Current density

Encyclopedia

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 ampere

s per square metre

.

is described by the current density:

where

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.

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

units of newtons per coulomb (N C

s per metre

(V m

A more fundamental approach to calculation of current density is based upon:

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:

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.

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

:

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.

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

,

Hence:

Because this relation is valid for any volume, independent of size or location,

This relation is called the continuity equation

.

Density

The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

of flow of a conserved charge

Charge (physics)

In physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.-Formal definition:...

. Usually the charge is the electric charge, in which case the associated current density is the electric current

Electric current

Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

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 ampere

Ampere

The ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...

s per square metre

Square metre

The square metre or square meter is the SI derived unit of area, with symbol m2 . It is defined as the area of a square whose sides measure exactly one metre...

.

## Definition

Electric current is a coarse, average quantity that tells what is happening in an entire wire. The distribution of flow of chargeElectric charge

Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

is described by the current density:

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 densityCharge densityThe linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...

(SI unit: coulombs per cubic metreCubic metreThe cubic metre is the SI derived unit of volume. It is the volume of a cube with edges one metre in length. An alternative name, which allowed a different usage with metric prefixes, was the stère...

), and - v
_{d}(r, t) is the particles' average drift velocityDrift velocityThe drift velocity is the average velocity that a particle, such as an electron, attains due to an electric field. It can also be referred to as Axial Drift Velocity since particles defined are assumed to be moving along a plane. In general, an electron will 'rattle around' in a conductor at the...

at position r at time t (SI unit: metreMetreThe metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

s per secondSecondThe second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

)

## Importance

Current density is important to the design of electrical and electronicElectronics

Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...

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

Moore's Law

Moore's law describes a long-term trend in the history of computing hardware: the number of transistors that can be placed inexpensively on an integrated circuit doubles approximately every two years....

.

At high frequencies, current density can increase because the conducting region in a wire becomes confined near its surface, the so-called skin effect

Skin effect

Skin effect is the tendency of an alternating electric current to distribute itself within a conductor with the current density being largest near the surface of the conductor, decreasing at greater depths. In other words, the electric current flows mainly at the "skin" of the conductor, at an...

.

High current densities have undesirable consequences. Most electrical conductors have a finite, positive resistance

Electrical resistance

The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

, making them dissipate power

Power (physics)

In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...

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

Electromigration

Electromigration is the transport of material caused by the gradual movement of the ions in a conductor due to the momentum transfer between conducting electrons and diffusing metal atoms. The effect is important in applications where high direct current densities are used, such as in...

. In superconductors

Superconductivity

Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...

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

Maxwell's equations

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

), which relates current density to magnetic field

Magnetic field

A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

.

In special relativity

Special relativity

Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

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:where E is the electric field

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

and σ is the electrical conductivity.

Conductivity σ is the reciprocal (inverse) of electrical resistivity

Resistivity

Electrical resistivity is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electric charge. The SI unit of electrical resistivity is the ohm metre...

and has the SI units of siemens

Siemens (unit)

The siemens is the SI derived unit of electric conductance and electric admittance. Conductance and admittance are the reciprocals of resistance and impedance respectively, hence one siemens is equal to the reciprocal of one ohm, and is sometimes referred to as the mho. In English, the term...

per metre

Metre

The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

(S m

^{−1}), and E has the SISi

Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

units of newtons per coulomb (N C

^{−1}) or, equivalently, voltVolt

The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...

s per metre

Metre

The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

(V m

^{−1}).A more fundamental approach to calculation of current density is based upon:

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

Linear response function

A linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response...

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

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

in space and time then results in:

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

In many materials, for example, in crystalline materials, the conductivity is a tensor

Tensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

, 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 areaSurface area

Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces...

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

Surface integral

In mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral...

:

where the current is in fact the integral of the dot product

Dot product

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

of the current density vector and the differential of the directed surface element dA, in other words, the net flux

Flux

In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...

of the current density vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

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: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

Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...

,

Hence:

Because this relation is valid for any volume, independent of size or location,

This relation is called the continuity equation

Continuity equation

A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...

.

## In practice

- In the domain of electrical wiring (isolated copper), maximum current density can vary from 4A/mm
^{2}for a wire isolated from free air to 6A/mm^{2}for a wire in free air. However, for compact designs (e.g. windings of SMPS transformersSwitched-mode power supplyA switched-mode power supply is an electronic power supply that incorporates a switching regulator in order to be highly efficient in the conversion of electrical power...

) the value might be as low as 2A/mm^{2}. If the wire is carrying high frequency currents, depending on its diameter, the skin effectSkin effectSkin effect is the tendency of an alternating electric current to distribute itself within a conductor with the current density being largest near the surface of the conductor, decreasing at greater depths. In other words, the electric current flows mainly at the "skin" of the conductor, at an...

may affect the distribution of the current across the section by concentrating the current on the surface of the conductor. This skin effectSkin effectSkin effect is the tendency of an alternating electric current to distribute itself within a conductor with the current density being largest near the surface of the conductor, decreasing at greater depths. In other words, the electric current flows mainly at the "skin" of the conductor, at an...

plays an important role in Switched-mode power supplySwitched-mode power supplyA switched-mode power supply is an electronic power supply that incorporates a switching regulator in order to be highly efficient in the conversion of electrical power...

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/mm
^{2}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/µm
^{2}at 25°C for 180 nm technology. Above the maximum current density, apart from the joule effect, some other effects like electromigrationElectromigrationElectromigration is the transport of material caused by the gradual movement of the ions in a conductor due to the momentum transfer between conducting electrons and diffusing metal atoms. The effect is important in applications where high direct current densities are used, such as in...

appear in the micrometer scale.

- In biological systems, ion channelIon channelIon channels are pore-forming proteins that help establish and control the small voltage gradient across the plasma membrane of cells by allowing the flow of ions down their electrochemical gradient. They are present in the membranes that surround all biological cells...

s 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 capacitanceCapacitanceIn electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

, 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/cm
^{2}. High current densities can be more than 4000A/cm^{2}.

## External links

## See also

- Hall effectHall effectThe Hall effect is the production of a voltage difference across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current...
- Quantum Hall effect
- SuperconductivitySuperconductivitySuperconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...
- Electron mobilityElectron mobilityIn solid-state physics, the electron mobility characterizes how quickly an electron can move through a metal or semiconductor, when pulled by an electric field. In semiconductors, there is an analogous quantity for holes, called hole mobility...
- Drift velocityDrift velocityThe drift velocity is the average velocity that a particle, such as an electron, attains due to an electric field. It can also be referred to as Axial Drift Velocity since particles defined are assumed to be moving along a plane. In general, an electron will 'rattle around' in a conductor at the...

- Effective mass
- Electrical resistanceElectrical resistanceThe electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
- Sheet resistanceSheet resistanceSheet resistance is a measure of resistance of thin films that are namely uniform in thickness. It is commonly used to characterize materials made by semiconductor doping, metal deposition, resistive paste printing, and glass coating. Examples of these processes are: doped semiconductor regions...

- Speed of electricitySpeed of electricityThe speed of electricity refers to the movement of electrons through a conductor in the presence of potential and an electric field. The finite velocity of the electric field thus requires consideration, and may even become the dominating factor in the electrical phenomena: In the conduction of...
- Electrical conduction
- Green–Kubo relations
- Green's function (many-body theory)Green's function (many-body theory)In many-body theory, the term Green's function is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators....