AcademyInterference
Academy
Two-Source Light Interference
Level 1 - Physics topic page in Interference.
Principle
Two coherent light sources, such as the two slits in Young's experiment, form bright and dark fringes because the path difference varies across the screen.
Notation
\(d\)
source or slit separation
\(\mathrm{m}\)
\(L\)
distance from sources to screen
\(\mathrm{m}\)
\(\theta\)
angle from the central axis
\(\mathrm{rad}\)
\(y_m\)
screen position of order m
\(\mathrm{m}\)
\(\lambda\)
wavelength in the medium
\(\mathrm{m}\)
Method
Derivation 1: Path difference
For a distant screen, the path difference from two slits separated by \(d\) is approximately \(d\\sin\\theta\).
Path difference
\[\Delta r=d\sin\theta\]
Small-angle screen position
\[\sin\theta\approx\tan\theta\approx\frac{y}{L}\]
Derivation 2: Bright fringes
Bright fringes occur where the waves arrive in phase.
Constructive condition
\[d\sin\theta=m\lambda\]
Screen position
\[y_m\approx\frac{m\lambda L}{d}\]
Derivation 3: Dark fringes
Dark fringes occur where the waves arrive half a cycle out of phase.
Destructive condition
\[d\sin\theta=\left(m+\frac12\right)\lambda\]
Fringe spacing
\[\Delta y=\frac{\lambda L}{d}\]
Rules
Bright fringes
\[d\sin\theta=m\lambda\]
Dark fringes
\[d\sin\theta=\left(m+\frac12\right)\lambda\]
Bright-fringe position
\[y_m\approx\frac{m\lambda L}{d}\]
Fringe spacing
\[\Delta y\approx\frac{\lambda L}{d}\]
Examples
Question
A double slit has
\[d=0.20\,\mathrm{mm}\]
\[L=2.0\,\mathrm{m}\]
and \[\lambda=500\,\mathrm{nm}\]
Find the bright-fringe spacing.Answer
\[\Delta y=\frac{\lambda L}{d}=\frac{(500\times10^{-9})(2.0)}{0.20\times10^{-3}}=5.0\times10^{-3}\,\mathrm{m}\]
Checks
- The central bright fringe is \(m=0\).
- Use the wavelength in the medium where the light travels.
- The small-angle formula is valid only when \(y\\ll L\).
- Larger slit spacing gives smaller fringe spacing.