11.3 Current, resistance, and Ohm’s law

Charges in motion are a current, and a current in a conductor is what an electric field produces when charges are free to move. This lesson defines current density, derives Ohm’s law from the microscopic drift of charge carriers, and prepares the currents that will source magnetic fields in the next lesson.

Current density and drift

When charge carriers of number density $n$ and charge $q$ move with an average drift velocity $\mathbf{v}_\text{drift}$ , they carry a current density — charge crossing unit area per unit time —

\mathbf{J} \;=\; n q\,\mathbf{v}_\text{drift}.

where

$\mathbf{J}$: current density A/m²
$n$: carrier number density 1/m³
$q$: charge per carrier C
$\mathbf{v}_\text{drift}$: mean drift velocity m/s

In an applied field the carriers do not accelerate freely; they are scattered by the lattice or the surrounding medium and reach a terminal drift velocity proportional to the field. Writing $\mathbf{v}_\text{drift} = \mu\mathbf{E}$ defines the mobility $\mu$ , the drift velocity per unit field.

Ohm’s law, microscopically

Substituting the drift relation gives the local form of Ohm’s law:

\mathbf{J} \;=\; \sigma\mathbf{E}, \qquad \sigma = nq\mu,

with $\sigma$ the conductivity. This is the microscopic statement — current density proportional to field — from which the familiar $V = IR$ follows by integrating over a wire: for a uniform conductor of length $L$ and cross-section $A$ , $R = L/(\sigma A)$ .

E100 V/m

v_drift = μE7.00 μm/s

σ = nqμ1.12 S/m

J = σE112.1 A/m²

Field E = 100 V/m Mobility μ = 7.00 × 10⁻⁸ m²/(V·s) Density n = 1.0 × 10²⁶ m⁻³

A charged species in a fluid drifts at terminal velocity v_drift = μE, where μ is the mobility (drift velocity per unit field). The current density is J = ρ_q v_drift = nqμE = σE, with σ = nqμ the *conductivity*. This is the microscopic form of Ohm's law. For K+ ions in physiological saline, μ ~ 7×10⁻⁸ m²/(V·s); typical fields in cell membranes (10⁷ V/m) give drift speeds of metres per second — consistent with ion channels' ms-scale gating times.

The linearity of Ohm’s law is not fundamental but a consequence of frequent scattering: each carrier loses its drift memory so often that its average velocity stays proportional to the field. The mobility itself has a microscopic origin. For charge carriers moving through a viscous medium — ions in an electrolyte, for instance — the same drag coefficient that sets the diffusion coefficient sets the mobility, through the friction $\gamma$ of the transport chapter: $\mu = q/\gamma$ . Conductivity, diffusion, and viscous drag all trace back to the same collisions.

Continuity of charge

Charge is conserved, and the statement is identical in form to the conservation of mass from fluid mechanics:

\frac{\partial \rho}{\partial t} \;+\; \nabla\cdot\mathbf{J} \;=\; 0.

Charge accumulates only where current converges. In steady state $\partial\rho/\partial t = 0$ , so $\nabla\cdot\mathbf{J} = 0$ : steady currents flow in unbroken loops. Those steady currents are the sources of magnetism, the subject of the next lesson.