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 Incoming links: Carrier-current double counting in effective permittivity

Displacement current makes Ampère's law compatible with charge conservation. The macroscopic Maxwell equations are

\nabla \cdot D = ρ_{f}, \nabla \cdot B = 0, \nabla \times E = - \partial_{t} B, \nabla \times H = J_{f} + \partial_{t} D .

Taking the divergence of the last equation and using Gauss's law gives

\nabla \cdot J_{f} + \partial_{t} ρ_{f} = 0

. Omitting

\partial_{t} D

would wrongly require the conduction current to be divergence free during charging. The displacement-current density

\partial_{t} D

has current-density units, but need not represent free charges crossing a region.

The surface independence of Ampère circulation during capacitor charging illustrates the need directly. A surface spanning the circuit can cut the wire or pass between the plates. For ideal parallel plates,

Q = ϵ_{0} A E

and

I = \dot{Q} = ϵ_{0} A \dot{E} = \int_{gap} \partial_{t} D \cdot d A .

The corrected circulation of

H

is independent of that surface choice. Free conduction current stops at a plate; displacement current supplies the gap contribution.

In vacuum the two time-dependent curl equations support waves. Taking a curl of Faraday's law in a source-free region yields

\nabla^{2} E - μ_{0} ϵ_{0} \partial_{t}^{2} E = 0, c = (μ_{0} ϵ_{0})^{- 1/2} .

The corresponding magnetic equation is identical. For a plane wave,

k \cdot E = k \cdot B = 0

and

B = k \times E / ω

. The fields carry energy through the Poynting flux. Dropping displacement current everywhere would remove these freely propagating solutions.^[ref]

In a metal, conduction and displacement compete. For a uniform ohmic material with

J_{f} = σ E

, constant

ϵ, μ

\nabla^{2} E = μ ϵ \partial_{t}^{2} E + μ σ \partial_{t} E .

Use

e^{i k \cdot r - iω t}

k^{2} = μ ϵ ω^{2} + i μ σω, \frac{∣ J _{cond} ∣}{∣ J _{disp} ∣} = \frac{σ}{ω ϵ} .

When

σ ≫ ω ϵ

k ≃ (1 + i) / δ

with

δ = 2/ (μ σω)

: fields are screened within a skin depth and energy is dissipated by Joule heating. This quasistatic limit neglects displacement current locally compared with conduction; it does not invalidate the full Maxwell law. At sufficiently high frequency, conductivity and permittivity are frequency dependent, so a constant-

σ

approximation must be replaced by a material model.

In an unmagnetized cold plasma, electron inertia gives a cutoff. With immobile ions and negligible collisions,

m_{e} \dot{v} = - e E

and

J = - n_{e} e v

give

σ (ω) = i n_{e} e^{2} / (m_{e} ω)

ϵ_{eff} = ϵ_{0} (1 - \frac{ω _{p}^{2}}{ω ^{2}}), ω_{p}^{2} = \frac{n _{e} e ^{2}}{m _{e} ϵ _{0}}, ω^{2} = ω_{p}^{2} + c^{2} k^{2} .

Transverse waves propagate above

ω_{p}

and are evanescent below it. The group velocity is

c 1 - ω_{p}^{2} / ω^{2} < c

even though the phase velocity exceeds

c

ω_{p}

, distinct from transverse light waves. Collisions add damping, and pressure or magnetic fields modify these simple dispersions. To avoid carrier-current double counting in effective permittivity, once the carrier response is included in

ϵ_{eff}

it must not be counted a second time as a separate conductivity. Displacement current therefore connects circuit continuity, vacuum radiation, conductor screening and plasma propagation within one consistent field theory.^[ref]