Current density

Current density
Current density
Common symbols	j→,J
InSI base units	A m−2
Dimension	[IL−2]

Inelectromagnetism,current densityis the amount ofchargeper unit time that flows through a unit area of a chosencross section.^[1]Thecurrent density vectoris defined as avectorwhose magnitude is theelectric currentper cross-sectional area at a given point in space, its direction being that of the motion of the positive charges at this point. InSI base units,the electric current density is measured inamperespersquare metre.^[2]

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 $I A$ (SI unit:A) is theelectric currentflowing through $A$ ,thenelectric current density $j$ at $M$ is given by thelimit:^[3]

$j=\lim _{A\to 0}{\frac {I_{A}}{A}}=\left.{\frac {\partial I}{\partial A}}\right|_{A=0},$

with surface $A$ remaining centered at $M$ and orthogonal to the motion of the charges during the limit process.

Thecurrent 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 $v\,dt$ will flow through $dA$ .This charge is equal to $dq=\rho \,v\,dt\,dA,$ where $ρ$ is thecharge densityat $M$ .The electric current is $dI=dq/dt=\rho vdA$ ,it follows that the current density vector is the vector normal $dA$ (i.e. parallel to $v$ ) and of magnitude $dI/dA=\rho v$

$\mathbf {j} =\rho \mathbf {v}.$

Thesurface integralof $j$ over asurface $S$ ,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 ( $t 2 - t 1$ ):

$q=\int _{t_{1}}^{t_{2}}\iint _{S}\mathbf {j} \cdot \mathbf {\hat {n}} \,dA\,dt.$

More concisely, this is the integral of thefluxof $j$ across $S$ between $t 1$ and $t 2$ .

Thearearequired 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 anelectrical conductor,the area is the cross-section of the conductor, at the section considered.

Thevector areais a combination of the magnitude of the area through which the charge carriers pass, $A$ ,and aunit vectornormal to the area, $\mathbf {\hat {n}}.$ The relation is $\mathbf {A} =A\mathbf {\hat {n}}.$

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

If the current density $j$ passes through the area at an angle $θ$ to the area normal $\mathbf {\hat {n}},$ then

$\mathbf {j} \cdot \mathbf {\hat {n}} =j\cos \theta$

where $\cdot$ is thedot productof the unit vectors. That 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 isnocurrent density actually passingthroughthe area in the tangential direction. Theonlycomponent of current density passing normal to the area is the cosine component.

Importance

Current density is important to the design of electrical andelectronicsystems.

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, asintegrated circuitsare reduced in size, despite the lower current demanded by smallerdevices,there is a trend toward higher current densities to achieve higher device numbers in ever smallerchipareas. SeeMoore'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 theskin effect.

High current densities have undesirable consequences. Most electrical conductors have a finite, positiveresistance,making them dissipatepowerin the form of heat. The current density must be kept sufficiently low to prevent the conductor from melting or burning up, theinsulating materialfailing, or the desired electrical properties changing. At high current densities the material forming the interconnections actually moves, a phenomenon calledelectromigration.Insuperconductorsexcessive 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.^[4]^[5]

The current density is an important parameter inAmpère's circuital law(one ofMaxwell's equations), which relates current density tomagnetic field.

Inspecial relativitytheory, charge and current are combined into a4-vector.

Calculation of current densities in matter

Free currents

Charge carriers which are free to move constitute afree currentdensity, 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 $t$ ,thedistributionofchargeflowing is described by the current density:^[6]

$\mathbf {j} (\mathbf {r},t)=\rho (\mathbf {r},t)\;\mathbf {v} _{\text{d}}(\mathbf {r},t)$

where

$j (r, t)$ is the current density vector;
$v d (r, t)$ is the particles' averagedrift velocity(SI unit:m∙s⁻¹);
$\rho (\mathbf {r},t)=q\,n(\mathbf {r},t)$ $\rho (\mathbf {r} ,t)=q\,n(\mathbf {r} ,t)$ is thecharge density(SI unit: coulombs percubic 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:

$\mathbf {j} =\sigma \mathbf {E}$

where $E$ is theelectric fieldand $σ$ is theelectrical conductivity.

Conductivity $σ$ is thereciprocal(inverse) of electricalresistivityand has the SI units ofsiemensper metre (S⋅m⁻¹), and $E$ has the SI units ofnewtonsper coulomb (N⋅C⁻¹) or, equivalently,voltsper metre (V⋅m⁻¹).

