# Material equations of electrodynamics

The material equations describe the effects of external electromagnetic fields on matter within the framework of the theory of electrodynamics . For media at rest they consist of the equations

SI units cgs units
${\ displaystyle {\ vec {D}} = \ varepsilon _ {0} {\ vec {E}} + {\ vec {P}}}$ ${\ displaystyle {\ vec {D}} = {\ vec {E}} + 4 \ pi {\ vec {P}}}$
${\ displaystyle {\ vec {H}} = {\ frac {1} {\ mu _ {0}}} {\ vec {B}} - {\ vec {M}}}$ ${\ displaystyle {\ vec {H}} = {\ vec {B}} - 4 \ pi {\ vec {M}}}$

which link the microscopic with the macroscopic Maxwell equations , and the material dependencies for the polarization and magnetization listed below , which can be expressed in a frequently encountered approximate form and a more general form. ${\ displaystyle {\ vec {P}}}$ ${\ displaystyle {\ vec {M}}}$

The electric flux density and the magnetic field strength are only auxiliary fields that were introduced in order to be able to maintain the structure of the Maxwell equations of the vacuum in matter. The physically relevant measurands are the electric field strength and the magnetic flux density . ${\ displaystyle {\ vec {D}}}$ ${\ displaystyle {\ vec {H}}}$ ${\ displaystyle {\ vec {E}}}$ ${\ displaystyle {\ vec {B}}}$

## Derivation and explanation

The material equations arise from the microscopic Maxwell equations using the following approach:

1. Charges are considered to be the sum of free and electrically induced charges (polarization charges). Polarization charges are sources of the polarization field. (Magnetically induced charges do not occur.)
2. Currents are viewed as the sum of free and electrically or magnetically induced currents. Changes in the polarization field or eddies in the magnetization field cause induced currents.

The following macroscopic Maxwell equations contain only averaged quantities, i.e. H. the sizes may vary locally. A macroscopic measurement always means an averaging both over time and place (microscopic fluctuations are smoothed). This experimental inadequacy justifies the approach of using only averaged quantities. The following is based on the microscopic equations and then inferred from the macroscopic equations without explicitly specifying the averaging processes. A possible averaging looks like this (spatial averaging):

${\ displaystyle {\ overline {u \ left ({\ vec {r}}, t \ right)}} = {\ frac {1} {V}} \ int \ limits _ {V} {u \ left ({ \ vec {r}} + {\ vec {r}} \, ', t \ right) \ mathrm {d}} ^ {3} r'}$

${\ displaystyle u \ left ({\ vec {r}}, t \ right)}$be the microscopic quantity (can be scalar, like charge density, or vector, like electric field). The size integrating over a volume of space around , which is microscopic in size, but macroscopically small. A volume of (1/10 mm) 3 contains a huge number of particles (of the order of 10 16 particles). With such large numbers of particles, the purely spatial averaging also smooths out the temporal fluctuations. ${\ displaystyle V}$${\ displaystyle {\ vec {r}}}$

It should be noted that the macroscopic Maxwell equations cannot be formulated in Lorentz covariant fashion , since they are only valid in the inertial system in which matter rests on average.

From Maxwell's equations, the law of induction and the magnetic monopole prohibition continue to apply unchanged in matter (here with averaged fields):

${\ displaystyle {\ vec {\ nabla}} \ times {\ vec {E}} = - {\ frac {\ partial} {\ partial t}} {\ vec {B}}}$     and     ${\ displaystyle {\ vec {\ nabla}} \ cdot {\ vec {B}} = 0}$

### Gauss law: electrical flux density

Matter usually consists of more or less mobile, electrically charged particles (charges). These can e.g. B. be the negatively charged electrons of the atomic shell and the positively charged nuclei of the atoms forming matter. An electric field causes an electric force on this , which shifts the opposite charges from their equilibrium positions against each other. The material is thereby polarized (dipole and higher moments are created) and in turn generates an electric field that is superimposed on the external one. The sources of the resulting E-field are the free charges (also called excess charges, such as the quasi-free conduction electrons of a metallic conductor, generate the external electric field) and the bound charges (also polarization charges). So the total charge density is . ${\ displaystyle {\ vec {E}}}$${\ displaystyle \ rho = \ rho _ {\ text {free}} + \ rho _ {\ text {P}}}$

