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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 11 · Topic 11.1

Charge Flow and Conventional Current

Describe electric current as the signed rate at which charge crosses a chosen surface. Connect macroscopic current to microscopic carrier motion, distinguish conventional current from electron drift, interpret current graphs, and apply charge conservation throughout a circuit.

Learning Goals

  • Calculate average and instantaneous current from transferred charge.
  • Use the slope of a \(Q\)-versus-\(t\) graph and the signed area under an \(I\)-versus-\(t\) graph.
  • Distinguish conventional current direction from the motion of negative charge carriers.
  • Derive and apply the microscopic model \(I=n|q|Av_d\).
  • Relate current density, wire area, carrier density, and drift speed.
  • Apply charge conservation to steady wires, junctions, and regions where charge temporarily accumulates.
\(I\)electric currentampere, \(\mathrm{C/s}\)
\(Q\)charge crossing a sectioncoulombs
\(v_d\)drift velocityaverage carrier motion
\(\vec J\)current density\(\mathrm{A/m^2}\)

1. Current Counts Charge Crossing a Surface

Choose an imaginary cross-section through a wire. Electric current describes how rapidly net charge passes through that section, not how much charge is stored in the wire.

\(I_{\mathrm{avg}}=\frac{\Delta Q}{\Delta t}\)

The SI unit is the ampere:

\(1\,\mathrm A=1\,\mathrm{C/s}\)

2. Instantaneous Current

If the rate varies, take the time derivative of the signed charge that has crossed the section:

\(I(t)=\frac{dQ}{dt}\)

A positive or negative sign reports direction relative to the chosen positive-current arrow. It does not mean that the physical carriers must have positive or negative charge.

Worked Example 1 · Average Current

Charge Passing During a Time Interval

A net charge of \(18\,\mathrm C\) crosses a wire section uniformly in \(6.0\,\mathrm s\).

\(I_{\mathrm{avg}}=\frac{18}{6.0}=3.0\,\mathrm A\)

Three coulombs cross the selected section each second on average.

3. Two Complementary Graph Relationships

\(I=\frac{dQ}{dt}\quad\Longleftrightarrow\quad\Delta Q=\int_{t_i}^{t_f}I(t)\,dt\)
  • The slope of a \(Q\)-versus-\(t\) graph is instantaneous current.
  • The signed area between an \(I\)-versus-\(t\) curve and the time axis is net transferred charge.
  • Area below the axis subtracts from net charge because the current direction has reversed.
  • Integrating \(|I|\) instead would count total charge throughput without directional cancellation.
Current accumulates charge over timeThe signed area \(\int I\,dt\) has units \(\mathrm{A\,s}=\mathrm C\) and equals the net charge crossing the chosen section.
Worked Example 2 · Differentiate \(Q(t)\)

Find Current from a Charge Function

The signed charge that has crossed a section is \(Q(t)=(3t^2+2t)\,\mu\mathrm C\), with \(t\) in seconds.

\(I(t)=\frac{dQ}{dt}=(6t+2)\,\mu\mathrm A\)
\(I(2.0\,\mathrm s)=14\,\mu\mathrm A\)
Worked Example 3 · Integrate \(I(t)\)

Find Charge from a Changing Current

From \(t=0\) to \(4.0\,\mathrm s\), a current is \(I(t)=2.0+0.50t\) amperes.

\(\Delta Q=\int_0^4(2.0+0.50t)\,dt=[2.0t+0.25t^2]_0^4=12\,\mathrm C\)

The same result is the area of a rectangle plus a triangle under the graph.

4. Conventional Current Direction

Conventional current points in the direction a positive charge would move. In an external circuit, it is drawn from the positive terminal of a source through the components toward the negative terminal.

5. Electron Motion in a Metal

Mobile electrons have negative charge, so their average drift is opposite the electric field and opposite conventional current. Either description predicts the same measured current when signs are handled consistently.

6. The Carrier Depends on the Material

MediumCommon mobile carriersDirection relative to conventional current
Metal wireElectronsElectron drift is opposite \(I\)
ElectrolytePositive and negative ionsBoth species can contribute to the same \(I\)
SemiconductorElectrons and holesContributions add using signed charge and velocity
Ionized gasElectrons and ionsOpposite carrier motions can reinforce one conventional current

Current direction is therefore a sign convention for net charge transport, not a claim about which microscopic particles are present.

7. Deriving the Microscopic Current Model

Consider a wire of cross-sectional area \(A\) containing \(n\) mobile carriers per cubic meter. During time \(\Delta t\), carriers with drift-speed magnitude \(v_d\) advance an average distance \(v_d\Delta t\).

