### The 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*

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- 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*

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#### 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*

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- 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 n
_{1}and n_{2 }is**θ**_{2}– θ_{1} - The maximum angle to see orders of maxima is when the beam is at right angles to the diffraction grating. This means θ = 90
^{o}and sin(θ) = 1

#### 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.

### Determining the Wavelength of Light

#### Method

- The wavelength of light can be determined by rearranging the grating equation to make the wavelength
*λ*the subject - The value of
*θ*, the angle to the specific order of maximum measured from the centre, can be calculated through trigonometry - The distance from the grating to the screen is marked as
*D* - The distance between the centre and the order of maxima (e.g. n = 2 in the diagram) on the screen is labelled as
*h –*the fringe spacing - Measure both these values with a ruler
- This makes a right-angled triangle with the angle
*θ*as the ratio of the*h/D = tanθ*

*The wavelength of light is calculated by the angle to the order of maximum*

- Remember to find the inverse of tan to find
*θ =*tan^{-1}*(h/D)* - This value of
*θ*can then be substituted back into the diffraction grating equation to find the value of the wavelength (with the corresponding order*n*)

#### Improving experiment and reducing uncertainties

- The fringe spacing can be subjective depending on its intensity on the screen. Take multiple measurements of
*h*(between 3-8) and finding the average - Use a Vernier scale to record
*h*, in order to reduce percentage uncertainty - Reduce the uncertainty in
*h*by measuring across all fringes and dividing by the number of fringes - Increase the grating to screen distance
*D*to increase the fringe separation (although this may decrease the intensity of light reaching the screen) - Conduct the experiment in a darkened room, so the fringes are clearer
- Use grating with more lines per mm, so values of
*h*are greater to lower percentage uncertainty