${\ displaystyle \ varepsilon _ {0} {\ vec {\ nabla}} \ cdot {\ vec {E}} = \ rho _ {\ text {free}} + \ rho _ {\ text {P}}}$

The polarization is introduced as a dipole density (mean electric dipole moment per volume), the sources of which are the polarization charges. The sum of the polarization charges of a body results in zero, hence also a neutral component. However, the charge distribution differs from zero locally, especially on the surface of the body (surface charge density): ${\ displaystyle {\ vec {P}}}$

${\ displaystyle \ rho _ {\ text {P}} = - {\ vec {\ nabla}} \ cdot {\ vec {P}}}$

Polarization causes an additional internal electric field , the generated with the outer, by the free charges field superimposed: . It is limited in the following to the external field: . ${\ displaystyle {\ vec {E}} _ {\ text {P}} = - {\ vec {P}} / \ varepsilon _ {0}}$${\ displaystyle {\ vec {E}} _ {\ text {free}}}$${\ displaystyle {\ vec {E}} = {\ vec {E}} _ {\ text {free}} + {\ vec {E}} _ {\ text {P}}}$${\ displaystyle {\ vec {E}} _ {\ text {free}} = {\ vec {E}} - {\ vec {E}} _ {\ text {P}} = {\ vec {E}} + {\ vec {P}} / \ varepsilon _ {0}}$

The macroscopic Maxwell equation only contains the free charges as sources:

${\ displaystyle {\ vec {\ nabla}} \ cdot \ underbrace {\ left (\ varepsilon _ {0} {\ vec {E}} + {\ vec {P}} \ right)} _ {\ vec {D }} = \ rho _ {\ text {free}} \ quad \ Rightarrow \ quad {\ vec {\ nabla}} \ cdot {\ vec {D}} = \ rho _ {\ text {free}}}$

The dielectric displacement field or electric flux density arises from the superposition of the electric field and the polarization field : ${\ displaystyle {\ vec {D}}}$

${\ displaystyle {\ vec {D}} = \ varepsilon _ {0} {\ vec {E}} + {\ vec {P}}}$

### Flux law: magnetic field strength

Electrons, atomic nuclei and the atoms and molecules made up of them each carry magnetic moments, comparable atomically small magnets (classic illustration with Bohr's atomic model: atomic electrons move on stationary circular orbits around the core. This circular current generates a magnetic moment perpendicular to the plane of the orbit.) of the moments are statistically distributed without an external field and are compensated on average. However, they can be aligned by an external magnetic induction, which creates an additional internal field that overlaps the external: the material magnetizes .

An external magnetic field (more precisely: magnetic flux density) generates not only free currents from unbound charge carriers, such as the quasi-free conduction electrons of a metallic conductor, but also magnetizing currents from bound charge carriers . These in turn generate the macroscopic magnetization field , which represents an average magnetic dipole moment per volume: ${\ displaystyle {\ vec {B}}}$${\ displaystyle {\ vec {J}} _ {\ text {free}}}$${\ displaystyle {\ vec {J}} _ {\ text {M}}}$${\ displaystyle {\ vec {M}}}$

${\ displaystyle {\ vec {J}} _ {\ text {M}} = {\ vec {\ nabla}} \ times {\ vec {M}}}$

Furthermore, there are so-called polarization currents that result from an electrical polarization that changes over time (electrically induced current): ${\ displaystyle {\ vec {P}}}$

${\ displaystyle {\ vec {J}} _ {\ text {P}} = {\ frac {\ partial {\ vec {P}}} {\ partial t}}}$

The total current density is made up of three components, which are all coupled together with the external magnetic field: ${\ displaystyle {\ vec {J}}}$

${\ displaystyle {\ vec {J}} = {\ vec {J}} _ {\ text {free}} + {\ vec {J}} _ {\ text {M}} + {\ vec {J}} _ {\ text {P}}}$

The flooding law is initially:

${\ displaystyle {\ frac {1} {\ mu _ {0}}} {\ vec {\ nabla}} \ times {\ vec {B}} = {\ vec {J}} _ {\ text {free} } + {\ vec {J}} _ {\ text {M}} + {\ vec {J}} _ {\ text {P}} + \ varepsilon _ {0} {\ frac {\ partial {\ vec { E}}} {\ partial t}}}$

This gives the macroscopic Maxwell equation:

