---
title: "The Fundamental Interactions"
book: "High School Physics"
subject: physics
language: en
chapter: 13
exercises: 15
source: https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions
---

# Chapter 13 — The Fundamental Interactions

Rub a balloon on your hair and it picks up scraps of paper — beating, with a few square centimetres of rubber, the gravitational pull of the entire planet. That easy victory is why [atoms](#def-g11-fundamental-interactions-ladder) hold together and why heavy nuclei eventually shatter ([Chapter 19](https://one-course.com/books/physics/2/en/chapter/19-the-nucleus-and-radioactivity#ch-g11-nucleus-radioactivity)). The universe runs on exactly four interactions; this chapter meets them, weighs them against each other, and assigns each its floor.

## 13.1 Matter from quarks to galaxies

**Definition 13.1 (The structure ladder).**

Matter is built in floors, each assembled from the one below: three *quarks* (pointlike, smaller than $10^{-18}\,\mathrm{m}$) bind into a *nucleon* — proton or neutron, about $10^{-15}\,\mathrm{m}$ across; nucleons pack into a *nucleus* ($10^{-15}\,\mathrm{m}$ to $10^{-14}\,\mathrm{m}$); a nucleus plus its electron cloud is an *atom* ($10^{-10}\,\mathrm{m}$); atoms bind into *molecules* and crystals ($10^{-9}\,\mathrm{m}$ up), which assemble into everyday objects ($1\,\mathrm{m}$), planets ($10^{7}\,\mathrm{m}$), planetary systems ($10^{11}\,\mathrm{m}$), galaxies ($10^{21}\,\mathrm{m}$) and the observable universe ($10^{26}\,\mathrm{m}$).

**Definition 13.2 (The four interactions).**

A *fundamental interaction* is a [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) not reducible to other [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force); there are exactly four: *gravitation*, the *electromagnetic interaction*, the [strong interaction](#def-g11-fundamental-interactions-strong) and the [weak interaction](#def-g11-fundamental-interactions-weak).

**Remark 13.3.**

The floors are separated by voids: an [atom](#def-g11-fundamental-interactions-ladder) is $10^{5}$ times wider than its [nucleus](#def-g11-fundamental-interactions-ladder) — scale the [nucleus](#def-g11-fundamental-interactions-ladder) up to a $1\,\mathrm{cm}$ marble and the electron cloud is a kilometre across. Something must hold these sparse floors together, and gravity is almost never it.

![The ladder of structure on a powers-of-ten axis: forty-four decades separate the quark from the observable universe. Sizes in metres.](https://one-course.com/images/onecourse/chapters/physics-2/g11-fundamental-interactions/fig-ea018b795ae7.svg)

*The ladder of structure on a powers-of-ten axis: forty-four decades separate the [quark](#def-g11-fundamental-interactions-ladder) from the observable universe. Sizes in [metres](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit).*

## 13.2 Electric charge

**Definition 13.4 (Electric charge).**

*Electric charge* is the property of matter on which the [electromagnetic interaction](#def-g11-fundamental-interactions-four) acts, as mass is the property [gravitation](#def-g11-fundamental-interactions-four) acts on. It is measured in *coulombs* ($\mathrm{C}$) and comes in *two signs*, positive and negative, which cancel when added; like charges repel, unlike charges attract.

**Definition 13.5 (Elementary charge).**

Charge is grained: every observed charge is a whole multiple of the *elementary charge* $e = 1.60 \times 10^{-19}\,\mathrm{C}$. The proton carries $+e$, the electron $-e$, the neutron $0$; a charge $q$ holds $N = \abs{q}/e$ elementary charges.

**Proposition 13.6 (Conservation of charge).**

The total charge of an isolated system never changes: charge is neither created nor destroyed, only transferred — in practice by electrons, the lightest carriers.

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

**Remark 13.7.**

No violation has ever been observed; the deep reason, a symmetry of electromagnetism, is given in the university volumes.

**Definition 13.8 (Charging by friction and by contact).**

Rubbing two insulators tears electrons off one and onto the other, leaving opposite charges of equal magnitude (*charging by friction*); touching a charged object to a neutral one shares the charge (*charging by contact*). Only electrons move; the total charge is conserved.

**Example 13.9 (Counting electrons).**

A balloon rubbed on hair acquires $q = -1.6 \times 10^{-8}\,\mathrm{C}$: it has *gained* $N = 1.6 \times 10^{-8} / 1.6 \times 10^{-19} = 1.0 \times 10^{11}$ electrons, and the hair is left with exactly $+1.6 \times 10^{-8}\,\mathrm{C}$.

