current density
/ KUR-ent DEN-sih-tee /
Current density tells you how crowded the flow of charge is, not just how much is flowing. Two wires can carry the same 2 amperes, but if one is thin and one is thick, the charge in the thin one is packed far more tightly. Picture the same river discharging through a narrow gorge versus a wide delta: the total flow is equal, but the water rushes far more intensely through the gorge. Current density captures that intensity.
Precisely, current density J is the current per unit cross-sectional area, J = I / A, and it is a vector pointing in the direction the current flows. Its unit is amperes per square metre (A/m^2). At the microscopic level it connects to the carriers by J = n q v_d, where n is the carrier number density, q the charge per carrier, and v_d the drift velocity. It also links a material's response to the field through the microscopic form of Ohm's law, J = sigma E, where E is the electric field and sigma is the conductivity of the material.
Current density matters because materials fail when the flow gets too concentrated, not simply when the total current is large: a thin fuse wire melts because the same current forced through a small area gives a huge J and huge heating. Engineers rate wires by how much current density they can safely carry, which is why high-current cables are thick. Thinking in terms of J, rather than total current, is what lets you compare wires of different sizes fairly.
A current of 2 A flowing through a wire of cross-section 1 mm^2 (which is 1 x 10^-6 m^2) has a current density J = I / A = 2 / 1e-6 = 2 x 10^6 A/m^2. Squeeze the same 2 A into a wire ten times thinner in area and J becomes ten times larger.
Same current, thinner wire, higher current density and more heating per unit volume.
Current is a scalar total; current density is a vector describing how that flow is spread over area. Equal currents in a thick and a thin wire have very different current densities.