${\ displaystyle {\ vec {\ nabla}} \ times \ underbrace {\ left ({\ frac {1} {\ mu _ {0}}} {\ vec {B}} - {\ vec {M}} \ right)} _ {\ vec {H}} = {\ vec {J}} _ {\ text {free}} + {\ frac {\ partial} {\ partial t}} \ underbrace {\ left (\ varepsilon _ {0} {\ vec {E}} + {\ vec {P}} \ right)} _ {\ vec {D}} \ quad \ Rightarrow \ quad {\ vec {\ nabla}} \ times {\ vec { H}} = {\ vec {J}} _ {\ text {free}} + {\ frac {\ partial} {\ partial t}} {\ vec {D}}}$

The magnetic field , also called magnetic field strength, is created from the superposition of the external magnetic field and the magnetizing field : ${\ displaystyle {\ vec {H}}}$

${\ displaystyle {\ vec {H}} = {\ frac {1} {\ mu _ {0}}} {\ vec {B}} - {\ vec {M}}}$

### Current density and conductivity

An electric field drives a flow of free charge carriers, the electric current , in electrical conductors . The electrical current density is determined by the electrical conductivity . ${\ displaystyle {\ vec {J}}}$ ${\ displaystyle \ sigma}$

## Material dependencies

### General form

The polarization and magnetization depend on the microscopic structure of the material. For a closer look one would have to use quantum mechanics or quantum statistics . In electrodynamics, more phenomenological approaches are used, which are coordinated with the experiment.

In general, polarization and magnetization are functionals of the fields, in the case of conductive materials also the current density:

${\ displaystyle {\ vec {P}} ({\ vec {r}}, t) = P \ left [{\ vec {E}} ({\ vec {r}} \, ', t \,') \ right] \ ,, \ quad {\ vec {M}} ({\ vec {r}}, t) = M \ left [{\ vec {B}} ({\ vec {r}} \, ', t \, ') \ right] \ ,, \ quad {\ vec {J}} ({\ vec {r}}, t) = J \ left [{\ vec {E}} ({\ vec {r} } \, ', t \,') \ right]}$

here must always apply for reasons of causality . ${\ displaystyle t \ geq t \, '}$

The material dependence of the polarization is described by the electrical susceptibility : ${\ displaystyle {\ vec {P}}}$ ${\ displaystyle {\ hat {\ chi}} _ {\ mathrm {e}}}$

${\ displaystyle {\ vec {P}} ({\ vec {r}}, t) = \ varepsilon _ {0} \ int d ^ {3} {\ vec {r}} \, '\ int _ {- \ infty} ^ {t} dt \, '\; {\ hat {\ chi}} _ {\ mathrm {e}} ({\ vec {r}}, {\ vec {r}} \,', t , t \, '; {\ vec {E}}) \, {\ vec {E}} ({\ vec {r}} \,', t \, ')}$

Analogous to the electrical case, the material dependence of the magnetization is described by the magnetic susceptibility : ${\ displaystyle {\ vec {M}}}$${\ displaystyle {\ hat {\ zeta}} _ {\ mathrm {m}}}$

${\ displaystyle {\ vec {M}} ({\ vec {r}}, t) = {\ frac {1} {\ mu _ {0}}} \ int d ^ {3} {\ vec {r} } \, '\ int _ {- \ infty} ^ {t} dt \,' \; {\ hat {\ zeta}} _ {\ mathrm {m}} ({\ vec {r}}, {\ vec {r}} \, ', t, t \,'; {\ vec {B}}) \, {\ vec {B}} ({\ vec {r}} \, ', t \,')}$

The size does not quite correspond to the definition of magnetic susceptibility , as it is common outside of physics in particular. The definition given here is, however, physically more sensible for the general case; in the approximated, simplified representation, both definitions coincide (see below). ${\ displaystyle {\ hat {\ zeta}} _ {\ mathrm {m}}}$ ${\ displaystyle {\ hat {\ chi}} _ {\ mathrm {m}}}$

For materials that conduct electrical current, the generalized Ohm's law with electrical conductivity applies : ${\ displaystyle {\ hat {\ sigma}}}$

${\ displaystyle {\ vec {J}} ({\ vec {r}}, t) = \ int d ^ {3} {\ vec {r}} \, '\ int _ {- \ infty} ^ {t } dt \, '\; {\ hat {\ sigma}} ({\ vec {r}}, {\ vec {r}} \,', t, t \, '; {\ vec {E}}) \, {\ vec {E}} ({\ vec {r}} \, ', t \,')}$

