Coherence, constructive and destructive interference, Young's double-slit experiment
AQA A-Level Physics 3
➕State and apply the principle of superposition
✅Explain constructive and destructive interference
🔗Define coherence and explain why it is needed for stable interference
🔭Describe Young's double-slit experiment and its significance
📐Use λ = ax/D to calculate wavelength from fringe spacing
🔬Describe the required practical for measuring wavelength of light
Principle of Superposition
Principle of superposition: When two or more waves meet at a point, the resultant displacement at that point is the algebraic sum of the individual displacements.
The waves pass through each other unaffected — superposition is temporary. After passing through the meeting point, each wave continues unchanged.
Constructive interference: Waves arrive in phase (path difference = nλ where n = 0, 1, 2...). Displacements add → resultant amplitude is maximum (sum of individual amplitudes for two equal-amplitude waves = 2A).
Destructive interference: Waves arrive in antiphase (path difference = (n + ½)λ). Displacements cancel → resultant amplitude is minimum (zero for two equal-amplitude waves).
Coherence: Two wave sources are coherent if they have a constant phase difference and the same frequency. Only coherent sources produce stable interference patterns.
If the phase difference between two sources changes randomly (as with two separate light bulbs), the interference pattern shifts rapidly and averages out — no visible pattern results.
To achieve coherence in light experiments, a single source is split into two (e.g. using a double slit). Both slits emit light from the same wavefronts, so they are automatically coherent with a fixed phase relationship.
A laser beam is an excellent coherent source because all photons are emitted in phase with the same frequency. This is why lasers produce clear diffraction and interference patterns.
For two sources to produce visible interference fringes, they must also have:
The same frequency (and ideally the same wavelength)
Similar amplitudes (for high contrast fringes)
Overlapping beams (waves must meet)
Young's Double-Slit Experiment
Thomas Young's experiment (1801) demonstrated the wave nature of light by producing an interference pattern. A single light source illuminates two narrow slits. Each slit acts as a secondary source of coherent waves that overlap and interfere on a screen.
Bright fringes (maxima) occur where waves arrive in phase: path difference = nλ.
Dark fringes (minima) occur where waves arrive in antiphase: path difference = (n + ½)λ.
Fringe spacing (distance between adjacent bright fringes):
w = λD / a
λ = wavelength (m); D = slit-to-screen distance (m); a = slit separation (m)
Variable
Effect on fringe spacing w
Increase D
w increases (fringes spread out)
Increase a (slit separation)
w decreases (fringes closer together)
Increase λ (longer wavelength)
w increases (fringes spread out)
Young's experiment was historically important because it proved light is a wave (Newtonian particle model predicted no interference). The small spacing of light fringes requires large D and small a for measurement.
Interference with Sound and Microwaves
Interference is a general wave phenomenon, not unique to light. The same principles apply to all waves:
Sound: Two loudspeakers connected to the same signal generator are coherent. Walking across the line in front of them, you hear alternating loud (constructive) and quiet (destructive) regions.
Microwaves: A microwave transmitter directed at a metal sheet with two slits produces a similar pattern — a receiver moved along a parallel line detects maxima and minima.
Observing interference from any type of wave is strong evidence that the wave model is correct for that type of radiation. Young's experiment with light was the decisive evidence against Newton's corpuscle theory.
In a double-slit experiment, the slit separation is 0.50 mm, the screen is 1.80 m away, and fringes are 1.98 mm apart. Calculate the wavelength of the light.
1w = λD/a → λ = wa/D
2λ = (1.98 × 10⁻³ × 0.50 × 10⁻³) / 1.80
3λ = (9.9 × 10⁻⁷) / 1.80 = 5.50 × 10⁻⁷ m
λ = 550 nm (green light)
Two sources emit sound at 850 Hz. The speed of sound is 340 m s⁻¹. Point P is 4.60 m from one source and 5.10 m from the other. Is the interference at P constructive or destructive?
1λ = v/f = 340/850 = 0.40 m
2Path difference = 5.10 − 4.60 = 0.50 m
3Number of wavelengths in path difference: 0.50/0.40 = 1.25 = 1¼ = (1 + ¼)
4Path difference = 1.25λ = (n + ¼)λ — this is NOT a whole number nor a half-integer → partial interference. But 1.25 = 5/4 — check: nλ = 0 or 0.4; (n+½)λ = 0.2, 0.6... Path diff 0.5 = (n+½)λ → 0.5/0.4 = 1.25 = 1.25 which equals (1+0.25)... Hmm 0.5 = 1.25 × 0.4 → (0.5/0.4) = 1.25: not n or n+0.5. Actually 0.50 m = 1.25λ — between constructive (n=1, 0.40m) and destructive (n+½=1.5, 0.60m) → partial. But let me recheck: destructive if pd = (n+0.5)λ. 0.50/0.40 = 1.25 — not a half-integer. Constructive if pd = nλ: 0.50/0.40 = 1.25 — not integer. So partial cancellation.
A student performs Young's double-slit experiment and measures fringe spacing by finding the distance across 10 bright fringes (10 gaps). The measurement is 22.0 mm and D = 1.50 m, a = 0.30 mm. Calculate λ.
110 fringe spacings = 22.0 mm → w = 22.0/10 = 2.20 mm = 2.20 × 10⁻³ m
2λ = wa/D = (2.20 × 10⁻³ × 0.30 × 10⁻³) / 1.50
3λ = 6.60 × 10⁻⁷ / 1.50 = 4.40 × 10⁻⁷ m
λ = 440 nm (violet light)
In Young's experiment, the slit separation is halved and the screen distance is doubled. By what factor does the fringe spacing change?
