# Displacement current

The displacement current is that part of the electrical current that is given by the change in the electrical flow over time. It was recognized by James Clerk Maxwell as a necessary additional term in Ampère's law .

## Meaning and context

1. The convection current is based on a common electrical and material current without the charge carriers, e.g. B. conduction electrons or ions , are bound to a rest position by a restoring force. Often the drive for movement is an electric field , see electrical conductivity , but also see diffusion current , thermoelectricity and Van-de-Graaff generator . In colloquial terms, electricity only means this component.${\ displaystyle I _ {\ mathrm {l}}}$
2. The displacement current corresponds to changes in the electrical flux density , which consists of two contributions: the formation or alignment of electrical dipoles in matter, see dielectric polarization , and the electrical field strength multiplied by the electrical field constant .${\ displaystyle I _ {\ mathrm {v}}}$

Mathematically, the total current can be expressed as the sum of both components as: ${\ displaystyle I}$

${\ displaystyle I = I _ {\ mathrm {l}} + I _ {\ mathrm {v}}}$.

As a result, a conceptual expansion of the Ampèreschen flow law is necessary, which the entire electric current in the form

${\ displaystyle I = \ int _ {A} \ left (\ sigma {\ vec {E}} + \ varepsilon {\ frac {\ partial {\ vec {E}}} {\ partial t}} \ right) \ cdot \ mathrm {d} {\ vec {A}}}$

expresses. The first term is the conduction current that is triggered by the electric field strength . The constant that occurs is the electrical conductivity of the medium ( conductor ) in which the line current flows. ${\ displaystyle E}$${\ displaystyle \ sigma}$

The second term is the displacement current with the rate of change of the field strength and the permittivity over time . The permittivity is the measure of the polarization possible in the medium. Displacement current is important in materials with high permittivity and low conductivity, i.e. non-conductors (insulators). A special case with no conductivity but weak permittivity is the empty space ( vacuum ): In it (apart from free charge carriers due to possible high field strengths) only displacement current flows. ${\ displaystyle \ varepsilon}$${\ displaystyle \ varepsilon _ {0}}$

The two material constants conductivity and permittivity are generally 2nd level tensors and also describe non-linear and non-isotropic dependencies of the total current on the field strength. For most materials, however, these constants can be thought of as scalars .

The classification, from when the conduction current prevails in a medium and this can therefore be called an electrical conductor, and from when the displacement current predominates, results from the values ​​of the two material constants and - because the displacement current is the time derivative of the field strength - the Angular frequency of the field. In general: ${\ displaystyle \ omega}$

${\ displaystyle \ sigma \ gg \ omega \ varepsilon}$ Line current dominant
${\ displaystyle \ sigma \ ll \ omega \ varepsilon}$ Displacement current dominant

Typical conductors such as copper or typical insulators such as some plastics ( PVC ) have material constants that are independent of the frequency. With conductors such as copper , the conduction current outweighs the displacement current up to very high frequencies (in the X-ray range , see plasma oscillation ). In contrast, with certain substances such as ion conductors ( salt water ), the material constants are strongly frequency-dependent. Then it depends on the frequency (rate of change of the electric field over time) whether the substance is to be regarded as a conductor or a non-conductor.

In the case of temporal harmonic (sinusoidal) changes in the same medium, the displacement current is always phase-shifted by 90 ° (π / 2) compared to the line current. In contrast, in a circuit that is interrupted by an insulator, the displacement current dominating in the insulator and the conduction current dominating in the electrical conductor are in phase with one another, and the two currents are practically equal in magnitude. This technically important case occurs with the capacitor in the sinusoidal alternating current 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 is primarily carried by the displacement current . Without the displacement current, no current conduction through the capacitor would be possible - although this current conduction is always limited to alternating currents (temporal change) due to the necessary rate of change in the electrical flow due to the displacement current.

