# Interference

> CIE A-Level Physics · 9702
> Source: https://www.owlsprep.com/study/cie-9702-u8-interference/

This module covers core principles of wave interference, a key superposition effect. You will learn conditions for sustained interference, apply key formulas, and interpret common interference patterns for CIE 9702 exams.

**Prerequisites:** [Principle of Superposition](https://www.owlsprep.com/study/cie-9702-u8-superposition-principle/); [Wave Properties](https://www.owlsprep.com/study/cie-9702-u7-wave-properties/)

## Learning objectives

- State the conditions required for sustained interference of waves
- Distinguish between constructive and destructive interference in terms of path difference
- Apply the double-slit formula to calculate fringe spacing and related quantities
- Use the diffraction grating equation to find wavelength or angle of maxima
- Explain thin film interference and account for phase change on reflection

## Interference and Coherence

**Interference** — The superposition of two or more coherent waves that results in a new wave pattern with varying amplitude.

*Example:* Two overlapping water waves create an alternating pattern of high and low waves.

For a sustained (stable, visible) interference pattern to be observed, two key conditions must be met:

- Waves must be **coherent**: they have a constant phase difference between them, and the same frequency.
- Waves have similar amplitude, so the difference between maximum and minimum intensity is clear.
- **Constructive interference**: Path difference = $n\lambda$, phase difference = $2n\pi$, maximum amplitude.
- **Destructive interference**: Path difference = $(n + \frac{1}{2})\lambda$, phase difference = $(2n+1)\pi$, minimum amplitude, where $n = 0, 1, 2...$

> **tip**
>
> CIE mark schemes almost always require both conditions for sustained interference to gain full marks when asked this question.

## Young's Double-Slit Interference

Young's double-slit experiment was the first definitive proof that light behaves as a wave, and remains one of the most commonly examined interference topics. A single coherent source is split into two coherent beams by two narrow parallel slits, which interfere on a distant screen to create evenly spaced bright and dark fringes.

**Fringe spacing** — The distance between two consecutive bright fringes (or two consecutive dark fringes) in the interference pattern.

*Notation:* w

$$w = \frac{\lambda D}{d}$$

Where $\lambda$ = wavelength of light, $d$ = separation between the two slits, $D$ = distance from slits to screen.

**Worked example:** A double-slit experiment uses light of wavelength 500 nm, slit separation of 0.2 mm, and the screen is 2.0 m from the slits. Calculate the fringe spacing.

1. Convert all values to SI units (meters):

   $$\lambda = 500 \times 10^{-9} \text{ m}, \quad d = 0.2 \times 10^{-3} \text{ m}, \quad D = 2.0 \text{ m}$$
2. Substitute into the double-slit formula:

   $$w = \frac{(500 \times 10^{-9})(2.0)}{0.2 \times 10^{-3}}$$
3. Calculate the final result:

   $$w = 5.0 \times 10^{-3} \text{ m} = 5.0 \text{ mm}$$

**Check your understanding**

Check your understanding of proportionality:

1. If the slit separation $d$ is doubled, what happens to fringe spacing $w$ (all other variables constant)?

   - Halved
   - Unchanged
   - Doubled
   - Quadrupled

   *Answer:* Halved

   *Why:* Correct: $w$ is inversely proportional to $d$, so doubling $d$ halves $w$.

*Calculator:* allowed

## Diffraction Grating Interference

A diffraction grating consists of hundreds or thousands of equally spaced parallel slits. It produces much sharper, brighter interference maxima than a double slit, and is used to accurately measure the wavelength of light.

**Diffraction Grating Equation** — Relationship between grating spacing $d$, angle of the $n$-th order maximum $\theta$, order $n$, and wavelength $\lambda$. Grating spacing $d = \frac{1}{N}$, where $N$ is lines per unit length.

*Notation:* d \sin\theta = n\lambda

**Worked example:** A diffraction grating has 500 lines per mm. Calculate the angle of the first order maximum for 600 nm light.

