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AP Course

AP Physics 2

Study AP Physics 2 Units 9–15, with the Fall 2026 radioactive-decay clarification applied and official exam updates linked separately.

Study Units 9–15 through original models, derivations, experiments, quizzes, and practice sets.

Lessons
Particle Motion Behind Temperature and PressureConnecting Gas Variables with the Ideal-Gas ModelHeat Flow and the Approach to Thermal BalanceEnergy Accounting for Thermodynamic ProcessesMaterial Response to Heating and Heat ConductionEntropy, Probability, and the Direction of Thermal ChangeCharge Interactions and Coulomb-Force ModelsTracking Charge During Contact and InductionMapping Electric Fields from Source ChargesEnergy of Configurations of ChargesElectric Potential and Equipotential ReasoningCapacitance and Energy Stored in Electric FieldsConserving Energy in Charged-Particle MotionCharge Flow and Conventional CurrentModeling Sources, Wires, and Loads in Simple NetworksHow Geometry and Material Set ResistanceElectrical Energy Transfer and PowerReducing Series-Parallel DC NetworksLoop Equations from Energy ConservationJunction Equations from Charge ConservationTransient Charging and Discharging in RC NetworksSources, Direction, and Strength of Magnetic FieldsMagnetic Forces and Curved Paths of ChargesForces Between Fields and Current-Carrying ConductorsChanging Magnetic Flux and Induced EMFReflection and Absorption at Material BoundariesLocating Images with Mirror Ray ModelsRefraction, Index, and Total Internal ReflectionLocating Images with Thin-Lens ModelsPulses, Wave Types, and Propagation SpeedPeriodic-Wave Measures and GraphsBoundary Changes and PolarizationElectromagnetic-Wave BehaviorFrequency Shifts from Relative MotionSuperposition, Interference, and Standing WavesDiffraction from Openings and EdgesTwo-Slit and Grating Pattern GeometryPhase Change and Thin-Film ColorQuantum Models and Dual Wave-Particle EvidenceEnergy Levels in a Hydrogen-Like AtomConnecting Spectral Lines to Energy TransitionsThermal Spectra and Blackbody CurvesPhoton Thresholds in the Photoelectric EffectPhoton-Electron Scattering and MomentumMass-Energy Accounting in Fission and FusionRandom Nuclear Decay, Activity, and Half-Life
Quizzes
Practice Problems Formula notes, diagrams, and practice sets will be added here.

AP Physics 2 · Unit 10 · Topic 10.4

Energy of Configurations of Charges

Describe a charge arrangement with one scalar energy instead of tracking every force along a path. Interpret the sign of electric potential energy, calculate changes as separations vary, assemble many-charge systems pair by pair, and connect lost potential energy to kinetic energy.

Learning Goals

  • Explain why electric potential energy belongs to a system of interacting charges.
  • Use the zero-energy reference at infinite separation and interpret positive or negative \(U\).
  • Calculate \(U\), \(\Delta U\), electric-force work, and slow external work for two point charges.
  • Find the energy of a many-charge configuration by summing every distinct pair once.
  • Apply conservation of energy to release, closest-approach, and assembly situations.
\(U\)electric potential energyjoule, J
\(W_{\mathrm e}\)work by electric force\(W_{\mathrm e}=-\Delta U\)
\(W_{\mathrm{ext}}\)slow external work\(W_{\mathrm{ext}}=\Delta U\)
\(r\)pair separationcenter to center

1. Energy Belongs to the Configuration

Electric potential energy is not stored in one isolated charge. It describes the interaction of at least two charges and depends on their relative positions. Moving either charge changes the same system energy.

Always identify the system: two charges, all charges in an arrangement, or charges plus an external device. The energy equation must match that boundary.

2. Electrostatic Force Is Conservative

For fixed source charges, the work done by the electrostatic force depends only on initial and final configurations, not on the route between them:

\(W_{\mathrm e}=-\Delta U=U_i-U_f\)

A closed trip returns to the same configuration, so \(\Delta U=0\) and the net electrostatic work is zero.

