# Current Density

In electricity , 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 that is the motion of the positive charges at this point, going its direction per cross-sectional area at a given point in space. In SI base units , electric flux density is measured in amperes per square meter.

## Definition

Let A (SI unit: m 2 ) be a small surface centered at a given point M and orthogonal to the motion of charges at M. If i a (SI unit: a ) is the electric current flowing through a , then the electric flux density j in m is given by the range :

{\displaystyle j=\lim \limits _{A\rightarrow 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 boundary process.

Current density vector J vector whose magnitude is the same as the electric flux density, and whose direction is the same as the movement of positive charges m .

At a given time t , if v is the velocity of charges on M , and dA is an infinitesimal surface centered at M and orthogonal to v , then during time d , only the charge contained in the volume formed by dA and I = dq / dt will flow through dA . This chargeis equal to || V || d t d a , where is the charge density at M , and the electric current at M is I = || V || DA . It follows that the current density vector can be expressed as:

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

The surface integral of j over the surface S , followed by an integral over the time period t 1 to t 2 , gives the total amount of charge flowing through the surface in that time ( t2 – t1 ):

{\displaystyle q=\int _{t_{1}}^{t_{2}}\iint _{S}\mathbf {j} \cdot \mathbf {\hat {n}} \,{\rm {d}}A{\rm {d}}t.}

More precisely, it is the integration of the flux of j across S between 1 and 2 .

To calculate the area , the flow is real or imaginary, flat or curved, either in the form of a cross-sectional area or a surface. For example, for charge carriers passing through an electric conductor , the field is the cross-section of the conductor over the section considered.

The vector field is a combination of the magnitude of the field through which the charge carriers pass, a , and a unit vector , normal to the field . is related.

\mathbf {\hat {n}}
\mathbf {A} =A\mathbf {\hat {n}}

The differential vector area is similarly derived from the definition above:

{\displaystyle d\mathbf {A} =dA\mathbf {\hat {n}} }

The current density then J passes through the region at an angle to the region normal , then

\mathbf {\hat {n}}

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

where is the dot product of the unit vectors. That is, the current passing through the surface density (i.e. normal to it) has the component j because , while the current passing tangent to the densely populated area has the component j sin , but there is no current density actually passing through the field in the tangent direction. The only component of the current density passing through the normal to the field is the cosine component.

## Significance

Current density is important for the design of electrical and electronic systems.

The performance of the circuit depends strongly on the designed current level, and the current density is then determined by the dimensions of the conducting elements. For example, as integrated circuits reduce in size, there is a trend towards higher current densities to achieve higher device numbers in smaller chip areas , despite lower current demands by smaller devices . See Moore’s law .

At high frequencies, the conducting region in a wire becomes confined near its surface, increasing the current density in this region. This is known as the skin effect .

Higher current densities have undesirable consequences. Most electrical conductors have a limited, positive resistance , so they dissipate power in the form of heat . The current density must be kept sufficiently low to prevent the conductor from melting or burning, failing the insulating material , or changing the desired electrical properties. At higher current densities the material forming the interconnection actually moves, a phenomenon known as electromigration . In superconductors , excessive current densities may generate a magnetic field strong enough to cause spontaneous loss of the superconductive property.

The analysis and observation of current densities are also used to investigate the underlying physics of the nature of solids, including not only metals, but also semiconductors and insulators. An elaborate theoretical formalism has developed to explain many of the fundamental observations.

Current density is an important parameter in Ampere’s circuit law ( one of Maxwell’s equations ), which relates the current density to the magnetic field.

In special relativity theory, charge and current are combined into one 4-vector.

## Calculation of current density in a substance

### Free currents

Charge carriers that are free to move constitute a free current density , given by expressions such as those in this section.

Electric current is a rough, average quantity that describes what is happening throughout the wire. At position r in time t , the distribution of charge – flowing current density is described by:

{\displaystyle \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’ average drift velocity (SI unit: m s – 1 ), and

{\displaystyle \rho (\mathbf {r} ,t)=q\,n(\mathbf {r} ,t)}

is charge density: (in which the SI unit is cubic meter per coulomb) n ( r ,  t ) unit volume (“number density”) (SI unit: number of particles per meter is -3 ), q is charge density of n with individual particles (SI unit: coulomb).

A general approximation to current density assumes that the current is proportional to the electric field, expressed as:

{\displaystyle \mathbf {j} =\sigma \mathbf {E} \,}

Where E is the electric field and is the electrical conductivity .

Conductivity is the reciprocal (inverse of electricity) resistivity and is the SI units of siemens per meter (S⋅m -1 ) , and e is the SI units of newtons per coulomb (NC -1 ) or, equivalently, volts per meter. (Vm -1 ).

