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| {{dablink|This page is about the electric current density in [[electromagnetism]]. For the probability current density in quantum mechanics, see [[Probability current]].}}
| | Andera is what you can contact her but she by no means truly liked that name. The favorite hobby for him and his kids is fashion and he'll be beginning something else along with it. Invoicing is my profession. I've always loved residing in Mississippi.<br><br>My blog post: [http://www.zavodpm.ru/blogs/glennmusserrvji/14565-great-hobby-advice-assist-allow-you-get-going real psychics] |
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| In [[electromagnetism]], and related fields in [[solid state physics]], [[condensed matter physics]] etc. '''current density''' is the [[electric current]] per unit area of cross section. It is defined as a [[vector (geometric)|vector]] whose magnitude is the [[electric current]] per cross-sectional area at a given point in space (i.e. it's a [[vector field]]). In [[SI unit]]s, the electric current density is measured in [[ampere]]s per [[square metre]].<ref>Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3</ref>
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| == Definition ==
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| '''Electric current density''' ''J'' is simply the [[electric current]] ''I'' (SI unit: [[ampere|A]]) per unit [[area]] ''A'' (SI unit: [[metre|m]]<sup>2</sup>). Its magnitude is given by the [[limit of a function|limit]]:<ref>Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, ISBN 0-7195-3382-1</ref>
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| :<math>J = \lim\limits_{A \rightarrow 0}\frac{I(A)}{A}</math>
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| For current density as a vector '''J''', the [[surface integral]] over a [[surface]] ''S'', followed by an integral over the time duration ''t''<sub>1</sub> to ''t''<sub>2</sub>, gives the total amount of charge flowing through the surface in that time (''t''<sub>2</sub> − ''t''<sub>1</sub>):
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| :<math>q=\int_{t_1}^{t_2}\iint_S \bold{J}\cdot\bold{\hat{n}}{\rm d}A{\rm d}t </math>
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| 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.
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| The [[vector area]] is a combination of the magnitude of the area through which the mass passes through, ''A'', and a [[unit vector]] normal to the area, <math>\bold{\hat{n}}</math>. The relation is <math>\bold{A} = A \bold{\hat{n}}</math>.
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| If the current density '''J''' passes through the area at an angle θ to the area normal <math>\bold{\hat{n}}</math>, then
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| :<math>\bold{J}\cdot\bold{\hat{n}}= J\cos\theta </math>
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| where '''·''' is the [[dot product]] of the unit vectors. This is, the component of current density passing through the surface (i.e. normal to it) is ''J'' cos θ, while the component of current density passing tangential to the area is ''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.
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| ==Importance==
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| Current density is important to the design of electrical and [[electronics|electronic]] systems.
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| 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 [[semiconductor devices|devices]], there is trend toward higher current densities to achieve higher device numbers in ever smaller [[semiconductor chip|chip]] areas. See [[Moore's law]].
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| At high frequencies, current density can increase because the conducting region in a wire becomes confined near its surface, the so-called [[skin effect]].
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| 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.
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| 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=http://books.google.com/books?id=dmRTFLpSGNsC&pg=PA316&dq=isbn=0-521-78285-6#PPA369,M1 |author=Richard P Martin |publisher=Cambridge University Press |year=2004 |isbn=0-521-78285-6}}</ref><ref name=Altland>{{cite book |title=Condensed Matter Field Theory |url=http://books.google.com/books?id=0KMkfAMe3JkC&pg=RA4-PA557&dq=isbn=978-0-521-84508-3#PRA2-PA378,M1 |author=Alexander Altland & Ben Simons |publisher=Cambridge University Press |year=2006 |isbn=978-0-521-84508-3}}</ref>
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| The current density is an important parameter in [[Ampère's circuital law]] (one of [[Maxwell's equations]]), which relates current density to [[magnetic field]].
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| In [[special relativity]] theory, charge and current are combined into a [[4-vector]].
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| ==Calculation of current densities in matter==
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| ===Free currents===
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| Charge carriers which are free to move constitute a [[free current]] density, which are given by expressions such as those in this section.