1w = λD/a. New w' = λ(2D)/(a/2) = 4λD/a = 4w
Fringe spacing increases by a factor of 4
1. Two waves of equal amplitude meet in antiphase. The resultant amplitude is:
Antiphase means phase difference = π (180°). By superposition, displacements cancel completely → resultant amplitude = 0. This is destructive interference.
2. What path difference gives the 2nd order maximum (bright fringe) in a double-slit experiment?
Maxima occur at path difference = nλ. For n = 2: path difference = 2λ.
3. Why must two sources be coherent to produce a stable interference pattern?
Coherence requires a constant phase difference (same frequency, fixed phase relationship). If the phase difference fluctuates randomly, constructive and destructive regions shift rapidly and average out — no stable pattern.
4. In a double-slit experiment with λ = 600 nm, a = 0.40 mm, D = 2.00 m. What is the fringe spacing?
w = λD/a = (600×10⁻⁹ × 2.00) / (0.40×10⁻³) = 1.2×10⁻³ / 4.0×10⁻⁴ = 3.0×10⁻³ m = 3.00 mm
5. Ten bright fringes span 15.0 mm in a double-slit experiment (D = 1.20 m, a = 0.50 mm). Calculate the wavelength in nm.
1. In a double-slit experiment, the fringe pattern is observed on a screen 2.50 m away. The slit separation is 0.25 mm. The student measures 5 fringe spacings as 14.0 mm. (a) Calculate λ. (b) Calculate the angle θ to the 3rd order maximum. (c) Comment on the assumption made in using w = λD/a.
(a) w = 14.0/5 = 2.80 mm. λ = wa/D = (2.80×10⁻³ × 0.25×10⁻³)/2.50 = 2.80×10⁻⁷ m = 280 nm — UV, not visible. More likely measurement error; recalculate: perhaps 5 spacings = 14mm → w = 2.8mm. If a = 0.25mm, D = 2.50m: λ = (2.8×10⁻³ × 0.25×10⁻³)/2.50 = 2.8×10⁻⁷ m. This seems low for visible light — perhaps the slit gap is 0.25mm but should be 0.025mm? Sticking with the calculation: λ ≈ 280 nm. (b) For 3rd max: sin θ = 3λ/a = 3×280×10⁻⁹/0.25×10⁻³ = 3.36×10⁻³ → θ = 0.19°. (c) The formula w = λD/a assumes the screen is much further from the slits than the slit separation (small angle approximation: sin θ ≈ tan θ ≈ θ). This is valid when D >> a and the fringe order n is small.
2. Explain why replacing the monochromatic light source in Young's experiment with white light produces a coloured fringe pattern rather than a black-and-white pattern, and why the central fringe remains white.
White light contains all visible wavelengths (approximately 400–700 nm). Each wavelength produces its own fringe pattern with fringe spacing w = λD/a — longer wavelengths (red) give wider fringes, shorter wavelengths (violet) give narrower fringes. The patterns are superimposed on the screen. At the central maximum (path difference = 0), all wavelengths interfere constructively simultaneously → the central fringe is white. For other orders (n ≥ 1), different wavelengths have their maxima at slightly different positions, so the colours spread out. The 1st order fringe shows a spectrum — violet innermost, red outermost. Higher orders overlap and the pattern becomes less distinct.
3. Two radio antennae 80 m apart transmit coherently at 3.0 × 10⁸ Hz. At what angle (to the perpendicular bisector) is the first destructive interference minimum? (c = 3.00 × 10⁸ m s⁻¹)
λ = c/f = 3.00×10⁸ / 3.0×10⁸ = 1.0 m. First minimum: path difference = λ/2 = 0.5 m. sin θ = λ/(2a) = 0.5/80 = 0.00625. θ = arcsin(0.00625) = 0.358° ≈ 0.36°. Very small angle — the beams overlap over a wide area, which is why radio waves can reach many locations.
Required Practical Young's Double-Slit Experiment
Objective: Measure the wavelength of monochromatic light using a double-slit and a distant screen.
Equipment
Laser (or bright monochromatic lamp with a single slit to create a coherent source)
Double-slit slide (slits ~0.5 mm apart)
Optical bench or ruler for measuring D (at least 1 m)
Screen (plain white paper on a stand)
Ruler or travelling microscope to measure fringe spacing
Method
Set up the laser (or source + single slit), double-slit, and screen in a straight line.
Measure and record D (distance from double-slit to screen) — use a metre rule.
Turn on the laser and observe bright fringes on the screen.
Mark the positions of several bright fringes. Measure the total width across n fringe spacings (e.g. 10) and divide to find w.
Repeat the measurement of fringe spacing and take a mean.
Use w = λD/a rearranged to λ = wa/D. The value of a (slit separation) is printed on the slide or measured with a travelling microscope.
Key Safety Points
NEVER look directly into the laser beam or its reflections. Ensure the laser is fixed and pointed away from people. Use class 2 (max 1 mW) lasers in educational settings. Place a beam stop behind the screen.
Sources of Error and How to Reduce Them
Error source
How to reduce
Measuring fringe spacing from a single pair of fringes (small measurement)
Measure across 10 fringes, divide by 10 — reduces uncertainty by factor of 10
Parallax when reading positions on screen
Use a travelling microscope for accurate fringe position measurement
Uncertainty in D (screen-to-slit distance)
Use a long D (e.g. 2 m) to spread fringes out; measure D with steel rule, not flexible tape
Slit not vertical / screen not parallel to slit
Check alignment carefully; the fringe pattern should be symmetric about the central maximum
Expected Results
For a red laser (λ ≈ 632 nm), a = 0.5 mm, D = 1.5 m: w = λD/a = 632×10⁻⁹ × 1.5 / (0.5×10⁻³) = 1.90 × 10⁻³ m ≈ 1.9 mm. Fringes should be clearly visible and equally spaced.