A more fundamental approach to calculation of current density is based upon: $\mathbf {j} (\mathbf {r},t)=\int _{-\infty }^{t}\left[\int _{V}\sigma (\mathbf {r} -\mathbf {r} ',t-t')\;\mathbf {E} (\mathbf {r} ',t')\;{\text{d}}^{3}\mathbf {r} '\,\right]{\text{d}}t'$

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, thelinear response functionfor the conductive behaviour in the material. See, for example, Giuliani & Vignale (2005)^[7]or Rammer (2007).^[8]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.

AFourier transformin space and time then results in: $\mathbf {j} (\mathbf {k},\ Omega )=\sigma (\mathbf {k},\ Omega )\;\mathbf {E} (\mathbf {k},\ Omega )$

where $σ (k, ω)$ is now acomplex function.

In many materials, for example, in crystalline materials, the conductivity is atensor,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.^[9]

Indielectricmaterials, there is a current density corresponding to the net movement ofelectric dipole momentsper unit volume, i.e. thepolarization $P$ :

$\mathbf {j} _{\mathrm {P} }={\frac {\partial \mathbf {P} }{\partial t}}$

Similarly withmagnetic materials,circulations of themagnetic dipole momentsper unit volume, i.e. themagnetization $M$ ,lead tomagnetization currents:^[10]

$\mathbf {j} _{\mathrm {M} }=\nabla \times \mathbf {M}$

Together, these terms add up to form thebound currentdensity in the material (resultant current due to movements of electric and magnetic dipole moments per unit volume):

$\mathbf {j} _{\mathrm {b} }=\mathbf {j} _{\mathrm {P} }+\mathbf {j} _{\mathrm {M} }$

Total current in materials

The total current is simply the sum of the free and bound currents: $\mathbf {j} =\mathbf {j} _{\mathrm {f} }+\mathbf {j} _{\mathrm {b} }$

Displacement current

There is also adisplacement currentcorresponding to the time-varyingelectric displacement field $D$ :^[11]^[12]

$\mathbf {j} _{\mathrm {D} }={\frac {\partial \mathbf {D} }{\partial t}}$

which is an important term inAmpere's circuital law,one of Maxwell's equations, since absence of this term would not predictelectromagnetic wavesto propagate, or the time evolution ofelectric fieldsin general.

Continuity equation

Since charge is conserved, current density must satisfy acontinuity equation.Here is a derivation from first principles.^[9]

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:

$\int _{S}{\mathbf {j} \cdot d\mathbf {A} }=-{\frac {d}{dt}}\int _{V}{\rho \;dV}=-\int _{V}{{\frac {\partial \rho }{\partial t}}\;dV}$

where $ρ$ is thecharge density,and $d A$ is asurface elementof the surface $S$ enclosing the volume $V$ .The surface integral on the left expresses the currentoutflowfrom the volume, and the negatively signedvolume integralon the right expresses thedecreasein the total charge inside the volume. From thedivergence theorem:

$\oint _{S}{\mathbf {j} \cdot d\mathbf {A} }=\int _{V}{{\boldsymbol {\nabla }}\cdot \mathbf {j} \;dV}$

Hence:

$\int _{V}{{\boldsymbol {\nabla }}\cdot \mathbf {j} \;dV}\ =-\int _{V}{{\frac {\partial \rho }{\partial t}}\;dV}$

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

$\nabla \cdot \mathbf {j} =-{\frac {\partial \rho }{\partial t}}$

and this relation is called thecontinuity equation.^[13]^[14]

In practice

Inelectrical wiring,the maximum current density (for a giventemperature rating) can vary from 4 A⋅mm⁻²for a wire with no air circulation around it, to over 6 A⋅mm⁻²for a wire in free air. Regulations forbuilding wiringlist the maximum allowed current of each size of cable in differing conditions. For compact designs, such as windings ofSMPS transformers,the value might be as low as 2 A⋅mm⁻².^[15]If the wire is carrying high-frequencyalternating currents,theskin effectmay affect the distribution of the current across the section by concentrating the current on the surface of theconductor.Intransformersdesigned for high frequencies, loss is reduced ifLitz wireis used for the windings. This is made of multiple isolated wires in parallel with a diameter twice theskin depth.The isolated strands are twisted together to increase the total skin area and to reduce theresistancedue to skin effects.

