Electric Charge and Coulomb's Law

Every phenomenon in this block — currents in wires, the force between magnets, the light emitted by a heated gas, the stability of the nucleus — traces back to a single property of matter: electric charge. This chapter builds that idea from the ground up and then quantifies the force between charges at rest, a result that Charles Coulomb established by torsion-balance experiment in the 1780s and that still carries his name.

The strategy mirrors the gravitation chapter of Block B. There, a $\dfrac{1}{r^2}$ law between point masses generated a field, a potential and an energy; here, a $\dfrac{1}{r^2}$ law between point charges will do exactly the same in the next chapters. The differences — charge comes in two signs, the electrostatic force can attract or repel, and it is roughly $10^{36}$ times stronger than gravity — are precisely what make electricity, rather than gravity, the force that holds atoms and solids together.

Reading note. Work Example 2 in Section 7 with a calculator and keep the answer in mind: the electrostatic repulsion between two protons a nucleus-width apart is a few tens of newtons — an enormous force on a subatomic particle, and the reason the strong nuclear force of Chapter 15 has to exist at all.

1. Electric Charge: A Fundamental Property of Matter

Charge is a property carried by the constituents of the atom: the proton ( $+e$ ), the electron ( $-e$ ), and — in bulk matter — nothing else that matters at this level. It has three experimental properties that must be taken as given.

The three laws of charge.

Two kinds. Charges are positive or negative; like charges repel, unlike charges attract.

Quantisation. Free charge always occurs in integer multiples of the elementary charge $e = 1.60\times 10^{-19}\ \text{C}$ : any observable charge is $Q = ne$ with $n$ an integer.

Conservation. In any isolated system the total charge is constant. Charge is never created or destroyed, only transferred.

The SI unit is the coulomb ( $\text{C}$ ), defined through electric current: one coulomb is the charge passing a point in a wire carrying one ampere for one second. One coulomb is a very large charge — it is the combined charge of about $6.24\times 10^{18}$ protons — so laboratory electrostatics deals in microcoulombs ( $\mu\text{C}$ ) and nanocoulombs.

A neutral object is not one with no charge; it is one with equal amounts of positive and negative charge, roughly $10^{9}\ \text{C}$ (a few billion coulombs) of each in a typical human body, cancelling exactly.

2. Conductors, Insulators and Methods of Charging

In a conductor — a metal, or an ionic solution — some charges (the conduction electrons of a metal) are free to move macroscopic distances. In an insulator — glass, plastic, dry air — every electron is bound to its parent atom and can be displaced only slightly.

Three ways to charge an object follow.

Charging by friction. Rubbing two different insulators transfers electrons from one to the other; the two are left with equal and opposite charge (conservation). Which way the electrons go is set empirically by the triboelectric series.
Charging by contact (conduction). A charged conductor touched to a neutral one shares its charge; both end with the same sign.
Charging by induction. A charged rod brought near a neutral conductor repels like charge to the far side and attracts unlike charge to the near side (in a metal only the electrons actually move), leaving an opposite charge facing the rod. Earthing the conductor while the rod is held near, then removing the earth and then the rod, leaves the conductor with a net charge opposite to the rod's — and the rod loses nothing.

Polarisation is the insulator analogue: the rod cannot move charge through the insulator, but it distorts every molecule slightly, producing a thin layer of opposite charge on the near face. This is why a charged rod attracts small neutral scraps of paper.

3. Coulomb's Law

Coulomb measured the force between two charged spheres as a function of their separation and their charges, and found:

Coulomb's law. The electrostatic force between two point charges is directed along the line joining them, is proportional to the product of the charges, and is inversely proportional to the square of their separation:
$F = k\,\frac{|q_1 q_2|}{r^{2}}$

The constant is written

k = \frac{1}{4\pi\varepsilon_0} = 8.99\times 10^{9}\ \frac{\text{N}\cdot\text{m}^{2}}{\text{C}^{2}}

where $\varepsilon_0 = 8.85\times 10^{-12}\ \dfrac{\text{C}^{2}}{\text{N}\cdot\text{m}^{2}}$ is the permittivity of free space. The $4\pi$ is a deliberate choice: it makes the field equations of the next chapter come out without a $4\pi$ , at the cost of putting one here.

