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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).
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 equations take this form with the International System of Quantities. When dealing with only nondispersive isotropic linear materials, Maxwell's equations are often modified to ignore bound charges by replacing the permeability and permittivity of free space with the
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:
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
6 Equation for linear materials. 7 Relation to Coulomb's law. ... where ∇ · E is the divergence of the electric field, ε 0 is the vacuum permittivity and ...
For linear algebraic equations, one can make 'nice' rules to rewrite the equations and unknowns. The equations can be linearly dependent. But in differential equations, and especially partial differential equations (PDEs), one needs appropriate boundary conditions, which depend in not so obvious ways on the equations.
The derivation of the Helmholtz equation from the Maxwell's equations is an approximation as one neglects the spatial and temporal derivatives of the permittivity and permeability of the medium. A new formalism of light beam optics has been developed, starting with the Maxwell's equations in a matrix form: a single entity containing all the ...