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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 10.5

Electric Potential and Equipotential Reasoning

Use a scalar map to describe electrical energy per unit charge. Calculate potential from point sources, connect voltage to work and particle energy, and read equipotential maps to infer field direction, relative strength, and conductor behavior.

Learning Goals

  • Distinguish electric potential \(V\), electric potential energy \(U\), and electric field \(\vec E\).
  • Calculate point-charge and multiple-source potentials with signed scalar addition.
  • Use \(\Delta U=q\Delta V\) and \(W_{\mathrm e}=-q\Delta V\) to track energy transfer.
  • Relate a uniform electric field to potential change and displacement.
  • Interpret equipotential spacing, direction, zero-work motion, and electrostatic conductors.
\(V\)electric potentialvolt, \(\mathrm{J/C}\)
\(\Delta V\)potential difference\(V_B-V_A\)
\(U\)potential energy\(U=qV\)
\(\vec E\)electric fieldpoints toward decreasing \(V\)

1. Potential Is Energy per Unit Charge

Electric potential assigns one scalar value to every point in space:

\(V=\frac{U}{q}\)

Because \(U\) is proportional to the charge placed at the point, dividing by \(q\) removes that dependence. Potential belongs to the source configuration and location, not to a particular test charge.

2. Voltage Is a Difference

Potential difference compares two locations:

\(\Delta V=V_B-V_A=\frac{\Delta U}{q}\)

Its SI unit is the volt: \(1\,\mathrm V=1\,\mathrm{J/C}\). A statement such as “the battery is 9 V” means its terminals differ in potential by 9 V.

3. The Zero of Potential Is a Choice

Only potential differences directly affect work and energy. Adding the same constant to every potential leaves \(\Delta V\), electric fields, forces, and observable motion unchanged.

For isolated point charges, the convenient convention is \(V(\infty)=0\). Ground is often assigned \(0\,\mathrm V\) in circuits and laboratories, but that is another reference choice—not a claim that Earth has no electric interactions.

4. Potential of One Point Charge

With zero potential at infinity, a source charge \(Q\) produces

\(V(r)=k_e\frac{Q}{r}\)

Potential is positive around a positive source and negative around a negative source. Unlike electric field, potential has no direction.

5. Distance Scaling

For a fixed point charge, \(|V|\propto1/r\). Doubling distance halves the potential magnitude. This differs from electric field, whose magnitude falls as \(1/r^2\).

The distance \(r\) is measured from the source to the observation point.

Worked Example 1 · One Source

Potential Near a Positive Charge

Find the potential \(0.20\,\mathrm m\) from a \(+3.0\,\mathrm{nC}\) point charge.

\(V=(8.99\times10^9)\frac{3.0\times10^{-9}}{0.20}\approx+1.35\times10^2\,\mathrm V\)

The answer is approximately \(+135\,\mathrm V\). No direction is attached to this scalar.

6. Potential of Many Point Charges

Each source contributes a signed scalar potential:

\(V(P)=k_e\sum_i\frac{Q_i}{r_i}\)

Add the values algebraically. Do not resolve components, and do not combine separated source charges before accounting for their different distances from \(P\).

Potential is a signed scalar landscapeFor an equal dipole, potential is positive near the positive source, negative near the negative source, and zero at the equidistant midpoint—even though the electric field there is not zero.
Worked Example 2 · Scalar Cancellation

Zero Potential Does Not Require Zero Field

At point \(P\), a \(+4.0\,\mathrm{nC}\) source is \(0.30\,\mathrm m\) away and a \(-2.0\,\mathrm{nC}\) source is \(0.15\,\mathrm m\) away.

\(V_P=k_e\left(\frac{4.0\times10^{-9}}{0.30}-\frac{2.0\times10^{-9}}{0.15}\right)=0\)

The scalar contributions cancel. Their field vectors need not cancel because field also depends on direction and \(1/r^2\).

7. Potential Difference Controls Energy Change

When charge \(q\) moves from A to B,

\(\Delta U=q\Delta V=q(V_B-V_A)\)
\(W_{\mathrm e}=-\Delta U=-q\Delta V\)

A positive charge lowers its potential energy by moving toward lower \(V\). A negative charge lowers its potential energy by moving toward higher \(V\). Both move spontaneously toward lower \(U\), not necessarily toward lower numerical potential.

Worked Example 3 · Electron Energy

A Negative Charge Moves to Higher Potential

An electron moves from \(0\,\mathrm V\) to \(+120\,\mathrm V\).