## 13.3 Coulomb’s law

**Theorem 13.10 (Coulomb’s law).**

Two point charges $q_1$ and $q_2$ a distance $d$ apart exert on each other [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) along the line joining them, of equal magnitudes and opposite directions — repulsive for like signs, attractive for unlike signs — with

$$
F = k\,\frac{\abs{q_1 q_2}}{d^{2}},
\qquad k = 8.99 \times 10^{9}\,\mathrm{N}\,\mathrm{m}^{2}/\mathrm{C}^{2}.
$$

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

**Remark 13.11.**

Coulomb established the law in 1785 by measuring the twist of a [thin](https://one-course.com/books/physics/2/en/chapter/10-lenses-images-and-the-eye#def-g11-lenses-and-eye-lens) torsion fibre; why the [exponent](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-scinot) is exactly $2$ is answered in the university volumes.

![Like charges repel (left), unlike charges attract (right); the two forces always have equal magnitudes and opposite directions.](https://one-course.com/images/onecourse/chapters/physics-2/g11-fundamental-interactions/fig-f64c3fb8adb8.svg)

![Like charges repel (left), unlike charges attract (right); the two forces always have equal magnitudes and opposite directions.](https://one-course.com/images/onecourse/chapters/physics-2/g11-fundamental-interactions/fig-e64a2bf6cd89.svg)

*Like charges repel (left), unlike charges attract (right); the two [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) always have equal magnitudes and opposite directions.*

**Remark 13.12 (A formal twin of gravitation).**

Coulomb’s law is, symbol for symbol, the law of universal [gravitation](#def-g11-fundamental-interactions-four) ([Chapter 4](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#ch-g10-universal-gravitation)) with charges in place of masses:

| source | mass $m$ | charge $q$ |
| --- | --- | --- |
| law | $F = G\,\dfrac{m_1 m_2}{d^2}$ | $F = k\,\dfrac{\abs{q_1 q_2}}{d^2}$ |
| [1.5ex] constant | $G = 6.67 \times 10^{-11}\,\mathrm{N}\,\mathrm{m}^{2}/\mathrm{kg}^{2}$ | $k = 8.99 \times 10^{9}\,\mathrm{N}\,\mathrm{m}^{2}/\mathrm{C}^{2}$ |
| sign | always attractive | attractive or repulsive |

The two differences drive everything below: electricity is overwhelmingly stronger, and it alone can cancel itself.

**Example 13.13 (Two protons, both forces at once).**

Two protons ($m_p = 1.67 \times 10^{-27}\,\mathrm{kg}$) sit $d = 1.0 \times 10^{-15}\,\mathrm{m}$ apart — nuclear neighbours. Electric repulsion:

$$
F_E = 8.99 \times 10^{9} \times
\frac{(1.60 \times 10^{-19})^{2}}{(1.0 \times 10^{-15})^{2}}
\approx 2.3 \times 10^{2}\,\mathrm{N}
$$

— the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) of a $23$-kg suitcase, carried by a particle of $10^{-27}$ [kilograms](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit). Gravitational attraction: $F_G = 6.67 \times 10^{-11} \times (1.67 \times 10^{-27})^{2} /
(1.0 \times 10^{-15})^{2} \approx 1.9 \times 10^{-34}\,\mathrm{N}$. The ratio $F_E / F_G = k e^{2} / (G m_p^{2}) \approx 1.2 \times 10^{36}$ is distance-independent ($d^{2}$ cancels): between elementary particles, gravity is $10^{36}$ times too weak to matter — at *any* distance.

## 13.4 The strong and weak interactions

The example leaves a scandal: a helium [nucleus](#def-g11-fundamental-interactions-ladder) holds two protons that repel with tens of [newtons](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) — and yet helium exists.

**Definition 13.14 (The strong interaction).**

The *strong interaction* binds [quarks](#def-g11-fundamental-interactions-ladder) into [nucleons](#def-g11-fundamental-interactions-ladder) and [nucleons](#def-g11-fundamental-interactions-ladder) into nuclei. At $10^{-15}\,\mathrm{m}$ it is attractive and far stronger than the electric repulsion (thousands of [newtons](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) between [nucleons](#def-g11-fundamental-interactions-ladder)); beyond a few $10^{-15}$ [metres](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) it vanishes: its *range* is about $10^{-15}\,\mathrm{m}$. It grips protons and neutrons alike and ignores charge.

![The tug-of-war inside a nucleus: at 10-15 m the strong attraction (thousands of newtons) beats the electric repulsion.](https://one-course.com/images/onecourse/chapters/physics-2/g11-fundamental-interactions/fig-d7a4ef8d7f07.svg)

*The tug-of-war inside a [nucleus](#def-g11-fundamental-interactions-ladder): at $10^{-15}$ m the strong attraction (thousands of [newtons](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force)) beats the electric repulsion.*

**Remark 13.15 (Why nuclei exist — and why heavy ones give way).**

The two [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) in the [nucleus](#def-g11-fundamental-interactions-ladder) scale differently. The strong glue is short-ranged: a [nucleon](#def-g11-fundamental-interactions-ladder) binds only its few neighbours within $10^{-15}\,\mathrm{m}$. The electric repulsion is long-ranged: every proton pushes on *every other* proton. As nuclei grow, repulsion accumulates faster than glue: heavy nuclei need extra neutrons (glue without charge), and beyond about $80$ protons even that fails — the heaviest nuclei live on borrowed time ([Chapter 19](https://one-course.com/books/physics/2/en/chapter/19-the-nucleus-and-radioactivity#ch-g11-nucleus-radioactivity)).

