# Young's double-slit experiment

> CIE A-Level Physics · Unit 8: Superposition
> Source: https://www.owlsprep.com/study/cie-9702-u8-young-s-double-slit-experiment/

This module covers Young's double-slit experiment, the classic demonstration of wave interference for light. You will learn the setup, formula for fringe spacing, and solve common exam questions on changing parameters.

**Prerequisites:** [Superposition principle](https://www.owlsprep.com/study/cie-9702-u8-superposition-principle/); [Coherent wave sources](https://www.owlsprep.com/study/cie-9702-u8-coherence-interference/)

## Learning objectives

- Describe the experimental setup for Young's double-slit experiment
- Derive and apply the fringe spacing formula
- Predict how changing parameters alters the interference pattern
- Identify common features of double-slit interference patterns

## Experimental Setup and Core Requirements

Young's double-slit experiment provided definitive evidence that light behaves as a wave, by producing a stable interference pattern that cannot be explained by particle theory.

**Coherent Sources** — Two wave sources that maintain a constant phase difference and have the same frequency. Coherence is required to produce a visible, stable interference pattern.

*Example:* A laser passed through a single slit produces coherent light incident on the double slits.

1. Monochromatic light source (produces light of a single wavelength)
2. Narrow single slit to create a coherent point source
3. Two identical, parallel, narrow double slits
4. Screen placed a large fixed distance from the double slits

> **info**
>
> CIE exams regularly ask you to describe the setup. Always include the single slit: it is required to create coherence before the double slits.

> **Exam tip:** Marking schemes award a separate mark for mentioning the single slit requirement.

## Fringe Spacing Formula and Derivation

**Derivation:** Derive the fringe spacing formula for Young's double-slit experiment

*Starting from:* Constructive interference occurs when path difference = $n\lambda$, for integer $n$. For small angles, $\sin\theta \approx \tan\theta$.

1. Let $a$ = slit separation, $D$ = distance to screen, $\Delta x$ = spacing between adjacent bright fringes. For adjacent fringes, path difference increases by $\lambda$.
2. From the path difference condition for constructive interference:
3. $$a \sin\theta = \lambda$$
4. For small angles, $\sin\theta \approx \tan\theta = \frac{\Delta x}{D}$
5. Substitute and rearrange:
6. $$a \frac{\Delta x}{D} = \lambda \implies \Delta x = \frac{\lambda D}{a}$$

*Conclusion:* The fringe spacing formula is $\Delta x = \frac{\lambda D}{a}$, where all quantities are measured in the same length unit.

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

*Notation:* \Delta x

**Worked example:** A Young's double-slit experiment has slit separation $0.5$ mm, screen distance $2.5$ m, and light of wavelength $600$ nm. Calculate the fringe spacing in mm.

1. Convert all quantities to SI units (metres):
2. $$a = 0.5 \times 10^{-3} \text{ m}, \quad D = 2.5 \text{ m}, \quad \lambda = 600 \times 10^{-9} \text{ m}$$
3. Substitute into the fringe spacing formula:
4. $$\Delta x = \frac{(600 \times 10^{-9})(2.5)}{0.5 \times 10^{-3}} = 3.0 \times 10^{-3} \text{ m}$$
5. Convert to millimetres as requested: $3.0 \times 10^{-3}$ m = $3.0$ mm.

> **Exam tip:** Unit conversion is the most commonly tested error here. Always check all units match before calculating.

## Effects of Changing Experimental Parameters

Exam questions very commonly ask you to predict how the interference pattern changes when one experimental parameter is altered. All changes follow directly from the formula $\Delta x = \frac{\lambda D}{a}$.

| Changed Parameter | Effect on $\Delta x$ | Effect on Pattern |
| --- | --- | --- |
| Increase slit separation $a$ | $\Delta x$ decreases | Fringes move closer together |
| Decrease screen distance $D$ | $\Delta x$ decreases | Fringes move closer together |
| Increase wavelength $\lambda$ | $\Delta x$ increases | Fringes move further apart |
| White light instead of monochromatic | $\Delta x$ varies with $\lambda$ | Central white fringe, coloured fringes either side |

**Worked example:** Original fringe spacing is $2.0$ mm. If slit separation is doubled and screen distance is halved, what is the new fringe spacing?

