Current density

{{Short description|Amount of charge flowing through a unit cross-sectional area per unit time}} {{about|electric current density|the quantum concept|Probability current density}} {{Infobox physical quantity | name = Current density | image = | caption = | symbols = {{math|{{vec|''j ''}}}}, {{math|'''J'''}} | baseunits = A m⁻² | dimension = {{dimanalysis|current=1|length=-2}} }} {{electromagnetism|Network}}

In [[electromagnetism]], '''current density''' is the [[electric current]] (or the amount of [[Electric charge|charge]] per unit time) that flows through a unit area of a chosen [[Cross section (geometry)|cross section]].<ref>{{Cite book|title=Fundamentals of physics | last1=Walker| first1=Jearl | date=2014 | publisher=Wiley | last2=Halliday| first2=David |last3=Resnick |first3=Robert |isbn=9781118230732 | edition=10th |location=Hoboken, NJ| page= 749 |oclc=950235056}}</ref> The '''current density vector''' is defined as a [[vector (geometric)|vector]] whose magnitude is the current density at a given point in space and whose direction is that of the motion of the positive charges at this point. In [[SI base unit]]s, the electric current density is measured in [[ampere]]s per meter square.<ref>{{cite book |title=Encyclopaedia of Physics |edition=2nd |first1=R.G. |last1=Lerner |first2=G.L. | last2=Trigg | publisher=VHC publishers | year=1991|isbn=0895737523}}</ref>

== Definition == Consider a small surface with area {{mvar|A}} (SI unit: [[metre|m]]²) centered at a given point {{mvar|M}} and orthogonal to the motion of the charges at {{mvar|M}}. If {{mvar|I{{sub|A}}}} (SI unit: [[ampere|A]]) is the [[electric current]] flowing through {{mvar|A}}, then '''electric current density''' {{mvar|j}} at {{mvar|M}} is given by the [[limit of a function|limit]]:<ref>{{cite book| title=Essential Principles of Physics| first1=P.M.|last1=Whelan| first2=M.J.|last2=Hodgeson| edition=2nd |year=1978 | publisher=John Murray|isbn=0719533821}}</ref><math display="block">j = \lim_{A \to 0} \frac{I_A}{A} = \left.\frac{\partial I}{\partial A} \right|_{A=0},</math>

with surface {{mvar|A}} remaining centered at {{mvar|M}} and orthogonal to the motion of the charges during the limit process.

The '''current density vector''' {{math|'''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 {{mvar|M}}.

At a given time {{mvar|t}}, if {{math|'''v'''}} is the velocity of the charges at {{mvar|M}}, and {{mvar|dA}} is an infinitesimal surface centred at {{mvar|M}} and orthogonal to {{math|'''v'''}}, then during an amount of time {{mvar|dt}}, only the charge contained in the volume formed by {{mvar|dA}} and <math>v \,dt</math> will flow through {{mvar|dA}}. This charge is equal to <math>dq = \rho \, v\, dt \, dA,</math> where {{mvar|ρ}} is the [[charge density]] at {{mvar|M}}. The electric current is <math>dI = dq/dt = \rho v \, dA</math>, it follows that the current density vector is the vector normal <math>dA</math> (i.e. parallel to {{math|'''v'''}}) and of magnitude <math>dI/dA = \rho v</math>

<math display="block">\mathbf{j} = \rho \mathbf{v}.</math>

The [[surface integral]] of {{math|'''j'''}} over a [[surface (mathematics)|surface]] {{mvar|S}}, followed by an integral over the time duration {{math|''t''₁}} to {{math|''t''₂}}, gives the total amount of charge flowing through the surface in that time ({{math|''t''₂ − ''t''₁}}):

<math display="block">q = \int_{t_1}^{t_2}\iint_S \mathbf{j}\cdot\mathbf{\hat{n}}\,dA \,dt. </math>

More concisely, this is the integral of the [[flux]] of {{math|'''j'''}} across {{mvar|S}} between {{math|''t''₁}} and {{math|''t''₂}}.