**Definition 13.16 (The weak interaction).**

The *weak interaction*, of [range](#def-g11-fundamental-interactions-strong) shorter still, lets a neutron turn into a proton (and back): it drives the $\beta$ radioactivity of [Chapter 19](https://one-course.com/books/physics/2/en/chapter/19-the-nucleus-and-radioactivity#ch-g11-nucleus-radioactivity).

## 13.5 Who rules which floor

**Method 13.17 (Auditing a scale).**

To find which interaction rules a structure of size $d$, check [range](#def-g11-fundamental-interactions-strong) and cancellation:

1. $d \leq 10^{-14}\,\mathrm{m}$ : the *[strong interaction](#def-g11-fundamental-interactions-strong)* rules — it beats the electric repulsion, and gravity is $10^{36}$ out.
2. [atoms](#def-g11-fundamental-interactions-ladder) to mountains ( $10^{-10}\,\mathrm{m}$ to $10^{4}\,\mathrm{m}$ ): the strong [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) is out of [range](#def-g11-fundamental-interactions-strong) ; the *[electromagnetic interaction](#def-g11-fundamental-interactions-four)* rules — bonds, cohesion, [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) , contact.
3. planets and beyond ( $d \geq 10^{6}\,\mathrm{m}$ ): matter is neutral to fantastic precision, so electricity cancels itself; mass has one sign and always adds — *[gravitation](#def-g11-fundamental-interactions-four)* rules the sky.

The [weak interaction](#def-g11-fundamental-interactions-weak) holds no floor: it binds nothing, but arbitrates transformations ($\beta$ decay).

![Who rules where: strong for nuclei, electromagnetic from atoms to mountains, gravitation for planets and beyond. Sizes in metres.](https://one-course.com/images/onecourse/chapters/physics-2/g11-fundamental-interactions/fig-6e3f873f7775.svg)

*Who rules where: strong for nuclei, electromagnetic from [atoms](#def-g11-fundamental-interactions-ladder) to mountains, [gravitation](#def-g11-fundamental-interactions-four) for planets and beyond. Sizes in [metres](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit).*

## 13.6 Exercises

**Exercise 13.1 ★.**

Give the [order of magnitude](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-oom) of each size and its floor of the ladder: proton $1.7 \times 10^{-15}\,\mathrm{m}$; water [molecule](#def-g11-fundamental-interactions-ladder) $2.8 \times 10^{-10}\,\mathrm{m}$; grain of sand $2 \times 10^{-4}\,\mathrm{m}$; Earth $1.3 \times 10^{7}\,\mathrm{m}$; Milky Way $9.5 \times 10^{20}\,\mathrm{m}$.

**Solution of Exercise 13.1.**

Proton: $10^{-15}$ ([nucleon](#def-g11-fundamental-interactions-ladder) floor); water [molecule](#def-g11-fundamental-interactions-ladder): $10^{-10}$ ([molecule](#def-g11-fundamental-interactions-ladder) floor — single [molecules](#def-g11-fundamental-interactions-ladder) sit at atomic sizes); sand: $10^{-4}$ (everyday matter); Earth: $10^{7}$ (planet); Milky Way: $10^{21}$ (galaxy). All in [metres](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit).

**Exercise 13.2 ★.**

A balloon rubbed on hair carries $q = -4.8 \times 10^{-9}\,\mathrm{C}$. Has it gained or lost electrons, and how many? What is the hair’s charge afterwards, and which law says so?

**Solution of Exercise 13.2.**

Negative charge: it *gained* $N = 4.8 \times 10^{-9}/1.6 \times 10^{-19} = 3.0 \times 10^{10}$ electrons. The hair carries $+4.8 \times 10^{-9}\,\mathrm{C}$, by conservation of charge ([Proposition 13.6](#prop-g11-fundamental-interactions-conservation)).

**Exercise 13.3 ★.**

Charges $q_1 = 2.0 \times 10^{-6}\,\mathrm{C}$ and $q_2 = -3.0 \times 10^{-6}\,\mathrm{C}$ sit $30\,\mathrm{cm}$ apart. Compute the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force); attractive or repulsive? Compare the [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) on $q_1$ and on $q_2$.

**Solution of Exercise 13.3.**

$F = 8.99 \times 10^{9} \times \dfrac{2.0 \times 10^{-6} \times 3.0 \times 10^{-6}}{0.30^2}
= \dfrac{5.39 \times 10^{-2}}{0.090} \approx 0.60\,\mathrm{N}$, attractive (unlike signs). The two [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) have the same magnitude, opposite directions.

