Assemble interaction energy without double counting

Define the assembly reference

For localized point charges, set interaction energy to zero when their pair separations are infinite. Bring charges in slowly with negligible final kinetic energy. The external work equals the change in interaction energy when no other energy transfer is included. Attraction can make that external work negative: the agent removes energy rather than supplies it.

Count each interaction once

Bringing in charge 1 costs no interaction energy with other charges yet. Charge 2 adds k q1 q2/r12. Charge 3 adds k q3(q1/r13 + q2/r23). Thus U = sum over i < j of k qi qj/rij. Alternatively U = (1/2) sum qi Vi, where Vi excludes the charge’s own potential. Omitting the half counts each pair twice; including self-potential of an ideal point charge introduces a singularity outside this model.

Separate source potential from pair energy

A single source charge produces V = kq/r relative to infinity. Multiple sources add as signed scalars, V = sum kqi/ri; no test-charge factor belongs in that source potential. For +2 nC and −1 nC each 1 m from an empty observation point, V = k(1 nC)/1 m ≈ 8.98755 V. Doubling only the negative source distance makes the result k(1.5 nC)/1 m ≈ 13.4813 V. A negative test charge then has negative potential energy at that point; it does not reverse the source potential.

Change geometry, not the reference

At fixed charges, uniform dilation by factor s divides every pair energy and the total by s. The total can vanish by cancellation even when individual pair forces and energies do not vanish. A zero system interaction energy does not imply every particle is at rest or every local electric field is zero.

Worked original case

Charges +2, −1 and +1 nC occupy (0,0), (1,0) and (0,1) m. In nJ, the three terms are −2k×10⁻⁹, +2k×10⁻⁹ and −k×10⁻⁹/sqrt(2). The total is approximately −6.35516 nJ; the first two cancel, not the third.

Investigation

Predict all three signed terms for third charges −1, 0 and +1 nC before checking. Double all separations at fixed charges. Compare sequential assembly in a different order and explain why total work stays fixed while intermediate work changes.

Changed-case practice

Sources and evidence boundary

Original teaching and cases checked 30 October 2026 against the current official framework, Unit 9 and OpenStax electric potential. All records are supplied or generated, not measured by a learner. No physical charging, battery connection, high-voltage equipment or particle experiment is requested. Retain predictions, derivations and critique in the course evidence workspace; public feedback is not independent mastery or performed laboratory time.