---
title: "Light as a Wave: Diffraction and Interference"
book: "High School Physics"
subject: physics
language: en
chapter: 22
exercises: 15
source: https://one-course.com/books/physics/2/en/chapter/22-light-as-a-wave-diffraction-and-interference
---

# Chapter 22 — Light as a Wave: Diffraction and Interference

Tilt a soap film toward a window: colors slide across it, though nothing in soap or water is colored. Point a [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) at a single hair: the wall behind shows not a [hair-thin](https://one-course.com/books/physics/2/en/chapter/10-lenses-images-and-the-eye#def-g11-lenses-and-eye-lens) shadow but a bright streak, barred with dark lines, thousands of times wider than the hair. Straight-marching rays ([Chapter 3](https://one-course.com/books/physics/2/en/chapter/3-refraction-of-light#ch-g10-refraction)) explain neither sight. This chapter promotes light to a full [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) — half a micrometre from crest to crest — then cashes the idea in: a [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) and a bare wall will measure a hair, a [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength), a CD’s grooves.

## 22.1 Light is a wave

**Definition 22.1 (Light waves).**

[Monochromatic light](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-white) — one pure color, which no prism can split further ([Chapter 2](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#ch-g10-light-spectra)) — is a [periodic](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period) *light wave*: it crosses vacuum at $c = 3.00 \times 10^{8}\,\mathrm{m}/\mathrm{s}$ and is characterized by its [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) $f$ or its *wavelength in vacuum* $\lambda = c/f$, the crest-to-crest distance of [Chapter 20](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#ch-g12-mechanical-waves). The eye responds from about $400\,\mathrm{nm}$ (violet) to $800\,\mathrm{nm}$ (red); [white light](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-white) mixes the whole [range](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-strong).

**Example 22.2 (Orders of magnitude).**

A helium–neon [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) emits red light of $\lambda = 633\,\mathrm{nm} =
6.33 \times 10^{-7}\,\mathrm{m}$, so $f = c/\lambda \approx 4.7 \times 10^{14}\,\mathrm{Hz}$: no electronics counts that fast — every [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) in this chapter will be measured by geometry. For scale, a hair (about $70\,\text{µ}\mathrm{m}$) is a hundred [wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) across. What [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave)? Not the air: an electric and [magnetic field](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-bfield), honestly described in the Year 2 volume; only the [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) matters below.

## 22.2 Diffraction

**Definition 22.3 (Diffraction).**

When a [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) meets an aperture or an obstacle of size $a$ comparable to its [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength), it spreads into the region that rays would leave dark. This spreading is *diffraction* — negligible while $\lambda/a$ is tiny, dominant as $a$ shrinks toward $\lambda$.

![Plane wavefronts (one per crest, apart) meet a slit of width a comparable to : the wave fans out — diffraction.](https://one-course.com/images/onecourse/chapters/physics-2/g12-light-as-wave/fig-a9d8249a91f0.svg)

*Plane [wavefronts](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavefront) (one per crest, $\lambda$ apart) meet a slit of width $a$ comparable to $\lambda$: the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) fans out — [diffraction](#def-g12-light-as-wave-diffraction).*

**Proposition 22.4 (Angular half-width of the spread).**

A slit of width $a$, lit with [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) $\lambda < a$, spreads the light about the incident direction within the angular half-width $\theta$ (radians); a screen at distance $D$ shows a central bright band of width $L$, twice $D \tan\theta \approx D\,\theta$ (small angle):

$$
\theta \approx \frac{\lambda}{a}, \qquad L = \frac{2\lambda D}{a}.
$$

**Proof.** *Admitted at this level.* ∎

**Remark 22.5 (Read the formula backwards).**

The *narrower* the slit, the *wider* the spread — the opposite of the ray prediction. The honest derivation (summing wavelets re-emitted across the slit) lives in the Year 2 volume; here the formula is a measuring tool.

**Example 22.6 (A laser through a slit).**

Red [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps), $\lambda = 633\,\mathrm{nm}$; slit $a = 0.10\,\mathrm{mm}$; screen at $D = 2.00\,\mathrm{m}$. Then $\theta = 6.33 \times 10^{-7} /
1.0 \times 10^{-4} = 6.3 \times 10^{-3}\,\mathrm{rad}$ — a third of a degree, but the central band on the wall is $L = 2 \times 6.33 \times 10^{-7} \times 2.00 /
1.0 \times 10^{-4} \approx 2.5\,\mathrm{cm}$ wide, flanked by dark lines and fainter side bands. Halve the slit and the band doubles.

