Current density

The current density $\mathbf{J}$ is a vector field whose magnitude is the current per unit area flowing through a surface oriented perpendicular to $\mathbf{J}$, and whose direction points along the local flow. Units are A/m². It’s the moving analog of Charge density: charge density tells you where charge sits, current density tells you where it’s headed and how fast.

For a volume charge density ${\rho }_v$ moving with mean velocity $\mathbf{u}$:

$\mathbf{J}={\rho }_v\mathbf{u}.$

This is the basic identity. A region with carrier density ${\rho }_v$ drifting at $\mathbf{u}$ pushes charge through any cross-section at the local rate ${\rho }_{v}u$ per unit area.

Total current through a surface

The current through a surface $S$ is the flux of $\mathbf{J}$:

$I={\int }_S\mathbf{J}\cdot d\mathbf{s}\text{(A)}.$

Only the component of $\mathbf{J}$ along the surface normal contributes — flow parallel to the surface carries no charge across it. For a flat surface with uniform $\mathbf{J}$ perpendicular to it, this collapses to the familiar $I=JS$.

Three flavors

Like Charge density, current density comes in volume, surface, and line versions:

Volume current density $\mathbf{J}$ (A/m²) — fills a 3D region. Used for current flowing through the bulk of a conductor.

Surface current density ${\mathbf{J}}_s$ (A/m) — current per unit width, flowing on a 2D sheet. Used for idealized “thin sheet” currents, the surface currents on a perfect conductor at high frequency, or on the inside surface of a coaxial outer shield.

Line current $I$ (A) — current confined to a 1D filament. The idealization used in Biot-Savart law integrals over a wire.

The pattern matches charge density: a thin tube of cross-section $A$ carrying volume current $\mathbf{J}$ acts as line current $I=JA$; a thin slab of thickness $\Delta$ acts as surface current ${\mathbf{J}}_s=\mathbf{J}\Delta$ in the $\Delta \to 0$ limit.

Where it appears

$\mathbf{J}$ is the source on the right side of Ampère’s law:

$\nabla \times \mathbf{H}=\mathbf{J}\text{(magnetostatics)},$

extended to $\nabla \times \mathbf{H}=\mathbf{J}+{\partial }_t\mathbf{D}$ in dynamics, where ${\partial }_t\mathbf{D}$ is the Displacement current — itself an “effective” current density that closes the loop in regions where free charges aren’t moving but the electric field is.

Inside a conductor, $\mathbf{J}$ is set by the Electric field via the point-form Ohm’s law $\mathbf{J}=\sigma \mathbf{E}$ (see Conduction current and Ohm’s law). Joule dissipation $\mathbf{E}\cdot \mathbf{J}=\sigma E^2$ gives the power lost to heat per unit volume.

Continuity equation

Charge conservation forces $\mathbf{J}$ and ${\rho }_v$ to be linked. If charge flows out of a region, the charge inside drops:

$\nabla \cdot \mathbf{J}=-\frac{\partial {\rho }_v}{\partial t}.$

This is the Charge continuity equation — net outflow of current from a volume equals the rate of charge decrease inside. In steady-state (${\partial }_t{\rho }_v=0$) this becomes $\nabla \cdot \mathbf{J}=0$: current “in” equals current “out” at every point, which is what makes DC circuit analysis work.

Worked example

A copper wire of radius $a=1$ mm carrying $I=1$ A uniformly distributed over its cross-section.

$J=\frac{I}{\pi a^2}=\frac{1}{\pi (10^{-3}{)}^2}\approx 3.2\times 10^5{\text{ A/m}}^2.$

With copper’s free-electron density $n\approx 8.5\times 10^{28}$ m⁻³ and ${\rho }_v=-ne\approx -1.36\times 10^{10}$ C/m³, the drift velocity is

$u=\frac{J}{|{\rho }_v|}=\frac{3.2\times 10^5}{1.36\times 10^{10}}\approx 2.4\times 10^{-5}\text{ m/s}.$

Drift velocity in metals is famously tiny — fractions of a mm/s — even though signals propagate near $c$. The mismatch is because every carrier in the wire moves at this drift speed simultaneously; the field that sets them moving propagates at light speed.

Versus current

$\mathbf{J}$ is a field; current $I$ is a scalar through a surface. Two wires of the same cross-section with the same current have the same $J$, but they may have wildly different shapes and field patterns. $\mathbf{J}$ is what enters Maxwell’s equations point-by-point; $I$ is what a meter reads.