Current Density

Ordinary electric current I tells us how much charge flows per second through a conductor, but it does not tell us how that flow is distributed across the conductor's cross-section. Two wires can carry the same current, yet one may be thin and the other thick — the charge is far more "crowded" in the thin wire.

Think of water flowing through pipes of different widths. The same volume of water per second (current) passing through a narrow pipe flows faster and more densely than through a wide pipe. Current density measures this "crowding effect" of charge flow — just as flow rate per unit pipe area describes water flow intensity.

The amount of electric current flowing per unit cross-sectional area of a conductor, measured at a point perpendicular to the direction of current flow.

For a conductor of uniform cross-sectional area A carrying current I perpendicular to the surface:

j = \[\frac {I}{A}\]

SI Unit: ampere per square metre (A/m²)
Dimensional Formula: [A L⁻²]

When the current is not uniformly distributed, or the area under consideration is inclined to the direction of flow, current density must be expressed as a vector integral:

\[I=\int\vec{j}\cdot d\vec{A}\]

• Only the component of \[\vec{j}\]​ parallel to the area vector \[d\vec{A}\] contributes to the current crossing that surface.
• If the current flows at an angle θ to the normal of the surface, effective current = j A cos⁡ θ.

A conductor carries a current of 0.5 A for 10 minutes. Find the total charge transported.

Solution:

Answer: 300 C

A current of 2 A flows through a conductor of cross-sectional area 4 × 10⁻⁶ m². Find the current density.

Solution:

Answer: 5 × 10⁵ A m⁻²