# 5.26 The Diffraction Grating Equation

## The Diffraction Grating Equation

• A diffraction grating is a plate on which there is a very large number of parallel, identical, close-spaced slits
• When monochromatic light is incident on a grating, a pattern of narrow bright fringes is produced on a screen

Diagram of diffraction grating used to obtain a fringe pattern

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

Diffraction grating equation for the angle of bright fringes

• Exam questions sometime state the lines per m (or per mm, per nm etc.) on the grating which is represented by the symbol N
• d can be calculated from N using the equation

#### 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

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 n2 is 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:

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

#### 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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