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Constitutive relation

The E and D fields are related by the permittivity of the material, ε.

For linear, homogeneous, isotropic materials E and D are proportional and constant throughout the region, there is no position dependence: D (r) = ε E (r). {\displaystyle \mathbf {D} (\mathbf {r})=\varepsilon \mathbf {E} (\mathbf {r}).}

For inhomogeneous materials, there is a position dependence throughout the material: D (r) = ε (r) E (r) {\displaystyle \mathbf {D} (\mathbf {r})=\varepsilon (\mathbf {r})\mathbf {E} (\mathbf {r})}

For anisotropic materials the E and D fields are not parallel, and so E and D are related by the permittivity tensor (a 2nd order tensor field), in component form: D i = ε i j E j {\displaystyle D_{i}=\varepsilon _{ij}E_{j}}

For non-linear media, E and D are not proportional. Materials can have varying extents of linearity, homogeneity and isotropy.