1. Calculate grating spacing $d$ in meters:

   $$d = \frac{1}{500} \text{ mm} = 2 \times 10^{-6} \text{ m}$$
2. First order maximum has $n=1$, rearrange for $\sin\theta$:

   $$\sin\theta = \frac{n\lambda}{d} = \frac{1 \times 600 \times 10^{-9}}{2 \times 10^{-6}} = 0.3$$
3. Calculate the angle:

   $$\theta = \sin^{-1}(0.3) \approx 17.5^\circ$$

*Calculator:* allowed

## Thin Film Interference

Thin film interference occurs when light reflects off the two parallel surfaces of a thin transparent film (e.g. soap bubbles, oil slicks, anti-reflection lens coatings). The two reflected waves travel different path lengths and interfere.

> **info**
>
> When light reflects off a medium with a higher refractive index, it undergoes a phase change of $\pi$, which is equivalent to an extra path difference of $\frac{\lambda}{2}$. This is a frequently tested detail.

**Worked example:** Explain why a soap bubble appears green when illuminated with white light.

1. White light contains all visible wavelengths. Light reflects from both the outer and inner surface of the soap bubble.
2. The path difference between the two reflected waves is twice the thickness of the bubble. One reflection (off the outer surface) gains an extra $\frac{\lambda}{2}$ path difference from phase change.
3. For green light, the total path difference meets the condition for constructive interference, so green light is strongly reflected. All other wavelengths undergo destructive interference, so the bubble appears green.

> **Exam tip**
>
> Always account for phase change on reflection in thin film problems, forgetting this will flip your result for constructive/destructive interference.

## Common pitfalls

- **Wrong:** Confusing slit separation $d$ with slit width $a$.
  - Why it fails: $d$ is the distance between two separate slits, while $a$ is the width of a single slit, they are used in different formulas.
  - Correct: Remember: $d$ = distance between (separation of) slits for double slit/grating, $a$ = width of one slit for single-slit diffraction.
- **Wrong:** Mixing units when substituting into interference formulas.
  - Why it fails: Wavelength is often given in nanometres (nm) and slit separation in millimetres (mm), mixing units gives a result that is orders of magnitude wrong.
  - Correct: Always convert all lengths to meters before substitution, then convert your final answer to the required unit.
- **Wrong:** Forgetting phase change on reflection in thin film interference problems.
  - Why it fails: Phase change adds an extra $\frac{\lambda}{2}$ path difference that swaps the conditions for constructive and destructive interference.
  - Correct: Always check if any reflection occurs off a higher refractive index medium, add the extra $\frac{\lambda}{2}$ path difference if this is the case.
- **Wrong:** Only stating coherence as a condition for sustained interference.
  - Why it fails: CIE mark schemes require both conditions to be stated to gain full marks.
  - Correct: Always state: waves must be coherent (constant phase difference, same frequency) and have similar amplitude for a sustained visible pattern.

## Cheatsheet

| Concept | Formula/Rule | Variable Description |
| --- | --- | --- |
| Double-slit fringe spacing | $w = \frac{\lambda D}{d}$ | $w$ = fringe spacing, $D$ = screen distance, $d$ = slit separation |
| Diffraction grating | $d \sin\theta = n\lambda$ | $d$ = grating spacing, $\theta$ = max angle, $n$ = order |
| Constructive interference | Path difference = $n\lambda$ | $n = 0, 1, 2...$ |
| Destructive interference | Path difference = $(n + \frac{1}{2})\lambda$ | $n = 0, 1, 2...$ |
| Phase change on reflection | $\Delta x = \frac{\lambda}{2}$ | Applies when reflecting off higher refractive index medium |

## What's next

Interference is a core superposition effect that underpins many key topics in A-level physics, from optical instrumentation to quantum mechanics. Mastering interference conditions and formulas is essential for understanding diffraction, which is closely linked, as well as standing waves and quantum phenomena like electron diffraction. After completing this topic, you can build on your knowledge to learn about diffraction patterns and extend your understanding of wave behaviour to quantum systems.

- [Diffraction](https://www.owlsprep.com/study/cie-9702-u8-diffraction/)
- [Young's double-slit experiment](https://www.owlsprep.com/study/cie-9702-u8-young-s-double-slit-experiment/)
- [Diffraction grating](https://www.owlsprep.com/study/cie-9702-u8-diffraction-grating/)

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/cie-9702-u8-interference/