![The single-slit experiment: a cone of half-angle /a paints a central band L = 2 D/a wide.](https://one-course.com/images/onecourse/chapters/physics-2/g12-light-as-wave/fig-e356d2ed5362.svg)

*The single-slit experiment: a [cone](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-cones) of half-angle $\theta
\approx \lambda/a$ paints a central band $L = 2\lambda D/a$ wide.*

**Proposition 22.7 (An obstacle diffracts like a slit).**

A [thin](https://one-course.com/books/physics/2/en/chapter/10-lenses-images-and-the-eye#def-g11-lenses-and-eye-lens) opaque obstacle of width $d$ — a wire, a hair — produces, outside the incident beam, the same pattern as a slit of width $d$ (Babinet’s principle, derived in the Year 2 volume).

**Proof.** *Admitted at this level.* ∎

**Method 22.8 (Measuring a tiny width).**

1. Stretch the hair across the [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) beam; set a screen at a measured distance $D$ (a few [metres](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) ).
2. Measure the central band’s width $L$ , dark line to dark line.
3. Since $L = 2\lambda D/d$ : $d = 2\lambda D/L$ — finer hair, larger pattern.

**Remark 22.9 (Diffraction limits every instrument).**

Light enters a camera, a telescope or an eye through an aperture of width $a$, so even a perfect [lens](https://one-course.com/books/physics/2/en/chapter/10-lenses-images-and-the-eye#def-g11-lenses-and-eye-lens) blurs a point into a spot of angular size about $\lambda/a$: closer details merge. A visible-light microscope thus resolves nothing much below $\lambda \approx
0.5\,\text{µ}\mathrm{m}$ (bacteria yes, viruses no); telescope mirrors are built wide for the same reason.

## 22.3 Interference

**Definition 22.10 (Coherent sources, path difference).**

Two [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) sources are *coherent* when they have the same [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) and vibrate in step — in practice, when they are two copies of one [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave), obtained by splitting it. For a point $M$ reached by both, the *path difference* is $\delta = S_2M - S_1M$.

**Proposition 22.11 (Interference conditions).**

Where two [coherent](#def-g12-light-as-wave-coherent) [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) of [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) $\lambda$ overlap they add — they *interfere*:

- $\delta = k\lambda$ ( $k$ integer): in step, reinforcing — *constructive* interference (bright);
- $\delta = (k + \tfrac12)\lambda$ : crest on trough, canceling — *destructive* interference (dark).

**Proof.** Shifting a [periodic wave](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-periodic) by whole [wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) changes nothing: crests pile on crests. Half a [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) more puts crest on trough: equal [waves](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) cancel. ∎

**Example 22.12 (Young’s two-slit experiment).**

Pierce two fine slits $S_1$, $S_2$, a fraction of a millimetre apart, in an opaque plate, and light both with one [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps): two [coherent sources](#def-g12-light-as-wave-coherent). On a screen a couple of [metres](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) away the overlap is striped: at the center $\delta = 0$, bright; climbing the screen, $\delta$ grows through $\lambda/2$ (dark), $\lambda$ (bright) … darkness where light added to light cancels.

![Young’s slits: the paths to M differ by = S_2M - S_1M; bright fringes (gray ticks) sit where = k.](https://one-course.com/images/onecourse/chapters/physics-2/g12-light-as-wave/fig-d0802d70671e.svg)

*Young’s slits: the paths to $M$ differ by $\delta = S_2M -
S_1M$; bright fringes (gray ticks) sit where $\delta = k\lambda$.*

**Definition 22.13 (Fringes and interfringe).**

The pattern’s alternating bright and dark bands are its *interference fringes*; the distance $i$ between consecutive bright centers is the *interfringe*.