For the top and bottom layers ofprinted circuit boards,the maximum current density can be as high as 35 A⋅mm⁻²with 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 thesemiconductorsfield, the maximum current densities for different elements are given by the manufacturer. Exceeding those limits raises the following problems:

TheJoule effectwhich increases the temperature of the component.
Theelectromigration effectwhich will erode the interconnection and eventually cause an open circuit.
The slowdiffusion effectwhich, if exposed to high temperatures continuously, will move metallic ions anddopantsaway 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.

Material	Temperature	Maximum current density
Copper interconnections (180 nmtechnology)	025 °C	1000 μA⋅μm⁻²(1000 A⋅mm⁻²)
	050 °C	0700 μA⋅μm⁻²0(700 A⋅mm⁻²)
	085 °C	0400 μA⋅μm⁻²0(400 A⋅mm⁻²)
	125 °C	0100 μA⋅μm⁻²0(100 A⋅mm⁻²)
Graphene nanoribbons^[16]	025 °C	0.1–10 × 10⁸A⋅cm⁻²(0.1–10 × 10⁶A⋅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 toelectromigrationand slowdiffusion.

Inbiological organisms,ion channelsregulate the flow ofions(for example,sodium,calcium,potassium) across themembranein allcells.The membrane of a cell is assumed to act like a capacitor.^[17] Current densities are usually expressed in pA⋅pF⁻¹(pico amperesperpico farad) (i.e., current divided bycapacitance). 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.^[18]

Ingas discharge lamps,such asflashlamps,current density plays an important role in the outputspectrumproduced. Low current densities producespectral line emissionand tend to favour longerwavelengths.High current densities produce continuum emission and tend to favour shorter wavelengths.^[19]Low current densities for flash lamps are generally around 10 A⋅mm⁻².High current densities can be more than 40 A⋅mm⁻².

References

^Walker, Jearl; Halliday, David; Resnick, Robert (2014).Fundamentals of physics(10th ed.). Hoboken, NJ: Wiley. p. 749.ISBN 9781118230732.OCLC 950235056.
^Lerner, R.G.; Trigg, G.L. (1991).Encyclopaedia of Physics(2nd ed.). VHC publishers.ISBN 0895737523.
^Whelan, P.M.; Hodgeson, M.J. (1978).Essential Principles of Physics(2nd ed.). John Murray.ISBN 0719533821.
^Richard P Martin (2004).Electronic Structure: Basic theory and practical methods.Cambridge University Press.ISBN 0521782856.
^Altland, Alexander; Simons, Ben (2006).Condensed Matter Field Theory.Cambridge University Press.ISBN 9780521845083.
^Woan, G. (2010).The Cambridge Handbook of Physics Formulas.Cambridge University Press.ISBN 9780521575072.
^Giuliani, Gabriele; Vignale, Giovanni (2005).Quantum Theory of the Electron Liquid.Cambridge University Press. p. 111.ISBN 0521821126.linear response theory capacitance OR conductance.
^Rammer, Jørgen (2007).Quantum Field Theory of Non-equilibrium States.Cambridge University Press. p. 158.ISBN 9780521874991.
^^a ^bGrant, I.S.; Phillips, W.R. (2008).Electromagnetism(2 ed.). John Wiley & Sons.ISBN 9780471927129.
^Herczynski, Andrzej (2013)."Bound charges and currents"(PDF).American Journal of Physics.81(3). the American Association of Physics Teachers: 202–205.Bibcode:2013AmJPh..81..202H.doi:10.1119/1.4773441.Archived fromthe original(PDF)on 2020-09-20.Retrieved2017-04-23.
^Griffiths, D.J. (2007).Introduction to Electrodynamics(3 ed.). Pearson Education.ISBN 978-8177582932.
^Tipler, P. A.; Mosca, G. (2008).Physics for Scientists and Engineers - with Modern Physics(6 ed.). W. H. Freeman.ISBN 978-0716789642.
^Tai L Chow (2006).Introduction to Electromagnetic Theory: A modern perspective.Jones & Bartlett. pp. 130–131.ISBN 0-7637-3827-1.
^Griffiths, D.J. (1999).Introduction to Electrodynamics(3rd ed.). Pearson/Addison-Wesley. p.213.ISBN 0-13-805326-X.
^A. Pressman; et al. (2009).Switching power supply design(3rd ed.). McGraw-Hill. p. 320.ISBN 978-0-07-148272-1.
^Murali, Raghunath; Yang, Yinxiao; Brenner, Kevin; Beck, Thomas; Meindl, James D. (2009). "Breakdown current density of graphene nanoribbons".Applied Physics Letters.94(24): 243114.arXiv:0906.4156.Bibcode:2009ApPhL..94x3114M.doi:10.1063/1.3147183.ISSN 0003-6951.S2CID 55785299.
^Fall, C. P.; Marland, E. S.; Wagner, J. M.; Tyson, J. J., eds. (2002).Computational Cell Biology.New York: Springer. p. 28.ISBN 9780387224596.
^Weir, E. K.; Hume, J. R.; Reeves, J. T., eds. (1993)."The electrophysiology of smooth muscle cells and techniques for studying ion channels".Ion flux in pulmonary vascular control.New York: Springer Science. p. 29.ISBN 9780387224596.
^"Xenon lamp photocathodes"(PDF).