The [[area]] 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, {{mvar|A}}, and a [[unit vector]] normal to the area, <math>\mathbf{\hat{n}}.</math> The relation is <math>\mathbf{A} = A \mathbf{\hat{n}}.</math>

The differential vector area similarly follows from the definition given above: <math> d\mathbf{A} = dA \mathbf{\hat{n}}.</math>

If the current density {{math|'''j'''}} passes through the area at an angle {{mvar|θ}} to the area normal <math>\mathbf{\hat{n}},</math> then

<math display="block">\mathbf{j}\cdot\mathbf{\hat{n}}= j\cos\theta </math>

where {{math|'''⋅'''}} is the [[dot product]] of the unit vectors. That is, the component of current density passing through the surface (i.e. normal to it) is {{math|''j'' cos ''θ''}}, while the component of current density passing tangential to the area is {{math|''j'' sin ''θ''}}, 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 [[electronics|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 circuit]]s are reduced in size, despite the lower current demanded by smaller [[semiconductor devices|devices]], there is a trend toward higher current densities to achieve higher device numbers in ever smaller [[semiconductor chip|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 [[Electrical resistance|resistance]], making them dissipate [[Power (physics)|power]] in the form of heat. The current density must be kept sufficiently low to prevent the conductor from melting or burning up, the [[electrical insulator|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 [[superconductivity|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.<ref name= Martin>{{cite book |title=Electronic Structure: Basic theory and practical methods |url=https://books.google.com/books?id=dmRTFLpSGNsC&pg=PA316 |author=Richard P Martin |publisher=Cambridge University Press |year=2004 |isbn=0521782856}}</ref><ref name=Altland>{{cite book |title=Condensed Matter Field Theory | url=https://books.google.com/books?id=0KMkfAMe3JkC&pg=RA4-PA557| first1=Alexander|last1=Altland| first2=Ben|last2=Simons |publisher=Cambridge University Press |year=2006 |isbn=9780521845083}}</ref>

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]].

==Calculation of current densities in matter==

===Free currents=== Charge carriers which are free to move constitute a [[Electric current|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 {{math|'''r'''}} at time {{mvar|t}}, the ''distribution'' of [[electric charge|charge]] flowing is described by the current density:<ref>{{cite book |title=The Cambridge Handbook of Physics Formulas |url=https://archive.org/details/cambridgehandboo0000woan |url-access=registration |last=Woan|first=G. |publisher=Cambridge University Press |year=2010 |isbn=9780521575072}}</ref>

<math display="block">\mathbf{j}(\mathbf{r}, t) = \rho(\mathbf{r},t) \; \mathbf{v}_\text{d} (\mathbf{r},t) </math>

where *{{math|'''j'''('''r''', ''t'')}} is the current density vector; *{{math|'''v'''_d('''r''', ''t'')}} is the particles' average [[drift velocity]] (SI unit: [[metre|m]]∙[[second|s]]⁻¹); *<math>\rho(\mathbf{r}, t) = q \, n(\mathbf{r},t) </math> is the [[charge density]] (SI unit: coulombs per [[cubic metre]]), in which **{{math|''n''('''r''', ''t'')}} is the number of particles per unit volume ("number density") (SI unit: m^&minus;3); **{{mvar|q}} is the charge of the individual particles with density {{mvar|n}} (SI unit: [[coulomb]]s).

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

<math display="block">\mathbf{j} = \sigma \mathbf{E} </math>

where {{math|'''E'''}} is the [[electric field]] and {{mvar|σ}} is the [[electrical conductivity]]. This is also an alternative representation of [[Ohm's law]].

Conductivity {{mvar|σ}} is the [[Reciprocal (mathematics)|reciprocal]] ([[invertible matrix|inverse]]) of electrical [[resistivity]] and has the SI units of [[Siemens (unit)|siemens]] per metre (S⋅m^&minus;1), and {{math|'''E'''}} has the SI units of [[newton (unit)|newton]]s per coulomb (N⋅C^&minus;1) or, equivalently, [[volt]]s per metre (V⋅m^&minus;1).

