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Electric susceptibility
 
 
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The electric susceptibility ?e of a dielectric
Dielectric

A dielectric is a nonconducting substance, i.e. an Insulator . The term was coined by William Whewell in response to a request from Michael Faraday....
 material is a measure of how easily it polarizes
Polarization density

In classical electromagnetism, the polarization density is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material....
 in response to an electric field
Electric field

In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field ....
. This, in turn, determines the electric permittivity
Permittivity

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 polarization in response to the field, and thereby reduce the total electric field inside the material....
 of the material and thus influences many other phenomena in that medium, from the capacitance of capacitors to the speed of light
Speed of light

The speed of light in an free space is an important physical constant usually written as c, with a value of 299,792,458 metres per second....
.

It is defined as the constant of proportionality (which may be a tensor
Tensor

A tensor is an object which extends the notion of Scalar , Vector , and Matrix . The term has slightly different meanings in mathematics and physics....
) relating an electric field
Electric field

In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field ....
 E to the induced dielectric
Dielectric

A dielectric is a nonconducting substance, i.e. an Insulator . The term was coined by William Whewell in response to a request from Michael Faraday....
 polarization density P such that

where is the electric permittivity
Permittivity

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 polarization in response to the field, and thereby reduce the total electric field inside the material....
 of free space.

The susceptibility of a medium is related to its relative permittivity
Permittivity

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 polarization in response to the field, and thereby reduce the total electric field inside the material....
  by

So in the case of a vacuum,

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

is, the polarization is a convolution
Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operator on two function s f and g, producing a third function that is typically viewed as a modified version of one of the original functions....
 of the electric field at previous times with time-dependent susceptibility given by .






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The electric susceptibility ?e of a dielectric
Dielectric

A dielectric is a nonconducting substance, i.e. an Insulator . The term was coined by William Whewell in response to a request from Michael Faraday....
 material is a measure of how easily it polarizes
Polarization density

In classical electromagnetism, the polarization density is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material....
 in response to an electric field
Electric field

In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field ....
. This, in turn, determines the electric permittivity
Permittivity

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 polarization in response to the field, and thereby reduce the total electric field inside the material....
 of the material and thus influences many other phenomena in that medium, from the capacitance of capacitors to the speed of light
Speed of light

The speed of light in an free space is an important physical constant usually written as c, with a value of 299,792,458 metres per second....
.

It is defined as the constant of proportionality (which may be a tensor
Tensor

A tensor is an object which extends the notion of Scalar , Vector , and Matrix . The term has slightly different meanings in mathematics and physics....
) relating an electric field
Electric field

In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field ....
 E to the induced dielectric
Dielectric

A dielectric is a nonconducting substance, i.e. an Insulator . The term was coined by William Whewell in response to a request from Michael Faraday....
 polarization density P such that

where is the electric permittivity
Permittivity

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 polarization in response to the field, and thereby reduce the total electric field inside the material....
 of free space.

The susceptibility of a medium is related to its relative permittivity
Permittivity

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 polarization in response to the field, and thereby reduce the total electric field inside the material....
  by

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
Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operator on two function s f and g, producing a third function that is typically viewed as a modified version of one of the original functions....
 of the electric field at previous times with time-dependent susceptibility given by . The upper limit of this integral can be extended to infinity as well if one defines for . An instantaneous response corresponds to Dirac delta function
Dirac delta function

The Dirac delta or Dirac's delta is a mathematics construct introduced by theoretical physicist Paul Dirac. Informally, it is a function representing an infinitely sharp peak bounding unit area: a function d that has the value 0 everywhere except at x = 0 where its value is infinity in such a way that its total integral is 1....
 susceptibility .

It is more convenient in a linear system to take the Fourier transform
Continuous Fourier transform

In mathematics, the Fourier transform is an operation that Transform one complex number-valued function of a real variable into another. The new function, often called the frequency domain representation of the original function, describes which frequencies are present in the original function....
 and write this relationship as a function of frequency. Due to the convolution theorem
Convolution theorem

In mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a convolution is the pointwise product of Fourier transforms....
, the integral becomes a simple product,

This frequency dependence of the susceptibility leads to frequency dependence of the permittivity. The shape of the susceptibility with respect to frequency characterizes the dispersion
Dispersion (optics)

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency.Media having such a property are termed dispersive media....
 properties of the material.

Moreover, the fact that the polarization can only depend on the electric field at previous times (i.e. for ), a consequence of causality
Causality

Causality denotes a necessary relationship between one event and another event which is the direct consequence of the first.While this informal understanding suffices in everyday use, the Philosophy analysis of how best to characterize causality extends over millennia....
, imposes Kramers–Kronig constraints on the susceptibility .

See also


  • Application of tensor theory in physics
    Application of tensor theory in physics

    Tensors are used in various parts of physics, both as abstract constructs in mathematical physics and for describing relations between physical quantity represented by matrices....
  • Magnetic susceptibility
    Magnetic susceptibility

    In electromagnetism the magnetic susceptibility is the degree of magnetization of a material in response to an applied magnetic field....
  • Maxwell's equations
    Maxwell's equations

    In electromagnetism, James Clerk Maxwell equations are a set of four partial differential equations that describe the properties of the electric field and magnetic field fields and relate them to their sources, charge density and current density....
  • Permittivity
    Permittivity

    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 polarization in response to the field, and thereby reduce the total electric field inside the material....
  • Clausius-Mossotti relation
    Clausius-Mossotti relation

    The Clausius?Mossotti equation is named after the Italian physicist Ottaviano-Fabrizio Mossotti, whose 1850 book analyzed the relationship between the dielectric constants of two different media, and the German physicist Rudolf Clausius, who gave the formula explicitly in his 1879 book in the context not of dielectric constants but of indices...
  • Linear response function
    Linear response function

    A linear response function describes the input-output relationshipof a signal transducer such as a radio turning electromagnetic waves into music...
  • Green–Kubo relations