"Point charge" means the charge's size is negligible compared with $r$ . For two uniformly charged spheres the law still holds exactly with $r$ measured centre-to-centre — the spherical-shell result borrowed from gravitation.

The direction is handled by inspection: the force on each charge points away from the other if the charges have the same sign, towards the other if opposite. The two forces form a Newton's-third-law pair — equal in magnitude, opposite in direction — even when the charges themselves are unequal.

4. The Vector Form and the Inverse-Square Dependence

Written as a vector, the force on charge $q_1$ due to charge $q_2$ is

\vec{F}_{12} = k\,\frac{q_1 q_2}{r^{2}}\,\hat{r}_{12}

where $\hat{r}_{12}$ is the unit vector pointing from $q_2$ towards $q_1$ . With signed charges substituted, a positive product gives a force along $\hat{r}_{12}$ (repulsion) and a negative product gives a force opposite to it (attraction) — the signs take care of the direction automatically.

The inverse-square dependence is the single most important feature. Doubling the separation quarters the force; tripling it divides the force by nine. A common exam task is a ratio: if $F_1$ acts at separation $r$ , then at separation $3r$ the force is $F_1/9$ , with no need to know $k$ , $q_1$ or $q_2$ .

5. The Principle of Superposition

Coulomb's law gives the force between two charges. For a charge in the presence of several others:

Superposition. The net electrostatic force on a charge is the vector sum of the forces exerted on it by every other charge individually, each computed from Coulomb's law as if the others were absent.

The other charges do not "screen" or modify each other's pull; the pair forces simply add as vectors.

Worked example — three charges in a line. Charges $q_1 = +3.0\ \mu\text{C}$ , $q_2 = -2.0\ \mu\text{C}$ and $q_3 = +4.0\ \mu\text{C}$ sit on a straight line at $x = 0$ , $x = 0.20\ \text{m}$ and $x = 0.50\ \text{m}$ . Find the net force on $q_2$ .

Force from $q_1$ on $q_2$ : separation $0.20\ \text{m}$ , product $(+3.0)(-2.0)$ negative, so attractive — $q_2$ is pulled towards $q_1$ , i.e. in the $-x$ direction.

F_{12} = \frac{(8.99\times 10^{9})(3.0\times 10^{-6})(2.0\times 10^{-6})}{(0.20)^{2}} = 1.35\ \text{N}\ \text{in the } -x \text{ direction}

Force from $q_3$ on $q_2$ : separation $0.30\ \text{m}$ , product $(-2.0)(+4.0)$ negative, so attractive — $q_2$ is pulled towards $q_3$ , i.e. in the $+x$ direction.

F_{32} = \frac{(8.99\times 10^{9})(2.0\times 10^{-6})(4.0\times 10^{-6})}{(0.30)^{2}} = 0.799\ \text{N}\ \text{in the } +x \text{ direction}

Net force: $F = 1.35 - 0.799 = 0.55\ \text{N}$ in the $-x$ direction (towards $q_1$ ).

6. Comparison with Newtonian Gravitation

The two inverse-square laws sit side by side:

F_E = \frac{1}{4\pi\varepsilon_0}\frac{|q_1 q_2|}{r^{2}} \qquad\qquad F_G = G\,\frac{m_1 m_2}{r^{2}}

Strength. For two protons, $\dfrac{F_E}{F_G} = \dfrac{k e^{2}}{G m_p^{2}} \approx 1.2\times 10^{36}$ . Gravity is utterly negligible on the atomic scale.
Sign. Mass is always positive, so gravity only attracts. Charge has two signs, so the electrostatic force does both — which allows charges to cancel and form neutral bulk matter, leaving gravity to dominate on astronomical scales.
Medium. $G$ is truly universal; $\varepsilon_0$ is replaced by $\varepsilon = \varepsilon_r\varepsilon_0$ in a material, and the force is reduced by the relative permittivity $\varepsilon_r$ .

