Primary & Middle School Physics · Grades 1–9

71From Atoms to Galaxies: Scales of the Universe

Nine years ago this book began with your five senses and a ladybug’s legs. It closes with a journey no senses can make alone: forty-one powers of ten, from the heart of an atom to the rim of the observable universe — with you, rather exactly, in the middle. One last chapter, one last secret owed from an old joke, and a ladder to stand on for all the physics to come.

71.1 Down: into matter

Definition 71.1 (Atoms)

The molecules of your seventh year are themselves built: every molecule is an assembly of atoms — nature’s alphabet, about a hundred kinds, whose combinations spell every substance. A water molecule: two hydrogen atoms and one oxygen. An atom, in turn, has architecture: a minuscule, heavy, positively charged nucleus at the center, and, far outside it, a cloud of electrons — almost weightless carriers of negative charge. Atom sizes run near $10^{-10}$ $\mathrm{m}$ ; their nuclei, near $10^{-15}$ — a hundred thousand times smaller again.

Proposition 71.2 (Matter is mostly emptiness)

Scale an atom up to a great stadium, and its nucleus is a pea at the center circle — the electrons, gnats in the highest stands: everything between is empty. The marble table, the iron anvil, your own hand: overwhelmingly void, their solidity an electric refusal — the atoms’ electron clouds repelling each other into the illusion of fullness. It is among physics’ most astonishing sentences, and it is the plain arithmetic of $10^{-10}$ against $10^{-15}$ .

Example 71.3 (The old joke, paid)

The electricity chapters owed you a small joke, and here it is. The marching charges in every wire of this book are electrons — and they are negative, so they march from the battery’s $-$ terminal to its $+$ : precisely opposite to the conventional direction fixed by physicists a century before anyone could ask the marchers. The convention was kept — every law of your circuits works perfectly with it — but the little irony stands: in every diagram you have drawn, the real crowd walks the other way.

Example 71.4 (And further down)

The nucleus has parts of its own, and their story — how they bind, why some nuclei split (the power station’s heat) or merge (the Sun’s) — runs physics down to $10^{-15}$ metres and beyond, into the smallest structures humans have probed. The High School volume opens the nucleus properly; the university years go further still. The ladder’s lower rungs are still being built.

71.2 Up: into the sky

Example 71.5 (The ascent)

Climb now, power of ten by power of ten, past the familiar landmarks of nine years. You, about $10^{0}$ $\mathrm{m}$ . The schoolyard, $10^{2}$ . The Earth, $10^{7}$ across — the marble of the sports-field model. The Moon’s distance, $4 \times 10^{8}$ ; the Sun’s, $1.5 \times 10^{11}$ — eight light-minutes. The Solar System’s breadth, roughly $10^{13}$ ; the nearest star, $4 \times 10^{16}$ — four light-years. The Milky Way, $10^{21}$ metres of city; the neighbor galaxy, $2 \times 10^{22}$ away; and the deepest surveyed sky, near $10^{26}$ metres — the light of its galaxies older than the Earth beneath your chair.

The ladder of scales, in metres: forty-one powers of ten from nucleus to deep sky — with the reader standing almost exactly at the middle rung. (The ant forgives our rounding.)

Method 71.6 (Thinking in orders of magnitude)

The physicist’s first tool on any new question:

express the quantity in scientific notation;
keep only the power of ten — the order of magnitude;
compare ladders, not digits: a star at $10^{16}$ metres against a planet at $10^{11}$ differs by five rungs — a hundred-thousandfold, whatever the leading digits say;
only then, if needed, sharpen the digits.

Nine years of judging-steps — “is this answer sensible?” — mature into this habit. The High School volume’s very first chapter builds its measurement culture upon it.

Example 71.7 (Ladder arithmetic)

How many atoms span a pencil’s width? Pencil, $10^{-2}$ $\mathrm{m}$ ; atom, $10^{-10}$ : eight rungs — a hundred million atoms, shoulder to shoulder. How many Earths to the Sun? $1.5 \times 10^{11}$ over $1.3 \times 10^{7}$ : about $10^{4}$ — ten thousand marbles of the sports-field model, laid in a row. The ladder answers in seconds what raw kilometre-counts bury in zeros.

71.3 The view from the middle rung

Remark 71.8 (What nine years built)

Look back down the ladder and count what you own. At $10^{-10}$ : molecules and atoms — with them, the three states, heat, sound’s relay, the current’s marchers. Around $10^{0}$ : forces and energy, light’s straight rays, the laws of circuits — the human-sized physics of nine years of experiments. From $10^{7}$ to $10^{22}$ : the spinning Earth, the tilted seasons, the falling Moon, one law of gravitation running orchard and galaxy alike. No other schooling subject hands you a single ladder from the inside of matter to the edge of the visible — and every rung was climbed with instruments, honesty, and arithmetic you can check yourself.

