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.

Definition

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:

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

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.

In special relativity theory, charge and current are combined into a 4-vector.

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:

:$\backslash mathbf\{J\}\; =\; \backslash sigma\; \backslash mathbf\{E\}\; \backslash ,$

where E is the electric field 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:

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

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:

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

where ''σ''(k, ''ω'') is now a complex function.

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 area S perpendicular to the flow can be calculated using a surface integral:

:$I=\backslash int\_S\{\; \backslash mathbf\{J\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\}\}$

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

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:

:$\backslash int\_S\{\; \backslash mathbf\{J\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\}\}\; =\; -\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}t\}\; \backslash int\_V\{\backslash rho\; \backslash ;\; \backslash mathrm\{d\}V\}\; =\; -\; \backslash int\_V\{\backslash left(\; \backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; t\}\; \backslash right)\; \backslash mathrm\{d\}V\}\backslash \; ,$

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,

:$\backslash int\_S\{\; \backslash mathbf\{J\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\}\}\; =\; \backslash int\_V\{(\backslash mathbf\{\backslash nabla\}\; \backslash cdot\; \backslash mathbf\{J\})\; \backslash mathrm\{d\}V\}\backslash \; .$

Hence:

:$\backslash int\_V\{(\backslash mathbf\{\backslash nabla\}\; \backslash cdot\; \backslash mathbf\{J\})\; \backslash mathrm\{d\}V\}\backslash \; =\; -\; \backslash int\_V\{\backslash left(\; \backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; t\}\; \backslash right)\; \backslash mathrm\{d\}V\}\backslash \; .$

Because this relation is valid for any volume, no matter how small, no matter where located:

:$\backslash nabla\; \backslash cdot\; \backslash mathbf\{J\}\; =\; -\; \backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; t\}\backslash \; ,$

which is called the continuity equation.

In practice

In the domain of electrical wiring (isolated copper), maximum current density can vary from 4A/mm² for a wire isolated from free air to 6A/mm² for a wire in free air. However, for for compact designs (e.g. windings of SMPS transformers) the value might be as low as 2A/mm². If the wire is carrying high frequency currents, depending on its diameter, the skin effect may affect the distribution of the current across the section by concentrating the current on the surface of the conductor. This skin effect plays an important role in Switched-mode power supply transformers where the wires carries high currents and high frequencies (between 10 kHz and 1 MHz). Often in those transformers, the windings are made of multiple isolated wires in parallel with a diameter twice the skin depth and which are twisted together to increase the total skin area and to reduce the impact of the skin effect.

In the domain of printed circuit boards, for TOP and BOTTOM layers, maximum current density can be as high as 35A/mm² with a copper thickness of 35 µm. Inner layers cannot dissipate as much power as outer layers; thus it is not a good idea to put high power lines in inner layers.

In the domain of semiconductors, the maximum current density is given by the manufacturer. A common average is 1mA/µm² at 25°C for 180 nm technology. Above the maximum current density, apart from the joule effect, some other effects like electromigration appear in the micrometer scale.

In biological systems, ion channels regulate the flow of ions (for example, sodium, calcium, potassium) across the membrane in all cells. Current density is measured in pA/pF (picoamperes per picofarad), that is, current divided by capacitance, a de facto measure of membrane area.

In gas discharge lamps, such as flashlamps, current density plays an important role in the output spectrum produced. Low current densities produce spectral line emission and tend to favor longer wavelengths. High current densities produce continuum emission and tend to favor shorter wavelengths. Low current densities for flash lamps are generally around 1000A/cm². High current densities can be more than 4000A/cm².

References

External links

A short explanation of the current density

See also

Hall effect

Quantum Hall effect

Superconductivity

Electron mobility

Drift velocity

Effective mass

Electrical resistance

Sheet resistance

Speed of electricity

Electrical conduction

Green–Kubo relations

Green's function (many-body theory)

Category:Fundamental physics concepts Category:Electromagnetism Category:Density

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