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The relative permittivity of a medium is related to its electric susceptibility, χ e, as ε r (ω) = 1 + χ e. In anisotropic media (such as non cubic crystals) the relative permittivity is a second rank tensor. The relative permittivity of a material for a frequency of zero is known as its static relative permittivity.
If a dielectric material is a linear dielectric, then electric susceptibility 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 [3] [4] =, where
The linear permittivity of a homogeneous material is usually given relative to that of free space, as a relative permittivity ε r (also called dielectric constant, although this term is deprecated and sometimes only refers to the static, zero-frequency relative permittivity).
Here ε is known as the relative permittivity tensor or dielectric tensor. Consequently, the refractive index of the medium must also be a tensor. Consider a light wave propagating along the z principal axis polarised such the electric field of the wave is parallel to the x-axis. The wave experiences a susceptibility χ xx and a permittivity ε ...
In terms of relative permeability, the magnetic susceptibility is χ m = μ r − 1. {\displaystyle \chi _{m}=\mu _{r}-1.} The number χ m is a dimensionless quantity , sometimes called volumetric or bulk susceptibility, to distinguish it from χ p ( magnetic mass or specific susceptibility) and χ M ( molar or molar mass susceptibility).
Thus = (+) = = where ε r = 1 + χ is the relative permittivity of the material, and ε is the permittivity. In linear, homogeneous, isotropic media, ε is a constant. However, in linear anisotropic media it is a tensor , and in nonhomogeneous media it is a function of position inside the medium.
The refractive index of electromagnetic radiation equals =, where ε r is the material's relative permittivity, and μ r is its relative permeability. [ 47 ] : 229 The refractive index is used for optics in Fresnel equations and Snell's law ; while the relative permittivity and permeability are used in Maxwell's equations and electronics.
where R is the molar refractivity, is the Avogadro constant, is the electronic polarizability, p is the density of molecules, M is the molar mass, and = / is the material's relative permittivity or dielectric constant (or in optics, the square of the refractive index).