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Maxwell's equations (mid-left) as featured on a monument in front of Warsaw University'sCentre of New Technologies

Maxwell's equations are a set of partial differential equations that, together with theLorentz force law, form the foundation ofclassical electrodynamics, classical optics, andelectric circuits. They underpin all electric, optical and radio technologies such as power generation, electric motors, wirelesscommunication, cameras, televisions, computers etc. Maxwell's equations describe how electricand magnetic fields are generated by charges,currents and changes of each other. One important consequence of the equations is that fluctuating electric and magnetic fields can propagate at the speed of light. Thiselectromagnetic radiation manifests itself in manifold ways from radio waves to light and X- or γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations, and first proposed that light is an electromagnetic phenomenon.

The equations have two major variants. The microscopic Maxwell equations have universal applicability but may be infeasible to calculate with. They relate the electric and magnetic fields to total charge and total current, including the complicated charges and currents in materials at the atomic scale. The "macroscopic" Maxwell equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic scale details. However, their use requires experimentally determining parameters for a phenomenological description of the electromagnetic response of materials.

The term "Maxwell's equations" is often used forequivalent alternative formulations. Versions of Maxwell's equations based on the electric andmagnetic potentials are preferred for explicitly solving the equations as a boundary value problem,analytical mechanics, or for use in quantum mechanics. The space-time formulations (i.e., onspace-time rather than space and time separately), are commonly used in high energy and gravitational physics because they make the compatibility of the equations with special and general relativity manifest.^{[note 1]} In fact, historically,Einstein developed special and general relativity to accommodate the absolute speed of light that drops out of the Maxwell equations with the principle that only relative movement has physical consequences.

Since the mid-20th century, it has been understood that Maxwell's equations are not exact but are a classical field theory approximation to the more accurate and fundamental theory of quantum electrodynamics. In many situations, though, deviations from Maxwell's equations are immeasurably small. Exceptions include nonclassical light,photon-photon scattering, quantum optics, and many other phenomena related to photons orvirtual photons.

In the electric and magnetic field formulation there are four equations. The twoinhomogeneous equations describe how the fields vary in space due to sources. Gauss's law describes how electric fields emanate from electric charges. Gauss's law for magnetism describes magnetic fields as closed field lines not due to magnetic monopoles. The two homogeneous equations describe how the fields "circulate" around their respective sources. Ampère's law with Maxwell's addition describes how the magnetic field "circulates" around electric currents and time varying electric fields , whileFaraday's law describes how the electric field "circulates" around time varying magnetic fields.

A separate law of nature, the Lorentz force law, describes how the electric and magnetic field act on charged particles and currents. A version of this law was included in the original equations by Maxwell but, by convention, is no longer.

The precise formulation of Maxwell's equations depends on the precise definition of the quantities involved. Conventions differ with the unit systems, because various definitions and dimensions are changed by absorbing dimensionful factors like the speed of light c. This makes constants come out differently. The most common form is based on conventions used when quantities measured using SI units, but other commonly used conventions are used with other units including Gaussian units based on the cgs system,^[1] Lorentz–Heaviside units (used mainly in particle physics), and Planck units(used in theoretical physics).

The vector calculus formulation below has become standard. It is mathematically much more convenient than Maxwell's original 20 equations and is due to Oliver Heaviside^[2][3] The differential and integral equations formulations are mathematically equivalent and are both useful. The integral formulation relates fields within a region of space to fields on the boundary and can often be used to simplify and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential equations are purely local and are a more natural starting point for calculating the fields in more complicated (less symmetric) situations, for example using finite element analysis.^[4] For formulations using tensor calculus or differential forms, see alternative formulations. For relativistically invariant formulations, seerelativistic formulations.

Name	Integral equations	Differential equations	Meaning
Gauss's law	{displaystyle {scriptstyle partial Omega }} {displaystyle mathbf {E} cdot mathrm {d} mathbf {S} ={frac {1}{varepsilon _{0}}}iiint _{Omega }rho ,mathrm {d} V}	{displaystyle nabla cdot mathbf {E} ={frac {rho }{varepsilon _{0}}}}	The electric flux leaving a volume is proportional to the charge inside.
Gauss's law for magnetism	{displaystyle {scriptstyle partial Omega }} {displaystyle mathbf {B} cdot mathrm {d} mathbf {S} =0}	{displaystyle nabla cdot mathbf {B} =0}	There are nomagnetic monopoles; the total magnetic flux through a closed surface is zero.
Maxwell–Faraday equation (Faraday's law of induction)	{displaystyle oint _{partial Sigma }mathbf {E} cdot mathrm {d} {boldsymbol {ell }}=-{frac {mathrm {d} }{mathrm {d} t}}iint _{Sigma }mathbf {B} cdot mathrm {d} mathbf {S} }	{displaystyle nabla times mathbf {E} =-{frac {partial mathbf {B} }{partial t}}}	The voltage induced in a closed circuit is proportional to the rate of change of the magnetic flux it encloses.
Ampère's circuital law (with Maxwell's addition)	{displaystyle oint _{partial Sigma }mathbf {B} cdot mathrm {d} {boldsymbol {ell }}=mu _{0}iint _{Sigma }mathbf {J} cdot mathrm {d} mathbf {S} +mu _{0}varepsilon _{0}{frac {mathrm {d} }{mathrm {d} t}}iint _{Sigma }mathbf {E} cdot mathrm {d} mathbf {S} }	{displaystyle nabla times mathbf {B} =mu _{0}left(mathbf {J} +varepsilon _{0}{frac {partial mathbf {E} }{partial t}}right)}	The magnetic field induced around a closed loop is proportional to the electric current plus displacement current (rate of change of electric field) it encloses.