**Exercise 13.4 ★.**

Two charges attract with a [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) $F_0$. What does it become if: the distance doubles; the distance is divided by $3$; one charge doubles; both charges *and* the distance double?

**Solution of Exercise 13.4.**

$F_0/4$; $9F_0$; $2F_0$; $F_0$ (numerator $\times 4$, denominator $\times 4$).

**Exercise 13.5 ★.**

Name the interaction responsible for: (a) the Moon orbiting the Earth; (b) the cohesion of a salt crystal; (c) the cohesion of a helium [nucleus](#def-g11-fundamental-interactions-ladder); (d) the $\beta$ decay of carbon-14; (e) the balloon of [Exercise 13.2](#exo-g11-fundamental-interactions-2) sticking to a wall.

**Solution of Exercise 13.5.**

(a) [gravitation](#def-g11-fundamental-interactions-four); (b) electromagnetic; (c) strong; (d) weak; (e) electromagnetic.

**Exercise 13.6 ★★.**

The two protons of a helium [nucleus](#def-g11-fundamental-interactions-ladder) sit $d = 2.0 \times 10^{-15}\,\mathrm{m}$ apart ($m_p = 1.67 \times 10^{-27}\,\mathrm{kg}$). Compute their electric repulsion, their gravitational attraction, and the ratio. What keeps the [nucleus](#def-g11-fundamental-interactions-ladder) whole?

**Solution of Exercise 13.6.**

$F_E = 8.99 \times 10^{9} \times (1.60 \times 10^{-19})^2/(2.0 \times 10^{-15})^2
= 2.30 \times 10^{-28}/4.0 \times 10^{-30} \approx 58\,\mathrm{N}$; $F_G = 6.67 \times 10^{-11} \times (1.67 \times 10^{-27})^2/4.0 \times 10^{-30}
\approx 4.7 \times 10^{-35}\,\mathrm{N}$. Ratio $F_E/F_G \approx 1.2 \times 10^{36}$. The [strong interaction](#def-g11-fundamental-interactions-strong) holds the [nucleus](#def-g11-fundamental-interactions-ladder), gravity is irrelevant.

**Exercise 13.7 ★★.**

In a hydrogen [atom](#def-g11-fundamental-interactions-ladder), the electron ($m_e = 9.11 \times 10^{-31}\,\mathrm{kg}$) sits $d = 5.3 \times 10^{-11}\,\mathrm{m}$ from the proton ($m_p = 1.67 \times 10^{-27}\,\mathrm{kg}$). Compute the electric and [gravitational forces](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-force) between them and their ratio. What holds [atoms](#def-g11-fundamental-interactions-ladder) together?

**Solution of Exercise 13.7.**

$F_E = 2.30 \times 10^{-28}/(5.3 \times 10^{-11})^2 = 2.30 \times 10^{-28}/2.81 \times 10^{-21}
\approx 8.2 \times 10^{-8}\,\mathrm{N}$; $F_G = 6.67 \times 10^{-11} \times 1.67 \times 10^{-27} \times 9.11 \times 10^{-31} /
2.81 \times 10^{-21} \approx 3.6 \times 10^{-47}\,\mathrm{N}$. Ratio $\approx 2.3 \times 10^{39}$: the [electromagnetic interaction](#def-g11-fundamental-interactions-four) holds [atoms](#def-g11-fundamental-interactions-ladder) together.

**Exercise 13.8 ★★.**

Sphere A carries $q_A = 8.0 \times 10^{-9}\,\mathrm{C}$; an identical sphere B is neutral. They are touched together, then separated to $10\,\mathrm{cm}$. What charge does each carry (check the conservation of charge)? Compute the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) between them — attractive or repulsive?

**Solution of Exercise 13.8.**

By contact the charge is shared: $q_A = q_B = 4.0 \times 10^{-9}\,\mathrm{C}$ (total still $8.0 \times 10^{-9}\,\mathrm{C}$). Then $F = 8.99 \times 10^{9} \times (4.0 \times 10^{-9})^2/0.10^2 =
1.4 \times 10^{-5}\,\mathrm{N}$, repulsive (like signs).

**Exercise 13.9 ★★.**

Two identical charges $q = 1.0 \times 10^{-6}\,\mathrm{C}$ repel with $0.90\,\mathrm{N}$: how far apart are they? Where does the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) drop to $0.10\,\mathrm{N}$?

**Solution of Exercise 13.9.**

$d = \sqrt{k q^2/F} = \sqrt{8.99 \times 10^{9} \times 1.0 \times 10^{-12}/0.90}
\approx 0.10\,\mathrm{m}$. A [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) $9$ times smaller needs a distance $3$ times larger: $d = 0.30\,\mathrm{m}$.

