In
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a
vector whose magnitude is the
electric current per cross-sectional area at a given point in space, its direction being that of the motion of the positive charges at this point. In
SI base units, the electric current density is measured in
amperes per
square metre.
Definition
Assume that ''A'' (SI unit:
m2) is a small surface centred at a given point ''M'' and orthogonal to the motion of the charges at ''M''. If ''I'' (SI unit:
A) is the
electric current flowing through ''A'', then electric current density ''j'' at ''M'' is given by the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
:
:
with surface ''A'' remaining centered at ''M'' and orthogonal to the motion of the charges during the limit process.
The current density vector j is the vector whose magnitude is the electric current density, and whose direction is the same as the motion of the positive charges at ''M''.
At a given time ''t'', if v is the velocity of the charges at ''M'', and ''dA'' is an infinitesimal surface centred at ''M'' and orthogonal to v, then during an amount of time ''dt'', only the charge contained in the volume formed by ''dA'' and will flow through ''dA''. This charge is equal to , where ''ρ'' is the
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system i ...
at ''M'', and the electric current at ''M'' is . It follows that the current density vector can be expressed as:
:
The
surface integral
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one ...
of j over a
surface ''S'', followed by an integral over the time duration ''t''
1 to ''t''
2, gives the total amount of charge flowing through the surface in that time ():
:
More concisely, this is the integral of the
flux of j across ''S'' between ''t''
1 and ''t''
2.
The
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
required to calculate the flux is real or imaginary, flat or curved, either as a cross-sectional area or a surface. For example, for charge carriers passing through an
electrical conductor, the area is the cross-section of the conductor, at the section considered.
The
vector area is a combination of the magnitude of the area through which the charge carriers pass, ''A'', and a
unit vector normal to the area,
. The relation is
.
The differential vector area similarly follows from the definition given above:
.
If the current density j passes through the area at an angle ''θ'' to the area normal
, then
:
where ⋅ is the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
of the unit vectors. That is, the component of current density passing through the surface (i.e. normal to it) is , while the component of current density passing tangential to the area is , but there is ''no'' current density actually passing ''through'' the area in the tangential direction. The ''only'' component of current density passing normal to the area is the cosine component.
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 a trend toward higher current densities to achieve higher device numbers in ever smaller
chip areas. See
Moore's law.
At high frequencies, the conducting region in a wire becomes confined near its surface which increases the current density in this region. This is known as the
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, the
insulating material failing, or the desired electrical properties changing. 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
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
), which relates current density to
magnetic field.
In
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
theory, charge and current are combined into a
4-vector.
Calculation of current densities in matter
Free currents
Charge carriers which are free to move constitute a
free current density, which are given by expressions such as those in this section.
Electric current is a coarse, average quantity that tells what is happening in an entire wire. At position r at time ''t'', the ''distribution'' of
charge flowing is described by the current density:
:
where j(r, ''t'') is the current density vector, v
d(r, ''t'') is the particles' average
drift velocity (SI unit:
m∙
s−1), and
:
is the
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system i ...
(SI unit: coulombs per
cubic metre), in which ''n''(r, ''t'') is the number of particles per unit volume ("number density") (SI unit: m
−3), ''q'' is the charge of the individual particles with density ''n'' (SI unit:
coulombs).
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 and ''σ'' is the
electrical conductivity.
Conductivity ''σ'' is the
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
(
inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when a ...
) of electrical
resistivity and has the SI units of
siemens per
metre
The metre ( British spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pre ...
(S⋅m
−1), and E has the
SI units of
newtons per
coulomb (N⋅C
−1) or, equivalently,
volts per
metre
The metre ( British spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pre ...
(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 for the conductive behaviour in the material. See, for example, Giuliani & Vignale (2005)
or Rammer (2007).
The integral extends over the entire past history up to the present time.
The above conductivity and its associated 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 behaviour.
Polarization and magnetization currents
Currents arise in materials when there is a non-uniform distribution of charge.
In
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the ma ...
materials, there is a current density corresponding to the net movement of
electric dipole moments per unit volume, i.e. the
polarization P:
:
Similarly with
magnetic materials, circulations of the
magnetic dipole moments per unit volume, i.e. the
magnetization M, lead to
magnetization currents:
:
Together, these terms add up to form the
bound current density in the material (resultant current due to movements of electric and magnetic dipole moments per unit volume):
:
Total current in materials
The total current is simply the sum of the free and bound currents:
:
Displacement current
There is also a
displacement current corresponding to the time-varying
electric displacement field
In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in ...
D:
:
which is an important term in
Ampere's circuital law, one of Maxwell's equations, since absence of this term would not predict
electromagnetic waves to propagate, or the time evolution of
electric fields in general.
Continuity equation
Since charge is conserved, current density must satisfy a
continuity equation. Here is a derivation from first principles.
The net flow out of some volume ''V'' (which can have an arbitrary shape but fixed for the calculation) must equal the net change in charge held inside the volume:
:
where ''ρ'' is the
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system i ...
, 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:
:
This relation is valid for any volume, independent of size or location, which implies that:
:
and this relation is called the
continuity equation.
