Electric Potential Energy
Choose a reference and count interaction pairs once when finding electric potential energy.
AP Physics 2 Topic 10.4 · College Board CED printed p. 49 / PDF p. 56 · See this topic in the course outline
Force describes a push at an instant. Potential energy describes a configuration and the work needed to assemble it.
For another explanation, read The Electric Potential and Potential Energy. Use the corrected algebraic point-charge energy discussion to connect configuration changes with external and electric work.
Before you start
- Work and conservative forces
- Signed charge and separation
Choose the system and the reference
Electric potential energy belongs to a configuration of interacting charges. For two point charges, choose U = 0 at infinite separation. Slowly bringing them from infinity to separation r, with no change in kinetic energy, requires external work equal to the change in potential energy: . Unlike a force magnitude, this expression keeps the signs of both charges.
Like-charge U is positive: assembling the pair works against repulsion. Opposite-charge U is negative: attraction can do positive work as the charges approach, so a slow external agent does negative work. In either case . If the charges speed up during assembly, external work is instead ΔK + ΔU; “work to assemble” equals U only for the stated slow, no-kinetic-change procedure.
| Pair | r | 2r | As r grows without bound |
|---|---|---|---|
| Like signs | +U₀ | +U₀/2 | Approaches zero from above |
| Unlike signs | −U₀ | −U₀/2 | Approaches zero from below |
A graph of U against r is an inverse-distance curve, rather than an inverse-square curve. Moving a repelling pair apart lowers U; moving an attracting pair apart raises U.
Count each pair exactly once
For three charges, the interactions are pairs 12, 13, and 23. The scalar sum is . For four charges there are six unique pairs. Do not double count “1 on 2” and “2 on 1”: they describe one interaction energy.
When one charge moves and all others stay fixed, only pairs involving that charge change energy. This gives a fast and physically clear way to find ΔU. The AP calculation scope is configurations of four or fewer point charges; computing extended distributions generally needs mathematics beyond this unit.
Three charges, three terms
Three charges +2.0 nC, +2.0 nC, and −2.0 nC occupy an equilateral triangle of side 0.10 m. Find the assembly energy relative to infinite separation.
Select all three charges as the system and list the three unique pairs.
Why this step? Potential energy is a property of the complete configuration.
Each pair magnitude is (8.99 × 10⁹)(2.0 × 10⁻⁹)²/0.10 = 3.60 × 10⁻⁷ J. The pair signs are +, −, and −.
Why this step? Each term keeps charge-product sign; energy terms add as scalars.
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Why this step? Count each pair once and use the common triangle separation.
Answer and check
U = −3.60 × 10⁻⁷ J. A slow external assembly agent has this negative net work: the electric interaction releases energy overall.
Assembly work needs a reference and a speed condition
Fix one point charge and begin with the second infinitely far away. Choose U = 0 for that separated configuration. An external agent brings the second charge to a finite separation while keeping its kinetic-energy change negligible. For this quasistatic assembly, W_ext = ΔU = kq₁q₂/r. The agent’s force balances the electric force closely, so force directions and work signs can be reasoned before calculation.
For two positive charges, repulsion opposes approach: the agent does positive work and U becomes positive. For opposite signs, attraction assists approach: the agent restrains the motion, does negative work, and U becomes negative. Electric work is −ΔU. If the charge is released and gains kinetic energy, the complete relation is W_ext = ΔK + ΔU; identifying assembly work with U is no longer justified without the negligible-ΔK condition. A fixed charge and its support have no displacement, so that support does no mechanical work in this model.
Positive and negative assembly work
A +2.0 nC charge is fixed in vacuum. An agent slowly brings a second charge from infinity to r = 0.10 m. Compare second charges +1.0 nC and −1.0 nC. Speeds remain negligibly small; k = 8.99 × 10⁹ N·m²/C².
- For +1.0 nC, final U = k(+2.0 × 10⁻⁹)(+1.0 × 10⁻⁹)/0.10 = +1.798 × 10⁻⁷ J. Initial U at infinity is zero.
- The agent moves inward against outward repulsion, so W_ext = +1.798 × 10⁻⁷ J; electric work is the negative of that.
- For −1.0 nC, final U = −1.798 × 10⁻⁷ J. The agent restrains inward attraction, doing W_ext = −1.798 × 10⁻⁷ J while electric work is positive.
- Both U curves scale as 1/r, not 1/r². As r grows, the like-sign curve tends to zero from above and the opposite-sign curve tends to zero from below.
Answer and check
Slow assembly work is +0.1798 μJ for like signs and −0.1798 μJ for unlike signs, with U = 0 at infinity.
Try it from memory
Write your answer and reason before opening the worked answer. Your self-assessment is a study decision; it is not an automatically graded score.
Question 1
A +q and −q pair changes separation from r to 2r. Is ΔU positive or negative? What does the electric force’s work do?
Worked answer and reason
ΔU = +kq²/(2r), so electric work is −kq²/(2r). Separating attracting charges requires energy input if their speed is kept fixed.
The initial U is −kq²/r and the final U is less negative.
Question 2
Four charges have how many pair-energy terms? If only charge 1 moves, how many terms can change?
Worked answer and reason
Six unique pairs; three involve charge 1 and can change.
Choose each unordered pair once; unchanged distances leave their energy terms unchanged.
JavaScript enables saved drafts and self-assessment. You can always work on paper.
Catch the tempting mistake
- Negative energy means negative force magnitude.
- Energy is a signed scalar relative to a reference. A force magnitude is nonnegative, with direction treated separately.
Put it to work
Give the physical reason as well as the calculation. The worked solutions are on a separate page.