**Proposition 22.14 (The interfringe).**

For slits a distance $a$ apart and a screen at distance $D \gg a$, the point of the screen at distance $x$ from the center $O$ has [path difference](#def-g12-light-as-wave-coherent) $\delta \approx a\,x/D$ (admitted geometry); by [Proposition 22.11](#prop-g12-light-as-wave-conditions), bright fringes sit at $x_k = k\,\lambda D/a$, evenly spaced with [interfringe](#def-g12-light-as-wave-fringes)

$$
i = \frac{\lambda D}{a}.
$$

With $\lambda = 633\,\mathrm{nm}$, $a = 0.20\,\mathrm{mm}$, $D =
2.0\,\mathrm{m}$: $i = 6.3\,\mathrm{mm}$ — the setup magnifies $\lambda$ by $D/a = 10^4$. The geometry behind it is worked out in the Year 2 volume.

**Proof.** *Admitted at this level.* ∎

![Intensity across the screen in Young’s experiment: fringes i = D/a apart, under a schematic overall envelope.](https://one-course.com/images/onecourse/chapters/physics-2/g12-light-as-wave/fig-c9f9fba97e33.svg)

*Intensity across the screen in Young’s experiment: fringes $i = \lambda D/a$ apart, under a schematic overall envelope.*

**Method 22.15 (Measuring a wavelength).**

1. Send the unknown [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) through a double slit of known spacing $a$ ; place the screen at a measured distance $D$ .
2. Measure the span of $10$ [interfringes](#def-g12-light-as-wave-fringes) ; divide by $10$ : the [interfringe](#def-g12-light-as-wave-fringes) $i$ , with the ruler error shrunk tenfold.
3. Then $\lambda = i\,a/D$ .

**Remark 22.16 (Why two lamps never interfere).**

Two desk lamps lighting one wall produce no fringes: each emits short, uncorrelated [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) trains whose relative timing jumps randomly a billion times a second, so the fringes shift just as fast and the eye averages them into uniform light. Interference demands [coherence](#def-g12-light-as-wave-coherent) — one [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) split in two, as Young’s slits do; the [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) is the modern shortcut, one clean long [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave).

## 22.4 Colors painted by interference

**Definition 22.17 (Iridescence).**

Colors that shift with the viewing angle — soap bubbles, oil slicks, peacock feathers, the back of a CD — are *iridescence*: no pigment, but interference in [white light](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-white), reinforcing some [wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) and canceling others.

**Example 22.18 (Soap films and oil slicks).**

Light striking a soap film reflects partly off its front face, partly off its back: two [coherent](#def-g12-light-as-wave-coherent) copies whose [path difference](#def-g12-light-as-wave-coherent) is set by the thickness and the viewing angle. Each thickness returns some [wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) reinforced, others canceled: the film reflects a color that slides as it drains and [thins](https://one-course.com/books/physics/2/en/chapter/10-lenses-images-and-the-eye#def-g11-lenses-and-eye-lens). An oil slick on wet asphalt plays the same trick.

**Remark 22.19 (Monochromatic versus white light).**

In [white light](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-white) every [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) draws its own Young fringes with its own $i = \lambda D/a$: at the center all colors are bright (one white fringe); a few fringes out the patterns drift apart — iridescent edges, then blur. [Monochromatic light](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-white) gives crisp fringes without end: hence the [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) in every sharp measurement here.

## 22.5 Exercises

**Exercise 22.1 ★.**

Compute the [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) of red light ($\lambda = 700\,\mathrm{nm}$) and of violet light ($400\,\mathrm{nm}$). Which is higher?

**Solution of Exercise 22.1.**

$f = c/\lambda$: red $3.00 \times 10^{8}/7.0 \times 10^{-7} \approx
4.3 \times 10^{14}\,\mathrm{Hz}$; violet $\approx 7.5 \times 10^{14}\,\mathrm{Hz}$ — violet is higher (shorter [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength), same speed $c$).

**Exercise 22.2 ★.**

A green [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) ($\lambda = 532\,\mathrm{nm}$) crosses a slit of width $a = 0.20\,\mathrm{mm}$: compute $\theta$, then the central-band width on a screen $1.5\,\mathrm{m}$ away.

**Solution of Exercise 22.2.**

$\theta = 5.32 \times 10^{-7}/2.0 \times 10^{-4} = 2.7 \times 10^{-3}\,\mathrm{rad}$; $L = 2\theta D = 2 \times 2.66 \times 10^{-3} \times 1.5 \approx
8.0\,\mathrm{mm}$.