1. Original: $\Delta x_1 = \frac{\lambda D_1}{a_1} = 2.0$ mm
2. New parameters: $a_2 = 2a_1$, $D_2 = \frac{D_1}{2}$
3. $$\Delta x_2 = \frac{\lambda D_2}{a_2} = \frac{\lambda (D_1/2)}{2a_1} = \frac{1}{4} \Delta x_1$$
4. New fringe spacing = $\frac{2.0}{4} = 0.5$ mm

## Key Pattern Features

Double-slit interference patterns have consistent features you need to recall for exams:

- The central fringe is always bright (zero path difference = constructive interference)
- All bright fringes are approximately equally spaced for small angles
- Bright fringe intensity gradually decreases as you move away from the central fringe
- Dark fringes have near-zero intensity from complete destructive interference

> **note**
>
> CIE often tests the difference between double-slit and single-slit patterns: double-slit fringes are equally spaced, while single-slit has a wide bright central maximum.

**Check your understanding**

Check your understanding:

1. Which change increases fringe spacing in Young's experiment?

   - Increase slit separation $a$
   - Decrease screen distance $D$
   - Increase wavelength $\lambda$
   - Use white light instead of monochromatic

   *Answer:* Increase wavelength $\lambda$

   *Why:* Correct. From $\Delta x = \frac{\lambda D}{a}$, only increasing wavelength increases fringe spacing. All other options decrease or do not uniformly increase spacing.

## Common pitfalls

- **Wrong:** Forgetting to convert all quantities to the same unit before calculation
  - Why it fails: Wavelength is usually given in nanometres and slit separation in millimetres, so mixing units gives an incorrect order of magnitude
  - Correct: Convert all lengths to metres (or all to millimetres) before substituting into the formula
- **Wrong:** Dividing the total distance across $n$ fringes by $n$ to get $\Delta x$
  - Why it fails: There are $n-1$ gaps between $n$ fringes, so dividing by $n$ underestimates fringe spacing
  - Correct: Divide the total distance across $n$ fringes by $(n-1)$ to get the fringe spacing
- **Wrong:** Claiming the central fringe is dark in a double-slit pattern
  - Why it fails: Zero path difference at the centre gives constructive interference, so it must be bright
  - Correct: Always state that the central fringe is bright for double-slit interference
- **Wrong:** Omitting the single slit when describing the setup for a laser source
  - Why it fails: Even lasers require the single slit to expand the beam and uniformly illuminate the double slits
  - Correct: Always include the single slit in your description, regardless of the light source
- **Wrong:** Confusing fringe spacing with the distance from the centre to the $n$th fringe
  - Why it fails: $\Delta x$ is the gap between adjacent fringes, not the position of a single fringe
  - Correct: Remember that the position of the $n$th fringe is $x_n = n\Delta x$, so $\Delta x = \frac{x_n}{n}$

## Cheatsheet

| Quantity | Symbol | Key Rule |
| --- | --- | --- |
| Fringe spacing | $\Delta x$ | $\Delta x = \frac{\lambda D}{a}$ |
| Slit separation | $a$ | Distance between two double slits |
| Screen distance | $D$ | Distance from slits to screen |
| Increase $a$ |  | $\Delta x$ decreases, fringes closer |
| Increase $D$ |  | $\Delta x$ increases, fringes further apart |
| Increase $\lambda$ |  | $\Delta x$ increases, fringes further apart |
| Central fringe |  | Always bright for double-slit |

## What's next

Young's double-slit experiment is the foundation of wave interference and superposition topics in CIE A-level Physics. The fringe spacing formula and coherence requirements you learned here directly apply to diffraction gratings, a common follow-on topic that is heavily tested in both multiple choice and written papers. These principles also underpin experimental techniques for measuring the wavelength of light, and explain other superposition effects like thin film interference. Understanding double-slit interference also helps you distinguish between interference and diffraction effects, which is a common exam theme.

- [Diffraction](https://www.owlsprep.com/study/cie-9702-u8-diffraction/)
- [Diffraction grating](https://www.owlsprep.com/study/cie-9702-u8-diffraction-grating/)
- [Current of electricity](https://www.owlsprep.com/study/cie-9702-u9-overview/)

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