IB DP Physics: Superposition, double-slit interference, single-slit diffraction and gratings (DP Physics) – Revision Notes

Quick recall for C.3 Wave phenomena (second half) in the IB Physics guide, first assessment 2025. Superposition and Young’s double slit are for SL and HL. Single-slit diffraction, modulation and gratings are HL only. For full explanations and worked examples, read the study guide. Then test yourself with the practice questions. Other links: IB DP Physics hub, checklist and the free diagnostics.

Key definitions

• Superposition: where waves or pulses meet, the resultant displacement is the sum of the individual displacements (signs included).
• Coherent sources: same frequency and a constant phase difference.
• Path difference: difference in distance from the two sources to a point.
• Fringe separation s: distance between adjacent bright (or adjacent dark) fringes.
• Order n: the whole number counting maxima outward from the central one (n = 0).

Formulas

Result	Equation	Level
Constructive (sources in phase)	path difference = nλ	SL and HL
Destructive (sources in phase)	path difference = (n + ½)λ	SL and HL
Young’s double slit	s = λD / d	SL and HL
First single-slit minimum	θ = λ / b	HL only
Grating maxima	nλ = d sin θ	HL only

Symbols: d = slit separation, D = slits to screen, b = slit width, θ in radians for θ = λ/b.

Method boxes

Interference at a point

1. Find both distances to the point; subtract to get the path difference.
2. Divide by λ.
3. Whole number: constructive. Whole number plus a half: destructive. Anything else: in between.

Young’s fringes

1. Convert units to metres (mm and nm).
2. Use s = λD/d, or rearrange for λ, D or d.
3. For several fringes, divide the measured total by the number of spacings first.

Single slit

1. θ = λ/b gives the angle from the axis to the first minimum.
2. Central maximum full width on a screen = 2Dθ.
3. Narrower slit: wider, dimmer pattern.

Slit modulation

1. Fringes from d sit inside an envelope from b.
2. Missing order = d/b, when this is a whole number.
3. Bright fringes inside the central envelope = 2(d/b) − 1.

Grating

1. d = 1/(lines per metre).
2. sin θ = nλ/d. If this exceeds 1, that order does not exist.
3. Maximum order is the largest whole n below d/λ.
4. Total maxima = 2 × (maximum order) + 1.

Superposition reminders

• Add displacements with their signs; a pulse is not changed by passing through another.
• Constructive: displacements add to a larger resultant. Destructive: they cancel partly or fully.
• For equal-amplitude waves in phase the resultant amplitude is double, so the intensity is four times that of one wave (intensity depends on amplitude squared).
• Energy is not lost at a minimum: it is redistributed to the maxima.
• Use the words “in phase” and “in antiphase” when you explain a point, and say how the path difference produces them.
• Young’s experiment supports the wave model of light, because particles in straight lines would not give alternating bright and dark bands.

Must-know distinctions

Idea	Two slits	Grating (many slits)
Maxima	Wide, fairly gentle	Narrow, sharp, bright
Main use	Show wave nature, measure λ roughly	Measure λ accurately, produce spectra
Equation	s = λD/d (small angles)	nλ = d sin θ

Idea	Double slit	Single slit
Key length	d (separation)	b (width)
Pattern	Equal-spaced fringes	Wide centre, weaker sides
Making it smaller	Smaller d: wider fringes	Smaller b: wider, dimmer pattern

White light: a white central maximum, then spectra with violet nearest the centre and red furthest. Higher orders overlap.

Worked reminders

Fringes. λ = 589 nm, d = 0.45 mm, D = 1.35 m: s = (589 × 10⁻⁹ × 1.35) / (0.45 × 10⁻³) = 1.77 × 10⁻³ m.

Single slit. λ = 500 nm, b = 0.12 mm, D = 2.0 m: first minimum at distance Dλ/b = 8.33 mm from the centre, so the central maximum is 16.7 mm wide.

Grating. 700 lines per mm gives d = 1.43 × 10⁻⁶ m. A second-order maximum at 40.0° gives λ = d sin θ / 2 = 459 nm.

Sketches and descriptions to be able to give

Double-slit pattern (ideal, narrow slits). Equally spaced bright fringes of equal brightness, with the central one on the axis. Spacing s = λD/d.

Single-slit pattern (HL). Intensity against position: a tall central peak, with minima at ±λ/b, then much lower secondary peaks, each about half as wide as the central one. A wider slit squeezes the pattern inward and raises the peak.

