Academy

Current Density

Current density J is the current per unit cross-sectional area. It is a vector in the direction of conventional current flow. SI unit: A m⁻².
Class 12
Derivation

Definition

Current density J⃗\vec{J} at a point is the current per unit area perpendicular to the flow:

J⃗=IAn^(uniform flow)\vec{J} = \frac{I}{A}\hat{n} \quad \text{(uniform flow)}

For non-uniform flow, the current through an area element dA⃗d\vec{A}:

dI=J⃗⋅dA⃗dI = \vec{J} \cdot d\vec{A}

Total current through a surface:

I=∫SJ⃗⋅dA⃗\boxed{I = \int_S \vec{J} \cdot d\vec{A}}

Relation to drift velocity

From I=neAvdI = neAv_d for a conductor with carrier density nn:

J=IA=nevdJ = \frac{I}{A} = nev_d

In vector form: J⃗=nev⃗d\vec{J} = ne\vec{v}_d

Ohm's law in microscopic form

Combining with the drift velocity expression vd=eEτ/mv_d = eE\tau/m:

J=ne⋅eEτm=ne2τmE=σEJ = ne \cdot \frac{eE\tau}{m} = \frac{ne^2\tau}{m}E = \sigma E J⃗=σE⃗\vec{J} = \sigma\vec{E}

This is the microscopic (point-by-point) form of Ohm's law, more fundamental than V=IRV = IR.

Note
$\vec{J} = \sigma\vec{E}$ holds locally at every point inside the conductor. The macroscopic $V = IR$ is obtained by integrating this relation over the conductor geometry.