# Electric current density

Physical size
Surname Electric current density
Formula symbol ${\ displaystyle {\ vec {J}}}$, ,${\ displaystyle {\ vec {j}}}$${\ displaystyle {\ vec {S}}}$
Size and
unit system
unit dimension
SI A · m -2 I · L −2
Gauss ( cgs ) Stata · cm -2 L -1/2 · M 02/01 · T -2
esE ( cgs ) Stata · cm -2 L -1/2 · M 02/01 · T -2
emE ( cgs ) Aba · cm -2 L -3/2 · M 02/01 · T -1

The electric current density ( symbol (so in), also or ) indicates how densely packed an electric current flows. It thus also indicates the load on a conductor by the current. ${\ displaystyle {\ vec {J}}}$${\ displaystyle {\ vec {j}}}$${\ displaystyle {\ vec {S}}}$

The current density is defined as the ratio of the current intensity to a cross-sectional area available to the current through which the current passes perpendicularly. ${\ displaystyle I}$ ${\ displaystyle A}$

Furthermore, in non-homogeneous flow fields, the current density can be used to indicate how the current is distributed point by point over the cross-sectional area. Such applications relate, for example, to gas discharges and electron beams as well as the loading of electrodes and hot cathodes .

## definition

In classical physics:

${\ displaystyle I = \ int \ limits _ {A} {\ vec {J}} \ cdot \ mathrm {d} {\ vec {A}}}$

The vector is perpendicular to the associated surface element. If the current density is evenly distributed over the cross-sectional area (e.g. if direct current flows through a metallic conductor), the definition is simplified to . The scalar product is reduced under simple model calculations for vertically-carrying surface (in the picture) to the elementary product : . ${\ displaystyle \ mathrm {d} {\ vec {A}}}$${\ displaystyle I = {\ vec {J}} \ cdot {\ vec {A}}}$${\ displaystyle I = J \, A}$

Current in a conductor with the cross-sectional area ${\ displaystyle A}$

With

${\ displaystyle V}$ a considered volume,
${\ displaystyle Q}$the total electrical charge in this volume,
${\ displaystyle n}$the charge carrier density (number of charge carriers per volume),
${\ displaystyle e}$the charge of a single charge carrier ( elementary charge ; 1.60 · 10 −19 As),
${\ displaystyle \ rho = n \, e = \ mathrm {d} Q / \ mathrm {d} V}$the space charge density ,
${\ displaystyle x}$ the location coordinate in the direction of flow,
${\ displaystyle t}$ the time,
${\ displaystyle v = \ mathrm {d} x / \ mathrm {d} t}$the mean drift speed of the charge carriers,
${\ displaystyle I = \ mathrm {d} Q / \ mathrm {d} t}$ the current strength (charge per time)

results from an arrangement as in the figure with a current flowing evenly over the cross-sectional area and flowing in the x-direction (perpendicular to the marked yz-plane)

${\ displaystyle J = {\ frac {I} {A}} = {\ frac {1} {A}} \; {\ frac {\ mathrm {d} Q} {\ mathrm {d} t}} = { \ frac {1} {A}} \; {\ frac {\ mathrm {d} Q} {\ mathrm {d} V}} \; {\ frac {\ mathrm {d} V} {\ mathrm {d} t}} = {\ frac {1} {A}} \; \ rho \ cdot {\ frac {A \; \ mathrm {d} x} {\ mathrm {d} t}} = \ rho \; v}$ .

The current density is a vector quantity whose direction corresponds to that of the velocity vector of positive charge carriers: ${\ displaystyle {\ vec {v}}}$

${\ displaystyle {\ vec {J}} = \ rho \; {\ vec {v}} = n \; e \; {\ vec {v}}}$ .

## Applications

### Calculations

With regard to the electrical current, the current strength is preferably used in practical electrical engineering for bills ,

for example, one chooses the notation for Ohm's law${\ displaystyle I = G \ U}$
with the electrical conductance , the electrical voltage .${\ displaystyle G = 1 / R}$${\ displaystyle U}$

In contrast, in theoretical electrical engineering, the current density is usually used,

for example, one chooses the notation for Ohm's law ${\ displaystyle {\ vec {J}} = \ sigma \ {\ vec {E}}}$
with the electrical conductivity , the electrical field strength .${\ displaystyle \ sigma}$${\ displaystyle {\ vec {E}}}$

For example, the vector current density is used in Maxwell's equations and in the continuity equation of electrodynamics.