3. Choosing the Zero and Building the Two-Charge Formula

For point charges, infinite separation is the convenient reference because their interaction vanishes there: \(U(\infty)=0\). Bringing \(q_2\) slowly from infinity to distance \(r\) from fixed \(q_1\) changes the system energy by the work of the balancing external force.

\(U(r)=k_e\frac{q_1q_2}{r}\qquad\text{with }U(\infty)=0\)

The charge product keeps its algebraic sign. Unlike the force-magnitude formula, the energy formula must not replace \(q_1q_2\) with \(|q_1q_2|\).

4. Positive Energy: Like Charges

If \(q_1q_2>0\), then \(U>0\). An external agent must do positive work to push like charges closer slowly. If released, the charges repel, separation grows, and \(U\) falls toward zero while kinetic energy can increase.

5. Negative Energy: Opposite Charges

If \(q_1q_2<0\), then \(U<0\). The configuration has less energy than the separated reference state. Positive external work is required to pull the pair apart to infinity; the pair is energetically bound in this ideal model.

The sign comes from the charge productBoth curves approach the zero reference as separation grows. Moving like charges closer raises \(U\); moving unlike charges closer lowers \(U\).
Worked Example 1 · Like Charges

Energy Above the Separated Reference

Charges \(+2.0\,\mu\mathrm C\) and \(+3.0\,\mu\mathrm C\) are \(0.50\,\mathrm m\) apart.

\(U=(8.99\times10^9)\frac{(2.0\times10^{-6})(3.0\times10^{-6})}{0.50}\approx+0.108\,\mathrm J\)

The positive sign means this configuration lies \(0.108\,\mathrm J\) above the infinite-separation reference.

Worked Example 2 · Opposite Charges

Negative Configuration Energy

Charges \(+5.0\,\mathrm{nC}\) and \(-2.0\,\mathrm{nC}\) are separated by \(0.10\,\mathrm m\).

\(U=(8.99\times10^9)\frac{(5.0\times10^{-9})(-2.0\times10^{-9})}{0.10}\approx-8.99\times10^{-7}\,\mathrm J\)

The negative sign does not mean energy was calculated incorrectly. It indicates an attractive, bound configuration relative to infinity.

6. A Change in Separation Requires \(\Delta U\)

When a pair moves from \(r_i\) to \(r_f\), subtract the initial energy from the final energy:

\(\Delta U=U_f-U_i=k_eq_1q_2\left(\frac1{r_f}-\frac1{r_i}\right)\)

Do not report \(U_f\) when the question asks for a change. The initial configuration may already contain substantial positive or negative energy.

Worked Example 3 · Work and Energy Change

Push Like Charges Closer Slowly

Charges \(+4.0\,\mathrm{nC}\) and \(+6.0\,\mathrm{nC}\) move slowly from \(0.40\,\mathrm m\) apart to \(0.10\,\mathrm m\) apart.

\(\Delta U=(8.99\times10^9)(24\times10^{-18})\left(\frac1{0.10}-\frac1{0.40}\right)\approx+1.62\times10^{-6}\,\mathrm J\)

For a slow move with negligible kinetic-energy change, \(W_{\mathrm{ext}}=+1.62\,\mu\mathrm J\). The electric force does \(W_{\mathrm e}=-1.62\,\mu\mathrm J\).

7. Three Work Statements That Must Stay Separate

  • Electric-force work: \(W_{\mathrm e}=-\Delta U\).
  • Slow external work: \(W_{\mathrm{ext}}=\Delta U\) when \(\Delta K\approx0\).
  • Net work: \(W_{\mathrm{net}}=\Delta K\).

If the charge accelerates during the move, do not automatically set external work equal to \(\Delta U\).

8. Energy Conservation for an Isolated Charge System

When no energy crosses the chosen boundary and only conservative electric forces do work,

\(K_i+U_i=K_f+U_f\)

A decrease in electric potential energy becomes an equal increase in total kinetic energy. If several charges move, \(K\) means the sum of their kinetic energies.