Gaussian units are a popular system of units, that are part of the centimetre–gram–second system of units (cgs). When using cgs units it is conventional to use a slightly different definition of electric field E_cgs = c⁻¹ E_SI. This implies that the modified electric and magnetic field have the same units (in the SI convention this is not the case making dimensional analysis of the equations different: e.g. for an electromagnetic wave in vacuum {displaystyle |mathbf {E} |_{mathrm {SI} }=|c|_{mathrm {SI} }|mathbf {B} |_{mathrm {SI} }}, ). The CGS system uses a unit of charge defined in such a way that the permittivity of the vacuum ε₀ = 1/4πc, hence μ₀ = 4π/c. These units are sometimes preferred over SI units in the context of special relativity,^[5]:vii since when using them, the components of the electromagnetic tensor, the Lorentz covariant object describing the electromagnetic field, have the same unit without constant factors. Using these different conventions, the Maxwell equations become:^[6]

Equations in Gaussian units convention

Name	Integral equations	Differential equations	Meaning
Gauss's law	{displaystyle {scriptstyle partial Omega }} {displaystyle mathbf {E} cdot mathrm {d} mathbf {S} =4pi iiint _{Omega }rho ,mathrm {d} V}	{displaystyle nabla cdot mathbf {E} =4pi rho }	The electric flux leaving a volume is proportional to the charge inside.
Gauss's law for magnetism	{displaystyle {scriptstyle partial Omega }} {displaystyle mathbf {B} cdot mathrm {d} mathbf {S} =0}	{displaystyle nabla cdot mathbf {B} =0}	There are nomagnetic monopoles; the total magnetic flux through a closed surface is zero.
Maxwell–Faraday equation (Faraday's law of induction)	{displaystyle oint _{partial Sigma }mathbf {E} cdot mathrm {d} {boldsymbol {ell }}=-{frac {1}{c}}{frac {d}{dt}}iint _{Sigma }mathbf {B} cdot mathrm {d} mathbf {S} }	{displaystyle nabla times mathbf {E} =-{frac {1}{c}}{frac {partial mathbf {B} }{partial t}}}	The voltage induced in a closed circuit is proportional to the rate of change of the magnetic flux it encloses.
Ampère's law (with Maxwell's extension)	{displaystyle oint _{partial Sigma }mathbf {B} cdot mathrm {d} {boldsymbol {ell }}={frac {1}{c}}left(4pi iint _{Sigma }mathbf {J} cdot mathrm {d} mathbf {S} +{frac {d}{dt}}iint _{Sigma }mathbf {E} cdot mathrm {d} mathbf {S} right)}	{displaystyle nabla times mathbf {B} ={frac {1}{c}}left(4pi mathbf {J} +{frac {partial mathbf {E} }{partial t}}right)}	The magnetic field integrated around a closed loop is proportional to the electric current plus displacement current (rate of change of electric field) it encloses.

Symbols in bold represent vector quantities, and symbols in italics represent scalarquantities, unless otherwise indicated.

The equations introduce the electric field, E, a vector field, and the magnetic field,B, a pseudovector field, each generally having a time and location dependence. The sources are

• the electric charge density (charge per unit volume), ρ, and

• the electric current density (current per unit area), J.

The universal constants appearing in the equations are

• the permittivity of free space, ε₀, and

• the permeability of free space, μ₀.

In the differential equations,

• the nabla symbol, ∇, denotes the three-dimensional gradient operator,

• the ∇⋅ symbol denotes the divergence operator,

• the ∇× symbol denotes the curl operator.