A more fundamental approach to calculation of current density is based upon: <math display="block">\mathbf{j} (\mathbf{r}, t) = \int_{-\infty}^t \left[ \int_{V} \sigma(\mathbf{r}-\mathbf{r}', t-t') \; \mathbf{E}(\mathbf{r}', t') \; \text{d}^3 \mathbf{r}' \, \right] \text{d}t' </math>

indicating the lag in response by the time dependence of {{mvar|σ}}, and the non-local nature of response to the field by the spatial dependence of {{mvar|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)<ref name=Giuliani>{{cite book |title=Quantum Theory of the Electron Liquid |last1=Giuliani|first1=Gabriele|last2=Vignale|first2=Giovanni |page=111 |url=https://archive.org/details/quantumtheoryofe0000giul/page/111 | quote=linear response theory capacitance OR conductance. |isbn=0521821126 |publisher = Cambridge University Press |year=2005}}</ref> or Rammer (2007).<ref name=Rammer>{{cite book |title=Quantum Field Theory of Non-equilibrium States |last=Rammer | first = Jørgen |page=158 |url=https://books.google.com/books?id=A7TbrAm5Wq0C&pg=PR6 |isbn=9780521874991 |publisher=Cambridge University Press |year=2007}}</ref> 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: <math display="block">\mathbf{j} (\mathbf{k}, \omega) = \sigma(\mathbf{k}, \omega) \; \mathbf{E}(\mathbf{k}, \omega) </math>

where {{math|''σ''('''k''', ''ω'')}} is now a [[Complex function#Complex functions|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. <!--please leave the following sections alone, don't delete them... -->

===Polarization and magnetization currents=== Currents arise in materials when there is a non-uniform distribution of charge.<ref name="Electromagnetism 2008">{{cite book| title=Electromagnetism| edition=2| first1=I.S.|last1=Grant| first2=W.R.|last2=Phillips| publisher=John Wiley & Sons| year=2008| isbn=9780471927129}}</ref> In [[dielectric]] materials, there is a current density corresponding to the net movement of [[electric dipole moment]]s per unit volume, i.e. the [[polarization density|polarization]] {{math|'''P'''}}:

<math display="block">\mathbf{j}_\mathrm{P} = \frac{\partial \mathbf{P}}{\partial t} </math>

Similarly with [[magnetic materials]], circulations of the [[magnetic dipole moment]]s per unit volume, i.e. the [[magnetization]] {{math|'''M'''}}, lead to [[magnetization current]]s:<ref>{{Cite journal| last=Herczynski| first=Andrzej| date=2013| title=Bound charges and currents| url=http://www.bc.edu/content/dam/files/schools/cas_sites/physics/pdf/herczynski/AJP-81-202.pdf| journal=American Journal of Physics| publisher=the American Association of Physics Teachers| volume=81| issue=3| pages=202–205| doi=10.1119/1.4773441| bibcode=2013AmJPh..81..202H| access-date=2017-04-23| archive-date=2020-09-20| archive-url=https://web.archive.org/web/20200920175641/https://www.bc.edu/content/dam/files/schools/cas_sites/physics/pdf/herczynski/AJP-81-202.pdf| url-status=dead}}</ref>

<math display="block">\mathbf{j}_\mathrm{M} = \nabla\times\mathbf{M} </math>

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):

<math display="block">\mathbf{j}_\mathrm{b} = \mathbf{j}_\mathrm{P}+\mathbf{j}_\mathrm{M} </math>

===Total current in materials===

The total current is simply the sum of the free and bound currents: <math display="block">\mathbf{j} = \mathbf{j}_\mathrm{f} + \mathbf{j}_\mathrm{b} </math>

===Displacement current=== There is also a [[displacement current]] corresponding to the time-varying [[electric displacement field]] {{math|'''D'''}}:<ref>{{cite book| title=Introduction to Electrodynamics| edition=3| first1=D.J.|last1=Griffiths| publisher=Pearson Education| year=2007| isbn=978-8177582932}}</ref><ref>{{cite book|title=Physics for Scientists and Engineers - with Modern Physics| edition=6| first1=P. A.|last1=Tipler| first2=G.|last2=Mosca| year=2008| publisher=W. H. Freeman| isbn=978-0716789642}}</ref>

<math display="block">\mathbf{j}_\mathrm{D} = \frac{\partial \mathbf{D}}{\partial t} </math>

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 field]]s in general.