- Problem 7 · energy versus separation
- Problem 8 · unique-pair sum and symbolic change
- Problem 17 · source symmetry and energy synthesis
Explain and apply the model
Try these problems before opening the solutions. Show the physical model, your calculation or sketch, and the reason your conclusion follows.
Bring a pair together from infinity
A +3.0 nC point charge is held fixed. A second point charge initially infinitely far away is brought to r = 0.20 m in vacuum. Compare second charges +2.0 nC and −2.0 nC in separate trials. Choose pair U = 0 at infinity and use k = 8.99 × 10⁹ N·m²/C².
(a) For an external quasistatic assembly with negligible kinetic-energy change, determine the final U, the work done by the external agent from infinity, and the electric-force work in each trial. Explain each external-work sign by relating the agent’s force to the displacement.
(b) Construct U-versus-r graphs for the two pairs on common quantitative axes. Plot at least r = 0.20, 0.40 and 0.80 m, label values and units, and describe the r → ∞ limit. Distinguish this scaling from the force magnitude.
(c) In a different opposite-sign trial, remove the agent and release the second charge from rest at infinity in the idealized limit. At r = 0.20 m, calculate ΔK and W_ext. Explain why the negative final U is not the external work for this free motion.
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For the teacher
- Ask students to narrate the sign of external work before multiplying numbers.
- Have them list pair labels to expose double counting.
Source objectives and scope
- 10.4.A — Describe the electric potential energy of a system.
As the methods to calculate the electric potential energy due to extended charge distributions exceed the scope of the course, AP Physics 2 only requires that students calculate the electric potential energy of configurations of four or fewer point charges.
Suggested skills are teaching guidance. The course applies all science practices across its objectives.
Science references for this teaching
- College Board AP Physics 2 Course and Exam Description, Unit 10 (PDF pages 50–61) — Official topic order, learning objectives, all essential-knowledge clauses, algebra-based scope boundaries; equations visually inspected in local source pages 50,54,56–61.
- OpenStax University Physics Volume 2, 8.5 Molecular Model of a Dielectric — Independent verification that bound polarization leaves a dielectric net neutral and its induced field opposes the free-charge field. All lesson prose and problems are original.
- Corrected original reading: Charge and Matter — Use the corrected charge and Coulomb-law explanations to compare signed charges and pair forces.
- Corrected original reading: Conductors and Insulators — Use the corrected charge-mobility discussion to explain contact, polarization, and grounding with a system boundary.
- Corrected original reading: The Electric Field — Use the corrected point-source and vector-superposition discussion to practice fields and forces. General quantitative fields within insulators are outside the AP core.
- Corrected original reading: The Electric Potential and Potential Energy — Use the corrected algebraic point-charge energy discussion to connect configuration changes with external and electric work.
- Corrected original reading: Calculating Potentials — Use the corrected point-charge scalar sums to practice potential.
- Corrected original reading: Equipotentials, Field lines, and Gradients — Use the corrected contour maps and algebraic field estimates here; calculus-gradient material is an optional extension beyond the AP core.
- Corrected original reading: Capacitors — Use the corrected parallel-plate geometry discussion to compare C, Q, and voltage.
- Corrected original reading: Stored Energy and Dielectrics — Use the corrected energy and dielectric discussion with an explicit fixed-Q or fixed-voltage constraint.
Related source reading
These alternative explanations and examples retain their source context. Draft placement does not add accepted exam coverage.
- electric potential reference · alternative source explanation
Glossary and deeper reading · 10 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Potential energy
Potential energy describes energy associated with the configuration of interacting parts of a system. Its numerical zero is a chosen reference, while differences in potential energy determine physical energy changes. For example, separating Earth and a ball changes their gravitational potential energy even though the energy belongs to their interaction.
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Mechanical work
Mechanical work is energy transferred by a force through displacement of its point of application. For a constant force on a particle, it depends on the force component along the particle’s displacement, and it can be positive, negative, or zero. For example, a magnetic force perpendicular to a particle’s displacement does zero work on that particle.
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Electric charge
Electric charge is a conserved property that determines electric interactions. It can be positive or negative and is measured in coulombs; ordinary net charges arise from an imbalance of charged particles. For example, transferring electrons between initially neutral objects leaves one negatively charged and the other positively charged.
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Conservative force
A conservative force does work that depends only on the starting and ending configurations. A potential-energy function can describe it, and its work around a closed path is zero. For example, near-Earth gravity does the same work between two heights whether a ball follows a vertical or a curved path.
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Kinetic energy
Kinetic energy is energy associated with motion relative to a chosen reference frame. For a nonrelativistic translating object it is one half of its mass times speed squared. For example, doubling a cart’s speed quadruples its translational kinetic energy when its mass stays fixed.
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Force
A force describes an interaction that can change an object’s momentum. It is a vector measured in newtons, so both size and direction matter when forces are added. For example, a rope can pull a cart while the floor exerts friction in another direction.
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Scalar
A scalar is a quantity described by a value and unit without a spatial direction. A scalar can have a sign when its definition or reference permits one; scalar does not mean always positive. For example, mass is a scalar, whereas the velocity of that mass also needs a direction.
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Electric force
Electric force is the interaction of an electric field with a charged object. For point charges at rest, like signs repel and opposite signs attract, with direction along their separation. For example, an electron and proton exert equal-magnitude opposite forces on one another despite having different masses.
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Displacement
Displacement is the vector change in position from the start of an interval to its end. It depends on those endpoints rather than the length or shape of the path traveled. For example, walking around a closed track and returning to the start gives zero displacement but a nonzero distance traveled.
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Magnitude
Magnitude is the nonnegative size of a vector quantity. It does not specify direction, and it differs from a signed component. For example, a westward velocity can have a negative x component while its speed, the velocity magnitude, stays positive.