Remark 71.9 (What waits above)

And the unfinished business is the best of it. What law exactly binds force to motion? What is light, that it should race at $3 \times 10^{8}$ metres a second and carry colors inside it? What splits inside the nucleus — and what forges sunlight? Why does the tally of useful energy always run downhill? Each question has been honestly raised and honestly deferred in these pages; each has an answer, and the answers are among the finest things our species knows. They begin in the High School volume, one year from now. Bring the ladder — and the habit of checking everything.

71.4 Exercises

Exercise 71.1 ★

Order by size: molecule, atom, nucleus, ant. Give each its power of ten in metres.

Solution

Solution of Exercise 71.1.

Nucleus ( $10^{-15}$ $\mathrm{m}$ ), atom ( $10^{-10}$ ), molecule (around $10^{-9}$ ), ant ( $10^{-2}$ , or $10^{-3}$ for a modest one).

Exercise 71.2 ★

Describe the atom’s architecture — what sits at the center, what surrounds it, and the two sizes that make the stadium picture.

Solution

Solution of Exercise 71.2.

A minuscule, heavy, positively charged nucleus at the center; a cloud of nearly weightless negative electrons far outside it. The sizes: atom near $10^{-10}$ $\mathrm{m}$ , nucleus near $10^{-15}$ — the hundred-thousandfold gap that makes the pea-in-a-stadium.

Exercise 71.3 ★

Pay the old joke forward: who really marches in a wire, which way — and why do all the circuit laws survive the revelation?

Solution

Solution of Exercise 71.3.

Electrons — negative, marching from $-$ to $+$ : opposite to the convention. The laws survive because they never depended on who marches: currents, junction sums, Ohm’s portraits and power bills read identically with the arrow kept by worldwide agreement.

Exercise 71.4 ★

What is an order of magnitude? Give the order of magnitude of: your height; the classroom’s length; the Earth’s diameter.

Solution

Solution of Exercise 71.4.

The nearest power of ten. Height: $10^{0}$ $\mathrm{m}$ ; classroom: $10^{1}$ ; Earth’s diameter: $10^{7}$ .

Exercise 71.5 ★

By how many rungs of the ladder do these differ: atom and nucleus? you and the Earth? the Sun’s distance and the nearest star’s?

Solution

Solution of Exercise 71.5.

Atom to nucleus: five rungs. You to the Earth: seven. Sun’s distance ( $10^{11}$ ) to nearest star ( $4 \times 10^{16}$ ): five to six rungs — a few hundred thousandfold.

Exercise 71.6 ★

“The table is mostly empty space.” Defend the sentence with the two atomic numbers — and explain what makes the table feel solid anyway.

Solution

Solution of Exercise 71.6.

The atom is $10^{-10}$ $\mathrm{m}$ of mostly nothing around a $10^{-15}$ nucleus: pack atoms into a table and the void comes with them. Solidity is the electron clouds’ electric refusal to overlap — emptiness, firmly defended.

Exercise 71.7 ★

Ladder arithmetic: about how many atoms, laid in a row, span a millimetre?

Solution

Solution of Exercise 71.7.

Millimetre, $10^{-3}$ ; atom, $10^{-10}$ : seven rungs — about ten million atoms in the row.

Exercise 71.8 ★

Locate on the ladder, with a power of ten each: a light-year; the Milky Way; the deepest surveyed sky.

Solution

Solution of Exercise 71.8.

Light-year: $10^{16}$ $\mathrm{m}$ (nine and a half thousand million million). Milky Way: $10^{21}$ . Deepest surveyed sky: near $10^{26}$ .

Exercise 71.9 ★★

The stadium model: if the nucleus is a $1\,\mathrm{cm}$ pea, how wide is the atom-stadium? ( $10^{5}$ ratio — convert to metres and judge against a real stadium.)

Solution

Solution of Exercise 71.9.

$10^{5}$ peas: $0.01 \times 10^{5} = 1000\,\mathrm{m}$ — a kilometre-wide “stadium”: grander than any real one, but the picture holds — pea at the center spot, and the next peas a kilometre away.

Exercise 71.10 ★★

A drop of water holds about $10^{21}$ molecules. Using the sea-of-drops comparison of your seventh-year chapter — or your own estimate of drops in all the oceans (about $10^{21}$ litres, $10^{5}$ drops each) — weigh the old claim that a drop holds more molecules than the seas hold drops.

Solution

Solution of Exercise 71.10.

Drops in the seas: about $10^{21} \times 10^{5} = 10^{26}$ — so the old claim, taken literally, overreached: the seas hold more drops than a drop holds molecules by several rungs. The honest version — a drop’s molecules outnumber anything countable in a lifetime — survives; ladders keep even beloved claims honest.

Exercise 71.11 ★★

Your body stands near the ladder’s middle: about how many rungs down to the atom, and up to the deepest sky? What does this symmetry say about the reach of nine years of schooling?

Solution

Solution of Exercise 71.11.

