Editing Maxwell's equations (section)

== Macroscopic formulation ==
The above equations are the microscopic version of Maxwell's equations, expressing the electric and the magnetic fields in terms of the (possibly atomic-level) charges and currents present. This is sometimes called the "general" form, but the macroscopic version below is equally general, the difference being one of bookkeeping.

The microscopic version is sometimes called "Maxwell's equations in vacuum": this refers to the fact that the material medium is not built into the structure of the equations, but appears only in the charge and current terms. The microscopic version was introduced by Lorentz, who tried to use it to derive the macroscopic properties of bulk matter from its microscopic constituents.<ref name="MiltonSchwinger2006">{{cite book|author1=Kimball Milton|author2=J. Schwinger|title=Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators|date=18 June 2006|publisher=Springer Science & Business Media|isbn=978-3-540-29306-4}}</ref>{{rp|5}}

"Maxwell's macroscopic equations", also known as '''Maxwell's equations in matter''', are more similar to those that Maxwell introduced himself.

{| class="wikitable" 
|-
! scope="col" style="width: 15em;" | Name
! scope="col" | [[Integral]] equations<br/> (SI)
! scope="col" | [[Partial differential equation|Differential]] equations<br/> (SI)
! scope="col" | Differential equations<br/> (Gaussian system)
|-
| Gauss's law
| {{oiint
   | intsubscpt = <math>{\scriptstyle \partial \Omega }</math>
   | integrand  = <math>\mathbf{D}\cdot\mathrm{d}\mathbf{S} = \iiint_\Omega \rho_\text{f} \,\mathrm{d}V</math>
  }}
| <math>\nabla \cdot \mathbf{D} = \rho_\text{f}</math>
| <math> \nabla \cdot \mathbf{D} = 4\pi\rho_\text{f}</math>
|-
| Ampère–Maxwell law
| <math>
\begin{align}
\oint_{\partial \Sigma} & \mathbf{H} \cdot \mathrm{d}\boldsymbol{\ell} = \\
      & \iint_{\Sigma} \mathbf{J}_\text{f} \cdot \mathrm{d}\mathbf{S} + \frac{d}{dt} \iint_{\Sigma} \mathbf{D} \cdot \mathrm{d}\mathbf{S} \\
\end{align}
</math>
| <math>\nabla \times \mathbf{H} = \mathbf{J}_\text{f} + \frac{\partial \mathbf{D}} {\partial t}</math>
| <math> \nabla \times \mathbf{H} = \frac{1}{c} \left(4\pi\mathbf{J}_\text{f} + \frac{\partial \mathbf{D}} {\partial t} \right)</math>
|-
| Gauss's law for magnetism
| {{oiint
   | intsubscpt = <math>{\scriptstyle \partial \Omega }</math>
   | integrand  = <math>\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 0</math>
  }}
| <math>\nabla \cdot \mathbf{B} = 0</math>
| <math>\nabla \cdot \mathbf{B} = 0</math>
|-
| Maxwell–Faraday equation (Faraday's law of induction)
| <math>\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell}  = - \frac{d}{dt} \iint_{\Sigma} \mathbf B \cdot \mathrm{d}\mathbf{S} </math>
| <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
| <math>\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}</math>
|-
|}

In the macroscopic equations, the influence of bound charge {{math|''Q''<sub>b</sub>}} and bound current {{math|''I''<sub>b</sub>}} is incorporated into the [[electric displacement field|displacement field]] {{math|'''D'''}} and the [[magnetizing field]] {{math|'''H'''}}, while the equations depend only on the free charges {{math|''Q''<sub>f</sub>}} and free currents {{math|''I''<sub>f</sub>}}. This reflects a splitting of the total electric charge ''Q'' and current ''I'' (and their densities {{mvar|ρ}} and '''J''') into free and bound parts:
<math display="block">\begin{align}
  Q &= Q_\text{f} + Q_\text{b} = \iiint_\Omega \left(\rho_\text{f} + \rho_\text{b} \right) \, \mathrm{d}V = \iiint_\Omega \rho \,\mathrm{d}V, \\
  I &= I_\text{f} + I_\text{b} = \iint_\Sigma \left(\mathbf{J}_\text{f} + \mathbf{J}_\text{b} \right) \cdot \mathrm{d}\mathbf{S} = \iint_\Sigma \mathbf{J} \cdot \mathrm{d}\mathbf{S}.
\end{align}</math>

The cost of this splitting is that the additional fields {{math|'''D'''}} and {{math|'''H'''}} need to be determined through phenomenological constituent equations relating these fields to the electric field {{math|'''E'''}} and the magnetic field {{math|'''B'''}}, together with the bound charge and current.

See below for a detailed description of the differences between the microscopic equations, dealing with ''total'' charge and current including material contributions, useful in air/vacuum;<ref group="note" name="Effective_charge">In some books—e.g., in U. Krey and A. Owen's Basic Theoretical Physics (Springer 2007)—the term ''effective charge'' is used instead of ''total charge'', while ''free charge'' is simply called ''charge''.</ref>
and the macroscopic equations, dealing with ''free'' charge and current, practical to use within materials.