**Exercise 13.10 ★★.**

Two protons in neighbouring [molecules](#def-g11-fundamental-interactions-ladder) sit $d = 1.0 \times 10^{-10}\,\mathrm{m}$ apart. Compute their electric repulsion. What does the [strong interaction](#def-g11-fundamental-interactions-strong) contribute at this distance? Which interaction binds [molecules](#def-g11-fundamental-interactions-ladder)?

**Solution of Exercise 13.10.**

$F = 2.30 \times 10^{-28}/(1.0 \times 10^{-10})^2 = 2.3 \times 10^{-8}\,\mathrm{N}$. The [strong interaction](#def-g11-fundamental-interactions-strong) contributes nothing: $10^{-10}$ m is $10^{5}$ times its [range](#def-g11-fundamental-interactions-strong). [Molecules](#def-g11-fundamental-interactions-ladder) are bound by the [electromagnetic interaction](#def-g11-fundamental-interactions-four).

**Exercise 13.11 ★★.**

Each of two coins holds about $10^{23}$ electrons. Move one electron in a *million* between them and set the coins $1.0\,\mathrm{m}$ apart. Compute each coin’s charge, then the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force). The [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) of what mass equals it? Conclude on the neutrality of everyday matter.

**Solution of Exercise 13.11.**

$q = 10^{23}/10^{6} \times 1.6 \times 10^{-19} = 1.6 \times 10^{-2}\,\mathrm{C}$ on each coin. $F = 8.99 \times 10^{9} \times (1.6 \times 10^{-2})^2/1.0^2 \approx
2.3 \times 10^{6}\,\mathrm{N}$ — the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) of $2.3 \times 10^{6}/9.81 \approx
2.3 \times 10^{5}\,\mathrm{kg}$, a loaded freight train, from a millionth of the electrons. Everyday matter must be (and is) neutral to far better than one part in a million.

**Exercise 13.12 ★★★.**

A uranium [nucleus](#def-g11-fundamental-interactions-ladder) has diameter $1.4 \times 10^{-14}\,\mathrm{m}$ and $92$ protons. Compute the repulsion between two protons at opposite ends. Can the [strong interaction](#def-g11-fundamental-interactions-strong) bind this pair — why not? How many repelling proton pairs are there? Why do heavy nuclei need extra neutrons, and why beyond some size can even neutrons not save them?

**Solution of Exercise 13.12.**

$F = 2.30 \times 10^{-28}/(1.4 \times 10^{-14})^2 = 2.30 \times 10^{-28}/1.96 \times 10^{-28}
\approx 1.2\,\mathrm{N}$ — still enormous at this scale. No: they are $14$ times the strong [range](#def-g11-fundamental-interactions-strong) apart; each is glued only to its nearest neighbours. Pairs: $\frac{92 \times 91}{2} = 4186$, *all* repelling at any distance, while the glue does not grow with size. Neutrons add attraction without repulsion and dilute the protons; but repulsion grows with the square of the proton number and glue only with the number of neighbours, so beyond about $80$ protons no neutron budget balances the books — such nuclei decay ([Chapter 19](https://one-course.com/books/physics/2/en/chapter/19-the-nucleus-and-radioactivity#ch-g11-nucleus-radioactivity)).

**Exercise 13.13 ★★★.**

Two balls of mass $1.0\,\mathrm{g}$ hang from $50\,\mathrm{cm}$ threads tied to the same point. Given the same charge $q$, they settle $6.0\,\mathrm{cm}$ apart. Each ball is in equilibrium under its [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight), the thread’s [tension](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) and the electric [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force): show that $F = mg\tan\theta$ ($\theta$: angle of thread to vertical) and that here $\tan\theta \approx 0.060$. Compute $F$, then $q$, then the number of [elementary charges](#def-g11-fundamental-interactions-elementary) moved.

**Solution of Exercise 13.13.**

Equilibrium: [vertically](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) $T\cos\theta = mg$, horizontally $T\sin\theta = F$, so $F = mg\tan\theta$. Here $\sin\theta = 3.0/50 = 0.060$, and for small angles $\tan\theta \approx \sin\theta = 0.060$. Then $F = 1.0 \times 10^{-3} \times 9.81 \times 0.060 \approx 5.9 \times 10^{-4}\,\mathrm{N}$; $q = d\sqrt{F/k} = 0.060 \times \sqrt{5.9 \times 10^{-4}/8.99 \times 10^{9}}
\approx 1.54 \times 10^{-8}\,\mathrm{C}$; $N = 1.54 \times 10^{-8}/1.6 \times 10^{-19} \approx
9.6 \times 10^{10}$ [elementary charges](#def-g11-fundamental-interactions-elementary).

