Coulomb’s Law: Inverse-Square Electrostatic Force

Coulomb’s Law

The electrostatic force between two point charges is

$$F=\frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r^2}$$

where:

• $q_1$ and $q_2$ are the charges,
• $r$ is the distance between them,
• $\epsilon_0$ is the permittivity of free space.

How this form arises

The law can be understood from two key experimental facts:

1. Dependence on charge: the force increases proportionally with the amount of charge on each body.
2. Inverse-square dependence on distance: when the separation doubles, the force becomes one-fourth as large.

These observations combine into

$$F \propto \frac{q_1 q_2}{r^2}$$

To turn this proportionality into an equation, we introduce a constant of proportionality:

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

In SI units, this constant is

$$k = \frac{1}{4\pi\epsilon_0}$$

so the final result becomes

$$F=\frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r^2}$$

Interpretation

• If $q_1 q_2 > 0$, the force is repulsive.
• If $q_1 q_2 < 0$, the force is attractive.
• The force acts along the line joining the two charges.

Visual meaning

This visualization shows two charges and the force between them. You can change:

• the value of the charges,
• their separation,
• whether they attract or repel.

As distance increases, the force arrows shrink rapidly according to the inverse-square law.