\(\Delta U=q\Delta V=(-1.602\times10^{-19})(120)=-1.92\times10^{-17}\,\mathrm J\)

The electric force does \(+1.92\times10^{-17}\,\mathrm J\) of work. If the electron starts from rest and losses are negligible, it gains \(120\,\mathrm{eV}\) of kinetic energy.

8. Positive and Negative Test Charges

ChargeDirection of electric forceSpontaneous trend
\(q>0\)Along \(\vec E\)Toward lower \(V\) and lower \(U\)
\(q<0\)Opposite \(\vec E\)Toward higher \(V\) but lower \(U=qV\)

9. Potential Change in a Uniform Electric Field

For displacement \(\vec d\) through a uniform field, only the component parallel to the field changes potential:

\(\Delta V=-\vec E\cdot\vec d=-Ed\cos\theta\)

The minus sign means potential decreases when displacement is along \(\vec E\). For motion directly across a uniform field,

\(E=\frac{|\Delta V|}{d}\)

This relationship also shows \(1\,\mathrm{V/m}=1\,\mathrm{N/C}\).

Worked Example 4 · Parallel Plates

Voltage Spacing Sets Field Strength

Large parallel plates differ by \(600\,\mathrm V\) and are \(3.0\,\mathrm{cm}\) apart. Ignore edge effects.

\(E=\frac{600}{0.030}=2.0\times10^4\,\mathrm{V/m}\)

The electric field points from the higher-potential plate toward the lower-potential plate.

Equipotentials cross field lines at right anglesBetween ideal parallel plates, equal voltage steps have equal spacing, so the field is uniform and points from high potential toward low potential.

10. What an Equipotential Means

An equipotential line in two dimensions, or surface in three dimensions, contains points with the same \(V\). Moving any charge along one equipotential gives

\(\Delta V=0\quad\Rightarrow\quad\Delta U=q\Delta V=0\quad\Rightarrow\quad W_{\mathrm e}=0\)

The charge may travel a nonzero distance, but the electric force does no net work along that displacement.

11. Why Equipotentials Are Perpendicular to \(\vec E\)

If an electric-field component lay along an equipotential, moving along it would produce nonzero work and change \(V\), contradicting the definition. Therefore, electrostatic field lines cross equipotentials at \(90^\circ\).

12. Spacing Reveals Field Strength

When neighboring contours differ by the same voltage, closer spacing means a larger potential change per distance and therefore a stronger field. Wider spacing means a weaker field.

\(E_{\perp}\approx\frac{|\Delta V|}{\Delta s}\)
Worked Example 5 · Reading a Contour Map

Estimate Field from Equipotential Spacing

Adjacent equipotential lines differ by \(20\,\mathrm V\) and are \(5.0\,\mathrm{mm}\) apart near point P.

\(E\approx\frac{20}{5.0\times10^{-3}}=4.0\times10^3\,\mathrm{V/m}\)

The direction is perpendicular to the contours toward decreasing labels. This is a local estimate because spacing may change elsewhere.

13. Equipotentials of Familiar Sources

  • A point charge has spherical equipotential surfaces; planar cross-sections appear as concentric circles.
  • An ideal uniform field has parallel, evenly spaced equipotential planes for equal voltage steps.
  • An equal dipole has a zero-potential plane midway between the charges.
  • Multiple sources create contours from the scalar sum, not by simply overlaying circles unchanged.
Worked Example 6 · Radius of an Equipotential

Locate a Point-Charge Equipotential Surface

At what radius from a \(+9.0\,\mathrm{nC}\) point charge is \(V=+300\,\mathrm V\)?

\(r=\frac{k_eQ}{V}=\frac{(8.99\times10^9)(9.0\times10^{-9})}{300}\approx0.270\,\mathrm m\)

Every point \(0.270\,\mathrm m\) from the source belongs to the same spherical equipotential.

14. Electric Field from a Potential Map

The field points in the direction of the steepest decrease in potential and is perpendicular to equipotential contours:

\(\vec E=-\nabla V\)

AP Physics 2 problems usually apply this through contour spacing or a uniform-field relation rather than a multivariable derivative calculation.

15. Potential from Field Information

Potential difference is the negative accumulated field component along a path:

\(V_B-V_A=-\int_A^B\vec E\cdot d\vec \ell\)

Electrostatic fields are conservative, so every path between the same endpoints gives the same \(\Delta V\).