**Exercise 13.14 ★★★.**

Data: $M_E = 5.97 \times 10^{24}\,\mathrm{kg}$, $M_M = 7.35 \times 10^{22}\,\mathrm{kg}$, Earth–Moon distance $d = 3.84 \times 10^{8}\,\mathrm{m}$. Compute the Earth–Moon [gravitational force](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-force). Gravity is switched off: what charges $+Q$ on the Earth and $-Q$ on the Moon keep the same attraction? How many electrons is $Q$, and what do they weigh ($m_e = 9.11 \times 10^{-31}\,\mathrm{kg}$)? Comment.

**Solution of Exercise 13.14.**

$F = 6.67 \times 10^{-11} \times 5.97 \times 10^{24} \times 7.35 \times 10^{22} /
(3.84 \times 10^{8})^2 = 2.93 \times 10^{37}/1.47 \times 10^{17} \approx 2.0 \times 10^{20}\,\mathrm{N}$. Setting $kQ^2/d^2 = F$: $Q = d\sqrt{F/k} = 3.84 \times 10^{8} \times \sqrt{2.0 \times 10^{20}/8.99 \times 10^{9}}
\approx 5.7 \times 10^{13}\,\mathrm{C}$. That is $N = 5.7 \times 10^{13}/1.6 \times 10^{-19} \approx 3.6 \times 10^{32}$ electrons, of mass $3.6 \times 10^{32} \times 9.11 \times 10^{-31} \approx 3.2 \times 10^{2}\,\mathrm{kg}$: a few hundred [kilograms](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of electrons could do the work of $7.35 \times 10^{22}\,\mathrm{kg}$ of Moon — electricity’s strength and gravity’s weakness in one number.

**Exercise 13.15 ★★★.**

In a salt crystal, Na$^{+}$ and Cl$^{-}$ ions (charges $\pm e$) sit $d = 2.8 \times 10^{-10}\,\mathrm{m}$ apart; a Cl$^{-}$ ion has mass $5.9 \times 10^{-26}\,\mathrm{kg}$. Compute the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) between neighbouring ions and the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) of a Cl$^{-}$ ion; give the ratio. Then explain how gravity wins at large scales — no mountain tops $10^{4}\,\mathrm{m}$ — though each bond beats it by fifteen orders of magnitude.

**Solution of Exercise 13.15.**

$F = 2.30 \times 10^{-28}/(2.8 \times 10^{-10})^2 = 2.30 \times 10^{-28}/7.84 \times 10^{-20}
\approx 2.9 \times 10^{-9}\,\mathrm{N}$; [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) $P = 5.9 \times 10^{-26} \times 9.81 \approx 5.8 \times 10^{-25}\,\mathrm{N}$; ratio $F/P \approx 5 \times 10^{15}$. Each bond has a *fixed* strength, but [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) grows with the whole mass above: pile up rock and the load on the bottom layer grows without limit while its bonds do not. Around $10^{4}\,\mathrm{m}$ of rock the load crushes the electric bonds — gravity wins not by strength per particle but by *adding up* when charge cancels.

## 13.7 Problem: The audit of the four forces

**Problem 13.1.**

Weekend problem — why the world neither collapses nor flies apart: the four interactions audited floor by floor, down to matter’s neutrality to two parts in $10^{18}$

[Atoms](#def-g11-fundamental-interactions-ladder) do not implode, nuclei mostly hold, the Moon neither crashes nor escapes; this problem audits who deserves the credit. Data: $e = 1.60 \times 10^{-19}\,\mathrm{C}$, $k = 8.99 \times 10^{9}\,\mathrm{N}\,\mathrm{m}^{2}/\mathrm{C}^{2}$, $G = 6.67 \times 10^{-11}\,\mathrm{N}\,\mathrm{m}^{2}/\mathrm{kg}^{2}$, $m_p = 1.67 \times 10^{-27}\,\mathrm{kg}$, $m_e = 9.11 \times 10^{-31}\,\mathrm{kg}$, $M_E = 5.97 \times 10^{24}\,\mathrm{kg}$, $M_M = 7.35 \times 10^{22}\,\mathrm{kg}$, Earth–Moon distance $3.84 \times 10^{8}\,\mathrm{m}$.

**Part I — The ladder.**

1. List the floors of the ladder from [quark](#def-g11-fundamental-interactions-ladder) to galaxy, one [order of magnitude](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-oom) of size each.
2. Scale model: the [nucleus](#def-g11-fundamental-interactions-ladder) becomes a $1\,\mathrm{cm}$ marble; how wide is the [atom](#def-g11-fundamental-interactions-ladder) ?
3. What fraction of an [atom](#def-g11-fundamental-interactions-ladder) ’s volume does its [nucleus](#def-g11-fundamental-interactions-ladder) occupy?
4. A hair is $70\,\text{µ}\mathrm{m}$ thick. How many [atoms](#def-g11-fundamental-interactions-ladder) span it?
5. How many [powers](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-power) of ten from the [quark](#def-g11-fundamental-interactions-ladder) ceiling ( $10^{-18}\,\mathrm{m}$ ) to a galaxy ( $10^{21}\,\mathrm{m}$ )?

