Assemble a signed three-charge energy ledger

Establish the model and evidence

Electric potential energy belongs to a system of interacting charges. Choosing zero when charges are infinitely far apart gives U=kq₁q₂/r for a pair. Same-sign pairs have positive energy and opposite-sign pairs negative energy. This sign is distinct from the always-nonnegative magnitude of force. It describes how much energy an external agent supplies or removes during a quasistatic assembly.

Three charges have three unique pairs: 1–2, 1–3 and 2–3. Add each pair once. Summing the potential due to all other charges at every charge and multiplying by its charge would count each interaction twice unless corrected. A pair ledger makes the bookkeeping visible without requiring a new formula or calculus.

The activity uses an equilateral triangle so every separation is the same side length. Charges +2 nC and +3 nC are fixed while the third charge can change sign. In the continuous formula, changing the negative third charge can make the total positive, zero or negative; zero occurs at −1.2 nC. This control samples only integer nanocoulombs, so it brackets that zero rather than displaying it exactly. None of the pair forces vanish at the zero-total-energy setting. A zero total energy is not a stable-equilibrium criterion and does not mean the individual interactions disappear.

For slow assembly with unchanged kinetic energy, external work equals the system energy change and electric work has the opposite sign. If the external work is negative, the agent must remove energy rather than supply it. If charges are released instead, their kinetic energies can change; the fixed-geometry model does not predict the resulting many-body trajectories.

Worked evidence

For charges +2, +3 and −1 nC with side 0.20 m, the pair energies are 269.7, −89.9 and −134.85 nJ. Their sum is +44.95 nJ. Doubling every separation halves each pair energy and the total; it quarters each pair force magnitude.

Predict and investigate

Will doubling all separations change potential energy by the same factor as force?

Retain side 0.20 and 0.40 m with third charge −1 nC. Check each pair and the total. Then use third charge −2 nC and explain why the total changes sign without a sign change in every pair.

Transfer to changed conditions

Two charges +2 nC and −3 nC are moved quasistatically from 0.30 m to 0.15 m. Find ΔU, electric work and external work.

Compare the worked transfer

Initial U=−179.8 nJ and final U=−359.6 nJ, so ΔU=−179.8 nJ. Electric work is +179.8 nJ; external work is −179.8 nJ when kinetic energy is unchanged.

Explain the boundary

Why must an energy ledger count pairs once, and why does a zero total not prove zero force?

Sources and authorship

Physical relationship reference. Current AP Physics 2 framework, checked 30 October 2026. These cases, diagrams and numerical records are original. All experiments are bounded browser calculations or synthetic data, not measured laboratory evidence. Do not construct or handle charged capacitors, high-voltage sources or particle-beam equipment for these activities.