16. Conductors in Electrostatic Equilibrium

A conductor and its surface are equipotential in electrostatic equilibrium. If two surface points had different potentials, mobile charges would move. Inside the conducting material \(\vec E=\vec0\), so \(V\) is constant throughout—not necessarily zero.

Just outside the surface, the field is perpendicular to the conductor. Grounding fixes the conductor to Earth's chosen reference potential.

Worked Example 7 · Four Sources at Equal Distance

Use Scalar Addition at the Center of a Square

At the corners of a square of side \(0.40\,\mathrm m\) are charges \(+2.0\), \(+2.0\), \(-1.0\), and \(-1.0\,\mathrm{nC}\). The center is \(r=0.40/\sqrt2\approx0.283\,\mathrm m\) from every corner.

\(V_{\text{center}}=k_e\frac{(2+2-1-1)\times10^{-9}}{0.283}\approx+63.6\,\mathrm V\)

Because all distances match, the signed charges can be summed in the numerator. This shortcut would fail at an off-center point.

17. A Topographic-Map Analogy

Potential contours resemble equal-elevation lines. Close contour spacing marks a steep electrical “slope,” and the electric field points downhill most steeply. The analogy must stop there: a negative charge naturally accelerates electrically “uphill” in \(V\) because its potential energy is \(U=qV\).

18. Zero Potential Versus Zero Field

  • \(V=0\) can occur when positive and negative scalar contributions cancel while \(\vec E\ne0\).
  • \(\vec E=0\) throughout a region means \(V\) is constant there, but the constant need not be zero.
  • At one isolated point, \(\vec E=0\) says the local slope is zero; it does not alone determine the value of \(V\).

19. Reliable Potential Workflow

  1. State the reference and label the observation point.
  2. Measure a separate distance from every source.
  3. Add signed scalar potentials.
  4. Use \(\Delta V=V_f-V_i\) before calculating energy or work.
  5. For a map, point \(\vec E\) perpendicular to contours toward lower \(V\).
  6. Check units: volts, joules, or volts per meter are not interchangeable.

20. Common Traps

  • Treating potential as a vector.
  • Removing negative source-charge signs from \(V=k_eQ/r\).
  • Using \(1/r^2\) for point-charge potential.
  • Confusing \(V\) with \(U=qV\).
  • Claiming negative charges move toward lower \(V\).
  • Assuming \(V=0\) requires \(\vec E=0\).
  • Drawing field lines parallel to equipotentials.
  • Using \(E=|\Delta V|/d\) where the field is not approximately uniform.
Mastery Check

1. The distance from a positive point charge doubles. What happens to \(V\) and \(E\)?

Show reasoning and answer

\(V\propto1/r\), so it becomes one half. \(E\propto1/r^2\), so it becomes one fourth.

2. An electron moves through \(\Delta V=-50\,\mathrm V\). Does its potential energy rise or fall?

Show reasoning and answer

\(\Delta U=q\Delta V\). Both \(q\) and \(\Delta V\) are negative, so \(\Delta U>0\); its potential energy rises by \(50\,\mathrm{eV}\).

3. Why is no electrostatic work done along one equipotential?

Show reasoning and answer

The endpoints have \(\Delta V=0\), so \(\Delta U=q\Delta V=0\) and \(W_{\mathrm e}=-\Delta U=0\).

4. Adjacent contours differ by the same voltage. Where is the field strongest?

Show reasoning and answer

Where the contours are closest. The same \(|\Delta V|\) over a smaller perpendicular distance gives a larger \(E\).

5. Can a conductor in electrostatic equilibrium be at \(+200\,\mathrm V\) while its internal electric field is zero?

Show reasoning and answer

Yes. Zero internal field means potential is constant throughout the conductor, not that the constant must be zero.

6. A uniform field points east. Which way does potential decrease, and which way does a negative charge accelerate?

Show reasoning and answer

Potential decreases eastward because \(\vec E\) points toward lower \(V\). A negative charge experiences force opposite \(\vec E\), so it accelerates westward toward higher \(V\) but lower \(U\).

Investigation: Measure a Scalar Potential Map

Use the open PhET Charges and Fields simulation. Place one positive and one negative source at recorded coordinates. Predict the zero-potential line, then sample potential on a square grid and connect points with equal values. Add field sensors only after drawing the contours. Test whether field arrows cross the contours at right angles and whether closer contours correspond to larger measured field.

Repeat after doubling one source charge. Record which symmetry is lost, how the zero-potential contour shifts, and why the scalar potential calculation remains simpler than vector-field addition.

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