Under time reversal are and even, but , and odd. Polarization and magnetization are therefore compatible with time reversal and thus describe reversible processes. Ohm's law is not invariant under time reversal and thus describes irreversible processes: The field energy of the electric field is converted into kinetic energy of the charges, which is partially transferred to the material as Joule heat through impacts . This leads to an increase in the entropy of the material and this is not reversible according to the 2nd law of thermodynamics . ${\ displaystyle {\ vec {P}}}$${\ displaystyle {\ vec {E}}}$${\ displaystyle {\ vec {M}}}$${\ displaystyle {\ vec {B}}}$${\ displaystyle {\ vec {J}}}$

These general material dependencies are valid for non-linear, anisotropic and spatially and temporally inhomogeneous media.

• Non-linear behavior of the medium means that the susceptibilities are dependent on the fields or , see also non-linear optics${\ displaystyle {\ vec {E}}}$${\ displaystyle {\ vec {B}}}$
• If the medium is anisotropic, the susceptibilities must be regarded as tensors (for example in crystals).
• If the reaction of the medium depends not only on the time of observation , but also on the history of the material, i.e. a previous point in time , then it is a question of temporal inhomogeneity (see also hysteresis ).${\ displaystyle t}$${\ displaystyle t '}$
• Spatial inhomogeneity means that the reaction of the medium is not the same everywhere, but can change from point to point ( e.g. material with Weiss domains (magnetism), layered structures, but strictly speaking any spatially limited material).
• Time dependency leads to dispersion .

### Simplified form

In many applications, however, approximations for these complex relationships can be justified, and one often finds the following simplified representation for linear , spatially and temporally homogeneous media:

${\ displaystyle {\ vec {D}} = \ varepsilon {\ vec {E}}}$
${\ displaystyle {\ vec {H}} = {\ frac {1} {\ mu}} {\ vec {B}}}$
${\ displaystyle {\ vec {J}} = \ sigma {\ vec {E}}}$

The last equation represents Ohm's law with electrical conductivity . ${\ displaystyle \ sigma}$

As a rule, however, at least the time dependency ( frequency dispersion ) cannot be neglected, so that the quantities i. A. Functions of the frequency of the corresponding electromagnetic fields are. Furthermore, these quantities have a tensor character in non-isotropic media . ${\ displaystyle \ varepsilon, \ mu, \ sigma}$

#### Explanation

The material dependence of the polarization is described by the electrical susceptibility . In the linear, homogeneous approximation, the integral is omitted and the relationship is simplified to a multiplication: ${\ displaystyle {\ vec {P}}}$ ${\ displaystyle \ chi _ {\ mathrm {e}}}$

${\ displaystyle {\ vec {P}} = \ varepsilon _ {0} \ chi _ {e} {\ vec {E}}}$     results in:
${\ displaystyle {\ vec {D}} = \ varepsilon _ {0} {\ vec {E}} + {\ vec {P}} = \ varepsilon _ {0} \ underbrace {\ left (1+ \ chi _ {e} \ right)} _ {\ varepsilon _ {r}} {\ vec {E}} = \ underbrace {\ varepsilon _ {0} \ varepsilon _ {r}} _ {\ varepsilon} {\ vec {E }} = \ varepsilon {\ vec {E}}}$
${\ displaystyle \ varepsilon = \ varepsilon _ {0} \ varepsilon _ {r} \,}$  is the permittivity and     is the permittivity number.${\ displaystyle \ varepsilon _ {r} = 1 + \ chi _ {e} \,}$

Analogous to the electrical case, the material dependence of the magnetization is described by the magnetic susceptibility . Again, the relationship is simplified in the approximation: ${\ displaystyle {\ vec {M}}}$ ${\ displaystyle \ zeta _ {\ mathrm {m}}}$

${\ displaystyle {\ vec {M}} = {\ frac {1} {\ mu _ {0}}} \ zeta _ {\ mathrm {m}} {\ vec {B}}}$     results in:
${\ displaystyle {\ vec {H}} = {\ frac {1} {\ mu _ {0}}} {\ vec {B}} - {\ vec {M}} = {\ frac {1} {\ mu _ {0}}} \ underbrace {\ left (1- \ zeta _ {\ mathrm {m}} \ right)} _ {1 / \ mu _ {r}} {\ vec {B}} = {\ frac {1} {\ mu _ {0} \ mu _ {r}}} {\ vec {B}} = {\ frac {1} {\ mu}} {\ vec {B}}}$

