Lesson 11 of 11 · 15 min

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Must-know facts

20 facts

1A wavefront is a surface of constant phase; energy flows at right angles to it, along the rays.
2A point source gives spherical wavefronts; far away a small patch is a plane wavefront.
3Huygens: every point of a wavefront is a source of secondary wavelets; the forward envelope is the new wavefront.
4Refraction by wavefronts: sin i/sin r = v₁/v₂ = n₂/n₁, which is Snell's law n₁ sin i = n₂ sin r.
5Bending towards the normal means slower light in the second medium, as Foucault found in 1850.
6On refraction speed and wavelength change in the same ratio; frequency never changes.
7The energy a wave carries depends on its amplitude, not its speed.
8Critical angle: sin ic = n₂/n₁; beyond it, total internal reflection.
9Coherent sources keep a constant phase difference and give a steady pattern.
10I = 4I₀cos²(φ/2): 4I₀ at maxima, zero at minima; incoherent sources give 2I₀ everywhere.
11Path difference nλ: bright; (n + ½)λ: dark. A path difference of λ is a phase difference of 2π.
12Independent sources jump in phase about every 10⁻¹⁰ s, so they cannot show interference.
13Young locked two slits in phase by feeding both from one pinhole.
14Young's fringes: bright at x = nDλ/d, dark at (n + ½)Dλ/d, equally spaced by β = Dλ/d.
15Single slit of width a: central maximum at θ = 0, minima at θ ≈ nλ/a, weaker maxima near (n + ½)λ/a.
16Interference and diffraction redistribute energy; they neither create nor destroy it.
17Light is transverse; natural light is unpolarised.
18One polaroid passes linearly polarised light at half the unpolarised intensity, whatever its orientation.
19Malus' law: I = I₀cos²θ; crossed polaroids pass nothing.
20Polaroid between crossed polaroids: I = (I₀/4) sin²2θ, largest at θ = π/4.

Common traps

Where marks are lost

Saying the frequency of light changes when it enters glass or water.

Frequency is set by the source and never changes; speed and wavelength both fall by the factor n.

Believing light that slows down in glass has lost energy.

The energy carried depends on amplitude, not on speed.

Crediting the wave model with the prediction that light is faster in a denser medium.

That was the corpuscular model; the wave model predicts slower light, and Foucault's 1850 result agreed.

Expecting fringes from two separate bulbs shining through two slits.

Independent sources are incoherent; their intensities add to 2I₀ with no pattern. Young fed both slits from one source.

Writing the maximum intensity from two equal coherent sources as 2I₀.

At a bright fringe the amplitudes add, 2a, so I = 4I₀. The average over the pattern is 2I₀.

Using nλ as the condition for a bright band in single-slit diffraction.

For a single slit, a sin θ ≈ nλ gives the dark bands; secondary maxima are near (n + ½)λ/a.

Applying Malus' law straight to unpolarised light, so the first polaroid passes I cos²θ.

The first polaroid halves unpolarised light whatever its angle; cos²θ applies from the second polaroid on.

Using cos θ instead of cos²θ in Malus' law.

The field component is E cos θ; intensity goes as field squared, so cos²θ.

Thinking sound can be polarised because it shows interference and diffraction.

Polarisation needs a transverse wave; sound in air is longitudinal.

Formulas

8 to know

Refraction from wavefronts

sin i/sin r = v₁/v₂ = n₂/n₁

n = c/v; Snell's law n₁ sin i = n₂ sin r.

Wavelength in a medium

λ₁/λ₂ = v₁/v₂

Frequency is unchanged.

Critical angle

sin ic = n₂/n₁

Light going from the denser medium 1 to the rarer medium 2.

Two-source intensity

I = 4I₀ cos²(φ/2)

Phase difference φ = (2π/λ) × path difference; incoherent sources give 2I₀.

Young's fringes

x(bright) = nDλ/d, x(dark) = (n + ½)Dλ/d

Fringe spacing β = Dλ/d.

Single-slit minima

θ ≈ nλ/a

n = ±1, ±2, ...; secondary maxima near (n + ½)λ/a.

Malus' law

I = I₀ cos²θ

I₀ is the polarised intensity after the first polaroid, half the unpolarised intensity.

Polaroid between crossed polaroids

I = (I₀/4) sin²2θ

Maximum I₀/4 at θ = π/4.

Key terms

16 terms

Wavefront: A surface on which every point vibrates in the same phase.
Plane wavefront: A flat wavefront, as from a very distant source or a point source at a lens's focus.
Secondary wavelets: Small waves imagined to spread from every point of a wavefront in Huygens' construction.
Huygens principle: The new wavefront is the forward envelope of secondary wavelets from the old one.
Refractive index: n = c/v, the ratio of the speed of light in vacuum to that in the medium.
Critical angle: Angle of incidence in the denser medium for which the refracted ray grazes the boundary.
Coherent sources: Sources whose phase difference stays constant in time.
Incoherent sources: Sources whose phase difference changes rapidly and randomly; their intensities simply add.
Fringes: The alternate bright and dark bands of an interference or diffraction pattern.
Fringe spacing: Distance between neighbouring bright (or dark) fringes, Dλ/d in Young's experiment.
Diffraction: Spreading of a wave around edges and through openings into the geometrical shadow.
Unpolarised light: Light whose electric vector points in random, rapidly changing directions across the beam.
Linearly polarised light: Light whose electric vector oscillates along one fixed line.
Polaroid: A sheet of aligned long-chain molecules that passes only the field component along its pass-axis.
Pass-axis: The direction, perpendicular to the aligned molecules, along which a polaroid transmits the electric field.
Malus' law: Polarised light through a polaroid at angle θ has intensity I₀cos²θ.

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