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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 10.7

Conserving Energy in Charged-Particle Motion

Predict how charged particles speed up, slow down, stop, or escape by tracking kinetic energy and electric potential energy. Use voltage and scalar energy methods when a force-by-force calculation would be longer, while keeping charge signs, system boundaries, and model limits explicit.

Learning Goals

  • Apply \(K_i+U_i=K_f+U_f\) when only electrostatic forces transfer energy within the chosen system.
  • Relate voltage to energy through \(\Delta U=q\Delta V\) and \(\Delta K=-q\Delta V\).
  • Predict the motion of positive and negative particles without confusing potential \(V\) with potential energy \(U\).
  • Calculate particle speeds from voltage changes and express microscopic energies in electron-volts.
  • Use point-source and multiple-source potentials to analyze turning points, closest approach, and escape conditions.
  • Recognize when external work, collisions, radiation, or relativistic motion require a broader model.
\(K\)kinetic energy\(\frac12mv^2\)
\(U\)electric potential energy\(U=qV\)
\(V\)electric potentialsource property, \(\mathrm{J/C}\)
\(W_{\mathrm e}\)electric-force work\(-\Delta U=\Delta K\)

1. Begin with the System

Potential energy belongs to an interacting system, not to an isolated particle. A useful electrostatic system contains the moving particle and the source charges that create the field. If the source configuration stays fixed and no outside mechanism adds or removes energy, mechanical energy is conserved.

2. The Core Conservation Statement

\(K_i+U_i=K_f+U_f\)
\(\Delta K=-\Delta U\)

A decrease in electric potential energy becomes an equal increase in kinetic energy. Energy conservation predicts speed, but it does not by itself determine the direction of a two-dimensional trajectory.

3. From Voltage to Kinetic Energy

Potential is electric potential energy per unit charge, so moving a particle of charge \(q\) from point \(i\) to point \(f\) changes its potential energy by

\(\Delta U=U_f-U_i=q(V_f-V_i)=q\Delta V\)

For electrostatic work only, the work-energy theorem gives

\(W_{\mathrm e}=\Delta K=-\Delta U=-q\Delta V=q(V_i-V_f)\)

Combining this with \(K=\frac12mv^2\) produces the nonrelativistic speed relation

\(v_f=\sqrt{v_i^2+\frac{2q(V_i-V_f)}{m}}\)

The quantity inside the square root must be nonnegative. A negative result signals that the proposed final point is inaccessible with the stated initial energy.

Energy changes form without changing the totalFor an isolated electrostatic system, the loss in \(U\) matches the gain in \(K\). The same bar model works for either charge sign when \(U=qV\) is used correctly.

4. Positive Charges

For \(q>0\), potential energy and potential have the same sign trend. A freely released positive particle accelerates toward lower \(V\), where \(U=qV\) is lower and \(K\) is higher.

\(q>0,\quad V_f0\)

5. Negative Charges

For \(q<0\), multiplying by the negative charge reverses the trend. A freely released electron accelerates toward higher \(V\), yet its \(U=qV\) still decreases.

\(q<0,\quad V_f>V_i\quad\Rightarrow\quad\Delta K>0\)
Worked Example 1 · Positive Charge

Released from Rest Across a Potential Change

A \(+2.0\,\mu\mathrm C\) particle of mass \(0.012\,\mathrm{kg}\) is released from rest at \(+120\,\mathrm V\) and reaches \(-30\,\mathrm V\).

\(\Delta K=-q(V_f-V_i)=-(2.0\times10^{-6})(-150)=3.0\times10^{-4}\,\mathrm J\)
\(v_f=\sqrt{\frac{2K_f}{m}}=\sqrt{\frac{2(3.0\times10^{-4})}{0.012}}\approx0.224\,\mathrm{m/s}\)

The positive particle moves toward lower potential and loses electric potential energy.