**Part II — The electric floor.**

6. In a hydrogen [atom](#def-g11-fundamental-interactions-ladder) the electron sits $d = 5.3 \times 10^{-11}\,\mathrm{m}$ from the proton: compute the electric [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) between them.
7. Compute the [gravitational force](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-force) between them, and the ratio of the two. Which one holds the [atom](#def-g11-fundamental-interactions-ladder) together?
8. What changes if gravity is switched off inside [atoms](#def-g11-fundamental-interactions-ladder) ? And if electricity is?
9. Why are the “contact” [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) of everyday life — the floor pushing on your feet, [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) — electromagnetic in disguise?
10. Each of two coins holds about $10^{23}$ electrons. Transfer one electron in a *billion* between them, set them $1.0\,\mathrm{m}$ apart: compute the charges and the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) . Conclusion?
11. Which interaction rules the floors from $10^{-10}\,\mathrm{m}$ to $10^{4}\,\mathrm{m}$ , and why does the [strong interaction](#def-g11-fundamental-interactions-strong) not compete?

**Part III — The nuclear floor.**

12. Two protons sit $1.0 \times 10^{-15}\,\mathrm{m}$ apart in a [nucleus](#def-g11-fundamental-interactions-ladder) : compute their electric repulsion.
13. Unrestrained, what acceleration would this [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) give a proton? Compare with $g$ .
14. List the three properties the [strong interaction](#def-g11-fundamental-interactions-strong) needs to explain why nuclei exist, why neutrons are gripped too, and why nuclei are tiny.
15. In a uranium [nucleus](#def-g11-fundamental-interactions-ladder) (diameter $1.4 \times 10^{-14}\,\mathrm{m}$ , $92$ protons), compute the repulsion between two protons at opposite ends. Can the [strong interaction](#def-g11-fundamental-interactions-strong) bind that pair directly? Explain why repulsion wins as nuclei grow, and what the neutrons are for.
16. In one sentence, as promised: what does the [weak interaction](#def-g11-fundamental-interactions-weak) do?

**Part IV — The astronomical floor, and the verdict.**

17. Compute the [gravitational force](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-force) between the Earth and the Moon.
18. The Earth contains about $1.8 \times 10^{51}$ protons (and as many electrons), the Moon about $2.2 \times 10^{49}$ . What fraction $f$ of each body’s proton charge, left uncancelled, would make the electric [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) match the gravitational one?
19. Why does [gravitation](#def-g11-fundamental-interactions-four) , the weakling of question 7, rule astronomy? Two reasons.
20. Verdict — one sentence per floor ( [nucleus](#def-g11-fundamental-interactions-ladder) , [atom](#def-g11-fundamental-interactions-ladder) to mountain, planet and up): what holds it together? Then answer the title: why does the world neither collapse nor fly apart?

**Solution of Problem 13.1.**

**1.** [Quark](#def-g11-fundamental-interactions-ladder) $< 10^{-18}\,\mathrm{m}$; [nucleon](#def-g11-fundamental-interactions-ladder) $10^{-15}\,\mathrm{m}$; [nucleus](#def-g11-fundamental-interactions-ladder) $10^{-15}$–$10^{-14}\,\mathrm{m}$; [atom](#def-g11-fundamental-interactions-ladder) $10^{-10}\,\mathrm{m}$; [molecule](#def-g11-fundamental-interactions-ladder) $10^{-9}\,\mathrm{m}$; everyday object $1\,\mathrm{m}$; planet $10^{7}\,\mathrm{m}$; planetary system $10^{11}\,\mathrm{m}$; galaxy $10^{21}\,\mathrm{m}$.

**2.** Scale factor $10^{5}$: the [atom](#def-g11-fundamental-interactions-ladder) is $1\,\mathrm{cm} \times 10^{5} = 1\,\mathrm{km}$ wide.

**3.** $(10^{-15}/10^{-10})^3 = 10^{-15}$: matter is $99.999\,999\,999\,999\,9\,\%$ empty.

**4.** $7 \times 10^{-5}/10^{-10} = 7 \times 10^{5}$ [atoms](#def-g11-fundamental-interactions-ladder) — about a million across one hair.

**5.** From $10^{-18}$ to $10^{21}$: $39$ [powers](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-power) of ten.

**6.** $F_E = 8.99 \times 10^{9} \times (1.60 \times 10^{-19})^2 /
(5.3 \times 10^{-11})^2 \approx 8.2 \times 10^{-8}\,\mathrm{N}$.

**7.** $F_G = 6.67 \times 10^{-11} \times 1.67 \times 10^{-27} \times
9.11 \times 10^{-31}/(5.3 \times 10^{-11})^2 \approx 3.6 \times 10^{-47}\,\mathrm{N}$; ratio $F_E/F_G \approx 2.3 \times 10^{39}$. Electricity holds the [atom](#def-g11-fundamental-interactions-ladder); gravity is a spectator.

