Movatterモバイル変換

[0]ホーム

Jump to content

Current density

Edit links

From Wikipedia, the free encyclopedia

Amount of charge flowing through a unit cross-sectional area per unit time

This article is about electric current density. For the quantum concept, seeProbability current density.

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

Electromagnetism

Electricity Magnetism Optics History Computational Textbooks Phenomena
Electrostatics Charge density Conductor Coulomb law Electret Electric charge Electric dipole Electric field Electric flux Electric potential Electrostatic discharge Electrostatic induction Gauss's law Insulator Permittivity Polarization Potential energy Static electricity Triboelectricity
Magnetostatics Ampère's law Biot–Savart law Gauss's law for magnetism Magnetic dipole Magnetic field Magnetic flux Magnetic scalar potential Magnetic vector potential Magnetization Permeability Right-hand rule
Electrodynamics Bremsstrahlung Cyclotron radiation Displacement current Eddy current Electromagnetic field Electromagnetic induction Electromagnetic pulse Electromagnetic radiation Faraday's law Jefimenko equations Larmor formula Lenz's law Liénard–Wiechert potential London equations Lorentz force Maxwell's equations Maxwell tensor Poynting vector Synchrotron radiation
Electrical network Alternating current Capacitance Current density Direct current Electric current Electric power Electrolysis Electromotive force Impedance Inductance Joule heating Kirchhoff's laws Network analysis Ohm's law Parallel circuit Resistance Resonant cavities Series circuit Voltage Watt Waveguides
Magnetic circuit AC motor DC motor Electric machine Electric motor Gyrator–capacitor Induction motor Linear motor Magnetomotive force Permeance Reluctance (complex) Reluctance (real) Rotor Stator Transformer
Covariant formulation Electromagnetic tensor Electromagnetism and special relativity Four-current Four-potential Mathematical descriptions Maxwell equations in curved spacetime Relativistic electromagnetism Stress–energy tensor
Scientists Ampère Biot Coulomb Davy Einstein Faraday Fizeau Gauss Heaviside Helmholtz Henry Hertz Hopkinson Jefimenko Joule Kelvin Kirchhoff Larmor Lenz Liénard Lorentz Maxwell Neumann Ohm Ørsted Poisson Poynting Ritchie Savart Singer Steinmetz Tesla Thomson Volta Weber Wiechert
v t e

Inelectromagnetism,current density is the amount ofcharge per unit time that flows through a unit area of a chosencross section.^[1] Thecurrent density vector is defined as avector whose magnitude is theelectric 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. InSI base units, the electric current density is measured inamperes per meter square.^[2]

Definition

[edit]

Consider a small surface with areaA (SI unit:m²) centered at a given pointM and orthogonal to the motion of the charges atM. IfI_A (SI unit:A) is theelectric current flowing throughA, thenelectric current densityj atM is given by thelimit:^[3]

Current density at a point in a conductor is the ratio of the current at that point to the area of cross-section of the conductor at that point,provided area is held normal to the direction of flow of current. And is given by.. $j=\lim _{A\to 0}{\frac {I_{A}}{A}}=\left.{\frac {\partial I}{\partial A}}\right|_{A=0},$

with surfaceA remaining centered atM and orthogonal to the motion of the charges during the limit process.

Thecurrent density vectorj is the vector whose magnitude is the electric current density, and whose direction is the same as the motion of the positive charges atM.

At a given timet, ifv is the velocity of the charges atM, anddA is an infinitesimal surface centred atM and orthogonal tov, then during an amount of timedt, only the charge contained in the volume formed bydA and $v\,dt$ will flow throughdA. This charge is equal to $dq=\rho \,v\,dt\,dA,$ whereρ is thecharge density atM. The electric current is $dI=dq/dt=\rho vdA$ , it follows that the current density vector is the vector normal $d A {\displaystyle dA}$ (i.e. parallel tov) and of magnitude $dI/dA=\rho v$

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

Thesurface integral ofj over asurfaceS, followed by an integral over the time durationt₁ tot₂, gives the total amount of charge flowing through the surface in that time (t₂ −t₁):

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

More concisely, this is the integral of theflux ofj acrossS betweent₁ andt₂.

