🥋 The Move
Linear superposition + a clean path difference → phase map. For two slits/spots: \(\Delta = d\sin\theta\) (or \(d\,y/L\) at small angle), phase \(\phi = \dfrac{2\pi}{\lambda}\,\Delta\), and intensity \(I = I_{\max}\cos^2\!\left(\dfrac{\phi}{2}\right)\). Add single-slit envelope when apertures have finite width.
📘 Canonical Problem
Two narrow slits separated by distance \(d\) illuminate a screen a distance \(L\) away with monochromatic light of wavelength \(\lambda\). Find (i) the angular positions \(\theta_m\) and (ii) the fringe spacing \(\Delta y\) on the screen. Give the intensity pattern \(I(\theta)\).
- Given: \(d,\ L,\ \lambda,\ I_{\max}\)
- Find: \(\theta_m,\ \Delta y,\ I(\theta)\)
- Assume: small angles: \(\sin\theta\approx\tan\theta\approx\theta\), slits narrow (no envelope).
Sketch notes (no figure):
- Path difference \(\Delta = d\sin\theta\).
- Constructive: \(\Delta = m\lambda\). Destructive: \(\Delta=(m+\tfrac12)\lambda\).
- Intensity: \(I(\theta)=I_{\max}\cos^2\!\big(\pi d\sin\theta/\lambda\big)\).
🔎 Setup (Preview)
- Angles → screen. Small angle: \(y \approx L\theta\Rightarrow \Delta y \approx \lambda L/d\).
- Intensity law. Use \(I=I_{\max}\cos^2(\pi d\sin\theta/\lambda)\).
Full derivation, exam-speed forms, and traps at Black Belt.