The electric susceptibility χ_e of a dielectric material is a measure of how easily it polarizes in response to an electric field. This, in turn, determines the electric permittivity of the material and thus influences many other phenomena in that medium, from the capacitance of capacitors to the speed of light. A dielectric, or electrical insulator, is a substance that is highly resistant to electric current. ... In classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. ... It has been suggested that optical field be merged into this article or section. ... Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ... Various types of capacitors A capacitor is a device that stores energy in the electric field created between a pair of conductors on which equal but opposite electric charges have been placed. ... A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation...

It is defined as the constant of proportionality (which may be a tensor) relating an electric field E to the induced dielectric polarization density P such that In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... It has been suggested that optical field be merged into this article or section. ... A dielectric, or electrical insulator, is a substance that is highly resistant to electric current. ... In electrostatics, the polarization is the vector field that results from permanent or induced electric dipole moments in a dielectric material. ...

where is the electric permittivity of free space. Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ...

The susceptibility of a medium is related to its relative permittivity by Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ...

So in the case of a vacuum,

The electric displacement D is related to the polarization density P by

Dispersion and causality

In general, a material cannot polarize instantaneously in response to an applied field, and so the more general formulation as a function of time is

That is, the polarization is a convolution of the electric field at previous times with time-dependent susceptibility given by χ_e(Δt). The upper limit of this integral can be extended to infinity as well if one defines χ_e(Δt) = 0 for Δt < 0. An instantaneous response corresponds to Dirac delta function susceptibility χ_e(Δt) = δ(Δt). In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ...

It is more convenient in a linear system to take the Fourier transform and write this relationship as a function of frequency. Thanks to the convolution theorem, the integral disappears and one obtains In mathematics, the continuous Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the point-wise product of Fourier transforms. ...

This frequency dependence of the susceptibility leads to frequency dependence of the permittivity, which is sometimes known as material dispersion. Dispersion of a light beam in a prism. ...

Moreover, the fact that the polarization can only depend on the electric field at previous times (i.e. χ_e(Δt) = 0 for Δt < 0), a consequence of causality, imposes Kramers-Kronig constraints on the susceptibility χ_e(0). It has been suggested that this article be split into multiple articles accessible from a disambiguation page. ... In mathematics and physics, the Kramers-Kronig relations describe the relation between the real and imaginary part of a certain class of complex-valued functions. ...

See also

• Application of tensor theory in physics
• Magnetic susceptibility
• Maxwell's equations
• Permittivity
• Clausius-Mossotti relation