|In SI base units||A m−2|
In electromagnetism, 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.
Assume that A (SI unit: m 2) is a small surface centred at a given point M and orthogonal to the motion of the charges at M. If IA (SI unit: A) is the electric current flowing through A, then electric current densityj at M is given by the limit:
with surface A remaining centred at M and orthogonal to the motion of the charges during the limit process.
The current 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 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 I = dq / dt will flow through dA. This charge is equal to ρ ||v|| dt dA, where ρ is the charge density at M, and the electric current at M is I = ρ ||v|| dA. It follows that the current density vector can be expressed as:
The surface integral of j over a surface S, followed by an integral over the time duration t1 to t2, gives the total amount of charge flowing through the surface in that time (t2 − t1):
More concisely, this is the integral of the flux of j across S between t1 and t2.
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, 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 of 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 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.
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), which relates current density to magnetic field.
In special relativity theory, charge and current are combined into a 4-vector.
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, vd(r, t) is the particles' average drift velocity (SI unit: m∙s −1), and
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−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 (inverse) of electrical resistivity and has the SI units of siemens per metre (S⋅m−1), and E has the SI units of newtons 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:
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.
Currents arise in materials when there is a non-uniform distribution of charge.
In dielectric 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):
The total current is simply the sum of the free and bound currents:
There is also a displacement current corresponding to the time-varying electric displacement field 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.
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, 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:
This relation is valid for any volume, independent of size or location, which implies that:
and this relation is called the continuity equation.
In electrical wiring, the maximum current density can vary from 4 A⋅mm−2 for a wire with no air circulation around it, to 6 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 SMPS transformers, the value might be as low as 2 A⋅mm−2. 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 conductor. In transformers 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 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||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 to electromigration and slow diffusion.
In biological organisms, ion channels regulate the flow of ions (for example, sodium, calcium, potassium) across the membrane in all cells. The membrane of a cell is assumed to act like a capacitor.Current densities are usually expressed in pA⋅pF−1 (pico amperes per 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.
In gas discharge lamps, such as flashlamps, current density plays an important role in the output spectrum produced. Low current densities produce spectral line emission and tend to favour longer wavelengths. High current densities produce continuum emission and tend to favour shorter wavelengths. A⋅mm−2. High current densities can be more than 40 A⋅mm−2.Low current densities for flash lamps are generally around 10
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.
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A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically meant to describe electromagnetism and gravitation, two of the fundamental forces of nature.
In physics, charge conservation is the principle that the total electric charge in an isolated system never changes. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always conserved. Charge conservation, considered as a physical conservation law, implies that the change in the amount of electric charge in any volume of space is exactly equal to the amount of charge flowing into the volume minus the amount of charge flowing out of the volume. In essence, charge conservation is an accounting relationship between the amount of charge in a region and the flow of charge into and out of that region, given by a continuity equation between charge density and current density .
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m−3), at any point in a volume. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m−1), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.
In quantum mechanics, the probability current is a mathematical quantity describing the flow of probability in terms of probability per unit time per unit area. Specifically, if one describes the probability density as a heterogeneous fluid, then the probability current is the rate of flow of this fluid. This is analogous to mass currents in hydrodynamics and electric currents in electromagnetism. It is a real vector, like electric current density. The concept of a probability current is a useful formalism in quantum mechanics. The probability current is invariant under gauge transformation.
The method of image charges is a basic problem-solving tool in electrostatics. The name originates from the replacement of certain elements in the original layout with imaginary charges, which replicates the boundary conditions of the problem.
Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions in periodic systems. It was first developed as the method for calculating electrostatic energies of ionic crystals, and is now commonly used for calculating long-range interactions in computational chemistry. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. In this method, the long-range interaction is divided into two parts: a short-range contribution, and a long-range contribution which does not have a singularity. The short-range contribution is calculated in real space, whereas the long-range contribution is calculated using a Fourier transform. The advantage of this method is the rapid convergence of the energy compared with that of a direct summation. This means that the method has high accuracy and reasonable speed when computing long-range interactions, and it is thus the de facto standard method for calculating long-range interactions in periodic systems. The method requires charge neutrality of the molecular system in order to accurately calculate the total Coulombic interaction. A study of the truncation errors introduced in the energy and force calculations of disordered point-charge systems is provided by Kolafa and Perram.
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Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.
linear response theory capacitance OR conductance.