Current density
Current density | |
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Common symbols | j →, J |
In SI base units | A m−2 |
Dimension | [A L−2] |
Articles about |
Electromagnetism |
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In
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 IA (SI unit: A) is the electric current flowing through A, then electric current density j at M is given by the limit:[3]
with surface A remaining centered at M and orthogonal to the motion of the charges during the limit process.
The current 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 will flow through dA. This charge is equal to where ρ is the charge density at M. The electric current is , it follows that the current density vector is the vector normal (i.e. parallel to v) and of magnitude
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.
Importance
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
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
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 in Ampère's circuital law (one of Maxwell's equations), which relates current density to magnetic field.
In
Calculation of current densities in matter
Free currents
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:[6]
where
- j(r, t) is the current density vector;
- vd(r, t) is the particles' average drift velocity (SI unit: m∙s−1);
- 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
Conductivity σ is the
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)[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.
A Fourier transform in space and time then results in:
where σ(k, ω) is now a
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.
Polarization and magnetization currents
Currents arise in materials when there is a non-uniform distribution of charge.[9]
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
Together, these terms add up to form the
Total current in materials
The total current is simply the sum of the free and bound currents:
Displacement current
There is also a displacement current corresponding to the time-varying electric displacement field D:[11][12]
which is an important term in
Continuity equation
Since charge is conserved, current density must satisfy a continuity 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:
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:
Hence:
This relation is valid for any volume, independent of size or location, which implies that:
and this relation is called the continuity equation.[13][14]
In practice
In
For the top and bottom layers of
In the
- The Joule effect which increases the temperature of the component.
- The electromigration effect which will erode the interconnection and eventually cause an open circuit.
- The slow diffusion effect which, if exposed to high temperatures continuously, will move metallic ions and 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.
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 to electromigration and slow diffusion.
In
In
See also
- Hall effect
- Quantum Hall effect
- Superconductivity
- Electron mobility
- Drift velocity
- Effective mass
- Electrical resistance
- Sheet resistance
- Speed of electricity
- Electrical conduction
- Green–Kubo relations
- Green's function (many-body theory)
References
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linear response theory capacitance OR conductance.
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- ^ ISBN 9780471927129.
- doi:10.1119/1.4773441. Archived from the original(PDF) on 2020-09-20. Retrieved 2017-04-23.
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- ISBN 9780387224596.
- ^ "Xenon lamp photocathodes" (PDF).