AP Physics C: E&M Unit 4: Electric Circuits Notes

Introduction

Circuit analysis combines charge conservation at junctions with energy conservation around loops, then adds constitutive laws for each component.

Current measures charge transport, while voltage tracks energy transferred per charge. A schematic records electrical connections rather than physical distances, so the first task is always to identify nodes, branches, and closed paths.

Resistance, capacitance, sources, and meters provide component rules. Kirchhoff’s laws connect those local rules into a system; in an RC circuit the same laws become a differential equation that predicts the transient state.

Charge crossing a surface

current and current density

Material plus geometry

resistance

Node charge conservation

junction rule

Loop energy conservation

loop rule

Capacitance plus resistance

exponential transient

Current exam guidance: Aligned to the current 2026–27 AP Physics C: Electricity and Magnetism framework, with calculus-based field, circuit, and induction models, explicit assumptions, and experimental reasoning.

11.1

Electric Current

How do macroscopic current and microscopic carrier motion describe the same charge transport?

Short answer

Current is the signed rate $I=dQ/dt$ through a chosen cross section; microscopically it is the flux of current density, $I=\int_A\vec J\cdot d\vec A$ .

Know firstCharge conservation · Definite integrals · Vector flux

Current counts net crossing, not particle speed

Choose an orientation for a wire and an area normal. Positive current means net positive charge crosses in that direction. Conventional current therefore points opposite the drift of electrons in a metal. Although $\vec J$ is a vector, the current through an oriented surface is a signed scalar.

For uniform carriers, $\vec J=nq\vec v_d$ and $I=nqAv_d$ after consistent sign choices. The tiny drift velocity sits on top of much faster random microscopic motion. In steady series flow, current is not consumed by a resistor; different carrier density or cross-sectional area instead changes drift speed.

Physics model

Current is charge flow through an oriented area

Increase electron drift-speed magnitude in a wire with n = 8.0×10²⁸ m⁻³ and A = 1.00 mm². Conventional current points opposite electron drift and grows in proportion to its magnitude.

Electron drift speed1.5 mm/s

Drift speed: 1.5 mm/s
Current magnitude: 19.224 A
Electron motion: opposite conventional current

conducting wiremobile electronsconventional currentcross section

\begin{gathered}\vec J=nq\vec v_d\\I=n|q|Av_d\quad\text{(magnitude)}\end{gathered}

Read this figure without the graphic

Mobile electrons occupy a cylindrical wire and drift opposite a conventional-current arrow. A highlighted cross section marks the area used in the current integral.

Current quantity	Value
Drift speed	1.5 mm/s
Current magnitude	19.224 A
Electron motion	opposite conventional current
Model relation	$\begin{gathered}\vec J=nq\vec v_d\\I=n\|q\|Av_d\quad\text{(magnitude)}\end{gathered}$

Historic Weston electrical instrument with separate volts and amperes scales around one pointer. — Current becomes a reading only after the meter joins the circuit
The two scales show that current and potential difference are different measurements even when one instrument can report either quantity. To measure current, the instrument must be connected in series so the branch charge flow passes through it.
Use this visual to see
Connect electric current to a physical meter reading while distinguishing current measurement from voltage measurement.
What it cannot show
The close view shows one pointer between a volts scale and an amperes scale. It does not show the instrument connected to a circuit, identify the selected mode, or establish current direction, drift speed, or internal resistance.

Recover transported charge from a current history

System: Charge crossing one oriented wire section.
Frame: Laboratory frame.
Assumptions: $I(t)=0.4+0.2t$ A for $0\le t\le3$ s.
Givens: $I=dQ/dt$ .

Integrate current over the exact time interval.
Keep the sign of current in the integral.
Interpret the result relative to the selected surface normal.

Result

\Delta Q=\int_0^3(0.4+0.2t)\,dt=2.1\ \mathrm C

Units:

1\ \mathrm A=1\ \mathrm{C/s}

Boundary cases

A reversing current can give zero net transported charge despite nonzero total traffic.
Zero current means zero net drift, not zero microscopic motion.

The area under the current-versus-time graph is two point one coulombs.

Separate transport quantities

Quantity	Definition	Character
current $I$	$dQ/dt$	signed scalar for a path
current density $\vec J$	$nq\vec v_d$	vector field
transported charge	$\int I\,dt$	signed accumulation

Set a current convention

Choose the path direction or surface normal.
Identify carrier charge sign and drift direction.
Use flux for a nonuniform current density.
Integrate in time for transported charge.
Check C/s for current and A/m² for current density.

Keep these three ideas

Current is signed charge rate.
Current density carries spatial direction.
Conventional current opposes electron drift in metals.

Electric Circuits