Double slit with finite slit width (HL). The double-slit fringes under a single-slit envelope. The fringe heights follow the envelope curve; fringes vanish where the envelope reaches zero.

Grating with monochromatic light (HL). Thin, bright lines at θ = 0 and at each allowed order, with the lines further apart than for a coarser grating. A finer grating (more lines per mm) sends each order to a larger angle.

Grating with white light (HL). White centre; each order is a rainbow band. Violet is on the side nearer the centre.

Model sentences for explain questions

• “The sources are coherent because they are two parts of one wavefront, so the phase difference stays constant.”
• “At P the path difference is a whole number of wavelengths, so they arrive in phase, the displacements add and a maximum results.”
• “At P the path difference is an odd number of half wavelengths, so they arrive in antiphase and the displacements cancel.”
• “A narrower slit diffracts the light through a larger angle, so the central maximum is wider; less light passes through, so it is dimmer.”
• “The grating has many slits, so the maxima are narrower and brighter and the angles can be measured more precisely.”
• “Violet has the shortest wavelength, so sin θ is smallest and violet appears closest to the centre.”

Units and checks

• Convert first: 1 mm = 10⁻³ m, 1 nm = 10⁻⁹ m, 1 μm = 10⁻⁶ m.
• Lines per mm to spacing: d = 10⁻³ / N metres.
• Small angles in radians: θ = λ/b only works when θ is small.
• Check that s, θ and the number of orders are sensible: a fringe separation of metres means a unit slip.

Quick self-test

1. Two pulses of +6.0 cm and −2.5 cm coincide. What is the resultant displacement?
2. State two conditions for two sources to be coherent.
3. Sources in phase, λ = 0.40 m. A point is 3.2 m from one and 2.4 m from the other. What type of interference occurs?
4. Why do two separate torches not give fringes on a screen?
5. Find s for λ = 589 nm, d = 0.45 mm, D = 1.35 m, and state what happens to s if D is halved and d is doubled.
6. (HL) Green light is diffracted by a slit of width 0.12 mm. What happens to the central maximum if the slit is narrowed?
7. (HL) Find the width of the central maximum for λ = 500 nm, b = 0.12 mm, D = 2.0 m.
8. (HL) A double slit has d = 0.45 mm and b = 0.15 mm. Which order is missing, and how many bright fringes lie in the central envelope?
9. (HL) A grating has slit spacing 2.00 μm. For λ = 550 nm, find the angle of the first order and the highest order.
10. (HL) How many lines per millimetre give a first-order maximum at 20.0° for λ = 480 nm?
11. (HL) White light meets a grating. Describe the central maximum and the first-order maximum.

Answers

1. +6.0 + (−2.5) = +3.5 cm.
2. Same frequency (wavelength) and a constant phase difference.
3. Path difference = 0.8 m = 2λ: constructive.
4. Each torch emits light with random, rapidly changing phase, so the phase difference between them is not constant. The pattern averages out.
5. s = 1.77 mm. With D halved and d doubled, s falls to a quarter: 0.442 mm.
6. It becomes wider and less intense.
7. 2Dλ/b = 2 × 2.0 × 500 × 10⁻⁹ / 0.12 × 10⁻³ = 16.7 mm (1.67 cm).
8. d/b = 3: the third order is missing; 2(3) − 1 = 5 bright fringes.
9. sin θ = 550 × 10⁻⁹ / 2.00 × 10⁻⁶ = 0.275, θ = 16.0°. d/λ = 3.64, so the highest order is 3.
10. d = 480 × 10⁻⁹ / sin 20.0° = 1.40 × 10⁻⁶ m, giving 713 lines per mm (to 3 s.f.).
11. The central maximum is white. The first order is a spectrum, violet nearest the centre and red furthest.

Where marks are usually lost

• Using nλ for dark fringes and (n + ½)λ for bright ones.
• Forgetting to change mm and nm to metres.
• Mixing up slit width b and separation d.
• Giving the half-width of the central maximum as its full width.
• Writing “narrower slit, narrower pattern”.
• Saying the grating equation applies without checking that sin θ is at most 1.
• Writing “bright fringes get brighter” without linking it to the envelope.
• Calling the white central maximum coloured, or putting red nearest the centre.
• Explaining coherence without constant phase difference.

Official syllabus

Physics guide, first assessment 2025, International Baccalaureate Organization, sub-topic C.3 Wave phenomena.