6. Electron-Volts Are Energy Units

One electron-volt is the magnitude of energy change for one elementary charge moving through one volt:

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

A singly charged particle gaining energy across \(250\,\mathrm V\) gains \(250\,\mathrm{eV}\). An ion of charge magnitude \(3e\) crossing the same accelerating voltage gains \(750\,\mathrm{eV}\). Electron-volts measure energy, not electric potential.

Worked Example 2 · Electron Acceleration

An Electron Moves Toward Higher Potential

An electron starts from rest and moves from \(0\,\mathrm V\) to \(+250\,\mathrm V\).

\(\Delta U=(-e)(250\,\mathrm V)=-250\,\mathrm{eV}\)
\(K_f=250\,\mathrm{eV}=4.01\times10^{-17}\,\mathrm J\)
\(v_f=\sqrt{\frac{2K_f}{m_e}}\approx9.38\times10^6\,\mathrm{m/s}\)

The electron moves to higher potential but lower potential energy. Its speed is about \(0.031c\), so the classical result is a reasonable approximation here.

Worked Example 3 · Nonzero Initial Speed

Do Not Discard the Initial Kinetic Energy

A \(+5.0\,\mathrm{nC}\), \(2.0\,\mathrm{mg}\) particle moves at \(3.0\,\mathrm{m/s}\) from \(80\,\mathrm V\) to \(20\,\mathrm V\).

\(K_i=\frac12(2.0\times10^{-6})(3.0)^2=9.0\times10^{-6}\,\mathrm J\)
\(\Delta K=-q\Delta V=-(5.0\times10^{-9})(-60)=3.0\times10^{-7}\,\mathrm J\)
\(K_f=9.3\times10^{-6}\,\mathrm J,\quad v_f=\sqrt{\frac{2K_f}{m}}\approx3.05\,\mathrm{m/s}\)

The voltage change adds to the kinetic energy already present; it does not replace it.

7. Uniform Fields Between Parallel Plates

Ignoring edge effects, parallel plates produce an approximately uniform field:

\(E=\frac{|\Delta V|}{d},\quad F=|q|E,\quad a=\frac{|q|E}{m}\)

The energy gained across the full gap depends on \(|q\Delta V|\), not separately on plate spacing. Reducing \(d\) at fixed voltage increases \(E\), force, and acceleration, but the particle travels a shorter distance and reaches the same ideal final kinetic energy.

Worked Example 4 · Two Consistent Methods

Accelerate a Proton Through 600 V

A proton starts from rest between plates separated by \(3.0\,\mathrm{cm}\) with a \(600\,\mathrm V\) potential difference.

\(K_f=e(600\,\mathrm V)=600\,\mathrm{eV}=9.61\times10^{-17}\,\mathrm J\)
\(v_f=\sqrt{\frac{2K_f}{m_p}}\approx3.39\times10^5\,\mathrm{m/s}\)

Alternatively, \(E=600/0.030=2.0\times10^4\,\mathrm{V/m}\), then \(a=eE/m_p\), and \(v_f^2=2ad\) gives the same result. Energy is usually the shorter route.

8. Motion Near One Point Source

Taking \(U=0\) at infinite separation, a moving charge \(q\) interacting with a fixed point source \(Q\) has

\(U(r)=k_e\frac{Qq}{r}\)
\(\frac12mv_i^2+k_e\frac{Qq}{r_i}=\frac12mv_f^2+k_e\frac{Qq}{r_f}\)

Like charges have \(U>0\) and convert potential energy to kinetic energy as they separate. Opposite charges have \(U<0\); separating them toward infinity requires an input of energy.

Worked Example 5 · Point-Source Repulsion

Use Radius Instead of Path Length

A \(+3.0\,\mathrm{nC}\), \(2.0\,\mathrm{mg}\) particle starts from rest \(0.10\,\mathrm m\) from a fixed \(+2.0\,\mu\mathrm C\) source and moves to \(0.30\,\mathrm m\).

