IB DP Physics: HL复习笔记9.3.4 The Diffraction Grating Equation

The Diffraction Grating Equation

  • The angles at which the maxima of intensity (constructive interference) are produced can be deduced by the diffraction grating equation:

8.2.3.1-Grating-equation

  • The lines per m (or per mm, per nm etc.) on the grating is usually represented by the symbol N
  • Therefore, the spacing between each slit, d, can be calculated from N using the equation:

Screenshot-2022-01-04-at-10.48.19-am

Angular Separation

  • The angular separation of each maxima is calculated by rearranging the grating equation to make θ the subject
  • The angle θ is taken from the centre meaning the higher orders are at greater angles

8.2.3.1-Angular-separation

Angular separation

  • The angular separation between two angles is found by subtracting the smaller angle from the larger one
  • The angular separation between the first and second maxima n1 and nis θ2 – θ1

Orders of Maxima

  • The maximum angle to see orders of maxima is when the beam is at right angles to the diffraction grating
    • This means θ = 90o and sin θ = 1
  • The highest order of maxima visible is therefore calculated by the equation:

Screenshot-2022-01-06-at-3.03.50-pm

  • Note that since n must be an integer, if the value is a decimal it must be rounded down
    • E.g If n is calculated as 2.7 then n = 2 is the highest order visible

Worked Example

An experiment was set up to investigate light passing through a diffraction grating with a slit spacing of 1.7 µm. The fringe pattern was observed on a screen. The wavelength of the light is 550 nm.Worked-Example-Diffraction-Grating

Calculate the angle α between the two second-order lines.

8.2.3.1-Worked-example-diffraction-grating-equation-2-e1616684019706

Derivation of the Diffraction Grating Equation

  • When light passes through the slits of the diffraction grating, the path difference at the zeroth order maximum is zero
  • At the first-order maxima (n = 1), there is constructive interference, hence the path difference is λ
    • Therefore, at the nth order maxima, the path difference is equal to nλ

Diffraction-Grating-Equation

Using this diagram and trigonometry, the diffraction grating equation can be derived

  • Using trigonometry, an expression for the first order maxima can be written:

Diffraction-Grating-Derivation-1

  • Where:
    • θ = the angle between the normal and the maxima
    • λ = the wavelength of the light (m)
    • d = the slit separation (m)
  • This means, for n = 1:

Diffraction-Grating-Derivation-2

  • Similarly, for n = 2, where the path difference is 2λ:

Diffraction-Grating-Derivation-3

  • Therefore, in general, where the path difference is nλ:

Diffraction-Grating-Derivation-4

  • A small rearrangement leads to the equation for the diffraction grating:

d sin θn = 

Exam Tip

Take care that the angle θ is the correct angle taken from the centre and not the angle taken between two orders of maxima.

 

 

 

 

 

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