In the integral equations,

• Ω is any fixed volume with closed boundary surface ∂Ω, and

• Σ is any fixed surface with closed boundary curve ∂Σ,

Here a fixed volume or surface means that it does not change over time. The equations are correct, complete and a little easier to interpret with time-independent surfaces. However, since the surface is time-independent, we can bring the differentiation under the integral sign in Faraday's law:

The Maxwell's equations can be formulated with possibly time dependent surfaces and volumes by substituting the lefthand side with the righthand side in the integral equation version of the Maxwell equations.

• {displaystyle {scriptstyle partial Omega }} is a surface integral over the surface ∂Ω, (the loop indicates that the boundary surface is closed)

• {displaystyle iiint _{Omega }} is a volume integral over the volume Ω,

• {displaystyle iint _{Sigma }} is a surface integral over the surface Σ,

• {displaystyle oint _{partial Sigma }} is a line integral around the curve ∂Σ (the loop indicates that the boundary curve is closed).

• The volume integral over Ω of the total charge density ρ, is the total electric charge Q contained in Ω:

• The net electric current I is the surface integral of the electric current density Jpassing through a fixed surface, Σ:

The equivalence of the differential and integral formulations are a consequence of theGauss divergence theorem and the Kelvin–Stokes theorem.

Volume Ω and its closed boundary ∂Ω, containing (respectively enclosing) a source(+) and sink (−) of a vector field F. Here, F could be the Efield with source electric charges, but not the B field which has no magnetic charges as shown. The outward unit normal isn.

The "sources of the fields" (i.e. their divergence) can be determined from the surface integrals of the fields through the closed surface ∂Ω. E.g. theelectric flux is

{displaystyle {scriptstyle partial Omega }} {displaystyle mathbf {E} cdot mathrm {d} mathbf {S} =iiint _{Omega }nabla cdot mathbf {E} ,mathrm {d} V}

where the last equality uses the Gauss divergence theorem. Using the integral version of Gauss's equation we can rewrite this to

Since Ω can be chosen arbitrary, e.g. as an arbitrary small ball with arbitrary centre, this implies that the integrand must be zero, which is the differential equations formulation of Gauss equation up to a trivial rearrangement. Gauss's law for magnetism in differential equations form follows likewise from the integral form by rewriting the magnetic flux

{displaystyle {scriptstyle partial Omega }} {displaystyle mathbf {B} cdot mathrm {d} mathbf {S} =iiint _{Omega }nabla cdot mathbf {B} ,mathrm {d} V=0}

Surface Σ with closed boundary∂Σ. F could be the E or Bfields. Again, n is the unit normal. (The curl of a vector field doesn't literally look like the "circulations", this is a heuristic depiction).

The "circulation of the fields" (i.e their curls) can be determined from the line integrals of the fields around the closed curve ∂Σ. E.g. for the magnetic field

where we used the Kelvin-Stokes theorem. Using the modified Ampere law in integral form and the writing the time derivative of the flux as the surface integral of the partial time derivative of E we conclude that

Since Σ can be chosen arbitrary, e.g. as an arbitrary small, arbitrary oriented, and arbitrary centred disk, we conclude that the integrand must be zero. This is Ampere's modified law in differential equations form up to a trivial rearrangement. Likewise, the Faraday law in differential equations form follows from rewriting the integral form using the Kelvin-Stokes theorem.

The line integrals and curls are analogous to quantities in classical fluid dynamics: the circulation of a fluid is the line integral of the fluid's flow velocity field around a closed loop, and the vorticity of the fluid is the curl of the velocity field.

Gauss's law describes the relationship between a static electric field and the electric charges that cause it: The static electric field points away from positive charges and towards negative charges. In the field line description, electric field lines begin only at positive electric charges and end only at negative electric charges. 'Counting' the number of field lines passing through a closed surface, therefore, yields the total charge (including bound charge due to polarization of material) enclosed by that surface divided by dielectricity of free space (the vacuum permittivity). More technically, it relates the electric flux through any hypothetical closed "Gaussian surface" to the enclosed electric charge.

Gauss's law for magnetism: magnetic field lines never begin nor end but form loops or extend to infinity as shown here with the magnetic field due to a ring of current.

Gauss's law for magnetism states that there are no "magnetic charges" (also called magnetic monopoles), analogous to electric charges.^[7] Instead, the magnetic field due to materials is generated by a configuration called a dipole. Magnetic dipoles are best represented as loops of current but resemble positive and negative 'magnetic charges', inseparably bound together, having no net 'magnetic charge'. In terms of field lines, this equation states that magnetic field lines neither begin nor end but make loops or extend to infinity and back. In other words, any magnetic field line that enters a given volume must somewhere exit that volume. Equivalent technical statements are that the sum total magnetic fluxthrough any Gaussian surface is zero, or that the magnetic field is a solenoidal vector field.

In a geomagnetic storm, a surge in the flux of charged particles temporarily alters Earth's magnetic field, which induces electric fields in Earth's atmosphere, thus causing surges in electrical power grids. Artist's rendition; sizes are not to scale.