**Exercise 22.3 ★.**

A doorway is $0.80\,\mathrm{m}$ wide: compute $\lambda/a$ for a $425\,\mathrm{Hz}$ voice ($v = 340\,\mathrm{m}/\mathrm{s}$) and for $550\,\mathrm{nm}$ light. Why do you hear, but not see, around the corner?

**Solution of Exercise 22.3.**

Voice: $\lambda = 340/425 = 0.80\,\mathrm{m}$, so $\lambda/a = 1.0$ — maximal [diffraction](#def-g12-light-as-wave-diffraction), [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) floods the corridor. Light: $\lambda/a \approx 5.5 \times 10^{-7}/0.80 \approx 7 \times 10^{-7}$ — negligible spread: light keeps straight, so the corner hides it.

**Exercise 22.4 ★.**

Two loudspeakers driven by one generator emit, in step, [sound](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-sound) of [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) $2.0\,\mathrm{cm}$. [Path difference](#def-g12-light-as-wave-coherent) at $M$: $6.0\,\mathrm{cm}$; at $N$: $5.0\,\mathrm{cm}$. Loud or silent at each? Justify.

**Solution of Exercise 22.4.**

At $M$: $\delta = 6.0\,\mathrm{cm} = 3\lambda$, [constructive](#prop-g12-light-as-wave-conditions) — loud. At $N$: $\delta = 5.0\,\mathrm{cm} = (2 + \tfrac12)\lambda$, [destructive](#prop-g12-light-as-wave-conditions) — (nearly) silent.

**Exercise 22.5 ★.**

Young’s slits: $\lambda = 633\,\mathrm{nm}$, $a = 0.25\,\mathrm{mm}$, $D = 1.8\,\mathrm{m}$. Compute the [interfringe](#def-g12-light-as-wave-fringes). How many bright fringes fit in the screen’s central $2.0\,\mathrm{cm}$?

**Solution of Exercise 22.5.**

$i = \lambda D/a = 6.33 \times 10^{-7} \times 1.8/2.5 \times 10^{-4} \approx
4.6\,\mathrm{mm}$. Within $\pm1.0\,\mathrm{cm}$: $|k| \leq 10/4.6 = 2.2$, so $k = -2, \dots, 2$: five bright fringes.

**Exercise 22.6 ★★.**

A hair in a $\lambda = 650\,\mathrm{nm}$ beam, screen at $1.6\,\mathrm{m}$, gives a central band $2.8\,\mathrm{cm}$ wide. Compute the hair’s diameter.

**Solution of Exercise 22.6.**

$d = 2\lambda D/L = 2 \times 6.5 \times 10^{-7} \times 1.6/0.028 \approx
74\,\text{µ}\mathrm{m}$.

**Exercise 22.7 ★★.**

An unknown [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) through Young’s slits ($a = 0.20\,\mathrm{mm}$, $D = 1.50\,\mathrm{m}$): ten [interfringes](#def-g12-light-as-wave-fringes) span $4.7\,\mathrm{cm}$. Find $\lambda$ and the color. Why measure ten rather than one?

**Solution of Exercise 22.7.**

$i = 4.7\,\mathrm{mm}$, so $\lambda = i\,a/D = 4.7 \times 10^{-3} \times
2.0 \times 10^{-4}/1.5 \approx 6.3 \times 10^{-7}\,\mathrm{m} = 630\,\mathrm{nm}$: red. Measuring ten [interfringes](#def-g12-light-as-wave-fringes) divides the ruler’s reading error by ten.

**Exercise 22.8 ★★.**

Two identical desk lamps light the same wall; the beams overlap, yet no fringes appear, ever. Which [coherence](#def-g12-light-as-wave-coherent) condition ([Definition 22.10](#def-g12-light-as-wave-coherent)) fails, and how does Young’s arrangement repair it?

**Solution of Exercise 22.8.**

The lamps have (broadly) matching frequencies but are *not in step*: each emits short [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) trains with randomly jumping phase, so the sources are incoherent and the instantaneous fringes shift billions of times per second — the eye sees the average, uniform. Young lights both slits from *one* [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave): the two copies inherit every phase jump together and stay in step.

**Exercise 22.9 ★★.**

On a two-slit pattern ($a = 0.30\,\mathrm{mm}$, $D = 1.4\,\mathrm{m}$), the first and seventh bright-fringe centers are $18.0\,\mathrm{mm}$ apart: find the [interfringe](#def-g12-light-as-wave-fringes), then $\lambda$.

