Constitutive equations in electromagnetism and related areas In both
classical and
quantum physics, the precise dynamics of a system form a set of
coupled differential equations, which are almost always too complicated to be solved exactly, even at the level of
statistical mechanics. In the context of electromagnetism, this remark applies to not only the dynamics of free charges and currents (which enter Maxwell's equations directly), but also the dynamics of bound charges and currents (which enter Maxwell's equations through the constitutive relations). As a result, various approximation schemes are typically used. For example, in real materials, complex transport equations must be solved to determine the time and spatial response of charges, for example, the
Boltzmann equation or the
Fokker–Planck equation or the
Navier–Stokes equations. For example, see
magnetohydrodynamics,
fluid dynamics,
electrohydrodynamics,
superconductivity,
plasma modeling. An entire physical apparatus for dealing with these matters has developed. See for example,
linear response theory,
Green–Kubo relations and
Green's function (many-body theory). These complex theories provide detailed formulas for the constitutive relations describing the electrical response of various materials, such as
permittivities,
permeabilities,
conductivities and so forth. It is necessary to specify the relations between
displacement field D and
E, and the
magnetic H-field H and
B, before doing calculations in electromagnetism (i.e. applying Maxwell's macroscopic equations). These equations specify the response of bound charge and current to the applied fields and are called constitutive relations. Determining the constitutive relationship between the auxiliary fields
D and
H and the
E and
B fields starts with the definition of the auxiliary fields themselves: :\begin{align} \mathbf{D}(\mathbf{r}, t) &= \varepsilon_0 \mathbf{E}(\mathbf{r}, t) + \mathbf{P}(\mathbf{r}, t) \\ \mathbf{H}(\mathbf{r}, t) &= \frac{1}{\mu_0} \mathbf{B}(\mathbf{r}, t) - \mathbf{M}(\mathbf{r}, t), \end{align} where
P is the
polarization field and
M is the
magnetization field which are defined in terms of microscopic bound charges and bound current respectively. Before getting to how to calculate
M and
P it is useful to examine the following special cases.
Without magnetic or dielectric materials In the absence of magnetic or dielectric materials, the constitutive relations are simple: :\mathbf{D} = \varepsilon_0\mathbf{E} ,\quad \mathbf{H} = \mathbf{B}/\mu_0 where
ε0 and
μ0 are two universal constants, called the
permittivity of
free space and
permeability of free space, respectively.
Isotropic linear materials In an (
isotropic) linear material, where
P is proportional to
E, and
M is proportional to
B, the constitutive relations are also straightforward. In terms of the polarization
P and the magnetization
M they are: :\mathbf{P} = \varepsilon_0\chi_e\mathbf{E} ,\quad \mathbf{M} = \chi_m\mathbf{H}, where
χe and
χm are the
electric and
magnetic susceptibilities of a given material respectively. In terms of
D and
H the constitutive relations are: :\mathbf{D} = \varepsilon\mathbf{E} ,\quad \mathbf{H} = \mathbf{B}/\mu, where
ε and
μ are constants (which depend on the material), called the
permittivity and
permeability, respectively, of the material. These are related to the susceptibilities by: :\varepsilon/\varepsilon_0 = \varepsilon_r = \chi_e + 1 ,\quad \mu / \mu_0 = \mu_r = \chi_m + 1
General case For real-world materials, the constitutive relations are not linear, except approximately. Calculating the constitutive relations from first principles involves determining how
P and
M are created from a given
E and
B. These relations may be empirical (based directly upon measurements), or theoretical (based upon
statistical mechanics,
transport theory or other tools of
condensed matter physics). The detail employed may be
macroscopic or
microscopic, depending upon the level necessary to the problem under scrutiny. In general, the constitutive relations can usually still be written: :\mathbf{D} = \varepsilon\mathbf{E} ,\quad \mathbf{H} = \mu^{-1}\mathbf{B} but
ε and
μ are not, in general, simple constants, but rather functions of
E,
B, position and time, and tensorial in nature. Examples are: {{bulleted list D_i = \sum_j \varepsilon_{ij} E_j ,\quad B_i = \sum_j \mu_{ij} H_j. \begin{align} \mathbf{P}(\mathbf{r}, t) &= \varepsilon_0 \int {\rm d}^3 \mathbf{r}'{\rm d}t'\; \hat{\chi}_e \left(\mathbf{r}, \mathbf{r}', t, t'; \mathbf{E}\right)\, \mathbf{E}\left(\mathbf{r}', t'\right) \\ \mathbf{M}(\mathbf{r}, t) &= \frac{1}{\mu_0} \int {\rm d}^3 \mathbf{r}'{\rm d}t' \; \hat{\chi}_m \left(\mathbf{r}, \mathbf{r}', t, t'; \mathbf{B}\right)\, \mathbf{B}\left(\mathbf{r}', t'\right), \end{align} in which the permittivity and permeability functions are replaced by integrals over the more general
electric and
magnetic susceptibilities. In homogeneous materials, dependence on other locations is known as
spatial dispersion. }} As a variation of these examples, in general materials are
bianisotropic where
D and
B depend on both
E and
H, through the additional
coupling constants ξ and
ζ: : \mathbf{D}=\varepsilon \mathbf{E} + \xi \mathbf{H} \,,\quad \mathbf{B} = \mu \mathbf{H} + \zeta \mathbf{E}. In practice, some materials properties have a negligible impact in particular circumstances, permitting neglect of small effects. For example: optical nonlinearities can be neglected for low field strengths; material dispersion is unimportant when frequency is limited to a narrow
bandwidth; material absorption can be neglected for wavelengths for which a material is transparent; and
metals with finite conductivity often are approximated at
microwave or longer wavelengths as
perfect metals with infinite conductivity (forming hard barriers with zero
skin depth of field penetration). Some man-made materials such as
metamaterials and
photonic crystals are designed to have customized permittivity and permeability.
Calculation of constitutive relations The theoretical calculation of a material's constitutive equations is a common, important, and sometimes difficult task in theoretical
condensed-matter physics and
materials science. In general, the constitutive equations are theoretically determined by calculating how a molecule responds to the local fields through the
Lorentz force. Other forces may need to be modeled as well such as lattice vibrations in crystals or bond forces. Including all of the forces leads to changes in the molecule which are used to calculate
P and
M as a function of the local fields. The local fields differ from the applied fields due to the fields produced by the polarization and magnetization of nearby material; an effect which also needs to be modeled. Further, real materials are not
continuous media; the local fields of real materials vary wildly on the atomic scale. The fields need to be averaged over a suitable volume to form a continuum approximation. These continuum approximations often require some type of
quantum mechanical analysis such as
quantum field theory as applied to
condensed matter physics. See, for example,
density functional theory,
Green–Kubo relations and
Green's function. A different set of
homogenization methods (evolving from a tradition in treating materials such as
conglomerates and
laminates) are based upon approximation of an inhomogeneous material by a homogeneous
effective medium (valid for excitations with
wavelengths much larger than the scale of the inhomogeneity). The theoretical modeling of the continuum-approximation properties of many real materials often rely upon experimental measurement as well. For example,
ε of an insulator at low frequencies can be measured by making it into a
parallel-plate capacitor, and
ε at optical-light frequencies is often measured by
ellipsometry.
Thermoelectric and electromagnetic properties of matter These constitutive equations are often used in
crystallography, a field of
solid-state physics. ==Photonics==