Editing Maxwell's equations (section)

=== Constitutive relations ===
{{main|Constitutive equation#Electromagnetism}}

In order to apply 'Maxwell's macroscopic equations', it is necessary to specify the relations between [[Electric displacement field|displacement field]] {{math|'''D'''}} and the electric field {{math|'''E'''}}, as well as the [[Magnetic field#H-field and magnetic materials|magnetizing]] field {{math|'''H'''}} and the magnetic field {{math|'''B'''}}. Equivalently, we have to specify the dependence of the polarization {{math|'''P'''}} (hence the bound charge) and the magnetization {{math|'''M'''}} (hence the bound current) on the applied electric and magnetic field. The equations specifying this response are called [[constitutive relation]]s. For real-world materials, the constitutive relations are rarely simple, except approximately, and usually determined by experiment. See the main article on constitutive relations for a fuller description.<ref name="Zangwill2013">{{cite book|author=Andrew Zangwill|title=Modern Electrodynamics|year=2013|publisher=Cambridge University Press|isbn=978-0-521-89697-9}}</ref>{{rp|44–45}}

For materials without polarization and magnetization, the constitutive relations are (by definition)<ref name=Jackson/>{{rp|2}}
<math display="block">\mathbf{D} = \varepsilon_0\mathbf{E}, \quad \mathbf{H} = \frac{1}{\mu_0}\mathbf{B},</math>
where {{math|''ε''<sub>0</sub>}} is the [[permittivity]] of free space and {{math|''μ''<sub>0</sub>}} the [[permeability (electromagnetism)|permeability]] of free space. Since there is no bound charge, the total and the free charge and current are equal.

An alternative viewpoint on the microscopic equations is that they are the macroscopic equations ''together'' with the statement that vacuum behaves like a perfect linear "material" without additional polarization and magnetization.
More generally, for linear materials the constitutive relations are<ref name="Zangwill2013"/>{{rp|44–45}}
<math display="block">\mathbf{D} = \varepsilon\mathbf{E}, \quad \mathbf{H} = \frac{1}{\mu}\mathbf{B},</math>
where {{math|''ε''}} is the [[permittivity]] and {{math|''μ''}} the [[permeability (electromagnetism)|permeability]] of the material. For the displacement field {{math|'''D'''}} the linear approximation is usually excellent because for all but the most extreme electric fields or temperatures obtainable in the laboratory (high power pulsed lasers) the interatomic electric fields of materials of the order of 10<sup>11</sup> V/m are much higher than the external field. For the magnetizing field <math>\mathbf{H}</math>, however, the linear approximation can break down in common materials like iron leading to phenomena like [[hysteresis]]. Even the linear case can have various complications, however.
* For homogeneous materials, {{math|''ε''}} and {{math|''μ''}} are constant throughout the material, while for inhomogeneous materials they depend on [[position vector|location]] within the material (and perhaps time).<ref name=Kittel2005>{{citation|last=Kittel|first=Charles|title=[[Introduction to Solid State Physics]]|publisher=John Wiley & Sons, Inc.|year=2005|location=USA|edition=8th|isbn=978-0-471-41526-8}}</ref>{{rp|463}}
* For isotropic materials, {{math|''ε''}} and {{math|''μ''}} are scalars, while for anisotropic materials (e.g. due to crystal structure) they are [[tensor]]s.<ref name="Zangwill2013"/>{{rp|421}}<ref name=Kittel2005/>{{rp|463}}
* Materials are generally [[dispersion (optics)|dispersive]], so {{math|''ε''}} and {{math|''μ''}} depend on the [[frequency]] of any incident EM waves.<ref name="Zangwill2013"/>{{rp|625}}<ref name=Kittel2005/>{{rp|397}}

Even more generally, in the case of non-linear materials (see for example [[nonlinear optics]]), {{math|'''D'''}} and {{math|'''P'''}} are not necessarily proportional to {{math|'''E'''}}, similarly {{math|'''H'''}} or {{math|'''M'''}} is not necessarily proportional to {{math|'''B'''}}. In general {{math|'''D'''}} and {{math|'''H'''}} depend on both {{math|'''E'''}} and {{math|'''B'''}}, on location and time, and possibly other physical quantities.

In applications one also has to describe how the free currents and charge density behave in terms of {{math|'''E'''}} and {{math|'''B'''}} possibly coupled to other physical quantities like pressure, and the mass, number density, and velocity of charge-carrying particles. E.g., the original equations given by Maxwell (see ''[[History of Maxwell's equations]]'') included [[Ohm's law]] in the form
<math display="block">\mathbf{J}_\text{f} = \sigma \mathbf{E}.</math>