In practice
In
electrical wiring, the maximum current density (for a given
temperature rating) can vary from 4 A⋅mm
−2 for a wire with no air circulation around it, to over 6 A⋅mm
−2 for a wire in free air. Regulations for
building wiring
Electrical wiring is an electrical installation of cabling and associated devices such as switches, distribution boards, sockets, and light fittings in a structure.
Wiring is subject to safety standards for design and installation. Allowable ...
list the maximum allowed current of each size of cable in differing conditions. For compact designs, such as windings of
SMPS transformers, the value might be as low as 2 A⋅mm
−2. If the wire is carrying high-frequency
alternating current
Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...
s, the
skin effect may affect the distribution of the current across the section by concentrating the current on the surface of the
conductor. In
transformer
A transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple circuits. A varying current in any coil of the transformer produces a varying magnetic flux in the transformer' ...
s designed for high frequencies, loss is reduced if
Litz wire is used for the windings. This is made of multiple isolated wires in parallel with a diameter twice the
skin depth. The isolated strands are twisted together to increase the total skin area and to reduce the
resistance due to skin effects.
For the top and bottom layers of
printed circuit boards, the maximum current density can be as high as 35 A⋅mm
−2 with a copper thickness of 35 μm. Inner layers cannot dissipate as much heat as outer layers; designers of circuit boards avoid putting high-current traces on inner layers.
In the
semiconductors field, the maximum current densities for different elements are given by the manufacturer. Exceeding those limits raises the following problems:
* The
Joule effect which increases the temperature of the component.
* The
electromigration effect which will erode the interconnection and eventually cause an open circuit.
* The slow
diffusion effect which, if exposed to high temperatures continuously, will move metallic ions and
dopants
A dopant, also called a doping agent, is a trace of impurity element that is introduced into a chemical material to alter its original electrical or optical properties. The amount of dopant necessary to cause changes is typically very low. Whe ...
away from where they should be. This effect is also synonym to ageing.
The following table gives an idea of the maximum current density for various materials.
Even if manufacturers add some margin to their numbers, it is recommended to, at least, double the calculated section to improve the reliability, especially for high-quality electronics. One can also notice the importance of keeping electronic devices cool to avoid exposing them to
electromigration and slow
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical ...
.
In
biological organism
In biology, an organism () is any living system that functions as an individual entity. All organisms are composed of cells (cell theory). Organisms are classified by taxonomy into groups such as multicellular animals, plants, and fungi ...
s,
ion channels regulate the flow of
ions (for example,
sodium
Sodium is a chemical element with the symbol Na (from Latin ''natrium'') and atomic number 11. It is a soft, silvery-white, highly reactive metal. Sodium is an alkali metal, being in group 1 of the periodic table. Its only stable ...
,
calcium
Calcium is a chemical element with the symbol Ca and atomic number 20. As an alkaline earth metal, calcium is a reactive metal that forms a dark oxide-nitride layer when exposed to air. Its physical and chemical properties are most similar ...
,
potassium
Potassium is the chemical element with the symbol K (from Neo-Latin '' kalium'') and atomic number19. Potassium is a silvery-white metal that is soft enough to be cut with a knife with little force. Potassium metal reacts rapidly with atmos ...
) across the
membrane in all
cells
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
. The membrane of a cell is assumed to act like a capacitor.
Current densities are usually expressed in pA⋅pF
−1 (
pico
Pico may refer to:
Places The Moon
* Mons Pico, a lunar mountain in the northern part of the Mare Imbrium basin
Portugal
* Pico, a civil parish in the municipality of Vila Verde
* Pico da Pedra, a civil parish in the municipality of Ribe ...
amperes per
pico
Pico may refer to:
Places The Moon
* Mons Pico, a lunar mountain in the northern part of the Mare Imbrium basin
Portugal
* Pico, a civil parish in the municipality of Vila Verde
* Pico da Pedra, a civil parish in the municipality of Ribe ...
farad) (i.e., current divided by
capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
). Techniques exist to empirically measure capacitance and surface area of cells, which enables calculation of current densities for different cells. This enables researchers to compare ionic currents in cells of different sizes.
In
gas discharge lamps, such as
flashlamps, current density plays an important role in the output
spectrum produced. Low current densities produce
spectral line
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to ident ...
emission and tend to favour longer
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
s. High current densities produce continuum emission and tend to favour shorter wavelengths.
Low current densities for flash lamps are generally around 10 A⋅mm
−2. High current densities can be more than 40 A⋅mm
−2.
See also
*
Hall effect
*
Quantum Hall effect
*
Superconductivity
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
*
Electron mobility
*
Drift velocity
*
Effective mass
*
Electrical resistance
The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallel ...
*
Sheet resistance
*
Speed of electricity
*
Electrical conduction
*
Green–Kubo relations
The Green–Kubo relations (Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for transport coefficients \gamma in terms of integrals of time correlation functions:
:\gamma = \int_0^\infty \left\langle \dot(t) \dot(0 ...
*
Green's function (many-body theory)
References
{{Authority control
Electromagnetism
Density