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 .

See also

• Particle flux density
• Diffusion current density

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 .