**Solution of Exercise 22.9.**

First to seventh $= 6i = 18.0\,\mathrm{mm}$, so $i = 3.0\,\mathrm{mm}$; $\lambda = i\,a/D = 3.0 \times 10^{-3} \times 3.0 \times 10^{-4}/1.4 \approx
6.4 \times 10^{-7}\,\mathrm{m} \approx 640\,\mathrm{nm}$.

**Exercise 22.10 ★★.**

A CD’s tracks form a grating of spacing $d = 1.6\,\text{µ}\mathrm{m}$. Admitting $d \sin\theta = k\lambda$ ($k$ integer), find the beam directions for $\lambda = 633\,\mathrm{nm}$. Highest order $k$?

**Solution of Exercise 22.10.**

$\sin\theta = k\lambda/d = 0.396\,k$: $k = 1$, $\theta = 23^\circ$; $k = 2$, $\sin\theta = 0.79$, $\theta = 52^\circ$; $k = 3$ would need $\sin\theta = 1.19 > 1$: impossible. Highest order $k = 2$.

**Exercise 22.11 ★★.**

A telescope of aperture $a = 10\,\mathrm{cm}$ at $\lambda =
550\,\mathrm{nm}$: compute its [diffraction](#def-g12-light-as-wave-diffraction) blur $\theta \approx
\lambda/a$. Can it separate two headlights $1.5\,\mathrm{m}$ apart, $100\,\mathrm{km}$ away?

**Solution of Exercise 22.11.**

$\theta = 5.5 \times 10^{-7}/0.10 = 5.5 \times 10^{-6}\,\mathrm{rad}$. The headlights subtend $1.5/1.0 \times 10^{5} = 1.5 \times 10^{-5}\,\mathrm{rad}$, about three times the blur: separated, just.

**Exercise 22.12 ★★★.**

A vertical soap film drains, thicker at the bottom, and shows horizontal colored bands creeping downward. Explain: why colors, why bands, why moving. (No computation.)

**Solution of Exercise 22.12.**

Front- and back-face reflections [interfere](#prop-g12-light-as-wave-conditions); in [white light](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-white) each thickness reinforces some [wavelengths](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) and cancels others, hence a color. The film’s thickness depends only on height, so equal-color points form horizontal bands. Draining [thins](https://one-course.com/books/physics/2/en/chapter/10-lenses-images-and-the-eye#def-g11-lenses-and-eye-lens) the film, so the thickness that painted a given color sits ever lower: the bands creep down.

**Exercise 22.13 ★★★.**

Young’s slits in [white light](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-white), $a = 0.20\,\mathrm{mm}$, $D = 1.5\,\mathrm{m}$: compute the [interfringe](#def-g12-light-as-wave-fringes) for violet ($400\,\mathrm{nm}$) and red ($750\,\mathrm{nm}$), then deduce what the screen shows at the center, a few millimetres out, and far out.

**Solution of Exercise 22.13.**

$i = \lambda D/a$: violet $4.0 \times 10^{-7} \times 1.5/2.0 \times 10^{-4} =
3.0\,\mathrm{mm}$; red $5.6\,\mathrm{mm}$. Center: all colors bright — one white fringe. A few millimetres out the color patterns have slid apart: iridescent fringes. Farther, maxima of all colors overlap everywhere: uniform white blur.

**Exercise 22.14 ★★★.**

A surveillance [satellite](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) carries a $2.4\,\mathrm{m}$ mirror at $250\,\mathrm{km}$. Using $\theta \approx \lambda/a$ at $550\,\mathrm{nm}$, find the smallest ground detail it resolves; judge the claim that it “reads newspapers over your shoulder”.

**Solution of Exercise 22.14.**

$\theta = 5.5 \times 10^{-7}/2.4 = 2.3 \times 10^{-7}\,\mathrm{rad}$; at $250\,\mathrm{km}$ that is $2.3 \times 10^{-7} \times 2.5 \times 10^{5} \approx 6\,\mathrm{cm}$. It can count cars and spot a person, but newsprint (millimetre letters) is twenty times below the [diffraction](#def-g12-light-as-wave-diffraction) limit: the claim is myth.

