Diffraction Grating Analysis

00:00 9 marks

Monochromatic light is incident normally on a diffraction grating having 500 lines per millimetre. The angle between the central maximum and the second-order maximum is measured to be 35.0°.

Part a

4 minutes 3 marks

Question

Calculate the wavelength of the light. \[3\]

Hint Hint

Hint

First, find the distance ‘d’ between adjacent lines on the grating in metres. Then use the diffraction grating formula \(nλ = d \sin θ\) with the given order (n=2) and angle.

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The wavelength of the light is approximately 574 nm.

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Part b

3 minutes 2 marks

Question

Determine the total number of maxima observed on a screen far from the grating. \[2\]

Hint Hint

Hint

The maximum possible diffraction angle is 90°. Use this in the grating equation \(nλ = d \sin θ\) to find the maximum possible integer value for the order ‘n’. Remember the total number includes the central maximum and maxima on both sides.

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The total number of maxima observed is 7.

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Part c

2 minutes 2 marks

Question

The light source is replaced by one emitting two distinct wavelengths, λ₁ and λ₂ = λ₁ + Δλ, where Δλ is small. Outline how the diffraction grating allows these two wavelengths to be distinguished in the second-order spectrum. \[2\]

Hint Hint

Hint

Consider the diffraction grating equation \(nλ = d \sin θ\). For a fixed order (n=2) and grating spacing (d), how does the angle \(θ\) change if the wavelength \(λ\) changes slightly?

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According to the grating equation \(nλ = d \sin θ\), for a fixed order (n=2) and grating spacing (d), the diffraction angle \(θ\) depends directly on the wavelength \(λ\). Since \(λ₁\) and \(λ₂\) are different, they will diffract at slightly different angles \(θ₁\) and \(θ₂\). This angular separation allows the two wavelengths to be observed as distinct maxima (resolved).

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Part d

3 minutes 2 marks

Question

The diffraction grating is now illuminated with white light. Describe the appearance of the pattern observed on the screen. \[2\]

Hint Hint

Hint

White light contains a continuous range of wavelengths (colours). Consider how the diffraction angle depends on wavelength for n=0 and for n>0.

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Complete Answer

A bright white central maximum (n=0) is observed at \(θ=0°\). On either side of the central maximum, continuous spectra are observed for each diffraction order (n=1, 2, 3…). Within each spectrum, violet light (shorter wavelength) is diffracted at the smallest angle, and red light (longer wavelength) is diffracted at the largest angle. Higher orders are wider and dimmer, and may overlap.

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