Count each charge pair exactly once
You will be able to: Sum unique pair energies and verify the result by sequential assembly.
How much energy does it take to assemble three charges?
Bring the first charge into empty space: there is no interaction energy yet. The second interacts with the first. The third interacts with both charges already present. Three charges create three unique pairs.
A useful starting point: Potential energy belongs to the pair of charges →
Words and symbols before equations
- Unique pair
- One interaction between two different charges, counted once.
- Assembly work
- External work to build a configuration slowly from infinite separation.
- Separation r_ij
- Distance between charges i and j.
- Fixed configuration
- Charge positions are specified and held; this calculation is not a motion simulation.
What this picture assumes
Three held point charges on an equilateral triangle; zero interaction energy at infinite separation. Unique pair energies only, with no point-charge self-energy. Positions are fixed; this is not a motion simulation.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Equilateral side 1 m. Total U = -0.009 J. Each pair appears once; the first charge has zero assembly interaction work.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
For three charges, U = kq₁q₂/r₁₂ + kq₁q₃/r₁₃ + kq₂q₃/r₂₃. Each term is signed. Do not add magnitudes or include a point charge’s divergent self-energy.
The same answer comes from bringing in each new charge against the potential of charges already assembled. The first costs zero, the second contributes the 1–2 pair, and the third contributes pairs 1–3 and 2–3.
Assembly order does not change the final electrostatic energy. If all q_iV_other,i terms are summed, every pair appears twice; use U = ½Σq_iV_other,i. This factor does not belong in the energy qV of a single probe in fixed source potential.
A worked example, step by step
Charges +1 μC, +1 μC and −1 μC occupy an equilateral triangle of side 1 m. Find total energy.
- List the three pairs: 1–2, 1–3, 2–3.
- Their energies are +0.009 J, −0.009 J and −0.009 J.
- Add signed values: U = −0.009 J.
- The positive pair costs energy to assemble, but the two attractive pairs lower the total by more.
Sum distinct pairs once; qV_other summed over every charge needs a factor of one half.
Can the total be zero although individual pair energies are nonzero?
Compare with an explanation
Yes. Positive and negative pair contributions can cancel.
Predict. Change one thing. Explain.
Change the sign of the third charge and compare the three pair bars. Change triangle size at fixed charges and predict how the total scales.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Equilateral side 1 m. Total U = -0.009 J. Each pair appears once; the first charge has zero assembly interaction work.
Three held point charges on an equilateral triangle; zero interaction energy at infinite separation. Unique pair energies only, with no point-charge self-energy. Positions are fixed; this is not a motion simulation.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant potential, energy, work, field-gradient or line-integral relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionPlace +1 μC at x = 0, −1 μC at x = 1 m, and +2 μC at x = 2 m. Calculate the three pair terms and total U.
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Compare with the answer and four-point rubric
- 1 point: U₁₂ = −0.009 J because r₁₂ = 1 m.
- 1 point: U₁₃ = +0.009 J because r₁₃ = 2 m.
- 1 point: U₂₃ = −0.018 J because r₂₃ = 1 m.
- 1 point: Total U = −0.018 J, counting each pair once.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1How many pairs for N charges?
N(N − 1)/2.
RECALL 2Does assembly order affect final electrostatic U?
No, if the final configuration and reference are the same.
RECALL 3Why omit self-energy here?
The model describes interactions between distinct point charges.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Count each charge pair exactly once
- U = Σ_(i<j) kq_iq_j/r_ij.
- For a complete finite configuration, U = ½Σq_iV_other,i.
- Exclude self-potentials from V_other.
Remember: Sum distinct pairs once; qV_other summed over every charge needs a factor of one half.
Conditions: Three held point charges on an equilateral triangle; zero interaction energy at infinite separation. Unique pair energies only, with no point-charge self-energy. Positions are fixed; this is not a motion simulation.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 2 (official Unit 9) · Objectives 9.1.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 9.1, objectives 9.1.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. This is E&M Unit 2: Electric Potential, numbered Unit 9 in the official combined Physics C sequence. Topics 9.1–9.3 retain their official identifiers. Models assume electrostatic fields and state whether source charges are fixed or free. Potential integrals use the specified rods, ring, arc and infinite line or cylinder. Infinite-line examples use a finite reference radius, not zero potential at infinity. The optional 3D equipotential surface uses height to represent volts, not a particle trajectory. Checked with the Fall 2026 clarifications. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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