Energy belongs to interacting charges
You will be able to: Calculate signed pair energy and count each interaction once.
What changes when you push like charges closer?
Pushing two positive charges together requires external work if you move them slowly. Their interaction can later supply kinetic energy when they are released. Electric potential energy describes the configuration, not a substance stored in one isolated charge.
A useful starting point: Fields inside and around conductors →
Words and symbols before equations
- Potential energy U
- Energy associated with the charge configuration, in J.
- Zero reference
- For point-charge pairs here, U=0 at infinite separation.
- Pair energy
- U_ij=kq_iq_j/r_ij, including charge signs.
- Quasistatic external work
- For negligible kinetic-energy change, W_ext=ΔU when only electric and external forces matter.
What this picture assumes
Vacuum point-charge pair +2 μC and ±1 μC. Potential energy zero at infinite separation. Signed U, in J, is a scalar. The graph does not include r=0, where the point model is singular.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- U=-0.018 J for +2 μC and −1 μC at r=1 m. Bringing unlike charges closer lowers U (more negative).
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
Like-sign pairs have positive U relative to infinity; opposite-sign pairs have negative U. Decreasing r raises U for like signs but lowers U for opposite signs. Negative potential energy is allowed because the zero is a chosen reference.
For several point charges, add the scalar energy of every distinct pair exactly once. Three charges have three pairs: 12, 13 and 23. Do not attach a direction arrow to these scalar energies.
Work by the electric interaction is W_electric=−ΔU. When an external agent moves charges slowly, its work has the opposite sign. If the charges accelerate, include kinetic energy before identifying external work with ΔU.
A worked example, step by step
Charges +2 μC and −1 μC move from 1 m separation to 0.5 m. Find ΔU and electric work.
- U_i=9×10⁹(−2×10⁻¹²)/1=−0.018 J.
- U_f=−0.036 J.
- ΔU=−0.018 J; the configuration loses potential energy.
- W_electric=+0.018 J. Slow external control would do −0.018 J of work.
Include charge signs in U. Sum distinct pairs once; do not confuse U with field-vector addition.
Can U be negative?
Compare with an explanation
Yes. For unlike charges with zero at infinity, the bound configuration has negative U.
Predict. Change one thing. Explain.
Change separation and the second charge’s sign. Compare the signed energy curve. Read how bringing the pair closer affects U, not merely its absolute magnitude.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
U=-0.018 J for +2 μC and −1 μC at r=1 m. Bringing unlike charges closer lowers U (more negative).
Vacuum point-charge pair +2 μC and ±1 μC. Potential energy zero at infinite separation. Signed U, in J, is a scalar. The graph does not include r=0, where the point model is singular.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant charge, field or energy 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 questionThree +1 μC charges form an equilateral triangle of side 1 m. (a) Count pairs. (b) Find one pair’s U. (c) Find total U. (d) Find external work to assemble slowly from infinity.
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Compare with the answer and four-point rubric
- 1 point: Three distinct pairs.
- 1 point: Each pair U=0.009 J.
- 1 point: Total U=0.027 J.
- 1 point: W_ext=+0.027 J if kinetic energy does not change.
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 1What is U’s zero here?
Infinite separation of the point charges.
RECALL 2Electric work sign?
W_electric=−ΔU.
RECALL 3Is pair energy a vector?
No. Add signed scalar values.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Energy belongs to interacting charges
- U_pair=kq₁q₂/r with zero at infinity.
- U_total=Σ distinct-pair energies.
- W_electric=−ΔU; quasistatic W_ext=ΔU.
Remember: Include charge signs in U. Sum distinct pairs once; do not confuse U with field-vector addition.
Conditions: Vacuum point-charge pair +2 μC and ±1 μC. Potential energy zero at infinite separation. Signed U, in J, is a scalar. The graph does not include r=0, where the point model is singular.
Refresh Kid · AP Physics 2 Unit 2 (official Unit 10) · Objectives 10.4.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.4, objectives 10.4.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Refresh Kid calls this the second AP Physics 2 unit; College Board numbers it Unit 10; the first unit in this course is official Unit 9. 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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