== Continuity equation == {{Main|Continuity equation}} Since charge is conserved, current density must satisfy a [[continuity equation]]. Here is a derivation from first principles.<ref name="Electromagnetism 2008"/>

The net flow out of some volume {{mvar|V}} (which can have an arbitrary shape but fixed for the calculation) must equal the net change in charge held inside the volume:

<math display="block">\int_S { \mathbf{j} \cdot d\mathbf{A}} = -\frac{d}{dt} \int_V{\rho \; dV} = - \int_V { \frac{\partial \rho}{\partial t}\;dV}</math>

where {{mvar|ρ}} is the [[charge density]], and {{math|''d'''''A'''}} is a [[surface integral|surface element]] of the surface {{mvar|S}} enclosing the volume {{mvar|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]]:

<math display="block">\oint_S { \mathbf{j} \cdot d\mathbf{A}} = \int_V {\boldsymbol{\nabla} \cdot \mathbf{j} \; dV}</math>

Hence:

<math display="block">\int_V {\boldsymbol{\nabla} \cdot \mathbf{j}\; dV} \ = - \int_V{ \frac{\partial \rho}{\partial t} \;dV}</math>

This relation is valid for any volume, independent of size or location, which implies that:

<math display="block">\nabla \cdot \mathbf{j} = - \frac{\partial \rho}{\partial t}</math>

and this relation is called the [[continuity equation]].<ref name=Chow>{{cite book |title=Introduction to Electromagnetic Theory: A modern perspective |author = Tai L Chow |publisher=Jones & Bartlett |url=https://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153 |isbn=0-7637-3827-1 |year=2006 |pages=130–131}}</ref><ref name=Griffiths>{{cite book |author=Griffiths, D.J. |title=Introduction to Electrodynamics |page=[https://archive.org/details/introductiontoel00grif_0/page/213 213] |publisher=Pearson/Addison-Wesley |year=1999 |isbn=0-13-805326-X |edition=3rd |url-access=registration |url=https://archive.org/details/introductiontoel00grif_0/page/213 }}</ref>

== In practice == In [[electrical wiring]], the maximum current density (for a given [[temperature rating]]) can vary from 4&nbsp;A⋅mm⁻² for a wire with no air circulation around it, to over 6&nbsp;A⋅mm⁻² for a wire in free air. Regulations for [[building wiring]] list the maximum allowed current of each size of cable in differing conditions. For compact designs, such as windings of [[Switched-mode power supply|SMPS transformers]], the value might be as low as 2&nbsp;A⋅mm⁻².<ref>{{cite book |author=A. Pressman |display-authors=etal |title=Switching power supply design |edition=3rd |publisher=McGraw-Hill |isbn=978-0-07-148272-1 |year=2009 |page=320}}</ref> If the wire is carrying high-frequency [[alternating current]]s, the [[skin effect]] may affect the distribution of the current across the section by concentrating the current on the surface of the [[electrical conductor|conductor]]. In [[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 [[Electrical resistance and conductance|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&nbsp;A⋅mm⁻² with a copper thickness of 35&nbsp;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 heating|Joule effect]] which increases the temperature of the component. * The [[electromigration|electromigration effect]] which will erode the interconnection and eventually cause an open circuit. * The slow [[diffusion|diffusion effect]] which, if exposed to high temperatures continuously, will move metallic ions and [[Doping (semiconductor)|dopants]] away from where they should be. This effect is also synonymous with ageing.

The following table gives an idea of the maximum current density for various materials.