\(K_f=k_eQq\left(\frac1{r_i}-\frac1{r_f}\right)\)
\(K_f=(8.99\times10^9)(2.0\times10^{-6})(3.0\times10^{-9})\left(10-\frac{10}{3}\right)=3.60\times10^{-4}\,\mathrm J\)
\(v_f=\sqrt{\frac{2K_f}{m}}\approx19.0\,\mathrm{m/s}\)

Because the electric force is conservative, only the initial and final radii matter.

The sign of \(Qq\) shapes the energy landscapeLike charges have positive \(U\) and an energy barrier at small separation. Unlike charges have negative \(U\) relative to infinite separation.

9. Turning Points and Accessible Regions

At any position,

\(K=E_{\mathrm{mech}}-U\ge0\)

A classical turning point occurs where \(K=0\), so \(U=E_{\mathrm{mech}}\). Regions where \(U>E_{\mathrm{mech}}\) would require negative kinetic energy and are inaccessible.

10. Escape Condition

With \(U(\infty)=0\), a particle can reach infinity only if its total mechanical energy is at least zero. For an attractive \(Qq<0\) interaction, the minimum launch kinetic energy from radius \(r_0\) is

\(K_{\mathrm{escape}}=-U(r_0)=k_e\frac{|Qq|}{r_0}\)
Worked Example 6 · Closest Approach

A Like Charge Approaches a Repulsive Source

A \(+2.0\,\mathrm{nC}\) particle approaches a fixed \(+4.0\,\mathrm{nC}\) source from very far away with \(K_i=4.0\times10^{-7}\,\mathrm J\). At closest approach, \(K_f=0\).

\(K_i=k_e\frac{Qq}{r_{\min}}\)
\(r_{\min}=\frac{(8.99\times10^9)(4.0\times10^{-9})(2.0\times10^{-9})}{4.0\times10^{-7}}\approx0.180\,\mathrm m\)
Worked Example 7 · Minimum Escape Energy

Separate Opposite Charges to Infinity

A \(+2.0\,\mathrm{nC}\) particle begins \(0.10\,\mathrm m\) from a fixed \(-5.0\,\mathrm{nC}\) source.

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

Exactly this launch energy gives zero speed infinitely far away. Any smaller launch energy cannot reach infinity in the ideal two-charge model.

11. Several Source Charges: Calculate Potential First

Potential is a scalar, so multiple fixed sources can be combined before analyzing the moving particle:

\(V(P)=k_e\sum_j\frac{Q_j}{r_j},\quad U(P)=qV(P)\)
\(K_f=K_i+q[V_i-V_f]\)

Every source charge keeps its algebraic sign, and every distance \(r_j\) is measured from that source to the observation point. Do not add potential magnitudes or use \(1/r^2\).

Worked Example 8 · Scalar Superposition

An Electron Moves Through a Two-Source Potential Map

At point A, the distances to \(+4.0\,\mathrm{nC}\) and \(-2.0\,\mathrm{nC}\) sources are \(0.20\,\mathrm m\) and \(0.10\,\mathrm m\). At point B they are \(0.10\,\mathrm m\) and \(0.30\,\mathrm m\).

\(V_A=k_e\left(\frac{4.0\times10^{-9}}{0.20}-\frac{2.0\times10^{-9}}{0.10}\right)=0\)
\(V_B=k_e\left(\frac{4.0\times10^{-9}}{0.10}-\frac{2.0\times10^{-9}}{0.30}\right)\approx+300\,\mathrm V\)

An electron released from rest at A gains approximately \(300\,\mathrm{eV}\) by reaching B because \(\Delta K=-q\Delta V=e(300\,\mathrm V)\).

12. What Energy Can and Cannot Predict

  • Energy gives speed or possible positions without following every force component.
  • It cannot determine whether the particle turns left or right in a two-dimensional field.
  • Combine energy with field direction, force, or momentum when trajectory information is required.