7. Worked Examples

Example 1 — counting electrons. A plastic rod carries a charge of $-45\ \text{nC}$ . How many excess electrons does it hold?

n = \frac{|Q|}{e} = \frac{45\times 10^{-9}}{1.60\times 10^{-19}} = 2.8\times 10^{11}\ \text{electrons}

A charge that reads as "tiny" still involves hundreds of billions of elementary charges — quantisation is invisible at this scale, which is why classical electrostatics can treat charge as continuous.

Example 2 — proton–proton repulsion in a nucleus. Estimate the electrostatic force between two protons separated by $r = 2.0\times 10^{-15}\ \text{m}$ .

F = \frac{(8.99\times 10^{9})(1.60\times 10^{-19})^{2}}{(2.0\times 10^{-15})^{2}} = \frac{(8.99\times 10^{9})(2.56\times 10^{-38})}{4.0\times 10^{-30}} = 58\ \text{N}

Fifty-eight newtons pushing two particles of mass $1.7\times 10^{-27}\ \text{kg}$ apart implies an acceleration of order $10^{28}\ \dfrac{\text{m}}{\text{s}^{2}}$ . Nuclei hold together only because the short-range strong force is stronger still at this separation.

Example 3 — the equilibrium point. Two charges $+q$ and $+4q$ are a distance $d$ apart. Where on the line between them can a third charge be placed so that it feels no net force?

Let the point be a distance $x$ from $+q$ . Balancing magnitudes:

\frac{kq\,Q}{x^{2}} = \frac{k(4q)\,Q}{(d-x)^{2}} \;\Rightarrow\; (d-x)^{2} = 4x^{2} \;\Rightarrow\; d - x = 2x \;\Rightarrow\; x = \frac{d}{3}

The equilibrium sits one-third of the way from the smaller charge — closer to the weaker one, as expected, and independent of the sign or size of $Q$ .

Common Exam Pitfalls

Putting the sign of the charge into the magnitude formula and then also assigning a direction. Use $F = k|q_1 q_2|/r^{2}$ for the size, then decide "attract or repel" separately from the signs. Doing both double-counts.
Adding Coulomb forces as scalars. Superposition is a vector sum. Forces from charges on different sides, or at an angle, must be resolved into components first.
Forgetting to square the separation, or squaring only the number and not the power of ten: $(0.30)^{2} = 0.090$ , and $(3.0\times 10^{-6})$ is not squared unless the formula says so.
Using a radius instead of a centre-to-centre distance for charged spheres, or a surface-to-surface gap.
Treating $\varepsilon_0$ and $k$ as interchangeable. They are reciprocally related through $k = 1/(4\pi\varepsilon_0)$ ; substituting one where the other belongs is out by a factor of $4\pi$ .
Assuming the larger charge feels the larger force. By Newton's third law the two forces in a pair are equal in magnitude regardless of how different the charges are.

Summary

Charge is quantised ( $Q = ne$ , $e = 1.60\times 10^{-19}\ \text{C}$ ), conserved in every isolated system, and comes in two signs; like repels, unlike attracts.
Conductors allow charge to move freely; insulators do not. Objects are charged by friction, contact or induction; insulators near a charge become polarised.
Coulomb's law: $F = k\,\dfrac{|q_1 q_2|}{r^{2}}$ with $k = \dfrac{1}{4\pi\varepsilon_0} = 8.99\times 10^{9}\ \dfrac{\text{N}\cdot\text{m}^{2}}{\text{C}^{2}}$ and $\varepsilon_0 = 8.85\times 10^{-12}\ \dfrac{\text{C}^{2}}{\text{N}\cdot\text{m}^{2}}$ .
The force acts along the line of centres, obeys Newton's third law, and falls off as $\dfrac{1}{r^{2}}$ .
Superposition: the net force on a charge is the vector sum of the individual Coulomb forces from every other charge.
Electrostatics and gravitation share the inverse-square form; the electrostatic force is $\sim 10^{36}$ times stronger and can attract or repel, which is why it governs the structure of atoms and matter.