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 was obtained by measuring the distances from the diffraction grating to the positions of the first order images (s1 and s2). The observer records the angle θ1 to the left from the grating and θ2 to the right. The angle ϕ then can be calculated as:

$ϕ = θ_1 + θ_2$

This involves making precise measurements with the spectrometer.

To calculate the wavelength of the light, we can use the diffraction grating formula:

$nλ = d imes ext{sin}(θ)$

Where:

• n = order of the image (n = 1 for first order)
• d = distance between grating lines = ( \frac{1}{number ; of ; lines ; per ; mm} = \frac{1}{500 ; mm} = 2 \times 10^{-6} ; m )
• θ = angle measured, which is given as 17.05°

Substituting the values:

• Convert the angle θ to radians for calculation: θ = 17.05°
• Calculate sin(θ): $sin(17.05°) = 0.292$
• Now plug in the values:

$λ = \frac{d \times sin(θ)}{n} = \frac{(2 \times 10^{-6}) \times 0.292}{1} = 5.86 \times 10^{-7} m$

The angle ϕ would decrease if the diffraction grating is replaced with one having 80 lines per mm. This is because a finer grating (more lines per unit length) produces narrower diffraction patterns, thus resulting in smaller angles for the same order of diffraction.

The grating with 500 lines per mm would provide a more accurate value for λ. This is due to the fact that larger measurements (i.e., larger angles) lead to smaller percentage errors in calculations, as the resolution of measurement improves with a higher number of lines per mm.

If the source of monochromatic light is replaced with white light, the student would observe a spectrum of colors instead of a single diffraction pattern. This phenomenon occurs because white light is composed of multiple wavelengths, each of which is diffracted at different angles, resulting in a continuum of color across the screen.