Formula & Calculator
Young's Double-Slit Equation
Position of bright fringes on a screen in a double-slit interference experiment.
Interpretation
y = mλL/d. Fringe position on a screen from Young's double-slit experiment. y is distance from centre, m is order, λ wavelength, L screen distance, d slit separation.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| y | Fringe position | m |
| m | Fringe order | |
| λ | Wavelength | m |
| L | Screen distance | m |
| d | Slit separation | m |
What it means
This formula gives the position of bright fringes (constructive interference) in Young’s double-slit experiment. It is a classic demonstration of the wave nature of light. The spacing between fringes is Δy = λL/d. This experiment is used to measure the wavelength of light, to demonstrate interference, and to illustrate wave‑particle duality. It is also used in optical testing and in coherence measurements. The equation is derived from the path difference between the two slits. Understanding this is fundamental for wave optics and for interpreting interference patterns.
Worked example
Young's Double‑Slit – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| m | 1 |
| λ (nm) | 500 |
| L (m) | 1.0 |
| d (mm) | 0.20 |
| Parameter | Value |
|---|---|
| m | 2 |
| y (mm) | 5.76 |
| L (m) | 1.2 |
| d (mm) | 0.25 |
Common mistakes
- Double‑slit interference: y = mλL/d – the position of the m‑th bright fringe on a screen.
- m: Order number (0, ±1, ±2, …) – m=0 is the central maximum.
- L: Distance from the slits to the screen – must be much larger than d for the small‑angle approximation.
- d: Slit separation – in the same units as λ.
- Units: y, L, d in metres, λ in metres – y in the same unit as L.
- Assumption: Small angles (sinθ ≈ tanθ ≈ θ) – for large angles, use the exact formula y = L·tanθ.
Applications
Young's double‑slit equation, y = mλL/d, gives the positions (y) of bright fringes on a screen placed at distance L from two slits separated by distance d, for wavelength λ and order m. This classic experiment demonstrates the wave nature of light and is fundamental to wave optics. Engineers use it to measure the wavelength of light, to design interference filters, and to study the coherence of light sources. It also has applications in optical metrology and in testing optical systems. By analysing fringe patterns, professionals can determine the characteristics of light sources and optical components. This equation is a cornerstone of optics education and research.
- Measurement of light wavelength using interference patterns
- Optical metrology for surface flatness and displacement
- Design of interferometers for precision measurements
- Testing optical components and systems
- Educational demonstration of wave interference
Frequently Asked Questions
It gives the position of bright fringes on a screen in a double‑slit interference experiment: y = mλL/d.
y = distance from the central maximum to the m‑th bright fringe.
m = order number (0, ±1, ±2, …).
λ = wavelength.
L = distance from slits to screen.
d = slit separation.
Δy = λL/d. It is constant for small angles and depends only on λ, L, and d.
From the path difference between the two slits: δ = d sinθ ≈ d (y/L) for small angles. Constructive interference occurs when δ = mλ, giving y = mλL/d.
The fringes become closer together (Δy decreases) because Δy ∝ 1/d.
Slit separation d = 0.5 mm, distance to screen L = 2.0 m, wavelength λ = 600 nm. Fringe spacing Δy = (600×10⁻⁹ × 2.0) / (0.5×10⁻³) = 2.4×10⁻³ m = 2.4 mm.
- Forgetting the small‑angle approximation; use exact formula if angles are not small.
- Confusing fringe spacing with fringe position.
- Not converting units consistently.
Bright fringes occur at y = mλL/d (constructive). Dark fringes occur at y = (m+½)λL/d (destructive).
The intensity follows a cos² envelope, modified by the single‑slit diffraction pattern, giving a series of bright and dark fringes with varying intensity.
By measuring Δy, L, and d, one can calculate λ = Δy · d / L.