Current_density

Current density

This page is mainly about the electric current density in electromagnetism. For the probability current density in quantum mechanics, see Current density (quantum mechanics).

Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved quantities. It is defined as a vector whose magnitude is the current per cross-sectional area.

In SI units, the electric current density is measured in amperes per square metre.

1 Definition
2 Divergence of current density
3 In practice
4 References
5 External links

Definition

Electric current is a coarse, average quantity that tells what is happening in an entire wire. If we want to describe the distribution of the charge flow, we use the concept of the current density:

$\mathbf{J}=nq\mathbf{v}_d=\rho \mathbf{v}_d \!\$

where

$\mathbf{J} \!\$ is the current density vector (SI unit amperes per square metre)

$n \!\$ is the particle density in count per volume (SI unit m^-3)

$q \!\$ is the individual particles' charge (SI unit coulombs)

$\rho = nq \!\$ is the charge density (SI unit coulombs per cubic metre)

$\mathbf{v}_d \!\$ is the particles' average drift velocity (SI unit metres per second)

The current density can also be defined as:

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

where

$\mathbf{E}$ is the electric field strength and

σ

is the electrical conductivity.

The current through a surface S can be calculated by the following relation:

$I=\int_S{ \mathbf{J} \cdot \mathrm{d}\mathbf{A}}$

– where the current is in fact the integral of the dot product of the current density vector and the differential surface element $\mathrm{d} \mathbf{A} \$ , i.e. the net flux of the current density vector field flowing through the surface S.

The current density is an important parameter in Ampère's circuital law (one of Maxwell's equations), which show the direct link between current density and magnetic field.

Current density is an important consideration in the design of electrical and electronic systems. 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, or the insulating material failing. In superconductors excessive current density may generate a strong enough magnetic field to cause spontaneous loss of the superconductive property.

Divergence of current density

From the divergence theorem,

$\int_S{ \mathbf{J} \cdot \mathrm{d}\mathbf{A}} = \int_V{(\mathbf{\nabla} \cdot \mathbf{J}) \mathrm{d}V}$

since charge is conserved,

$\int_V{(\nabla \cdot \mathbf{J}) \mathrm{d}V} = -\frac{\mathrm{d}}{\mathrm{d}t} \int_V{\rho \; \mathrm{d}V} = - \int_V{\left( \frac{\partial \rho}{\partial t} \right) \mathrm{d}V}$

Since this is valid for any volume,

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

which is also called the continuity equation.^[1]

In practice

In the domain of electrical wiring (isolated copper), maximum current density can vary from 4A/mm² for a wire isolated from free air to 6A/mm² for a wire at free air.
If the wire is carrying high frequency currents (above 100kHz) the skin effect may affect the distribution of the current across the section by concentrating the current on the surface of the conductor.

In the domain of printed board, for TOP and BOTTOM layers, maximum current density can be as high as 35A/mm² with a copper thickness of 35um. Inner layers cannot dissipate as much power as outer layers, thus it is not a good idea to put high power lines in inner layers.

In the domain of semiconductor, The maximum current density is given by the manufacturer. But a common average is 1mA/um (180nm technology) (Ampere per width of the line). Above the maximum current density apart joule effect, some other effects like electromigration appears in the micrometer scale.

In biological systems, ion channels regulate the flow of ions (e.g., sodium, calcium, potassium) across the membrane in all cells. Current density is measured in pA/pF (picoamperes per picofarad), that is, current divided by capacitance.

References

^ Griffiths, D.J., Introduction to Electrodynamics, page 213, Prentice-Hall International, 1999.

External links

[1] - A short explanation of the current density

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Contents

Definition

Divergence of current density

In practice

References

External links