A student used a narrow beam of monochromatic light and a diffraction grating to determine λ, the wavelength of the monochromatic light - Leaving Cert Physics - Question 3 - 2018

The angle ϕ between the first order images was obtained by measuring the distance between the first order images on the screen (s2) and the distance from the grating to the screen (s1). Using trigonometry,

• The angle can be calculated using $heta = an^{-1} \left( \frac{d}{s_1} \right)$
  where d is the distance between the two first order images.

To calculate the wavelength, we utilize the diffraction formula:
$n \lambda = d \sin \theta$
Where:

• n = order of the image (1 for the first order),
• d = distance between the slits,
  Using the number of lines per mm, convert d to meters:
  $d = \frac{1}{500 \text{ lines/mm}} = 2 \times 10^{-6} \text{ m}$
  Substituting the known values into the formula:

Using θ, where ϕ = 34.1°:
$\sin(17.05°) = 0.293$
Thus,
$\lambda = \frac{d \sin \theta}{n} = 5.86 \times 10^{-7} \text{ m}$

Replacing the diffraction grating with one of 80 lines per mm would lead to an increase in the distance between the lines, which means that the angle ϕ would decrease. Lower line density results in smaller angles for the same order of diffraction, decreasing the spread of the beam on the screen.

The grating with 500 lines per mm would give a more accurate value for λ as it yields a larger angle ϕ, resulting in greater measurements and thus smaller percentage error. A higher line density also allows for a more precise determination of angles, making calculations more reliable.

If a source of white light is used, the student would observe a spectrum being produced rather than discrete order images. The white light would diffract into different wavelengths, creating a series of colors on the screen due to the varying angles of diffraction for each wavelength.