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| 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 [[electric charge|charge]] flowing is described by the current density:<ref>The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2</ref>
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| :<math>\mathbf{J}(\mathbf{r}, t) = \rho(\mathbf{r},t) \; \mathbf{v}_\text{d} (\mathbf{r},t) \,</math>
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| where '''J'''('''r''', ''t'') is the current density vector, '''v'''<sub>d</sub>('''r''', ''t'') is the particles' average [[drift velocity]] (SI unit: [[metre|m]]∙[[second|s]]<sup>−1</sup>), and
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| :<math>\rho(\mathbf{r}, t)= qn(\mathbf{r},t) </math>
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| is the [[charge density]] (SI unit: coulombs per [[cubic metre]]), in which ''n''('''r''', ''t'') is the number of particles per unit volume ("number density") (SI unit: m<sup>−3</sup>), ''q'' is the charge of the individual particles with density ''n'' (SI unit: [[coulomb]]s).
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| A common approximation to the current density assumes the current simply is proportional to the electric field, as expressed by:
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| :<math>\mathbf{J} = \sigma \mathbf{E} \, </math>
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| where '''E''' is the [[electric field]] and ''σ'' is the [[electrical conductivity]].
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| Conductivity ''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<sup>−1</sup>), and '''E''' has the [[SI]] units of [[newton (unit)|newton]]s per [[coulomb]] (N C<sup>−1</sup>) or, equivalently, [[volt]]s per [[metre]] (V m<sup>−1</sup>).
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| A more fundamental approach to calculation of current density is based upon:
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| :<math>\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') \, </math>
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| 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 or Rammer.<ref name=Giuliani>{{cite book |title=Quantum Theory of the Electron Liquid |author=Gabriele Giuliani, Giovanni Vignale |page=111 |url=http://books.google.com/books?id=kFkIKRfgUpsC&pg=PA538&dq=%22linear+response+theory%22+capacitance+OR+conductance#PPA111,M1 |isbn=0-521-82112-6 |publisher=Cambridge University Press |year=2005 }}</ref><ref name=Rammer>{{cite book |title=Quantum Field Theory of Non-equilibrium States |author=Jørgen Rammer |page=158 |url=http://books.google.com/books?id=A7TbrAm5Wq0C&pg=PR6&dq=%22linear+response+theory%22+capacitance+OR+conductance#PPA158,M1 |isbn=0-521-87499-8 |publisher=Cambridge University Press |year=2007}}</ref> The integral extends over the entire past history up to the present time.
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| The above conductivity and its associated current density reflect the fundamental mechanisms underlying charge transport in the medium, both in time and over distance. | |
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| A [[Fourier transform]] in space and time then results in:
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| :<math>\mathbf{J} (\mathbf{k}, \omega) = \sigma(\mathbf{k}, \omega) \; \mathbf{E}(\mathbf{k}, \omega) \,</math>
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| where ''σ''('''k''', ''ω'') is now a [[Complex function#Complex functions|complex function]].
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| 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.
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| <!--please leave the following sections alone, don't delete them... -->
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| ===Polarization and magnetization currents===
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| Currents arise in materials when there is a non-uniform distribution of charge.<ref>Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9</ref>
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| 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]] '''P''':
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| :<math>\mathbf{J}_\mathrm{P}=\frac{\partial \mathbf{P}}{\partial t} </math>
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| Similarly with [[magnetic materials]], circulations of the [[magnetic dipole moment]]s per unit volume, i.e. the [[magnetization]] '''M''' lead to volume [[magnetization current]]s:
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| :<math>\mathbf{J}_\mathrm{M}=\nabla\times\mathbf{M} </math>
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| Together, these terms form add up to the [[bound current]] density in the material (resultant current due to movements of electric and magnetic dipole moments per unit volume):
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| :<math>\mathbf{J}_\mathrm{b}=\mathbf{J}_\mathrm{P}+\mathbf{J}_\mathrm{M} </math>
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| ===Total current in materials===
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| The total current is simply the sum of the free and bound currents:
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| :<math>\mathbf{J} = \mathbf{J}_\mathrm{f}+\mathbf{J}_\mathrm{b} </math>
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| ===Displacement current===
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| There is also a [[displacement current]] corresponding to the time-varying [[electric displacement field]] '''D''':<ref>Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3</ref><ref>Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, ISBN 0-7167-8964-7</ref>
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| :<math>\mathbf{J}_\mathrm{D}=\frac{\partial \mathbf{D}}{\partial t} </math>
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| 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.
