Result

Energy of a pair

(E3.3.2) E3.3 · Electric Potential and Potential Energy Due to Point Charges →

U = k_{e} \frac{q _{1} q _{2}}{r _{12}}

Statement

The work needed to bring two charges from far apart to a separation r₁₂.

Symbols and units

$U$	potential energy of the pair	J
$q_{1}, q_{2}$	the two charges, with sign	C
$r_{12}$	their separation	m
$k_{e}$	Coulomb constant	8.99×10⁹ N·m²/C² (=1/4πε₀)

When it applies

Point charges at rest; sum over distinct pairs.

Don't use when

Must be summed over distinct pairs, not all combinations — three charges give three pair terms, not six.

Derivation

Place the first charge — nothing to work against, so no cost.

The second charge is moved into the potential the first one has already made:

U = q_{2} V_{1} = q_{2} (k_{e} \frac{q _{1}}{r _{12}})

Hence:

U = k_{e} \frac{q _{1} q _{2}}{r _{12}}

For a group, add one term for every distinct pair:

U = k_{e} i < j \sum \frac{q _{i} q _{j}}{r _{ij}}

Worked examples

Three charges at the corners of a triangle

Equal charges q, side a, three distinct pairs:

U = \frac{3 k _{e} q ^{2}}{a}

Positive U: work was required to assemble the configuration, and it would separate if released.

Watch out for

Sum over pairs, not over combinations — three charges give three terms, not six. Positive U means the arrangement would fly apart if released; negative means it is bound.

Connects to

Point charge Stored energy