Definition

Current density

J = \frac{I}{A} = n q v_{d}

Statement

Current per unit area — the local version of current, and a vector.

Symbols and units

$J$	current density	A/m²
$I$	current	A
$A$	cross-sectional area	m²
$n, q, v_{d}$	carrier density, charge per carrier, drift speed	m⁻³, C, m/s

When it applies

Steady current, uniform over the cross-section.

Derivation

Divide the current by the area it flows through:

J = \frac{I}{A}

Substitute the microscopic expression; the area cancels:

J = \frac{n q v _{d} A}{A} = n q v_{d}

Worked examples

Reducing the cross-section from 2.0 to 0.5 mm²

J \to 4 J, R \to 4 R, P = I^{2} R \to 4 P

Four times the dissipation in a conductor with less surface to lose heat. This is how a fuse is designed to fail.

Watch out for

Current density is what sets the heating. Halve the cross-section at the same current and J doubles — which is exactly how a fuse is designed to fail.

Connects to

Ohm's law Microscopic model