**8.** Without gravity: nothing measurable changes (it is $10^{39}$ below). Without electricity: no [atoms](#def-g11-fundamental-interactions-ladder), no [molecules](#def-g11-fundamental-interactions-ladder), no matter at all.

**9.** “Contact” is the electric repulsion between the electron clouds of [atoms](#def-g11-fundamental-interactions-ladder) brought to about $10^{-10}\,\mathrm{m}$: nothing ever touches. Floor reaction and [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) are Coulomb’s law in bulk.

**10.** $q = 10^{23} \times 10^{-9} \times 1.6 \times 10^{-19} =
1.6 \times 10^{-5}\,\mathrm{C}$ each; $F = 8.99 \times 10^{9} \times (1.6 \times 10^{-5})^2 \approx 2.3\,\mathrm{N}$ — a plainly visible [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) from one electron in a billion. Everyday matter is neutral to far better than $10^{-9}$.

**11.** The [electromagnetic interaction](#def-g11-fundamental-interactions-four). The strong [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) is out of [range](#def-g11-fundamental-interactions-strong) beyond about $10^{-14}\,\mathrm{m}$ — $10^{4}$ times smaller than a single [atom](#def-g11-fundamental-interactions-ladder).

**12.** $F = 8.99 \times 10^{9} \times (1.60 \times 10^{-19})^2 /
(1.0 \times 10^{-15})^2 \approx 2.3 \times 10^{2}\,\mathrm{N}$.

**13.** $a = F/m_p = 230/1.67 \times 10^{-27} \approx
1.4 \times 10^{29}\,\mathrm{m}/\mathrm{s}^{2}$, about $1.4 \times 10^{28}$ times $g$: no everyday [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) comes close.

**14.** Stronger than $230\,\mathrm{N}$ at $10^{-15}\,\mathrm{m}$ (nuclei exist); acting on protons and neutrons alike, blind to charge (neutrons are gripped); [range](#def-g11-fundamental-interactions-strong) about $10^{-15}\,\mathrm{m}$ (nuclei stay tiny — the glue cannot reach further).

**15.** $F = 2.30 \times 10^{-28}/(1.4 \times 10^{-14})^2 \approx
1.2\,\mathrm{N}$. No: $14$ times the [range](#def-g11-fundamental-interactions-strong). Repulsion acts across *all* $\frac{92\times91}{2} = 4186$ proton pairs; the glue binds only nearest neighbours, so growth favours repulsion. Neutrons add glue and spacing without adding repulsion — until, beyond about $80$ protons, no mixture balances and the [nucleus](#def-g11-fundamental-interactions-ladder) decays.

**16.** The [weak interaction](#def-g11-fundamental-interactions-weak) turns a neutron into a proton (and back), causing $\beta$ decay — it transforms; it never binds.

**17.** $F = 6.67 \times 10^{-11} \times 5.97 \times 10^{24} \times
7.35 \times 10^{22}/(3.84 \times 10^{8})^2 \approx 2.0 \times 10^{20}\,\mathrm{N}$.

**18.** $Q_E = 1.8 \times 10^{51} \times 1.6 \times 10^{-19} =
2.9 \times 10^{32}\,\mathrm{C}$, $Q_M = 3.5 \times 10^{30}\,\mathrm{C}$. Matching [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) means $f^2\,k\,Q_E Q_M = G M_E M_M$:

$$
f = \sqrt{\frac{2.93 \times 10^{37}}{8.99 \times 10^{9} \times 2.9 \times 10^{32}
\times 3.5 \times 10^{30}}} = \sqrt{3.2 \times 10^{-36}} \approx 1.8 \times 10^{-18} :
$$

an imbalance of *two parts in $10^{18}$* would cancel gravity.

**19.** Charge comes in two signs and cancels (question 18 shows how perfectly), so the stronger [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) switches itself off at large scales; mass has one sign, adds forever, and nothing screens it.

**20.** [Nucleus](#def-g11-fundamental-interactions-ladder): the [strong interaction](#def-g11-fundamental-interactions-strong) outmuscles the protons’ repulsion at $10^{-15}\,\mathrm{m}$. [Atom](#def-g11-fundamental-interactions-ladder) to mountain: the [electromagnetic interaction](#def-g11-fundamental-interactions-four) binds electrons to nuclei and [atoms](#def-g11-fundamental-interactions-ladder) to each other. Planet and up: [gravitation](#def-g11-fundamental-interactions-four), unopposed once charge cancels, holds planets, [orbits](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) and galaxies. The world does not collapse because each floor is propped by a binder stiffer than the load, and does not fly apart because matter is neutral to about $10^{-18}$ — so every scale has exactly one [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) in charge, and it is attractive where it must be.