However, for historical reasons it is found more often

${\ displaystyle {\ vec {B}} = \ mu _ {0} \ left ({\ vec {H}} + {\ vec {M}} \ right) = \ mu _ {0} \ underbrace {\ left (1+ \ chi _ {\ mathrm {m}} \ right)} _ {\ mu _ {r}} {\ vec {H}} = \ underbrace {\ mu _ {0} \ mu _ {r}} _ {\ mu} {\ vec {H}} = \ mu {\ vec {H}}}$
${\ displaystyle \ mu = \ mu _ {0} \ mu _ {r} \,}$  is the permeability and   permeability number .${\ displaystyle \ mu _ {r} = 1 + \ chi _ {m} \,}$

Both equations are equivalent to. The second form can only be set up in this approximation. ${\ displaystyle \ chi _ {\ mathrm {m}} = {\ frac {\ zeta _ {\ mathrm {m}}} {1- \ zeta _ {\ mathrm {m}}}}}$

## Material equations in moving media

If there is a constant relative movement between an observer and the surrounding, linear, isotropic and homogeneous medium, with the material constants μ ′ = μ ′ r ⋅μ 0 and ε ′ = ε ′ r ⋅ε 0 , the material equations must be expanded to include the constant the vacuum speed of light c 0 between different inertial systems into account. In contrast to Maxwell's equations, the material equations are not invariant to the Lorentz transformation . The deleted material constants relate to the moving system from the perspective of the stationary observer.

The field sizes involved are split into two components: If a general field size is used, then that field component is called that is normal to the velocity vector . describes the part that is parallel to the velocity vector. This results for the field components parallel to the movement: ${\ displaystyle {\ vec {F}}}$${\ displaystyle {\ vec {F}} _ {\ bot}}$${\ displaystyle {\ vec {v}}}$${\ displaystyle {\ vec {F}} _ {\ |}}$${\ displaystyle {\ vec {v}}}$

${\ displaystyle {\ vec {D}} _ {\ |} = \ varepsilon '{\ vec {E}} _ {\ |}}$
${\ displaystyle {\ vec {H}} _ {\ |} = {\ frac {1} {\ mu '}} {\ vec {B}} _ {\ |}}$

For the normal components there are more complicated expressions:

${\ displaystyle {\ vec {D _ {\ bot}}} = \ varepsilon '{\ frac {\ gamma ^ {2}} {n ^ {2}}} \ left ((n ^ {2} - \ beta ^ {2}) {\ vec {E _ {\ bot}}} + (n ^ {2} -1) {\ vec {v}} \ times {\ vec {B}} \ right)}$
${\ displaystyle {\ vec {H _ {\ bot}}} = {\ frac {\ gamma ^ {2}} {\ mu '}} \ left ((1-n ^ {2} \ beta ^ {2}) {\ vec {B _ {\ bot}}} + (n ^ {2} -1) {\ frac {{\ vec {v}} \ times {\ vec {E}}} {c_ {0} ^ {2 }}} \ right)}$

with the abbreviations:

${\ displaystyle \ beta = {\ frac {v} {c_ {0}}}}$,
${\ displaystyle \ gamma = {\ frac {1} {\ sqrt {1- \ beta ^ {2}}}}}$,

and the refractive index n :

${\ displaystyle n = c_ {0} {\ sqrt {\ mu '\ varepsilon'}}}$

It should be noted that with moving media, even with isotropic media, the vectors and and and are no longer parallel to each other. As a special case, with n = 1 as well as with the absolute value of the relative speed of v = 0, the additional terms disappear from the above equations, and the relationships shown in the introduction result. ${\ displaystyle {\ vec {D}}}$${\ displaystyle {\ vec {E}}}$${\ displaystyle {\ vec {H}}}$${\ displaystyle {\ vec {B}}}$

## credentials

• John David Jackson: Classical Electrodynamics . 4th, revised. Edition. de Gruyter, Berlin 2006, ISBN 978-3-11-018970-4 .
• Peter Halevi: Spatial dispersion in solids and plasmas . North-Holland, Amsterdam 1992, ISBN 978-0-444-87405-4 (English).
• Klaus Kark: Antennas and radiation fields - electromagnetic waves on lines, in free space and their radiation . 2nd Edition. Vieweg, 2006, ISBN 978-3-8348-0216-3 .