## Historical development

To derive the displacement current : S 1 (light blue) and S 2 (light red) have the same boundary ∂S, so that Ampère's law should lead to the same result for both surfaces. The
conduction current I flows through the circular area S 1 and generates a magnetic field, while no conduction current flows through the area S 2 and therefore there is no magnetic field. The result of Ampère's law would therefore depend on the shape of the surface. The introduction of the displacement current resolves this contradiction.

When Maxwell tried to combine the knowledge about electromagnetic phenomena that had been collated by other physicists such as Ampère and Faraday in Maxwell's equations , it became clear to him that Ampère's law about the generation of magnetic fields by currents could not be complete.

This fact becomes clear through a simple thought experiment. A current I would flow through a long wire with a capacitor in it . Ampère's law

${\ displaystyle \ oint _ {S} {\ vec {B}} \ cdot \; \ mathrm {d} {\ vec {s}} = \ mu _ {0} I}$

now states that the path integral of the magnetic field along any path around the wire is proportional to the current that flows through an area spanned by this path. Also the differential form

${\ displaystyle \ operatorname {red} \, {\ vec {B}} = \ mu _ {0} {\ vec {J}}}$

demands that the choice of this spanned area is arbitrary. Now the integration path has the simplest possible form, a circle around the longitudinal axis of the wire (denoted by ∂S in the graphic). The most natural choice of the area spanned by this circle is obviously the circular area S 1 . As expected, this circular area cuts the wire, so the current through the area is I. From the symmetry of the wire, it follows for the magnetic field of the long wire that its field lines are circular paths around the longitudinal axis.

Even if you “bulge” or “inflate” the surface at will, the same current will still flow through it - unless you expand it so that it runs between the two capacitor plates. Apparently no current flows through this area S 2 . Maxwell assumed that Ampère's law is not false, but only incomplete.

### resolution

No current flows through the capacitor, but the electric field and thus the electric flux change when the capacitor is charged (this refers to the electric field D without the influence of dielectric material; in the graphic, denoted by E ). Maxwell now defined a displacement current as the change in electrical flow through the given surface. The displacement current is therefore not a current in which charge is transported. Rather, it is a descriptive name for precisely this change in the electrical flow, since it obviously has the same effect as a real current.

## Mathematical derivation

### Integral form

Displacement current, the change in electrical flow through a surface , is defined by ${\ displaystyle A}$

 ${\ displaystyle I _ {\ mathrm {v}} = {\ frac {\ partial \ Psi} {\ partial t}}}$, (1)

where the electrical flow is defined by

 ${\ displaystyle \ Psi = \ iint _ {A} {\ vec {D}} \ cdot \ mathrm {d} {\ vec {A}}}$. (2)

The prefactor, consisting of the two dielectric constants, eliminates dielectric effects, since the electrical flux density , which remains unaffected by these and only comes from charges, applies

 ${\ displaystyle {\ vec {D}} = \ varepsilon \ varepsilon _ {0} {\ vec {E}} \ rightarrow I _ {\ mathrm {v}} = \ varepsilon \ varepsilon _ {0} {\ frac {\ partial} {\ partial t}} \ iint _ {A} {\ vec {E}} \ cdot \ mathrm {d} {\ vec {A}} = {\ frac {\ partial} {\ partial t}} \ iint _ {A} {\ vec {D}} \ cdot \ mathrm {d} {\ vec {A}}}$ (3)

with the dielectric constant of the vacuum and the constant of the corresponding matter.

