Current Density

Electric current is a coarse, average quantity that tells what is happening in an entire wire. The distribution of flow of charge is described by the current density:

$\mathbf{J}(\mathbf{r}, t) = qn(\mathbf{r},t) \; \mathbf{v}_d (\mathbf{r},t) = \rho(\mathbf{r},t) \; \mathbf{v}_d (\mathbf{r},t) \, ,$

where

J(r, t) is the current density vector at location r at time t (SI unit: amperes per square metre),

n(r, t) is the particle density in count per volume at location r at time t (SI unit: m⁻³),

q is the charge of the individual particles with density n (SI unit: coulombs)

ρ(r, t) = qn(r, t) is the charge density (SI unit: coulombs per cubic metre), and

v_d(r, t) is the particles' average drift velocity at position r at time t (SI unit: metres per second).

Importance

Current density is important to the design of electrical and electronic systems.
Circuit performance depends strongly upon the designed current level, and the current density then is determined by the dimensions of the conducting elements. For example, as integrated circuits are reduced in size, despite the lower current demanded by smaller devices, there is trend toward higher current densities to achieve higher device numbers in ever smaller chip areas. See Moore's law.
At high frequencies, current density can increase because the conducting region in a wire becomes confined near its surface, the so-called skin effect.
High current densities have undesirable consequences. 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. At high current densities the material forming the interconnections actually moves, a phenomenon called electromigration. In superconductors excessive current density may generate a strong enough magnetic field to cause spontaneous loss of the superconductive property.
The analysis and observation of current density also is used to probe the physics underlying the nature of solids, including not only metals, but also semiconductors and insulators. An elaborate theoretical formalism has developed to explain many fundamental observations^.
The current density is an important parameter in Ampère's circuital law (one of Maxwell's equations), which relates current density to magnetic field.

Approximate calculation of current density

A common approximation to the current density assumes the current simply is proportional to the electric field, as expressed by:

$\mathbf{J} = \sigma \mathbf{E} \,$

where E is the electric field strength and σ is the electrical conductivity.
Conductivity σ is the reciprocal (inverse) of electrical resistivity and has the SI units of siemens per metre (S m⁻¹), and E has the SI units of newtons per coulomb (N C⁻¹) or, equivalently, volts per metre (V m⁻¹).
A more fundamental approach to calculation of current density is based upon:

$\mathbf{J} (\mathbf{r}, t) = \int_{-\infty}^t \mathrm{d}t' \int \mathrm{d}^3\mathbf{r}' \; \sigma(\mathbf{r}-\mathbf{r}', t-t') \; \mathbf{E}(\mathbf{r}',\ t') \, ,$

indicating the lag in response by the time dependence of σ, and the non-local nature of response to the field by the spatial dependence of σ, both calculated in principle from an underlying microscopic analysis, for example, in the case of small enough fields, the linear response function for the conductive behavior in the material. See, for example, Giuliani or Rammer.The integral extends over the entire past history up to the present time.
As some reflection might indicate, the above conductivity and its associate current density reflect the fundamental mechanisms underlying charge transport in the medium, both in time and over distance.
A Fourier transform in space and time then results in:

$\mathbf{J} (\mathbf{k}, \omega) = \sigma(\mathbf{k}, \omega) \; \mathbf{E}(\mathbf{k}, \omega) \, ,$

where σ(k, ω) is now a complex function.
It should be noted that in many materials, for example, in crystalline materials, the conductivity is a tensor, and the current is not necessarily in the same direction as the applied field. Aside from the material properties themselves, the application of magnetic fields can alter conductive behavior.

Current through a surface

The current through a surface S can be calculated using a surface integral:

$I=\int_S{ \mathbf{J} \cdot \mathrm{d}\mathbf{A}}$

where the current is in fact the integral of the dot product of the current density vector and the differential surface element dA, in other words, the net flux of the current density vector field flowing through the surface S.

Continuity equation

Main article: Continuity equation

Because charge is conserved, the net flow out of a chosen volume must equal the net change in charge held inside the volume:

$\int_S{ \mathbf{J} \cdot \mathrm{d}\mathbf{A}} = -\frac{\mathrm{d}}{\mathrm{d}t} \int_V{\rho \; \mathrm{d}V} = - \int_V{\left( \frac{\partial \rho}{\partial t} \right) \mathrm{d}V}\ ,$

where ρ is the charge density per unit volume, and dA is a surface element of the surface S enclosing the volume V. The surface integral on the left expresses the current outflow from the volume, and the negatively signed volume integral on the right expresses the decrease in the total charge inside the volume. From the divergence theorem,

$\int_S{ \mathbf{J} \cdot \mathrm{d}\mathbf{A}} = \int_V{(\mathbf{\nabla} \cdot \mathbf{J}) \mathrm{d}V}\ .$