Current density

Current density J is the local, vector-valued version of electric current. Where an ammeter reading I tells you how much charge passes through a particular cross-section per second (a single number, in amperes), J at a point tells you how much charge per second passes through a unit area perpendicular to the flow direction at that point, in the direction of the flow. Its magnitude has units of amperes per square metre, and it is a vector pointing along the local direction of charge transport.

For a uniform current flowing through a wire of cross-sectional area A, J = I/A and points along the wire. For a copper wire carrying a kilo-amp through a square millimetre, J ≈ 10⁹ A/m² — a nearly inconceivable density that the wire survives only because the conduction electrons are moving at drift velocities of millimetres per second through a sea of about 10²⁹ free electrons per cubic metre. The microscopic decomposition is J = nqv, where n is the number density of charge carriers, q is each carrier's charge, and v is the average drift velocity.

The total current through any surface is just the integral of J·dA over that surface — the flux of current density through the surface, exactly analogous to the flux of electric field that appears in Gauss's law. This is why current-density is the natural variable for the differential form of Ampère's law (∇×B = μ₀J + …) and for the Biot–Savart integral over an extended source (∫J × r̂ / r² dV). Whenever you go from talking about a wire as a 1-dimensional thread to talking about an extended conductor or a continuous medium, current splits into current-density.