About ten rungs down to the atom ( $10^{0}$ to $10^{-10}$ ) and twenty-six up to the deep sky — close to the middle, tilted a little toward the small. Nine years of schooling put a reader within honest reach of both ends of everything measured.

Exercise 71.12 ★★★

The book’s last exercise. Choose any three chapters from your nine years — one from grades 1–3, one from 4–6, one from 7–9 — and write, for each, the single sentence you would tell your younger self starting that chapter: what it truly taught, seen from the top of the ladder. (There is no solution page for a letter to yourself — but write it; the course ahead will be glad you did.)

Solution

Solution of Exercise 71.12.

Answers are personal — the exercise is the letter itself. (One specimen, for form’s sake: to the grade-one reader of the shadows chapter — “the dark twin on the pavement will one day explain eclipses of the Sun”; to the grade-five reader of the energy chapter — “the chains you trace will become formulas with joules in them”; to the grade-eight reader of Ohm’s law — “the portraits you draw are the habit of testing laws, which is the whole secret”.)

71.5 Problem: The Journey by Powers of Ten

Problem 71.1

Weekend problem — the class films “The Journey by Powers of Ten”; one zoom, forty-one rungs; the script conference

The class’s film: one continuous zoom from a picnic blanket outward to the deep sky, then inward to the nucleus — one power of ten per second of film. You write the scientific script.

Part I — Outward. The zoom starts framing one metre of blanket: $10^{0}$ .

At which seconds does the frame first contain: the whole schoolyard ( $10^{2}$ ); the whole Earth ( $10^{7}$ ); the Moon’s orbit ( $10^{9}$ )?
Which second frames the Sun’s distance — and what does the narrator say sunshine’s age is, from the light chapter?
Between which seconds does the frame cross the great emptiness where the Solar System ends but the nearest star has not yet entered? What should the screen honestly show there?
The zoom ends at second $26$ : what fills the frame, and how old is the oldest light in it compared with the Earth?

Part II — Inward. The zoom reverses and dives into a leaf on the blanket.

At which seconds does the film pass: an ant ( $10^{-2}$ , generously); a cell ( $10^{-5}$ ); a molecule ( $10^{-9}$ ); an atom ( $10^{-10}$ )?
Between the atom’s electron cloud and its nucleus, the film must zoom five more seconds through — what? What does the narrator say about the stadium?
The film’s final frame, at $10^{-15}$ : its subject — and the narrator’s honest closing line about the rungs below.
Total the film’s length: how many seconds of outward, how many of inward, from the blanket’s metre?

Part III — The script conference.

A teacher objects that the film spends one second on the step from house to street and one second on the step from galaxy to galaxy-cluster — “wildly unequal steps!” Defend the equal seconds with the ladder’s logic.
The narrator wants one recurring line for both directions — something true at every rung about emptiness. Draft it (the Solar System and the atom must both fit under it).
The credits must name the two instruments that conquered the two directions. Name them, and the chapters of this book that introduced their principles.
The film’s last words belong to the course itself: write the two-sentence epilogue — nine years, one ladder, and what begins next year.

Solution

Solution of Problem 71.1.

1. Schoolyard: second $2$ ; the Earth: second $7$ ; the Moon’s orbit: second $9$ . 2. Second $11$ (at $1.5 \times 10^{11}$ $\mathrm{m}$ ); the narrator: “the sunshine on the blanket is eight minutes old.” 3. Roughly seconds $13$ to $16$ : the frame holds the whole Solar System as a dot and no star yet — honestly, near-black emptiness, seconds of it: the film’s truest scenes. 4. The deepest surveyed sky, $10^{26}$ $\mathrm{m}$ of galaxies — its oldest light older than the Earth itself. 5. Ant: second $2$ inward; cell: second $5$ ; molecule: second $9$ ; atom: second $10$ . 6. Through the atom’s own emptiness — five seconds of nothing between cloud and center: “if this atom were a stadium, we have left the highest stands and are still flying toward a pea.” 7. The nucleus — and honestly: “here our ladder ends, not because nature stops, but because these are the smallest rungs humans have measured; the building continues.” 8. Outward $26$ seconds; inward $15$ : a forty-one-second universe. 9. Each second multiplies the view tenfold — equal ratios, not equal additions: the ladder’s logic is that a rung is a factor, and by factors the street-step and the cluster-step are the same size. 10. For example: “However full it looks, almost everything here is empty — matter and sky alike are rare islands in wide nothing.” 11. The microscope (lenses chapter: two converging lenses over the small) and the telescope (same chapter, gathering the faint far) — the two directions’ keys, both ground from this book’s optics. 12. For example: “Nine years, one ladder: from a ladybug’s countable legs to galaxies older than the ground underfoot, every rung was set by measurement anyone may check. Next year the questions we left open — force and motion, the nature of light, the atom’s heart — begin to answer; bring the ladder.”

Terms defined in this chapter

Atoms

See all 393 terms in the glossary