  1. The contributing cylinder has volume \(Av_d\Delta t\).
  2. It contains \(nAv_d\Delta t\) mobile carriers.
  3. If each carrier has charge magnitude \(|q|\), the transported charge magnitude is \(\Delta Q=n|q|Av_d\Delta t\).
  4. Divide by \(\Delta t\):
\(I=n|q|Av_d\)

This scalar form gives magnitudes. Direction is supplied by the conventional-current arrow. A signed vector form is \(\vec J=nq\vec v_d\).

Slow electron drift can produce a steady conventional currentIn a metal, negative carriers drift opposite the conventional-current arrow. Their disordered microscopic motion is much faster than the small net drift.
Worked Example 4 · Count the Electrons

Convert Current to a Number of Carriers

A \(0.50\,\mathrm A\) current runs for \(2.0\,\mathrm{min}=120\,\mathrm s\).

\(Q=I\Delta t=(0.50)(120)=60\,\mathrm C\)
\(N=\frac{Q}{e}=\frac{60}{1.602\times10^{-19}}\approx3.75\times10^{20}\text{ electrons}\)

This is the number crossing one selected cross-section during the interval, not the number of electrons contained in the entire circuit.

Worked Example 5 · Drift Speed

A Large Current Does Not Require Fast Electron Drift

A metal wire carries \(2.0\,\mathrm A\). Its carrier density is \(n=8.5\times10^{28}\,\mathrm{m^{-3}}\), and its cross-sectional area is \(1.0\,\mathrm{mm^2}=1.0\times10^{-6}\,\mathrm{m^2}\).

\(v_d=\frac{I}{neA}=\frac{2.0}{(8.5\times10^{28})(1.602\times10^{-19})(1.0\times10^{-6})}\)
\(v_d\approx1.47\times10^{-4}\,\mathrm{m/s}=0.147\,\mathrm{mm/s}\)

There are enormously many carriers, so a small average drift can still transport substantial charge each second.

8. Why a Lamp Responds Quickly

When a switch closes, electrons do not need to travel from the battery to the lamp before the lamp responds. Mobile charges are already distributed throughout the conducting path. The circuit's electric field reconfigures through the system rapidly, and local electrons begin a small drift nearly throughout the circuit.

Do not confuse signal propagation with carrier drift. The field disturbance travels far faster than the average electron drift speed.

9. Current Density

For uniform current over a cross-section, current density is

\(J=\frac{I}{A}=n|q|v_d\)

The vector \(\vec J\) points with conventional current. If the same current passes through a narrower uniform wire, \(J\) and the needed drift speed are larger.

10. A Wire Can Carry Current and Remain Neutral

A metal contains a positively charged lattice and mobile electrons whose charges nearly cancel in each macroscopic segment. Current requires a tiny organized drift of the mobile electrons, not a large net charge throughout the wire.

Worked Example 6 · Current Density

Account for Circular Cross-Section

A wire of radius \(0.50\,\mathrm{mm}\) carries \(3.0\,\mathrm A\).

\(A=\pi r^2=\pi(5.0\times10^{-4})^2=7.85\times10^{-7}\,\mathrm{m^2}\)
\(J=\frac{3.0}{7.85\times10^{-7}}\approx3.82\times10^6\,\mathrm{A/m^2}\)

Using diameter in place of radius would make the area four times too large.

11. Charge Conservation in a Region

Surround a junction or wire segment with an imaginary boundary. The difference between incoming and outgoing current changes the charge stored inside:

\(\frac{dQ_{\mathrm{inside}}}{dt}=\sum I_{\mathrm{in}}-\sum I_{\mathrm{out}}\)

In steady state, charge no longer accumulates, so

\(\sum I_{\mathrm{in}}=\sum I_{\mathrm{out}}\)

This is charge conservation—not a statement that current is consumed by a bulb or resistor.

Worked Example 7 · Junction

Find an Unknown Branch Current

A steady \(4.2\,\mathrm A\) enters a junction. One outgoing branch carries \(1.5\,\mathrm A\).

\(4.2=1.5+I_2\quad\Rightarrow\quad I_2=2.7\,\mathrm A\)

If the outgoing total were temporarily smaller than the incoming current, charge would accumulate at the junction until fields and currents changed.

12. A Closed Conducting Path Is Required

A sustained current needs a complete path and a source that maintains an electric field. Opening a switch interrupts the conducting path. Brief redistribution or capacitor charging may occur, but there is no steady current around an open loop.