The same applies to the magnetic field unaffected by slide and paramagnetic effects

 ${\ displaystyle {\ vec {H}} = {\ dfrac {1} {\ mu _ {0}}} {\ vec {B}}}$. (4)

(This is a simplification. In matter, dia- and paramagnetism are taken into account, with magnetic permeability . In ferromagnetic materials, however, there is no longer a linear relationship. Because it is not relevant to the problem of this article, the one remains here Simplification to the vacuum.) ${\ displaystyle {\ vec {B}} = \ mu _ {\ mathrm {r}} \ mu _ {0} {\ vec {H}}}$ ${\ displaystyle \ mu _ {\ mathrm {r}}}$

It is also known that the (actual) current I through a conductor can be represented as the surface integral of a current density j :

 ${\ displaystyle I = \ iint _ {A} {\ vec {J}} \ cdot \ mathrm {d} {\ vec {A}}}$ (5)

With this preparation you get

 ${\ displaystyle I _ {\ mathrm {v}} \; {\ stackrel {(1)} {=}} \; {\ frac {\ partial \ Psi} {\ partial t}}}$ (6)
${\ displaystyle \; {\ stackrel {(3)} {=}} \; {\ frac {\ partial} {\ partial t}} \ iint _ {A} {\ vec {D}} \ cdot \ mathrm { d} {\ vec {A}} = \ iint _ {A} {\ frac {\ partial {\ vec {D}}} {\ partial t}} \ mathrm {d} {\ vec {A}}}$

This displacement current must now be inserted into Ampère's law quoted in the first section:

${\ displaystyle \ oint _ {S} {\ vec {B}} \ cdot \; \ mathrm {d} {\ vec {s}} = \ mu _ {0} (I + I _ {\ mathrm {v}} ) \ quad \ Leftrightarrow \ quad \ oint _ {S} {\ frac {1} {\ mu _ {0}}} {\ vec {B}} \ cdot \; \ mathrm {d} {\ vec {s} } = I + I _ {\ mathrm {v}}}$
${\ displaystyle {{\ stackrel {(4), (6)} {\ Leftrightarrow}} \ quad \ oint _ {S} {\ vec {H}} \ cdot \; \ mathrm {d} {\ vec {s }} = I + \ iint _ {A} {\ frac {\ partial {\ vec {D}}} {\ partial t}} \ mathrm {d} {\ vec {A}} \ quad {\ stackrel {(5 )} {\ Leftrightarrow}} \ quad \ oint _ {S} {\ vec {H}} \ cdot \; \ mathrm {d} {\ vec {s}} = \ iint _ {A} {\ vec {J }} \ cdot \ mathrm {d} {\ vec {A}} + \ iint _ {A} {\ frac {\ partial {\ vec {D}}} {\ partial t}} \ cdot \ mathrm {d} {\ vec {A}}}}$
${\ displaystyle \ Leftrightarrow \ quad \ oint _ {S} {\ vec {H}} \ cdot \; \ mathrm {d} {\ vec {s}} = \ iint _ {A} \ left ({\ vec { J}} + {\ frac {\ partial {\ vec {D}}} {\ partial t}} \ right) \ mathrm {d} {\ vec {A}}}$

thus reaching the integral form of Maxwell's fourth equation.

### Differential form

For the differential formulation, only the definition of a displacement current density for the displacement current analogous to the magnitude of the current density J of the actual current I is missing :

 ${\ displaystyle {\ vec {J}} _ {\ mathrm {v}} = {\ frac {\ partial {\ vec {D}}} {\ partial t}}}$. (7)

You get

${\ displaystyle \ operatorname {rot} {\ vec {B}} = \ mu _ {0} \ left ({\ vec {J}} _ {\ mathrm {l}} + {\ vec {J}} _ { \ mathrm {v}} \ right) \ quad \ Leftrightarrow \ quad \ operatorname {red} {\ frac {1} {\ mu _ {0}}} {\ vec {B}} = {\ vec {J}} _ {\ mathrm {l}} + {\ vec {J}} _ {\ mathrm {v}}}$
${\ displaystyle {\ stackrel {(4), (7)} {\ Leftrightarrow}} \ quad \ operatorname {red} {\ vec {H}} = {\ vec {J}} + {\ frac {\ partial { \ vec {D}}} {\ partial t}}}$,

the differential form of Maxwell's fourth equation.

## literature

• Adolf J. Schwab (Ed.): Conceptual world of field theory . Springer, 2002, ISBN 3-540-42018-5 .