Worked Example 4 · Released Like Charges

Convert Interaction Energy into Kinetic Energy

Two \(+1.0\,\mu\mathrm C\) charges begin at rest \(0.50\,\mathrm m\) apart. One is fixed; the other has mass \(0.010\,\mathrm{kg}\). Find the moving charge's speed when it is very far away.

\(U_i=k_e\frac{q^2}{r_i}=(8.99\times10^9)\frac{(1.0\times10^{-6})^2}{0.50}=1.80\times10^{-2}\,\mathrm J\)
\(\frac12mv^2=U_i\quad\Rightarrow\quad v=\sqrt{\frac{2U_i}{m}}\approx1.90\,\mathrm{m/s}\)

At infinite separation \(U_f=0\). The support holding the first charge fixed has zero displacement, so it does no mechanical work in this ideal setup.

9. Many Charges: Add Pair Energies as Scalars

For \(N\) point charges, calculate the energy of every distinct pair and add:

\(U_{\text{system}}=k_e\sum_{i

The condition \(i

Three charges make three distinct pairsCompute \(U_{12}\), \(U_{13}\), and \(U_{23}\) separately. Positive and negative contributions can partially or completely cancel in the scalar sum.

10. Assembly Method

  1. Bring the first charge from infinity. With no other charge present, no interaction work is required.
  2. Bring the second charge; add its interaction energy with the first.
  3. Bring the third; add its interactions with both earlier charges.
  4. Continue until every distinct pair has appeared once.

Because electrostatic force is conservative, the final total is independent of assembly order.

11. Count Pairs Before Calculating

The number of distinct pairs among \(N\) charges is

\(N_{\text{pairs}}=\frac{N(N-1)}2\)

Three charges give three pairs, four charges give six, and five give ten. This quick count catches omitted or double-counted interactions.

Worked Example 5 · Three-Charge Triangle

Sum Positive and Negative Pair Contributions

Charges \(q_1=+2.0\,\mathrm{nC}\), \(q_2=+3.0\,\mathrm{nC}\), and \(q_3=-4.0\,\mathrm{nC}\) form a triangle with \(r_{12}=0.30\,\mathrm m\), \(r_{13}=0.40\,\mathrm m\), and \(r_{23}=0.50\,\mathrm m\).

\(U=k_e\left(\frac{q_1q_2}{r_{12}}+\frac{q_1q_3}{r_{13}}+\frac{q_2q_3}{r_{23}}\right)\)
\(U=(8.99\times10^9)\left(20-20-24\right)\times10^{-18}\approx-2.16\times10^{-7}\,\mathrm J\)

The negative total means the complete configuration lies below the separated reference even though one pair contributes positive energy.

Worked Example 6 · Four Equal Charges on a Square

Group Equal Pair Distances

Four equal positive charges \(+q\) occupy the corners of a square of side \(a\). There are four edge pairs at distance \(a\) and two diagonal pairs at distance \(\sqrt2a\):

\(U=4\left(k_e\frac{q^2}{a}\right)+2\left(k_e\frac{q^2}{\sqrt2a}\right)=\frac{k_eq^2}{a}\left(4+\sqrt2\right)\)

Counting the six pairs first makes the geometry transparent and prevents treating the square as only four neighbor interactions.

12. Zero Total Energy Does Not Mean No Interaction

In a many-charge system, positive and negative pair energies can cancel so that \(U_{\text{system}}=0\). Individual forces and fields can still be nonzero. Energy is a scalar global property; force and field are local vectors.

13. Scaling the Entire Configuration

If every separation is multiplied by the same positive scale factor \(s\) while charges stay fixed, every pair energy is divided by \(s\):

\(U_{\text{new}}=\frac{U_{\text{old}}}{s}\)

This works because every term contains \(1/r_{ij}\). It does not apply when only some distances change.

Worked Example 7 · Closest Approach

Use a Turning Point

A \(+2.0\,\mu\mathrm C\) charge approaches an identical fixed charge from very far away with kinetic energy \(0.180\,\mathrm J\). At closest approach it momentarily stops, so \(K_f=0\).

