Coulomb's Law: Electric Force Between Charges

Goal

Derive the magnitude and direction of the electrostatic force between two point charges.

1) Symmetry and assumptions

For two point charges in empty space, the force must:

• lie along the line joining the charges,
• depend only on the separation distance $r$,
• be attractive for opposite charges and repulsive for like charges.

So the force vector must have the form

$$\vec F \propto \frac{1}{r^2}\,\hat r$$

where $\hat r$ is the unit vector along the line connecting the charges.

2) Inverse-square dependence

Experiment shows that the electric force weakens with distance exactly like gravity: doubling the distance makes the force four times smaller. This gives

$$|\vec F| \propto \frac{1}{r^2}.$$

3) Dependence on charge

The force should also increase with the amount of charge on each object, and be proportional to both charges:

$$|\vec F| \propto |q_1 q_2|.$$

Combining this with the distance dependence:

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

4) Introducing the constant of proportionality

We define the proportionality constant $k$ (or $\frac{1}{4\pi\varepsilon_0}$ in vacuum):

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

In vector form,

$$\vec F_{12} = k\frac{q_1 q_2}{r^2}\,\hat r_{12}.$$

The sign of $q_1 q_2$ determines whether the force is repulsive or attractive.

5) Final law

In vacuum,

$$\boxed{\vec F_{12} = \frac{1}{4\pi\varepsilon_0}\,\frac{q_1 q_2}{r^2}\,\hat r_{12}}$$

where

$$\frac{1}{4\pi\varepsilon_0} \approx 8.99\times10^9\ \text{N·m}^2/\text{C}^2.$$

What the visualization shows

• Two charges on a line.
• Electric field lines and a force arrow.
• A distance slider showing the inverse-square drop in force.
• Charge sign toggles to switch between attraction and repulsion.