AQA A-Level Physics Paper 1, June 2025: Question 15

Diffraction Grating Equation: Conditions & Derivation

📋 What this question tests

This question assesses AO1 (Knowledge and Understanding) of wave optics, specifically:

The physical principles underlying the derivation of the transmission diffraction grating formula: nλ = d sinθ .
The requirement for constructive interference (reinforcement) to form bright maxima on a distant screen.
Distinguishing between diffraction grating rules and Young's double-slit approximations (e.g. small-angle assumptions).
Precise definitions of variables such as slit spacing ( d ) versus line density ( N ).

Question 15 (Multiple Choice)

A student derives the equation nλ = d sinθ for the production of a diffraction pattern on a screen. This derivation requires that...

✅ Correct Answer

C: light from adjacent slits arrives at the screen in phase.

Award: 1 Mark (AO1)

💡 Key Knowledge

The grating equation gives the angles θ at which principal maxima occur.
For an intensity maximum, waves from adjacent slits must interfere constructively.
Constructive interference requires that waves arrive at the screen in phase (phase difference = 0 or a multiple of 2π radians).
This occurs when the optical path difference between adjacent slits equals a whole number of wavelengths: path difference = nλ .

Step-by-Step Derivation Breakdown

Understanding the geometry behind nλ = d sinθ

📐 Geometric Analysis of the Derivation

Incident Wavefront: Plane monochromatic coherent light of wavelength λ falls normally onto a diffraction grating.
Slit Separation: Adjacent slits are separated by centre-to-centre distance d .
Diffracted Rays: Consider parallel rays emerging from adjacent slits at an angle θ to the normal towards a bright fringe on a distant screen.
Path Difference Triangle: Dropping a perpendicular from one slit onto the ray from the neighbouring slit forms a right-angled triangle where the hypotenuse is d .
Opposite side = Path Difference = d sinθ
Phase Condition for Maxima: For a bright maximum to form, the light arriving at the screen must interfere constructively, meaning they arrive in phase.
Therefore: Path Difference = nλ where n = 0, 1, 2, ...
Combining both: nλ = d sinθ .

🖼️ Examiner Diagram Specification

If sketching this derivation in an exam, draw: two parallel slits separated by distance d ; two parallel rays departing at angle θ to the grating normal; a dashed line perpendicular to the rays forming a right-angled triangle; and the extra path distance clearly labelled as d sinθ or nλ .

Distractor Analysis & Exam Technique

Why the other options are strictly incorrect

❌ Option-by-Option Elimination

A is incorrect: The small angle approximation ( sinθ ≈ tanθ ≈ θ ) is used for Young's double-slit equation ( w = λD / s ), but never for diffraction gratings. Diffraction grating angles are often large (up to 90°), which is why sinθ is retained explicitly.
B is incorrect: d is the slit spacing (distance between adjacent slits in metres), not the number of slits per unit length. The number of slits per metre is N = 1 / d .
D is incorrect: Since sinθ ≤ 1 , we have nλ = d sinθ ≤ d . Thus, nλ ≤ d for any observable diffraction order. Option D says nλ ≥ d , which is physically impossible for diffraction orders (except the limit where θ = 90° ).

🧠 Exam Technique & Revision Tips

Always link "maximum/bright fringe" to "in phase": In wave optics questions, any condition for a principal maximum inherently relies on waves arriving in phase with zero phase difference modulo 2π.
Watch the formula sheet traps: Remember that d in the formula has units of metres (m). If a question says "500 lines per mm", convert immediately:
d = 1 / (500 × 10³ m⁻¹) = 2.0 × 10⁻⁶ m .
Maximum order calculations: To find the maximum observable order n_max , set sinθ = 1 , calculate n = d / λ , and round down to the nearest whole integer.