Electrostatic potential energy

The Coulomb force, like gravity, is a conservative force: the work it does does not depend on the path taken but only on the initial and final positions of the charges. Every conservative force has a corresponding potential energy, and Coulomb’s has a particularly simple form.

Key formula — Electrostatic potential energy

$\ev{E_{\text{pot}} = k\,\frac{q_1\,q_2}{r}}$ Positive if the charges repel each other (same sign), negative if they attract (opposite signs).

Note the difference from the force law: the potential energy goes as $1/r$, not $1/r^2$. And above all, unlike gravitational potential energy $mgh$ — always referred to a height and with mass always positive — here the signed product $q_1 q_2$ appears. This is the crucial novelty of electrostatics.

Can be positive or negative

Coulomb potential energy plays the same role as $E_{\text{pot,grav}}$, but with a decisive difference: it can be either positive or negative. Two opposite charges have negative potential energy (they are “bound”, energy must be supplied to pull them apart); two like charges have positive energy (they tend to flee each other, releasing energy as they move apart).

For a system of $N$ charges the total potential energy is the sum over all distinct pairs:

$E_{\text{pot,tot}} = \sum_{\text{pairs}} k\,\frac{q_i\,q_j}{r_{ij}}$

This energy is added to the energy table alongside $E_{\text{cin}}$, $E_{\text{pot,grav}}$ and $E_{\text{term}}$: from this point on it is possible to solve collision or motion problems between charged particles with the usual conservation of energy method, exactly as was done for springs and gravity.

Links

Topics: Electrostatics Concepts: Electrostatic potential energy · Coulomb’s law Skills: Conservation of energy

Related exercises: Work to separate two charges · Worked exercise — Three charges at the vertices of a triangle · Problem — Force between two 1-microcoulomb charges