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| == Continuity equation ==
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| {{main|Continuity equation}}
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| Since charge is conserved, current density must satisfy a [[continuity equation]]. Here is a derivation from first principles.<ref>Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9</ref>
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| 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:
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| :<math>\int_S{ \mathbf{J} \cdot \mathrm{d}\mathbf{A}} = -\frac{\mathrm{d}}{\mathrm{d}t} \int_V{\rho \; \mathrm{d}V} = - \int_V{ \frac{\partial \rho}{\partial t}\;\mathrm{d}V}</math>
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| where ''ρ'' is the [[charge density]], and ''d'''A''''' is a [[surface|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]]:
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| :<math>\int_S{ \mathbf{J} \cdot \mathrm{d}\mathbf{A}} = \int_V{\mathbf{\nabla} \cdot \mathbf{J}\; \mathrm{d}V}</math>
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| Hence:
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| :<math>\int_V{\mathbf{\nabla} \cdot \mathbf{J}\; \mathrm{d}V}\ = - \int_V{ \frac{\partial \rho}{\partial t} \;\mathrm{d}V}</math>
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| This relation is valid for any volume, independent of size or location, which implies that:
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| :<math>\nabla \cdot \mathbf{J} = - \frac{\partial \rho}{\partial t}</math>
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| 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=http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&dq=isbn=0-7637-3827-1#PPA204,M1 |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=213 |publisher=Pearson/Addison-Wesley |year=1999 |isbn=0-13-805326-X |edition=3rd Edition}}</ref>
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| == In practice ==
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| In [[electrical wiring]], the maximum current density can vary from 4A∙mm<sup>−2</sup> for a wire with no air circulation around it, to 6A∙mm<sup>−2</sup> 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 2A∙mm<sup>−2</sup>.<ref>A. Pressman et al., Switching power supply design, McGraw-Hill, ISBN 978-0-07-148272-1, page 320</ref> If the wire is carrying high frequency currents, 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 [[Coil|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.
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| For the top and bottom layers of [[printed circuit boards]], the maximum current density can be as high as 35A∙mm<sup>−2</sup> 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.
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| In [[semiconductors]], the maximum current density is given by the manufacturer. A common average is 1mA∙µm<sup>−2</sup> at 25°C for 180 nm technology. Above the maximum current density, apart from the [[Joule heating|joule effect]], some other effects like [[electromigration]] appear in the [[Metric prefix|micro]][[meter]] scale.
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| In [[biological organism]]s, [[ion channel]]s regulate the flow of [[ions]] (for example, [[sodium]], [[calcium]], [[potassium]]) across the [[Cell membrane|membrane]] in all [[Cell (biology)|cells]]. Current density is measured in pA∙pF<sup>−1</sup> ([[Metric prefix|pico]][[ampere]]s per [[Metric prefix|pico]][[farad]]), that is, current divided by [[capacitance]], a de facto measure of membrane area.{{Clarify|date=February 2009}}
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| 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>[https://kb.osu.edu/dspace/bitstream/1811/5654/1/V71N06_343.pdf Xenon lamp photocathodes]</ref> Low current densities for flash lamps are generally around 1000A∙cm<sup>−2</sup>. High current densities can be more than 4000A∙cm<sup>−2</sup>.
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| == References ==
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| {{Reflist|2}}
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| == External links ==
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| *[http://maxwell.byu.edu/~spencerr/websumm122/node46.html A short explanation of the current density]
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| ==See also==
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| {{Col-begin}}
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| {{Col-1-of-3}}
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| *[[Hall effect]]
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| *[[Quantum Hall effect]]
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| *[[Superconductivity]]
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| *[[Electron mobility]]
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| *[[Drift velocity]]
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| {{Col-2-of-3}}
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| *[[Effective mass (solid-state physics)|Effective mass]]
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| *[[Electrical resistance]]
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| *[[Sheet resistance]]
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| {{Col-3-of-3}}
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| *[[Speed of electricity]]
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| *[[Electrical conduction]]
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| *[[Green–Kubo relations]]
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| *[[Green's function (many-body theory)]]
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| {{col-end}}
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| [[Category:Concepts in physics]]
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| [[Category:Electromagnetism]]
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| [[Category:Density]]
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