### Current density in lines

The density of the conductor current in copper windings may not exceed 1.2 ... 6 A / mm 2 , depending on the application , so that no inadmissible heating occurs under continuous load . This is also referred to as current carrying capacity . In extreme cases, however, it can   rise to a melt current density of 3060 A / mm 2 . The heating in fuses is used to interrupt the current. In conductors, the maximum current strength according to VDE 0298-4: 2013-06, table 11 and column 5 is:

12 A with a cross-sectional area of ​​0.75 mm 2 ,
15 A at 1.0 mm 2 and
26 A at 2.5 mm 2 .

With a current density evenly distributed over the cross-section, the average speed in the conductor is the same . The typical electron density for conduction electrons in metallic solids is in the order of magnitude of = 10 28  m −3 . If one takes into account that in a positive half-oscillation of an alternating current, the mean current intensity is smaller than its effective value by the factor (  = form factor , with sine curve = 1.11), then with a current density of 6 A / mm 2 for a directional movement a mean speed of the order of 10 −3  m / s. The high speed of electrical communication is not based on the displacement of the electrons in the wire. ${\ displaystyle v = {\ frac {J} {ne}}}$${\ displaystyle n}$${\ displaystyle 1 / k_ {f}}$${\ displaystyle k_ {f}}$

In the case of alternating current , the skin effect must be observed, according to which the current density inside a conductor is lower than on the surface. For orientation, the depth is given for a decrease in the current density to 1 / e = 37%. In thick, solid aluminum or copper round conductors, it is around 10 mm at 50 Hz.

### Electroplating

In electroplating , the current density that is set for the coating is specified. The typical values ​​are between 0.5 and 5 A / dm 2 , which must be observed in order to e.g. B. to get good results with a galvanizing or nickel plating .

### Power sources

In the case of solar cells , one rather specifies a power density. It can be very roughly up to 150 W / m 2 . The electrical voltage at maximum power in the most common cells is around 0.5 V, so that a current density of up to 300 A / m 2 can result.

Correspondingly, fuel cells are also examined depending on their current densities, in particularly favorable cases up to about 1 A / cm 2 .

## Surface current density and line current

Analogous to the current density in a body, the current density can also be related to two-dimensional surfaces. This assumption is useful if you want to describe the surface conduction ( leakage current ) of electrical insulators . The total flow is the sum of the individual area flows. The surface current density is obtained by relating the total current to the width of the individual surface: ${\ displaystyle I}$ ${\ displaystyle K}$${\ displaystyle b}$

${\ displaystyle K = {\ frac {I} {b}}}$

The electrical current intensity can also be viewed as the sum of line currents at a point, from which Kirchhoff's first rule follows:

${\ displaystyle I (P) = \ sum _ {l} I_ {l}}$

## literature

• H. Lindner, H. Brauer, C. Lehmann: Pocket book of electrical engineering and electronics . 8th, revised edition, Fachbuchverlag Leipzig in Carl Hanser Verlag, Munich 2004, ISBN 3-446-22546-3 .

## Individual evidence

1. DIN 1304-1: 1994 Formula symbols - General formula symbols .
2. DIN EN 80000-6: 2008 Sizes and units - electromagnetism .
3. Wolfgang Demtröder : Experimentalphysik 2, electricity and optics .
4. DIN 41300-1: 1979 Small transformers - characteristic data
5. DIN 43671: 1975 busbars made of copper - dimensioning for continuous current
6. Erwin Böhmer: Elements of Applied Electronics
7. Melting current density is the current density at which the conductor temperature rises to melting temperature after 1/100 s load. Value according to Müller-Hildebrand
8. Eduard Vinaricky: Electrical contacts, materials and applications: Basics, technologies ... Springer DE, 2002, ISBN 3-642-56237-X , p. 395 ( limited preview in Google Book search).
9. Wolfgang Demtröder: Experimentalphysik 3. Atoms, molecules and solids
10. ^ Christian Gerthsen: Physics
11. Anne Bendzulla: From the component to the stack: Development and design of HT-PEFC stacks of the 5 kW class ; Dissertation Aachen 2010. ISBN 9783893366347 .