A more fundamental approach to the calculation of current density is based on:

{\displaystyle \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 , and the non-local nature of the field response by the spatial dependence of , both calculated in principle from a microscopic analysis underlying, for example, in the case of small enough fields, in the material Linear response functions for conductive behavior. See, for example, Giuliani and Vignale (2005)  or Ramer (2007).  The entire past history extends to the present time.

The above conductivity and its associated current density reflect the fundamental mechanism underlying charge transport in the medium, both in time and in distance.

A Fourier transform in space and time results in:

{\displaystyle \mathbf {j} (\mathbf {k} ,\omega )=\sigma (\mathbf {k} ,\omega )\;\mathbf {E} (\mathbf {k} ,\omega )\,}

where ( k ,  ) 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. In addition to the physical properties themselves, the application of a magnetic field can change the conductive behavior.

### Polarization and Magnetization Currents

Currents in a material arise when there is an uneven 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. polarization P :

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

Similarly with magnetic materials, the movement of magnetic dipole moments per unit volume, i.e. magnetization M, leads to magnetization currents:

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

Together, these terms combine to form the bound current density in the material (resulting in the movement of electric and magnetic dipole moments per unit volume):

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

### Total current in the material

The total current is simply the sum of the free and bound currents:

{\displaystyle \mathbf {j} =\mathbf {j} _{\mathrm {f} }+\mathbf {j} _{\mathrm {b} }}

### Displacement current

There is also a displacement current corresponding to the time-varying electric displacement field D :

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

which is an important term in Ampere’s circuitry, one of Maxwell’s equations, because without this term the propagation of electromagnetic waves or the time evolution of electric fields in general cannot be predicted.

## Continuity equation

Since charge is conserved, the current density must satisfy the continuity equation. Here’s the derivation from first principles. 

The net flux from some volume V (which may have an arbitrary shape but be fixed for calculation) must be equal to the net change of charge placed inside the volume:

{\displaystyle \int _{S}{\mathbf {j} \cdot \mathrm {d} \mathbf {A} }=-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}{\rho \;\mathrm {d} V}=-\int _{V}{{\frac {\partial \rho }{\partial t}}\;\mathrm {d} V}}

where is the charge density, and d is a surface element S of the surface enclosing volume V. The surface integral on the left expresses the current outflow from the volume , and the negatively signed volume on the right expresses the reduction in total charge inside the integral volume . Deviation Theorem from :

{\displaystyle \int _{S}{\mathbf {j} \cdot \mathrm {d} \mathbf {A} }=\int _{V}{\mathbf {\nabla } \cdot \mathbf {j} \;\mathrm {d} V}}

That’s why:

{\displaystyle \int _{V}{\mathbf {\nabla } \cdot \mathbf {j} \;\mathrm {d} V}\ =-\int _{V}{{\frac {\partial \rho }{\partial t}}\;\mathrm {d} V}}

This relation is valid for any quantity independent of size or space, which means that:

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

And this relation is called continuity equation.

## Use

In electrical wiring, the maximum current density can vary from 4 A⋅mm, -2 to 6 A⋅mm for a wire with no air circulation around it, -2 for a wire in free air. The rules for building wiring list the maximum permissible current of each size of cable under different circumstances. For compact designs, such as windings of SMPS transformers, the value can be as low as 2 A⋅mm -2 . If the wire is carrying high frequency currents, the skin effect can affect the distribution of current across the section by concentrating the current on the surface of the conductor. In transformers designed for high frequencies, the loss is reduced if the Litz wire is used for winding. It is made up of several separate wires in parallel with a diameter twice the depth of the skin. Individual strands are twisted together to increase the total skin area and reduce resistance due to skin impacts.

For the top and bottom layers of printed circuit boards, the maximum current density can be as high as 35Amm -2 and the copper thickness can be 35μm. The inner layers cannot shed as much heat as the outer layers; Circuit board designers avoid putting high-current markings on the inner layers.

In the semiconductors sector, the maximum current density for different elements is given by the manufacturer. Exceeding those limits creates the following problems:

• Joule effect which increases the temperature of the component.
• Electromigration effect which will eliminate the interconnection and eventually cause an open circuit.
• The slow diffusion effect, which, when constantly exposed to high temperatures, will move metal ions and dopants away from where they should be. This effect is also synonymous with aging.

The following table gives an idea of ​​the maximum current densities for different materials.

Even if manufacturers add some margin to their numbers, it is recommended that in order to improve reliability, especially for high-quality electronics, at least double the calculated section. Do it. One can also see the importance of pacifying electronic devices in the view 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 membranes 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 ampere per pico farad) (i.e., capacitance divided by current). Techniques exist to measure the capacitance and surface area of ​​cells empirically, which enables the 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, the current density plays an important role in the output spectrum produced. Low current densities generate spectral line emission and favor longer wavelengths. Higher current densities produce continuum emission and favor shorter wavelengths.  The low current densities for flash lamps are usually around 10 Amm -2 . High current densities can exceed 40 Amm −2 .