When light shines on a compact disc it acts as a diffraction grating causing diffraction and dispersion of the light - Leaving Cert Physics - Question 7 - 2009

To derive the diffraction grating formula, we begin with the grating setup:

• For constructive interference, the path difference is given by: $d imes ext{sin} heta = n imes ext{λ}$

Where:

• d = distance between grating lines (1/number of lines per mm)
• θ = angle of diffraction
• n = order of diffraction
• λ = wavelength of light

This relationship establishes how light interacts with the grating surface.

Using the formula, we have:

• Distance to the screen, L = 90 cm = 0.9 m
• Lines per mm = 80, hence d = 1/80 mm = 1.25 x 10^-5 m

The path difference for the third order image:

• The distance between third order images is given as 23.8 cm = 0.238 m.

Using the small angle approximation:

• $ext{sin}( heta) ext{ } \approx \text{ } \frac{ ext{opposite}}{ ext{hypotenuse}} = \frac{23.8/2}{90} = 0.264$

Substituting into the formula: $ext{λ} = \frac{d imes ext{sin} heta}{n} = \frac{(1.25 \times 10^{-5}) \times 0.264}{3} \approx 5.51 \times 10^{-7} m \text{ or } 551 \text{ nm.}$

To find the maximum number of images:

The angle will not exceed 90 degrees, hence:

• Using the formula: $n \leq \frac{d}{\text{λ}}$

From earlier,

• d for 80 lines/mm: d = $1.25 \times 10^{-5} m$ and λ found is $5.51 \times 10^{-7} m$.

Calculating: $n = \frac{(1.25 \times 10^{-5})}{(5.51 \times 10^{-7})} \approx 22.7$

Thus, the maximum number of images formed = 22 + 22 + 1 = 45.

The diffraction grating produces a spectrum because:

• Different colors of light have different wavelengths.
• When white light passes through the grating, different wavelengths are diffracted at different angles, causing the separation of light into distinct colors.

A spectrum is not formed at the central image (zero order) because:

• At zero order, all colors constructively interfere, resulting in white light.
• There is no path difference for light waves in the zero order, thus not allowing the separation into a spectrum.