Potential energy belongs to the pair of charges
You will be able to: Calculate signed pair energy and relate its change to electric and external work.
Why can an electric potential energy be negative?
Pushing two positive beads closer together takes effort. Letting opposite charges approach slowly allows an external agent to remove energy. The sign becomes meaningful after we choose zero energy at infinite separation.
A useful starting point: Electric field and force →
Words and symbols before equations
- Configuration energy U
- Energy associated with the arrangement of interacting charges, measured in joules (J).
- Reference zero
- The separation assigned U = 0; here the two charges infinitely far apart.
- External work W_ext
- Work by an agent moving the charges; equals ΔU only when kinetic energy stays fixed and no other energy transfer occurs.
- Electric work W_e
- Work by the electric force; W_e = −ΔU for the electrostatic interaction.
What this picture assumes
Two stationary point charges in vacuum, U = 0 at infinite separation. Quasistatic assembly has negligible kinetic-energy change. Positive radial force increases separation; graph height is energy, not physical altitude.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- U = -0.18 J; radial force -0.6 N. Slow assembly work from infinity equals U; electric work equals −U.
- 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 two stationary point charges in vacuum, U = kq₁q₂/r when U(∞) = 0. Retain the charge signs: equal signs give positive U and opposite signs negative U. Use k = 9.0 × 10⁹ N·m²/C² in this unit.
This is one interaction energy belonging to the pair, not an independent copy assigned to each charge. Negative U means the configuration is below the chosen separated reference; it does not mean negative kinetic energy.
Moving the charges quasistatically means so slowly that ΔK is negligible. Then W_ext = ΔU and W_e = −ΔU. For faster motion the external-work balance also includes ΔK. Force comes from the slope, F_r = −dU/dr, not from the sign of U alone.
A worked example, step by step
A +2 μC charge and a −3 μC charge are brought from infinite separation to 0.30 m, slowly. Find U and the external work.
- Convert the charges to +2 × 10⁻⁶ and −3 × 10⁻⁶ C.
- U = (9 × 10⁹)(−6 × 10⁻¹²)/0.30 = −0.18 J.
- The initial energy is zero, so ΔU = −0.18 J.
- W_ext = −0.18 J: the slow-moving agent removes energy; the electric force does +0.18 J of work.
Negative potential energy is allowed; kinetic energy cannot be negative. Never count a pair twice.
Does doubling r halve U or quarter it?
Compare with an explanation
It halves U, including its sign. Force magnitude instead scales as 1/r².
Predict. Change one thing. Explain.
Reverse one charge and vary separation. Predict the sign of U and whether bringing the pair closer raises or lowers it. Compare force with the slope of the energy curve.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
U = -0.18 J; radial force -0.6 N. Slow assembly work from infinity equals U; electric work equals −U.
Two stationary point charges in vacuum, U = 0 at infinite separation. Quasistatic assembly has negligible kinetic-energy change. Positive radial force increases separation; graph height is energy, not physical altitude.
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 questionTwo +1 μC charges move slowly from r = 1 m to r = 0.50 m. Find ΔU, electric work and external work.
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Compare with the answer and four-point rubric
- 1 point: U_i = 0.009 J and U_f = 0.018 J.
- 1 point: ΔU = +0.009 J.
- 1 point: W_e = −0.009 J.
- 1 point: With negligible ΔK, W_ext = +0.009 J.
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 1Where does pair energy reside conceptually?
In the interacting configuration, not twice in separate particles.
RECALL 2When is W_ext equal to ΔU?
For constant kinetic energy with no other work or energy transfer.
RECALL 3What does a negative U mean?
Energy below the chosen reference configuration.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Potential energy belongs to the pair of charges
- U_pair = kq₁q₂/r with U(∞) = 0.
- W_e = −ΔU.
- W_ext = ΔU for quasistatic transfer with no other work.
Remember: Negative potential energy is allowed; kinetic energy cannot be negative. Never count a pair twice.
Conditions: Two stationary point charges in vacuum, U = 0 at infinite separation. Quasistatic assembly has negligible kinetic-energy change. Positive radial force increases separation; graph height is energy, not physical altitude.
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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