Displacement current
Displacement current ( engl. displacement current) is a term from electrodynamics and represents the fact that the temporal change of an electric field or electric flux density is an integral part of the total electric current. The concept was developed by James Clerk Maxwell and extends the Ampère law by an additional Term
- 3.1 Integral form
- 3.2 Differential form
Meaning and context
The total electric current ( engl. true current or total current) in consists of two additive components:
Mathematically, the total electric current I express both components as:
This is a conceptual extension of the Ampère ampere law required that all the electric current in the form of
Expresses. Here, the first term represents the conduction current, which is triggered by the electric field strength E. The constant thereby occurring represents an expression for the electric conductivity in that the medium in which to propagate the line current. One calls such media usually electrical conductors.
The second term represents the displacement current with the time rate of change of the electric field strength and the dielectric conductivity represents the dielectric conductivity expresses the property of a medium for conducting the electrical flow. Therefore, the displacement current flows mainly in materials with good dielectric and poor electrical conductivity. This is called materials with those material constants and insulators. As a special form of non-existent electrical conductivity, but weak existing dielectric conductivity, while the empty space in phenomenon occurs (vacuum): In it spreads, apart from free electrical charge carriers due to high field strengths, only the displacement current from.
In the general case, the two material constants of the electrical conductivity or dielectric tensors 2nd stage and describe non-linear, non-isotropic dependencies of the total electric current of the electric field strength. This fact can, however, for the fundamental understanding initially be neglected and simplifies these constants are considered as two scalar that are specific to the medium of propagation of the respective current components.
The classification of the date from a medium of the line current is dominant, and this medium can be therefore referred to as an electrical conductor, and at what point in a medium of the displacement current is dominant, therefore, may be made of the sizes of the material constants, and, since the component of the displacement current is the time derivative of the electric field occurs, can be derived by the angular frequency of the electric field. General:
Typical electrical conductor such as copper or typical insulators such as some plastics (PVC ) have on the frequency independent material constants. In conductors such as copper predominates up to very high frequencies ( in the X-ray range, see plasma oscillation ) of the line current with respect to the displacement current. However, there are certain substances such as ionic conductors (salt water), which show in their material constants a strong frequency dependence. It will therefore depend on the particular frequency ( time rate of change of the electric field ) on whether the substance will be the conductors or non-conductors, is counted and which dominates the two current components therein.
As a special form of phase with temporal harmonic ( sinusoidal ) changes in the same medium, the displacement current with respect to the line current is always at a 90 ° ( π / 2). However, is in an electric circuit which is interrupted by an insulator, both the dominant in the insulator displacement current and the line current in the electrical conductor dominating each other in phase, and the two currents are substantially equal in magnitude. This technically important case makes its appearance in the capacitor in the sinusoidal AC circuit: The current in the lead wires and the capacitor plates ( electrical conductor ) is carried by the line current, the current through the dielectric ( insulator ) between the capacitor plates primarily by the displacement current. Without displacement current does not flow line would be possible through the capacitor - although this power line is limited by the displacement current due to the necessary time rate of change in the electrical flow always alternating currents ( time change ).
Historical development
Derivation of a contradiction
As Maxwell sought to unite the hitherto by other physicists such as Ampère and Faraday collated findings concerning electromagnetic phenomena in the Maxwell 's equations, he realized that the Ampère law on the generation of magnetic fields by currents could not be complete.
This fact is clear by a simple thought experiment. A current I flowed through a long wire, in which a capacitor is. The Ampère law
States that, the path integral of the magnetic field along any path about the wire is proportional to the current flowing through a plane defined by this path surface. The differential form
Requires that the choice of these spanned surface is arbitrary. Now the path of integration have the simplest possible form, a circle around the longitudinal axis of the wire ( in the graph with ∂ S denotes ). The natural choice of the plane defined by this circuit is obviously the area of circle S1. As expected, this circle intersecting the wire, thus the current through the area I from the symmetry of the wire is obtained according to the magnetic field of the long wire that its field lines are circles around the longitudinal axis.
Even if the surface any " bulge " or " inflates ", flows through it still the same current - unless they expand far enough out that it intersects the longitudinal axis of the wire between the two capacitor plates (area S2). There, of course, no current flows, so the magnetic field of the wire is zero, obviously in contradiction to the result just discussed. Maxwell assumed that the Ampère law is not wrong, but incomplete.
Resolution
Due to the capacitor no current flows, but the electric field and hence the electric flux changes during charging of the capacitor ( it is the electric field D without influences meant by dielectric matter, referred to in the graph with E). Maxwell defined now as a displacement current, the change in electrical flow through the given surface. The displacement current is therefore no current is transported in the cargo. Rather, it is a descriptive term for precisely this change of electric flux, as it apparently has the same effect as a real power.
Mathematical derivation
Integrals of the form
The displacement current, the change in electrical flow through a surface area A is defined by
,
(1)
The electric flux is defined by
.
(2)
The pre-factor, consisting of the two dielectric constants, this eliminates dielectric effects, as for the electric field, which remains unaffected by this and assume only charges applies
(3)
The dielectric constant of the vacuum and the rate of the corresponding matter.
Analogously, for the virgin of dia - and paramagnetic effects magnetic field
.
(4)
( This is a simplification. Applies in matter, considering slide and paramagnetism, the magnetic permeability. In ferromagnetic materials but applies no linear relationship more. Since it is not relevant to the problem of this article, so stay here the simplification of the vacuum. )
In addition, the (actual ) current I through a conductor as a surface integral of the current density is known, can be represented j:
(5)
With this preparation, we obtain
(6)
This displacement current must now be added to the quoted in the first section Ampere's law:
Thus the integral form of Maxwell's fourth equation is obtained.
Differential form
For the differential formulation, only the definition of a displacement current density for the displacement current is missing analogous to the amount of current density J of the actual current I:
.
(7)
Obtained
The differential form of Maxwell's fourth equation.