Keep source potential separate from pair energy

Define the source and measurement boundary

Potential is a scalar source property at a point, with units joules per coulomb. The potential created by several point charges is the signed sum kQ_i/r_i. Multiplying that potential by a newly introduced passive charge gives its interaction energy with those sources. It does not automatically include energy already stored in interactions among the original sources.

Derive the quantitative comparison

In this triangle, +2 nC is at (0,0), +1 nC at (L,0) and q3=a nC at (0,L). The first pair contributes 2k×10⁻¹⁸/L joules. The third-site potential before insertion is k×10⁻⁹[2+1/√2]/L volts. Adding q3 therefore contributes k×10⁻¹⁸[2a+a/√2]/L joules. Sum the three unordered pairs once; counting the energy of every charge in the potential of all the others would count each pair twice.

Test an alternative before claiming identification

Quasistatic external assembly work equals the interaction-energy change when final kinetic energy is negligible. Negative work means an external agent could extract energy under that idealization, not that a negative amount of physical energy was created. Doubling all separations halves every pair term. Total net charge alone does not determine energy: equal totals can have different pair products and geometry. The source self-energy at its own location is excluded, and infinity is the reference for this localized configuration.

Worked original case

At a=−1 nC and L=2 m, the dimensionless pair coefficient is 2−2−1/√2=−1/√2. The total is −k×10⁻¹⁸/(2√2)≈−3.17757941 nJ. The first positive pair stores positive energy, but adding the negative charge releases a larger amount.

Investigation

Compare third charges −3,0,+3 nC at one geometry. Retain first-pair energy and third-insertion work separately. Then double L. Explain why removing the third charge leaves nonzero first-pair energy even though the third insertion term vanishes.

Changed-case practice

Sources and evidence boundary

Original teaching, numerical cases and diagrams checked 30 October 2026 against the relevant primary textbook explanation. No textbook exercise, image or passage is reproduced. These are bounded synthetic records, not observed apparatus. No charging, wiring, capacitor handling, medical application or safety certification is requested. Preserve first predictions, contrasting settings and critiques in the course evidence workspace; public feedback is not independent mastery or performed laboratory time.