AcademyInterference
Academy
Coherence and Interference
Level 1 - Physics topic page in Interference.
Principle
Interference happens when waves overlap and their fields add. A stable light interference pattern requires coherent sources with a fixed phase relationship.
Notation
\(\Delta r\)
path difference
\(\mathrm{m}\)
\(\Delta\phi\)
phase difference at the observation point
\(\mathrm{rad}\)
\(\lambda\)
wavelength
\(\mathrm{m}\)
\(\phi_0\)
initial phase difference between sources
\(\mathrm{rad}\)
\(m\)
integer interference order
1
Method
Derivation 1: Superposition
Light interference is field superposition. The electric fields add first; intensity follows from the resultant field amplitude.
Field sum
\[E_{\mathrm{net}}=E_1+E_2\]
Intensity link
\[I\propto E_{\mathrm{net},0}^2\]
Derivation 2: Phase difference
Path difference changes phase because one wave travels farther than the other.
Path phase
\[\Delta\phi_{\mathrm{path}}=\frac{2\pi\Delta r}{\lambda}\]
Total phase
\[\Delta\phi=\phi_0+\frac{2\pi\Delta r}{\lambda}\]
Derivation 3: Maxima and minima
For equal-amplitude waves, maxima and minima are set by phase.
Constructive interference
\[\Delta\phi=2\pi m\]
Destructive interference
\[\Delta\phi=(2m+1)\pi\]
Rules
For in-phase sources, \(\\phi_0=0\).
Constructive path difference
\[\Delta r=m\lambda\]
Destructive path difference
\[\Delta r=\left(m+\frac12\right)\lambda\]
Phase from path
\[\Delta\phi=\frac{2\pi\Delta r}{\lambda}\]
Examples
Question
Two coherent in-phase light waves have
\[\Delta r=2\lambda\]
Classify the interference.Answer
Because
\[\Delta r=m\lambda\]
with \[m=2\]
the interference is constructive.Checks
- Coherence means the phase difference is stable long enough to observe a pattern.
- Interference conditions use path difference and any initial phase difference.
- Destructive interference is complete only when the amplitudes are equal.
- Add fields or amplitudes first, not intensities directly.