A diffraction grating is made up of slits of width 260 nm with separation 810 nm. The grating is illuminated by monochromatic plane waves of

Question

A diffraction grating is made up of slits of width 260 nm with separation 810 nm. The grating is illuminated by monochromatic plane waves of wavelength λ = 550 nm at normal incidence.

a. How many maxima are there in the full diffraction pattern?
b. What is the angular width of a spectral line observed in the first order if the grating has 1300 slits?

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Sapo 1 week 2021-07-22T08:06:57+00:00 1 Answers 1 views 0

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    2021-07-22T08:08:21+00:00

    Answer:

    Explanation:

    Given that,

    Grafting width is 260nm

    Separation is d= 810nm

    Wavelength is λ = 550 nm

    A. The condition for maximum grating is given as

    d Sinθ = mλ

    where,

    λ is the wavelength

    d is the operation between the two slit.

    The range is Sinθ is -1 and 1

    But since θ Is between 0-180°

    Then, 0≤Sinθ≤1

    Then, we must have d ≥ mλ

    So, d/λ ≤m

    Then, m ≥ d/λ

    m ≥ 810/550

    m ≥1.47

    So, we can take m = 1

    There are three maxima m = -1,0,1

    b. When N = 1300.

    For first order maxima m = 1

    Then, d Sinθ = mλ

    d Sinθ = λ

    The angular width is given as.

    ∆θ = λ/NdCosθ, where d Sinθ = λ

    ∆θ = dSinθ/NdCosθ

    ∆θ = tanθ/N, equation 1

    d Sinθ = λ

    Sinθ = λ / d

    θ = ArcSin(λ/d)

    θ = ArcSin(550/810)

    θ = 42.77°

    Substituting theta into equation 1

    ∆θ = tanθ/N

    ∆θ = tan(42.77)/1000

    ∆θ = 0.000925rad

    To degree 2πrad = 360°

    ∆θ = 0.000925rad × 360°/2πrad

    ∆θ = 0.053°

    The angular width of the spectral line is observed to be 0.053°

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