Summary & Master Memory Capsule

Master Formula Sheet

#	Result	Formula
1	Wavefront	surface of constant phase; energy travels perpendicular to it
2	Huygens	new wavefront = forward envelope of secondary wavelets
3	Refraction (wave)	$\frac{\sin i}{\sin r} = \frac{v_1}{v_2} = \frac{n_2}{n_1}$; $\lambda_{med} = \lambda/n$; $\nu$ fixed
4	Coherent intensity	$I = 4I_0\cos^2(\phi/2)$; bright $\Delta = n\lambda$ ($4I_0$), dark $(n+\tfrac12)\lambda$ (0)
5	Incoherent	$I = 2I_0$ (no fringes)
6	Unequal sources	$\frac{I_{max}}{I_{min}} = \left(\frac{a_1+a_2}{a_1-a_2}\right)^2$
7	Young bright/dark	$x = n\lambda D/d$ / $(n+\tfrac12)\lambda D/d$
8	Fringe width	$\beta = \lambda D/d$; angular $\theta = \lambda/d$
9	Single-slit minima	$a\sin\theta = n\lambda$; central width $2\lambda D/a$
10	Half rule	unpolarised through 1 polaroid: $I_0/2$
11	Malus' law	$I = I_0\cos^2\theta$
12	Three-polaroid	$\frac{I_0}{8}\sin^2 2\theta$ (unpolarised input), max $I_0/8$ at 45 degrees

Coherent vs Incoherent, and Interference vs Diffraction

Adding waves:

	Coherent	Incoherent
Phase difference	constant	randomly drifting
Combine	amplitudes (then square)	intensities directly
Bright/dark	$4I_0$ / 0	flat $2I_0$
Fringes?	yes	no

Interference vs diffraction (Feynman: 'only a question of usage'):

	Interference (Young)	Diffraction (single slit)
Sources	a few discrete coherent slits	continuous wavefront (one slit)
Key length	slit separation d	slit width a
Fringes	equally spaced, equal brightness	central max twice as wide, sides weaken
Bright/dark equation	bright $d\sin\theta = n\lambda$	dark $a\sin\theta = n\lambda$

The trap to never fall into: $d\sin\theta = n\lambda$ marks bright fringes; $a\sin\theta = n\lambda$ marks dark fringes (minima). Same algebra, opposite meaning.

Fringe-Width Variation Rules & Polarisation Toolkit

Young's $\beta = \lambda D/d$ responds to:

• $\lambda$ up (red vs blue): $\beta$ up.
• D up (screen farther): $\beta$ up.
• d up (slits apart): $\beta$ down.
• Immerse in medium n: $\lambda \to \lambda/n$, so $\beta \to \beta/n$.
• Central fringe: white in white light, wavelength-independent.

Polarisation:

• Light is transverse → can be polarised (sound cannot).
• One polaroid on unpolarised light: half intensity, output polarised, rotation irrelevant.
• Two polaroids: Malus $I = I_0\cos^2\theta$; crossed (90 degrees) = dark.
• Unpolarised through two: $\frac{I_0}{2}\cos^2\theta$.
• Three (crossed outer pair, middle at $\theta$): $\frac{I_0}{8}\sin^2 2\theta$, max $I_0/8$ at 45 degrees.
• Applications: sunglasses (cut horizontally polarised glare), LCDs, photography.

One-Glance Revision Flow

The chapter in seven steps:

1. Wave model wins: Huygens (1678), Young's interference (1801), Foucault's slower-in-water (1850), Maxwell's EM nature.
2. Huygens' principle: wavefront = constant phase; new wavefront = envelope of secondary wavelets.
3. Refraction/reflection from wavefronts: $\frac{\sin i}{\sin r} = \frac{v_1}{v_2}$; light slower in denser media; $\lambda$ shrinks, $\nu$ fixed; i = r.
4. Interference: coherent $4I_0$/0, incoherent $2I_0$; the coherence requirement.
5. Young's slits: $\beta = \lambda D/d$ — the formula that measured $\lambda$; know every variation.
6. Diffraction: single-slit minima $a\sin\theta = n\lambda$, central max twice as wide; interference vs diffraction is 'usage'.
7. Polarisation: transverse proof; half rule + Malus + three-polaroid $\frac{I_0}{8}\sin^2 2\theta$.

Morning-of-exam checklist: frequency never changes on refraction … $4I_0$ (coherent) vs $2I_0$ (incoherent) … $\beta = \lambda D/d$, shrinks by n in water … central fringe white … $d\sin\theta = n\lambda$ BRIGHT but $a\sin\theta = n\lambda$ DARK … central diffraction max is double width … half rule THEN Malus … three polaroids peak at 45 degrees ($I_0/8$). Go score.