**Exercise 22.15 ★★★.**

A Young’s setup shows $i = 4.0\,\mathrm{mm}$ in air, then is immersed in water ($n = 1.33$). Recalling [Chapter 3](https://one-course.com/books/physics/2/en/chapter/3-refraction-of-light#ch-g10-refraction): does the [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) change? The [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength)? Compute the new [interfringe](#def-g12-light-as-wave-fringes).

**Solution of Exercise 22.15.**

The [frequency](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-frequency) is fixed by the source: unchanged. The speed drops to $c/n$, so $\lambda' = \lambda/n$ and $i' = i/n = 4.0/1.33 \approx 3.0\,\mathrm{mm}$: the fringes tighten.

## 22.6 Problem: Measuring with Light

**Problem 22.1.**

Weekend problem — measuring with light: a laser, a ruler and a bare wall become a micrometre workshop gauging a slit, a human hair, an unknown wavelength and the grooves of two silver discs

Your kit: a red [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) ($\lambda = 633\,\mathrm{nm}$, from the label), a green [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) of unknown [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength), a calibrated slit $a = 100\,\text{µ}\mathrm{m}$, a double slit of spacing $a' =
0.250\,\mathrm{mm}$, a tape measure, two hairs, a CD and a DVD.

**Part I — Calibrating on the known slit.** Red [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps), slit, wall at $D = 3.00\,\mathrm{m}$.

1. Compute the angular half-width $\theta$ of the diffracted beam.
2. Show that the central band has width $L = 2\lambda D/a$ ; compute it.
3. You measure $L = 3.7\,\mathrm{cm}$ : compute the relative deviation. Is the method validated?
4. A second slit is half as wide, $50\,\text{µ}\mathrm{m}$ : predict its $L$ . State the rule linking slit width and pattern width.
5. Justify question 2’s step $\tan\theta \approx \theta$ : compare $\theta$ and $\tan\theta$ at this angle.

**Part II — The hair.** Same [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) and wall; a stretched hair replaces the slit.

6. Which admitted result lets you keep the slit formula for a hair, and what does it say?
7. The central band is $L = 5.4\,\mathrm{cm}$ wide: compute the hair’s diameter $d$ .
8. Your ruler reads $L$ to $\pm0.2\,\mathrm{cm}$ : give the resulting [range](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-strong) for $d$ , as $d \pm \Delta d$ .
9. Your friend’s hair gives $L = 7.6\,\mathrm{cm}$ : its diameter? Whose hair is finer, and why does finer mean wider?
10. Could a desk lamp replace the [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) ? Give two reasons.

**Part III — The green [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps)’s [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength).** Green [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps), double slit, wall at $D = 2.00\,\mathrm{m}$.

11. Why must both slits be lit by the *same* [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) ?
12. Why is the central fringe bright whatever the [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) ?
13. Ten [interfringes](#def-g12-light-as-wave-fringes) span $4.26\,\mathrm{cm}$ . Give $i$ , and the reason for measuring ten at once.
14. Deduce the [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) of the green [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) .
15. The maker’s label says $(532 \pm 10)\,\mathrm{nm}$ : consistent? What color do you expect?

**Part IV — Reading silver discs.** The red [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) hits a CD at normal incidence; its tracks act as a reflection grating obeying (admitted) $d \sin\theta = k\lambda$. On the wall at $D =
1.00\,\mathrm{m}$, each first-order beam makes a spot at $x$ from the central one.

16. For the CD, $x = 43\,\mathrm{cm}$ : compute $\theta$ . Why is the small-angle shortcut of Part I now forbidden?
17. Deduce the CD’s track spacing $d$ .
18. The DVD sends its first-order spot to $x = 1.65\,\mathrm{m}$ : compute its $\theta$ and its track spacing.
19. Compare the two spacings. Why does a DVD store more than a CD?
20. Close the notebook: list the lengths measured this weekend, and state in one sentence what made light a ruler for all.

**Solution of Problem 22.1.**

**1.** $\theta = \lambda/a = 6.33 \times 10^{-7}/1.00 \times 10^{-4} =
6.3 \times 10^{-3}\,\mathrm{rad}$.

**2.** Half-width $D\tan\theta \approx D\theta = \lambda D/a$, so $L = 2\lambda D/a = 2 \times 6.33 \times 10^{-7} \times
3.00/1.00 \times 10^{-4} = 3.8\,\mathrm{cm}$.

