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# Permittivity

In electromagnetism, the permittivity ε of a medium is the ratio D / E where D is the electric displacement in coulombs per square metre (C/m2) and E is the electric field strength in volts per metre (V/m). In the common case of an isotropic medium, D and E are parallel and ε is a scalar, but in more general anisotropic media this is not the case and ε is a rank-2 tensor (causing birefringence).

Permittivity is specified in farads per metre (F/m). It can also be defined as a dimensionless relative permittivity, or dielectric constant, normalized to the absolute vacuum permittivity ε0 = 8.854 10-12F/m.

When an electric field is applied, a current flows. The total current flowing in a real medium is in general made of two parts: a conduction current and a displacement one. A perfect dielectric is a material that shows displacement current only.

The permittivity ε and magnetic permeability μ of a medium together determine the velocity of electromagnetic radiation through that medium.

In a vacuum, these are given by

where μ0 is the magnetic constant, or permeability of free space, equal to 4π × 10-7 N·A-2, and c is the speed of light, 299,792,458 m/s.

In case of lossy medium (i.e. when the conduction currents are not negligible) the total current density flowing is:

where , σ is the conductivity (responsible for conduction current) of the medium and εd is the relative permittivity (responsible for displacement current).

In this formalism the complex permittivity ε* is defined as:

For realistic materials, both the real and imaginary parts of the permittivity are more complicated functions of frequency ω; since this leads to dispersion of signals containing multiple frequencies, such materials are called dispersive. This frequency dependence reflects the fact that a material's polarization does not respond instantaneously to an applied field—because the response must always be causal (come after the applied field), the dielectric function ε(ω) must have poles only for ω with positive imaginary parts, and ε(ω) therefore satisfies the Kramers-Kronig relations. However, in the narrow frequency ranges that are often studied in practice, the dielectric constants can often be approximated as frequency-independent.

At a given frequency, the imaginary part of ε leads to absorption loss if it is negative (in the above sign convention for frequency) and gain if it is positive. (More generally, one looks at the imaginary parts of the eigenvalues of the anisotropic dielectric tensor.)