[1] Walker, Jearl; Halliday, David; Resnick, Robert (2014).Fundamentals of physics(10th ed.). Hoboken, NJ: Wiley. p. 749.ISBN 9781118230732.OCLC 950235056.

[2] Lerner, R.G.; Trigg, G.L. (1991).Encyclopaedia of Physics(2nd ed.). VHC publishers.ISBN 0895737523.

[3] Whelan, P.M.; Hodgeson, M.J. (1978).Essential Principles of Physics(2nd ed.). John Murray.ISBN 0719533821.

[Martin-4] Richard P Martin (2004).Electronic Structure: Basic theory and practical methods.Cambridge University Press.ISBN 0521782856.

[Altland-5] Altland, Alexander; Simons, Ben (2006).Condensed Matter Field Theory.Cambridge University Press.ISBN 9780521845083.

[6] Woan, G. (2010).The Cambridge Handbook of Physics Formulas.Cambridge University Press.ISBN 9780521575072.

[Giuliani-7] Giuliani, Gabriele; Vignale, Giovanni (2005).Quantum Theory of the Electron Liquid.Cambridge University Press. p. 111.ISBN 0521821126.linear response theory capacitance OR conductance.

[Rammer-8] Rammer, Jørgen (2007).Quantum Field Theory of Non-equilibrium States.Cambridge University Press. p. 158.ISBN 9780521874991.

[Electromagnetism_2008-9] Grant, I.S.; Phillips, W.R. (2008).Electromagnetism(2 ed.). John Wiley & Sons.ISBN 9780471927129.

[10] Herczynski, Andrzej (2013)."Bound charges and currents"(PDF).American Journal of Physics.81(3). the American Association of Physics Teachers: 202–205.Bibcode:2013AmJPh..81..202H.doi:10.1119/1.4773441.Archived fromthe original(PDF)on 2020-09-20.Retrieved2017-04-23.

[11] Griffiths, D.J. (2007).Introduction to Electrodynamics(3 ed.). Pearson Education.ISBN 978-8177582932.

[12] Tipler, P. A.; Mosca, G. (2008).Physics for Scientists and Engineers - with Modern Physics(6 ed.). W. H. Freeman.ISBN 978-0716789642.

[Chow-13] Tai L Chow (2006).Introduction to Electromagnetic Theory: A modern perspective.Jones & Bartlett. pp. 130–131.ISBN 0-7637-3827-1.

[Griffiths-14] Griffiths, D.J. (1999).Introduction to Electrodynamics(3rd ed.). Pearson/Addison-Wesley. p.213.ISBN 0-13-805326-X.

[15] A. Pressman; et al. (2009).Switching power supply design(3rd ed.). McGraw-Hill. p. 320.ISBN 978-0-07-148272-1.

[MuraliYang2009-16] Murali, Raghunath; Yang, Yinxiao; Brenner, Kevin; Beck, Thomas; Meindl, James D. (2009). "Breakdown current density of graphene nanoribbons".Applied Physics Letters.94(24): 243114.arXiv:0906.4156.Bibcode:2009ApPhL..94x3114M.doi:10.1063/1.3147183.ISSN 0003-6951.S2CID 55785299.

[17] Fall, C. P.; Marland, E. S.; Wagner, J. M.; Tyson, J. J., eds. (2002).Computational Cell Biology.New York: Springer. p. 28.ISBN 9780387224596.

[18] Weir, E. K.; Hume, J. R.; Reeves, J. T., eds. (1993)."The electrophysiology of smooth muscle cells and techniques for studying ion channels".Ion flux in pulmonary vascular control.New York: Springer Science. p. 29.ISBN 9780387224596.

[19] "Xenon lamp photocathodes"(PDF).

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