Coulomb's Law: The Force Between Charges

From 'they repel' to a formula

In the 1780s Charles-Augustin de Coulomb measured the force between two small charged spheres using a delicate torsion balance — a fibre that twists by a measurable angle when the charges push or pull. Varying the charges and the distance, he found the pattern that now bears his name. It is the electrostatic analogue of Newton's law of gravity, and just as fundamental.

F = k\,\frac{|q_1\,q_2|}{r^2}

Coulomb's law: the force between two point charges is proportional to the product of the charges and inversely proportional to the square of their separation.

Here q_1 and q_2 are the two charges, r is the distance between them, and k is Coulomb's constant. The formula assumes point charges — objects small compared with r — but it also works exactly for the field outside any spherically symmetric charge, so charged spheres count too. Coulomb's law is the whole engine of electrostatics; almost everything later is built from it.

The constant, and how strong the force really is

k = \frac{1}{4\pi\varepsilon_0} \approx 8.99\times10^{9}\ \text{N}\cdot\text{m}^2/\text{C}^2

Coulomb's constant, written in terms of the permittivity of free space ε₀ ≈ 8.85×10⁻¹² C²/(N·m²).

The 4\pi looks fussy but pays off later (it makes Gauss's law tidy). What the size of k tells you is that the electric force is staggeringly strong. Two charges of one coulomb each, held one metre apart, would repel with about 9\times10^{9} newtons — the weight of a million tonnes. That is why you never see a coulomb of net charge sitting on an object; matter simply cannot hold that much separated charge together.

Direction: read it off the signs

The formula above uses |q_1 q_2| to give the magnitude. The direction is always along the straight line joining the two charges, and the sign of the product tells you which way: if q_1 q_2 > 0 (like charges) the force is repulsive, pushing them apart; if q_1 q_2 < 0 (opposite charges) it is attractive, pulling them together. And by Newton's third law the two charges always feel equal and opposite forces — even when their charges are wildly different in size.

A worked example

Problem. A charge q_1 = +3.0\ \mu\text{C} and a charge q_2 = -5.0\ \mu\text{C} are held 0.20\ \text{m} apart. Find the force between them.

Step 1 — convert units. q_1 = 3.0\times10^{-6} C, q_2 = 5.0\times10^{-6} C (magnitude), r = 0.20 m so r^2 = 0.040\ \text{m}^2. Step 2 — put numbers into Coulomb's law.

F = (8.99\times10^{9})\,\frac{(3.0\times10^{-6})(5.0\times10^{-6})}{0.040} \approx 3.4\ \text{N}

The magnitude of the force. Because the charges are opposite, it is attractive.

Step 3 — state direction and sense-check. The product q_1 q_2 < 0, so the force is attractive — each charge is pulled toward the other along the line joining them, with equal $3.4$ N on each. A 3.4 N pull is roughly the weight of a large apple; substantial for microcoulomb charges 20 cm apart, exactly as the strength of k warned us.

Many charges: the superposition rule

Real problems have more than two charges. The saving grace is that Coulomb forces simply add as vectors. To find the force on one charge from several others, compute the Coulomb force from each source separately — ignoring the others entirely — then add those force vectors tip-to-tail. Each pair acts as if the rest were not there.

\vec{F}_1 = \vec{F}_{1\leftarrow2} + \vec{F}_{1\leftarrow3} + \cdots = \sum_{i\neq 1}\vec{F}_{1\leftarrow i}

The principle of superposition for forces: the net force on charge 1 is the vector sum of the separate Coulomb forces from every other charge.