\(K_i+U_i=K_f+U_f\quad\Rightarrow\quad0.180=k_e\frac{q^2}{r_{\min}}\)
\(r_{\min}=\frac{(8.99\times10^9)(2.0\times10^{-6})^2}{0.180}\approx0.200\,\mathrm m\)

The turning point is not an equilibrium point; the repulsive force is nonzero and reverses the motion.

14. Electron-Volt as an Energy Unit

Microscopic electric energies are often measured in electron-volts:

\(1\,\mathrm{eV}=1.602\times10^{-19}\,\mathrm J\)

An electron-volt is energy, not electric potential and not charge. Convert only when the requested unit or physical scale makes it useful.

15. Force and Energy Have Different Distance Laws

For two point charges, \(|F|\propto1/r^2\), but \(|U|\propto1/r\). Force describes the local rate at which energy changes with position; it is incorrect to reuse the inverse-square dependence in the energy formula.

16. Reliable Configuration Workflow

  1. Define the charge system and the energy reference.
  2. Draw the configuration and label every \(q_i\) and \(r_{ij}\).
  3. Count the expected distinct pairs.
  4. Compute each signed pair contribution.
  5. Add scalars and keep units in joules.
  6. Use \(\Delta U\), work signs, or energy conservation as the question requires.

17. Common Traps

  • Using \(|q_1q_2|\) and losing the energy sign.
  • Calling \(U\) the energy of only one charge.
  • Using \(1/r^2\) instead of \(1/r\).
  • Reporting \(U_f\) instead of \(\Delta U\).
  • Confusing work by the electric force with work by an external agent.
  • Adding pair energies as vectors.
  • Counting pair \(ij\) and \(ji\) twice.
  • Assuming \(U=0\) implies every force is zero.
Mastery Check

1. Two equal positive charges move from separation \(r\) to \(2r\). How does their potential energy change?

Show reasoning and answer

Because \(U=k_eq^2/r\), doubling separation halves the positive energy: \(U_f=U_i/2\). Thus \(\Delta U=-U_i/2\).

2. What does \(U<0\) mean when the zero reference is infinite separation?

Show reasoning and answer

The configuration has less energy than the infinitely separated state. Positive external work is required to separate the system to infinity; opposite-sign interactions commonly produce this result.

3. The electric force does \(+4.0\,\mathrm{mJ}\) of work. What is \(\Delta U\)?

Show reasoning and answer

For a conservative electric force, \(W_{\mathrm e}=-\Delta U\). Therefore \(\Delta U=-4.0\,\mathrm{mJ}\).

4. How many pair-energy terms are required for five point charges?

Show reasoning and answer

\(N(N-1)/2=5(4)/2=10\) distinct pairs. Each must appear exactly once.

5. Can a three-charge configuration have \(U=0\) while the charges experience forces?

Show reasoning and answer

Yes. Positive and negative scalar pair-energy terms can cancel, while the vector forces at individual charge locations remain nonzero.

6. A pair is moved slowly and \(\Delta U=-2.0\,\mu\mathrm J\). What work is done by the external agent and by the electric force?

Show reasoning and answer

For negligible kinetic-energy change, \(W_{\mathrm{ext}}=\Delta U=-2.0\,\mu\mathrm J\). The electric force does the opposite work: \(W_{\mathrm e}=+2.0\,\mu\mathrm J\).

Investigation: Build an Energy Ledger

Choose three point charges and place them at measured coordinates on graph paper or in a spreadsheet. Calculate the three pair distances, then create a signed ledger containing \(U_{12}\), \(U_{13}\), and \(U_{23}\). Move only one charge through at least five positions. At each position, compare total \(U\), the predicted attractive or repulsive tendency, and the work an external agent would do during a slow move from the previous position.

As a check, repeat the assembly in a different order. The intermediate ledger entries will differ, but the final configuration energy must agree within rounding uncertainty.

Official curriculum reference: College Board AP Physics 2 course page. The explanation and worked example are independently written for this study site.