13. Direct and Alternating Current

Direct current keeps one conventional direction, even if its magnitude changes. Alternating current reverses direction periodically, so its signed \(I(t)\) alternates above and below zero. Conventional direction remains the reference at every instant.

14. Charge Flow Is Not Energy Flow

Current measures coulombs per second. Voltage measures energy transferred per coulomb. A charge carrier can move through a device, transfer energy to it, and continue through the circuit; the charge is not used up.

\(P=I\Delta V\)

This power relationship will be developed later in the unit. It already shows why current alone does not determine the rate of energy transfer.

Worked Example 8 · Ampere-Hours

Battery Capacity Is a Charge Quantity

An idealized battery rated \(2.4\,\mathrm{A\,h}\) corresponds to

\(Q=(2.4\,\mathrm A)(3600\,\mathrm s)=8.64\times10^3\,\mathrm C\)

If its voltage were an ideal constant \(12\,\mathrm V\), the associated energy estimate would be \(E=Q\Delta V\approx1.04\times10^5\,\mathrm J\). Ampere-hours alone specify charge capacity; voltage and operating conditions are needed to estimate usable energy.

15. Measuring Current Safely

An ammeter measures charge flow through itself, so a traditional ammeter is inserted in series with the branch being measured and is designed to have very low resistance. Connecting such a meter directly across a source can create a dangerously large current.

For study activities, use a simulation or instructor-approved low-voltage equipment with current limiting. Never open mains-powered equipment or place a current meter directly across a battery without an appropriate load and range setting.

16. Reliable Current Workflow

  1. Choose a cross-section and a positive-current direction.
  2. Convert all time units to seconds and areas to square meters.
  3. Decide whether the task needs an average, derivative, or integral.
  4. For carrier models, distinguish charge magnitude from direction.
  5. At a junction, define the boundary and list every incoming and outgoing current.
  6. Check units and whether the result describes charge, current, speed, density, or energy.

17. Common Traps

  • Calling current the amount of charge rather than its flow rate.
  • Assuming conventional current follows electron drift.
  • Using \(Q=It\) when current changes without integrating.
  • Reading graph height instead of area when asked for transferred charge.
  • Thinking current is consumed by a circuit component.
  • Confusing slow drift speed with rapid circuit response.
  • Forgetting to square the wire radius when calculating area.
  • Treating ampere-hours as energy without a voltage.
  • Connecting an ammeter in parallel across a source.
Mastery Check

1. \(24\,\mathrm C\) crosses a section in \(8.0\,\mathrm s\). Find the average current.

Show reasoning and answer

\(I_{\mathrm{avg}}=\Delta Q/\Delta t=24/8.0=3.0\,\mathrm A\).

2. An electron drifts left in a metal wire. Which direction is conventional current?

Show reasoning and answer

Right. Conventional current is opposite the drift direction of negative carriers.

3. A constant \(-2.0\,\mathrm A\) lasts \(3.0\,\mathrm s\). What is the signed transferred charge?

Show reasoning and answer

\(\Delta Q=I\Delta t=(-2.0)(3.0)=-6.0\,\mathrm C\). The negative sign means transfer opposite the chosen positive direction.

4. The same material carries the same current through a section with half the area. What happens to \(J\) and \(v_d\)?

Show reasoning and answer

\(J=I/A\) doubles. With \(n\) and \(|q|\) fixed, \(v_d=I/(n|q|A)\) also doubles.

5. Currents of \(2.0\,\mathrm A\) and \(0.80\,\mathrm A\) enter a steady junction. One branch carries \(1.1\,\mathrm A\) out. Find the other outgoing current.

Show reasoning and answer

Total incoming current is \(2.8\,\mathrm A\). Charge conservation gives \(2.8=1.1+I\), so \(I=1.7\,\mathrm A\).

6. Why can a neutral wire carry current?

Show reasoning and answer

The positive lattice and electrons still nearly balance in total charge. Current comes from a small net drift of mobile carriers through each cross-section, not from giving the whole wire a large net charge.

Investigation: Compare Electron Flow and Conventional Current

Use the open PhET Circuit Construction Kit: DC simulation. Build one low-voltage battery–resistor loop with a switch. Display electron motion, predict the conventional-current direction, and then switch the display to conventional current to test the prediction.

Place non-contact ammeters at several points in one series loop and compare readings. Then create a two-branch junction and test \(\sum I_{\mathrm{in}}=\sum I_{\mathrm{out}}\). Finally, open the switch and explain the response using both the closed-path requirement and charge conservation.

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