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

Thevector area is a combination of the magnitude of the area through which the charge carriers pass,A, and aunit vector normal 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 densityj passes through the area at an angleθ to the area normal $\mathbf {\hat {n}} ,$ then

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

where⋅ is thedot product of the unit vectors. That is, the component of current density passing through the surface (i.e. normal to it) isj cosθ, while the component of current density passing tangential to the area isj sinθ, but there isno current density actually passingthrough the area in the tangential direction. Theonly component of current density passing normal to the area is the cosine component.

Importance

[edit]

Current density is important to the design of electrical andelectronic 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, asintegrated circuits are 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 smallerchip areas. 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 dissipatepower in the form of heat. The current density must be kept sufficiently low to prevent the conductor from melting or burning up, theinsulating material failing, or the desired electrical properties changing. At high current densities the material forming the interconnections actually moves, a phenomenon calledelectromigration. Insuperconductors 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.^[4]^[5]

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

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

Calculation of current densities in matter

[edit]

Free currents

[edit]

Charge carriers which are free to move constitute afree 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 positionr at timet, thedistribution ofcharge flowing 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);
$\rho (\mathbf {r} ,t)=q\,n(\mathbf {r} ,t)$ 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 densityn (SI unit:coulombs).

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

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

whereE is theelectric field andσ is theelectrical conductivity.

Conductivityσ is thereciprocal (inverse) of electricalresistivity and has the SI units ofsiemens per metre (S⋅m⁻¹), andE has the SI units ofnewtons per coulomb (N⋅C⁻¹) or, equivalently,volts per 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 ofσ, and the non-local nature of response to the field by the spatial dependence ofσ, both calculated in principle from an underlying microscopic analysis, for example, in the case of small enough fields, thelinear response function for 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 transform in space and time then results in: $\mathbf {j} (\mathbf {k} ,\omega )=\sigma (\mathbf {k} ,\omega )\;\mathbf {E} (\mathbf {k} ,\omega )$

whereσ(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

[edit]

Currents arise in materials when there is a non-uniform distribution of charge.^[9]

Indielectric materials, there is a current density corresponding to the net movement ofelectric dipole moments per unit volume, i.e. thepolarizationP:

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

Similarly withmagnetic materials, circulations of themagnetic dipole moments per unit volume, i.e. themagnetizationM, lead tomagnetization currents:^[10]

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

Together, these terms add up to form thebound current density 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

[edit]

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

[edit]

There is also adisplacement current corresponding to the time-varyingelectric displacement fieldD:^[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 waves to propagate, or the time evolution ofelectric fields in general.

Continuity equation

[edit]

Main article: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 volumeV (which can have an arbitrary shape but fixed for the calculation) must equal the net change in charge held inside the volume:

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

whereρ is thecharge density, anddA is asurface element of the surfaceS enclosing the volumeV. The surface integral on the left expresses the currentoutflow from the volume, and the negatively signedvolume integral on the right expresses thedecrease in 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

[edit]

Inelectrical wiring, the maximum current density (for a giventemperature 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 forbuilding wiring list 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 effect may affect the distribution of the current across the section by concentrating the current on the surface of theconductor. Intransformers designed for high frequencies, loss is reduced ifLitz wire is 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 theresistance due to skin effects.

For the top and bottom layers ofprinted 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 thesemiconductors field, the maximum current densities for different elements are given by the manufacturer. Exceeding those limits raises the following problems:

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

Material	Temperature	Maximum current density
Copper interconnections (180 nm technology)	25 °C	1000 μA⋅μm⁻² (1000 A⋅mm⁻²)
	50 °C	700 μA⋅μm⁻²(700 A⋅mm⁻²)
	85 °C	400 μA⋅μm⁻²(400 A⋅mm⁻²)
	125 °C	100 μA⋅μm⁻²(100 A⋅mm⁻²)
Graphene nanoribbons^[16]	25 °C	0.1–10 × 10⁸ A⋅cm⁻² (0.1–10 × 10⁶ 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 toelectromigration and slowdiffusion.

Inbiological organisms,ion channels regulate the flow ofions (for example,sodium,calcium,potassium) across themembrane in allcells. The membrane of a cell is assumed to act like a capacitor.^[17]Current densities are usually expressed in pA⋅pF⁻¹ (pico amperes perpico 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 outputspectrum produced. Low current densities producespectral line emission and 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

[edit]

^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).