{| class=wikitable |- !Material !Temperature !! Maximum current density |- | rowspan="4" |Copper interconnections<br />([[180 nanometer|180&nbsp;nm]] technology) |{{0}}25&nbsp;°C || 1000&nbsp;μA⋅μm⁻² (1000&nbsp;A⋅mm⁻²) |- |{{0}}50&nbsp;°C || {{0}}700&nbsp;μA⋅μm⁻² {{0}}(700&nbsp;A⋅mm⁻²) |- |{{0}}85&nbsp;°C || {{0}}400&nbsp;μA⋅μm⁻² {{0}}(400&nbsp;A⋅mm⁻²) |- |125&nbsp;°C || {{0}}100&nbsp;μA⋅μm⁻² {{0}}(100&nbsp;A⋅mm⁻²) |- |[[Graphene nanoribbons]]<ref name="MuraliYang2009">{{cite journal| last1=Murali|first1=Raghunath| last2=Yang|first2=Yinxiao| last3=Brenner|first3=Kevin| last4=Beck|first4=Thomas| last5=Meindl|first5=James D.| title=Breakdown current density of graphene nanoribbons| journal=Applied Physics Letters| volume=94|issue=24| year=2009|pages=243114| issn=0003-6951| doi=10.1063/1.3147183| arxiv=0906.4156| bibcode=2009ApPhL..94x3114M| s2cid=55785299}}</ref> |{{0}}25&nbsp;°C || 0.1–10&nbsp;×&nbsp;10⁸&nbsp;A⋅cm⁻² (0.1–10&nbsp;×&nbsp;10⁶&nbsp;A⋅mm⁻²) |}

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]].

In [[biological organism]]s, [[ion channel]]s regulate the flow of [[ion]]s (for example, [[sodium]], [[calcium]], [[potassium]]) across the [[Cell membrane|membrane]] in all [[Cell (biology)|cells]]. The membrane of a cell is assumed to act like a capacitor.<ref>{{cite book |editor1-last=Fall |editor1-first=C. P. |editor2-last=Marland |editor2-first=E. S. |editor3-last=Wagner |editor3-first=J. M. |editor4-last=Tyson |editor4-first=J. J. |title=Computational Cell Biology |date=2002 |location=New York | publisher=Springer |isbn=9780387224596 |page=28 |url={{google books |plainurl=y |id=AdCTvbOzRywC|page=28}}}}</ref> Current densities are usually expressed in pA⋅pF⁻¹ ([[Metric prefix|pico]][[ampere]]s per [[Metric prefix|pico]][[farad]]) (i.e., current divided by [[capacitance]]). 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.<ref>{{cite encyclopedia |editor1-last=Weir |editor1-first=E. K. |editor2-last=Hume|editor2-first=J. R. |editor3-last=Reeves |editor3-first=J. T. | title= The electrophysiology of smooth muscle cells and techniques for studying ion channels | encyclopedia=Ion flux in pulmonary vascular control |date=1993 |publisher=Springer Science |location=New York |isbn=9780387224596 |page=29 |url={{google books |plainurl=y |id=ImHSBwAAQBAJ|page=29}}}}</ref>

In [[gas discharge lamp]]s, such as [[flashlamp]]s, current density plays an important role in the output [[spectroscopy|spectrum]] produced. Low current densities produce [[spectral line]] [[emission spectrum|emission]] and tend to favour longer [[wavelength]]s. High current densities produce continuum emission and tend to favour shorter wavelengths.<ref>{{cite web|url=https://kb.osu.edu/dspace/bitstream/1811/5654/1/V71N06_343.pdf|title=Xenon lamp photocathodes}}</ref> Low current densities for flash lamps are generally around 10&nbsp;A⋅mm⁻². High current densities can be more than 40&nbsp;A⋅mm⁻².

==See also== *[[Hall effect]] *[[Quantum Hall effect]] *[[Superconductivity]] *[[Electron mobility]] *[[Drift velocity]] *[[Effective mass (solid-state physics)|Effective mass]] *[[Electrical resistance]] *[[Sheet resistance]] *[[Speed of electricity]] *[[Electrical conduction]] *[[Green–Kubo relations]] *[[Green's function (many-body theory)]]

== References == {{Reflist|2}}

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[[Category:Electromagnetic quantities]] [[Category:Area-specific quantities]]