# Current density

Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved quantities. It is defined as a vector whose magnitude is the current per cross-sectional area.

In SI units, the electric current density is measured in amperes per square metre or coulomb per second per square metre.

## Definition

Electric current is a coarse, average quantity that tells what is happening in an entire wire. If we want to describe the distribution of the charge flow, we use the concept of the current density:

${\displaystyle \mathbf {J} =nq\mathbf {v} _{d}=\rho \mathbf {v} _{d}\!\ }$

where

${\displaystyle \mathbf {J} \!\ }$ is the current density vector (SI unit amperes per square metre)
${\displaystyle n\!\ }$ is the particle density in count per volume (SI unit m-3)
${\displaystyle q\!\ }$ is the individual particles' charge (SI unit coulombs)
${\displaystyle \rho =nq\!\ }$ is the charge density (SI unit coulombs per cubic metre)
${\displaystyle \mathbf {v} _{d}\!\ }$ is the particles' average drift velocity (SI unit metres per second)

The current through a surface S can be calculated by the following relation:

${\displaystyle I=\int _{S}{\mathbf {J} \cdot \mathrm {d} \mathbf {S} }}$

– where the current is in fact the integral of the dot product of the current density vector and the differential surface element ${\displaystyle \mathrm {d} \mathbf {S} \ }$, i.e. the net flux of the current density vector field flowing through the surface S.

The current density is an important parameter in Ampère's circuital law (one of Maxwell's equations), which show the direct link between current density and magnetic field.

Current density is an important consideration in the design of electrical and electronic systems. Most electrical conductors have a finite, positive resistance, making them dissipate power in the form of heat. The current density must be kept sufficiently low to prevent the conductor from melting or burning up, or the insulating material failing. In superconductors excessive current density may generate a strong enough magnetic field to cause spontaneous loss of the superconductive property.

## Divergence of current density

From the divergence theorem,

${\displaystyle \int _{S}{\mathbf {J} \cdot \mathrm {d} \mathbf {S} }=\int _{V}{(\mathbf {\nabla } \cdot \mathbf {J} )\mathrm {d} V}}$

since charge is conserved,

${\displaystyle \int _{V}{(\nabla \cdot \mathbf {J} )\mathrm {d} V}=-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}{\rho \;\mathrm {d} V}=-\int _{V}{\left({\frac {\partial \rho }{\partial t}}\right)\mathrm {d} V}}$

Since this is valid for any volume,

${\displaystyle \nabla \cdot \mathbf {J} =-{\frac {\partial \rho }{\partial t}}}$.

which is also called the continuity equation.[1]

## References

1. Griffiths, D.J., Introduction to Electrodynamics, page 213, Prentice-Hall International, 1999.