the mm-th bright fringe is at ym=m⋅Δyy_m = m \cdot \Delta y.
  3. Calculate the position of the 2nd minimum on the screen (in mm).
    ymin,2=(2m−1)λL2d∣m=2=3λL2d=3⋅550×10−9⋅2.02⋅0.40×10−3y_{min,2} = \dfrac{(2m-1)\lambda L}{2d}\bigg|_{m=2} = \dfrac{3\lambda L}{2d} = \dfrac{3 \cdot 550\times10^{-9} \cdot 2.0}{2 \cdot 0.40\times10^{-3}}
    Second minimum: y=(2−12)Δy=1.5×2.75=4.125 mmy = \left(2 - \frac{1}{2}\right)\Delta y = 1.5 \times 2.75 = \mathbf{4.125\,mm}. Minima occur halfway between adjacent maxima.
Result:Δy = 2.75 mm — y₃ = 8.25 mm — 2nd minimum = 4.13 mm.
TheoryOtticaFormula sheetPhysics 2

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