13. When Mechanical Energy Is Not Constant

If an external agent moves source charges, a battery changes the field, collisions create thermal energy, or another nonconservative interaction acts, include the transfer explicitly:

\(K_i+U_i+W_{\mathrm{ext}}=K_f+U_f+\Delta E_{\mathrm{other}}\)

Do not force \(K+U\) to remain constant when the chosen system exchanges energy with its surroundings.

14. Classical Model Limit

The expression \(K=\frac12mv^2\) is nonrelativistic. If a calculation gives a speed that is a substantial fraction of \(c\), use relativistic kinetic energy rather than accepting a result near or above the speed of light. AP Physics 2 problems normally choose values where the classical approximation is appropriate, but the result should still be checked.

15. Reliable Energy Workflow

  1. Choose the system and state whether external work is negligible.
  2. Mark initial and final positions, speeds, and potentials.
  3. Write \(\Delta V=V_f-V_i\) before inserting the charge sign.
  4. Use \(\Delta K=-q\Delta V\) or the full \(K+U\) equation.
  5. For point sources, set the potential reference and calculate each \(r_j\).
  6. Check \(K_f\ge0\), units, direction reasoning, and the nonrelativistic limit.

16. Common Traps

  • Confusing \(V\) in volts with \(U=qV\) in joules.
  • Dropping the negative sign in \(\Delta K=-q\Delta V\).
  • Claiming every particle accelerates toward lower potential.
  • Setting \(K_i=0\) when an initial speed is given.
  • Using \(1/r^2\) for point-charge potential energy.
  • Adding source potentials as vectors or stripping away source signs.
  • Treating eV as a voltage unit.
  • Using energy alone to infer a two-dimensional direction.
  • Ignoring external work or accepting an unphysical classical speed.
Mastery Check

1. A proton moves from \(+40\,\mathrm V\) to \(-10\,\mathrm V\). What is its kinetic-energy change?

Show reasoning and answer

\(\Delta V=-50\,\mathrm V\), so \(\Delta K=-e(-50\,\mathrm V)=+50\,\mathrm{eV}\). Its kinetic energy increases.

2. An electron moves from \(-20\,\mathrm V\) to \(+80\,\mathrm V\). Does it speed up or slow down if only the electric force acts?

Show reasoning and answer

\(\Delta U=q\Delta V=(-e)(+100\,\mathrm V)=-100\,\mathrm{eV}\). Therefore \(\Delta K=+100\,\mathrm{eV}\), so it speeds up.

3. Why can two different paths between the same electrostatic endpoints give the same final speed?

Show reasoning and answer

Electrostatic force is conservative. The potential difference and \(\Delta U=q\Delta V\) depend only on the endpoints, so the same initial kinetic energy produces the same final kinetic energy.

4. A result gives \(K_f<0\). What does that mean?

Show reasoning and answer

Kinetic energy cannot be negative. The particle lacks enough initial energy to reach the proposed point; it must turn earlier or receive external energy.

5. A \(+2e\) ion accelerates from rest through a potential drop of \(300\,\mathrm V\). How much kinetic energy does it gain?

Show reasoning and answer

The magnitude is \(|q\Delta V|=(2e)(300\,\mathrm V)=600\,\mathrm{eV}=9.61\times10^{-17}\,\mathrm J\).

6. If plate spacing is halved while voltage stays fixed, what happens to field, acceleration, and final energy across the full gap?

Show reasoning and answer

\(E=|\Delta V|/d\) and acceleration double. The distance halves, so the total work \(|q\Delta V|\) and ideal final kinetic energy remain unchanged.

Investigation: Test Energy Conservation on a Potential Map

Use the open PhET Charges and Fields simulation. Place fixed source charges, record the potential at two grid points, and calculate the predicted \(\Delta U=q\Delta V\) for both a positive and a negative test charge. Draw the expected direction of the electric force and identify which charge would gain kinetic energy when moving between the points.

Repeat with two different paths connecting the same endpoints. Explain why the potential difference is unchanged even though local field strength and direction vary along the paths. For an extension, create a line of potential measurements and sketch the energy landscape \(U(x)=qV(x)\) for both charge signs.

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