Electric Circuits Foundations: Charge Flow, Material Response, and Circuit Sources

Electric current

The rate at which electric charge passes through a chosen cross-section (“gate”) of a conductor or circuit element.

Average current

Current computed over a time interval: $I = \frac{\triangle Q}{\triangle t}$.

Instantaneous current

Current at a specific moment: I = dQ/dt.

Ampere (A)

Unit of current; 1 A means 1 coulomb of charge crosses a cross-section each second (1 A = 1 C/s).

Steady-state (DC) current continuity

In a steady DC situation, the same current flows through every cross-section of a single, unbranched wire (no significant charge pileup in the interior).

Drift velocity (vd)

The average velocity of charge carriers in a conductor due to an electric field; typically very small in metals even for large currents.

Electric field signal (in circuits)

The rapidly established electric field around a circuit that sets up steady current flow; it propagates much faster than electron drift speed.

Conventional current direction

By convention, current points in the direction positive charge would move (used in almost all AP circuit problems).

Electron flow direction

In metals, electrons (negative charges) drift opposite the direction of conventional current.

Current density (J)

Current per unit cross-sectional area at a point; for uniform flow J = I/A, and as a vector it points with conventional current (units A/m²).

Microscopic current model (counting formula)

For carrier density $n$, carrier charge magnitude $|q|$, cross-sectional area $A$, and drift speed $v_d$: $I = n|q|A v_d$.

Carrier number density (n)

Number of mobile charge carriers per unit volume (carriers/m³).

Carrier charge (q)

Charge of each carrier (coulombs); for electrons $q$ is negative, though many formulas use the magnitude $|q|$ for current magnitude.

Drift speed–area relationship

For fixed current $I$, drift speed $v_d$ increases when cross-sectional area $A$ decreases ($v_d \propto \frac{1}{A}$), consistent with continuity of charge flow.

Resistance (R)

A macroscopic measure of how strongly a component opposes current for a given potential difference across it (units ohms, Ω).

Ohm’s law (macroscopic)

Relationship for many circuit elements: $V = IR$, where $V$ is potential difference across the element.

Ohmic material/component

A material or device for which resistance R is effectively constant over the operating range, so V is proportional to I.

Resistivity (ρ)

A material property describing how strongly the material resists current flow at the microscopic level (units $\Omega \, m$).

Conductivity (σ)

Material property measuring how well a material conducts; σ = 1/ρ (units S/m).

Microscopic Ohm’s law

Field form relating current density to electric field in a conductor: $\tilde{J} = \rho \tilde{E}$.

Resistance–geometry relation

For a uniform wire of length $L$ and cross-sectional area $A$: $R = \frac{\rho L}{A}$ (separates material property $\rho$ from geometry).

Power dissipated in a resistor

Rate electrical energy is converted to thermal energy: $P = IV = I^{2}R = \frac{V^{2}}{R}$ (choice depends on what is held constant).

Electromotive force (EMF, ℰ)

Energy per unit charge supplied by a source (not a mechanical force): $\boldsymbol{\text{E}} = \frac{W}{q}$ (units volts, J/C).

Internal resistance (r)

A model of energy loss inside a real voltage source; represented as a resistor in series with an ideal EMF source.

Terminal voltage (Vterminal)

The measured potential difference across a source’s terminals under load; when delivering current, $V_{\text{terminal}} = \boldsymbol{\text{E}} - Ir$ (equals $\boldsymbol{\text{E}}$ if $I = 0$).