=== Bound charge and current ===
{{Main|Current density|Polarization density#Polarization density in Maxwell's equations|Magnetization#Magnetization current|l2=Bound charge|l3=Bound current}}
[[File:Polarization and magnetization.svg|thumb|300px|''Left:'' A schematic view of how an assembly of microscopic dipoles produces opposite surface charges as shown at top and bottom. ''Right:'' How an assembly of microscopic current loops add together to produce a macroscopically circulating current loop. Inside the boundaries, the individual contributions tend to cancel, but at the boundaries no cancelation occurs.]]

When an electric field is applied to a [[dielectric|dielectric material]] its molecules respond by forming microscopic [[electric dipole]]s – their [[atomic nucleus|atomic nuclei]] move a tiny distance in the direction of the field, while their [[electron]]s move a tiny distance in the opposite direction. This produces a ''macroscopic'' ''bound charge'' in the material even though all of the charges involved are bound to individual molecules. For example, if every molecule responds the same, similar to that shown in the figure, these tiny movements of charge combine to produce a layer of positive [[Bound charge#Bound charge|bound charge]] on one side of the material and a layer of negative charge on the other side. The bound charge is most conveniently described in terms of the [[polarization density|polarization]] {{math|'''P'''}} of the material, its dipole moment per unit volume. If {{math|'''P'''}} is uniform, a macroscopic separation of charge is produced only at the surfaces where {{math|'''P'''}} enters and leaves the material. For non-uniform {{math|'''P'''}}, a charge is also produced in the bulk.<ref>See {{cite book|author=David J. Griffiths|title=Introduction to Electrodynamics|url=https://archive.org/details/introductiontoel00grif_0|url-access=registration|edition=third|section=4.2.2|publisher=[[Prentice Hall]]|year=1999|isbn=9780138053260|author-link=David J. Griffiths}} for a good description of how {{math|'''P'''}} relates to the [[Bound charge#Bound charge|bound charge]].</ref>

Somewhat similarly, in all materials the constituent atoms exhibit [[magnetic moment|magnetic moments]] that are intrinsically linked to the [[gyromagnetic ratio|angular momentum]] of the components of the atoms, most notably their [[electron]]s. The [[magnetic field#Magnetic dipoles|connection to angular momentum]] suggests the picture of an assembly of microscopic current loops. Outside the material, an assembly of such microscopic current loops is not different from a macroscopic current circulating around the material's surface, despite the fact that no individual charge is traveling a large distance. These ''[[Bound current#Magnetization current|bound currents]]'' can be described using the [[magnetization]] {{math|'''M'''}}.<ref>See {{cite book|author=David J. Griffiths|title=Introduction to Electrodynamics|url=https://archive.org/details/introductiontoel00grif_0|url-access=registration|edition=third|section=6.2.2|publisher=[[Prentice Hall]]|year=1999|isbn=9780138053260}} for a good description of how {{math|'''M'''}} relates to the [[bound current]].</ref>

The very complicated and granular bound charges and bound currents, therefore, can be represented on the macroscopic scale in terms of {{math|'''P'''}} and {{math|'''M'''}}, which average these charges and currents on a sufficiently large scale so as not to see the granularity of individual atoms, but also sufficiently small that they vary with location in the material. As such, ''Maxwell's macroscopic equations'' ignore many details on a fine scale that can be unimportant to understanding matters on a gross scale by calculating fields that are averaged over some suitable volume.

=== Auxiliary fields, polarization and magnetization ===
The ''[[List of electromagnetism equations#Definitions|definitions]]'' of the auxiliary fields are:
<math display="block">\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}</math>
where {{math|'''P'''}} is the [[polarization density|polarization]] field and {{math|'''M'''}} is the [[magnetization]] field, which are defined in terms of microscopic bound charges and bound currents respectively. The macroscopic bound charge density {{math|''ρ''<sub>b</sub>}} and bound current density {{math|'''J'''<sub>b</sub>}} in terms of [[polarization density|polarization]] {{math|'''P'''}} and [[magnetization]] {{math|'''M'''}} are then defined as
<math display="block">\begin{align}
        \rho_\text{b} &= -\nabla\cdot\mathbf{P}, \\
  \mathbf{J}_\text{b} &= \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t}.
\end{align}</math>

If we define the total, bound, and free charge and current density by
<math display="block">\begin{align}
        \rho &= \rho_\text{b} + \rho_\text{f}, \\
  \mathbf{J} &= \mathbf{J}_\text{b} + \mathbf{J}_\text{f},
\end{align}</math>
and use the defining relations above to eliminate {{math|'''D'''}}, and {{math|'''H'''}}, the "macroscopic" Maxwell's equations reproduce the "microscopic" equations.

=== 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>