**3.** $(3.8 - 3.7)/3.8 \approx 3\%$: within a ruler’s precision — validated.

**4.** $L \propto 1/a$: halving $a$ doubles $L$, so $L = 7.6\,\mathrm{cm}$. Narrower obstacle-or-slit, wider pattern.

**5.** $\tan(6.33 \times 10^{-3}) = 6.330\,08 \times 10^{-3}$: they agree to about $1 \times 10^{-5}$ relative — the approximation is far better than any measurement here.

**6.** By [Proposition 22.7](#prop-g12-light-as-wave-hair): an obstacle of width $d$ diffracts like a slit of width $d$, so $L = 2\lambda D/d$ still holds.

**7.** $d = 2\lambda D/L = 2 \times 6.33 \times 10^{-7} \times
3.00/0.054 \approx 70\,\text{µ}\mathrm{m}$.

**8.** $L = 5.2\,\mathrm{cm} \to d = 73\,\text{µ}\mathrm{m}$; $L = 5.6\,\mathrm{cm} \to d = 68\,\text{µ}\mathrm{m}$: so $d = (70 \pm 3)\,\text{µ}\mathrm{m}$.

**9.** $d = 3.80 \times 10^{-6}/0.076 = 50\,\text{µ}\mathrm{m}$: the friend’s is finer — and $L \propto 1/d$, so the finer hair throws the wider pattern.

**10.** No: a lamp is white (each [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) paints a different pattern, and they blur) and incoherent-and-wide (no single clean beam to diffract). The [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) is [monochromatic](https://one-course.com/books/physics/2/en/chapter/2-light-spectra-and-the-message-of-light#def-g10-light-spectra-white) and directional.

**11.** Only copies of one [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) are [coherent](#def-g12-light-as-wave-coherent); two independent sources drift out of step and the fringes wash out ([Remark 22.16](#rem-g12-light-as-wave-lamps)).

**12.** At the center $\delta = 0 = 0 \times \lambda$ for every $\lambda$: [constructive](#prop-g12-light-as-wave-conditions) regardless of [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength).

**13.** $i = 4.26/10 = 4.3\,\mathrm{mm}$; one span of ten is read with the same ruler error as one [interfringe](#def-g12-light-as-wave-fringes), so the error on $i$ is divided by ten.

**14.** $\lambda = i\,a'/D = 4.26 \times 10^{-3} \times
2.50 \times 10^{-4}/2.00 = 5.33 \times 10^{-7} \approx 533\,\mathrm{nm}$.

**15.** $533\,\mathrm{nm}$ lies inside $(532 \pm 10)\,\mathrm{nm}$: consistent; that [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) is green.

**16.** $\tan\theta = 0.43/1.00$, so $\theta = 23^\circ$. Here $\tan\theta = 0.43$ while $\sin\theta = 0.40$: at $23^\circ$ the small-angle identification of $\sin$, $\tan$ and $\theta$ is off by almost $10\%$ — use the exact functions.

**17.** $d = \lambda/\sin\theta = 6.33 \times 10^{-7}/0.396 \approx
1.6\,\text{µ}\mathrm{m}$.

**18.** $\tan\theta = 1.65$, $\theta = 59^\circ$, $\sin\theta = 0.855$: $d = 6.33 \times 10^{-7}/0.855 \approx
0.74\,\text{µ}\mathrm{m}$.

**19.** $1.6/0.74 \approx 2.2$: DVD tracks are twice as dense, and its pits are correspondingly smaller (read with a [shorter-wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps)) — several times the data in the same $12\,\mathrm{cm}$ disc.

**20.** Hair $(70 \pm 3)\,\text{µ}\mathrm{m}$; friend’s hair $50\,\text{µ}\mathrm{m}$; green [laser](https://one-course.com/books/physics/2/en/chapter/11-color-and-light-sources#def-g11-color-light-sources-lamps) $533\,\mathrm{nm}$; CD tracks $1.6\,\text{µ}\mathrm{m}$; DVD tracks $0.74\,\text{µ}\mathrm{m}$ — a known [wavelength](https://one-course.com/books/physics/2/en/chapter/20-mechanical-waves#def-g12-mechanical-waves-wavelength) turns every pattern’s geometry into a length, so light of half a micrometre is a ruler graduated at half a micrometre.