The Maxwell-Faraday's equation version of Faraday's law describes how a time varying magnetic fieldcreates ("induces") an electric field.^[7] This dynamically induced electric field has closed field lines just as the magnetic field, if not superposed by a static (charge induced) electric field. This aspect of electromagnetic induction is the operating principle behind many electric generators: for example, a rotating bar magnet creates a changing magnetic field, which in turn generates an electric field in a nearby wire.

Magnetic core memory (1954) is an application of Ampère's law. Each core stores one bit of data.

Ampère's law with Maxwell's addition states that magnetic fields can be generated in two ways: byelectric current (this was the original "Ampère's law") and by changing electric fields (this was "Maxwell's addition").

Maxwell's addition to Ampère's law is particularly important: it makes the set of equations mathematically consistent for non static fields, without changing the laws of Ampere and Gauss for static fields.^[8] However, as a consequence, it predicts that a changing magnetic field induces an electric field and vice versa.^[7][9] Therefore, these equations allow self-sustaining "electromagnetic waves" to travel through empty space (see electromagnetic wave equation).

The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents,^{[note 2]} exactly matches the speed of light; indeed,light is one form of electromagnetic radiation (as are X-rays, radio waves, and others). Maxwell understood the connection between electromagnetic waves and light in 1861, thereby unifying the theories of electromagnetism and optics.

The lefthand side of the modified Ampere's law has zero divergence by the div-curl-identity. Therefore the right handside, Gauss's law and interchanging derivatives give

By the Gauss divergence theorem that means that the rate of change of the charge in a fixed volume equals the current flowing in or out of the boundary

{displaystyle {frac {d}{dt}}Q_{Omega }={frac {d}{dt}}iiint _{Omega }rho mathrm {d} V=-} {displaystyle {scriptstyle partial Omega }} {displaystyle {mathbf {J}}cdot {rm {d}}{mathbf {S}}=-I_{partial Omega }.}

In particular, in an isolated system the total charge is conserved.

This 3D diagram shows a plane linearly polarized wave propagating from left to right with the same wave equations where E = E₀ sin(−ωt + k ⋅ r) andB = B₀ sin(−ωt + k ⋅ r)

In a region with no charges (ρ = 0) and no currents (J = 0), such as in a vacuum, Maxwell's equations reduce to:

Taking the curl (∇×) of the curl equations, and using the curl of the curl identity∇ × (∇ × X) = ∇(∇·X) − ∇²X we obtain the wave equations

which identify

with the speed of light in free space. In materials with relative permittivity, ε_r, andrelative permeability, μ_r, the phase velocity of light becomes

which is usually^{[note 3]} less than c.

In addition, E and B are mutually perpendicular to each other and the direction of wave propagation, and are in phase with each other. A sinusoidal plane wave is one special solution of these equations. Maxwell's equations explain how these waves can physically propagate through space. The changing magnetic field creates a changing electric field through Faraday's law. In turn, that electric field creates a changing magnetic field through Maxwell's addition to Ampère's law. This perpetual cycle allows these waves, now known as electromagnetic radiation, to move through space at velocity, c.

The microscopic variant of Maxwell's equation is the version given above. It expresses the electric E field and the magnetic B field in terms of the total charge and total current present, including the charges and currents at the atomic level. The "microscopic" form is sometimes called the "general" form of Maxwell's equations. The macroscopic variant of Maxwell's equation is equally general, however, with the difference being one of bookkeeping.

The "microscopic" variant is sometimes called "Maxwell's equations in a vacuum". This refers to the fact that the material medium is not built into the structure of the equation; it does not mean that space is empty of charge or current. They are also known as the "Maxwell-Lorentz equations". Lorentz tried to use these equations to predict the macroscopic properties of bulk matter from the physical behavior of its microscopic constituents.^[10]:5

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

Name	Integral equations (SI convention)	Differentialequations (SI convention)	Differential equations (Gaussian convention)
Gauss's law	{displaystyle {scriptstyle partial Omega }} {displaystyle mathbf {D} cdot mathrm {d} mathbf {S} =iiint _{Omega }rho _{text{f}},mathrm {d} V}	{displaystyle nabla cdot mathbf {D} =rho _{text{f}}}	{displaystyle nabla cdot mathbf {D} =4pi rho _{text{f}}}
Gauss's law for magnetism	{displaystyle {scriptstyle partial Omega }} {displaystyle mathbf {B} cdot mathrm {d} mathbf {S} =0}	{displaystyle nabla cdot mathbf {B} =0}	{displaystyle nabla cdot mathbf {B} =0}
Maxwell–Faraday equation (Faraday's law of induction)	{displaystyle oint _{partial Sigma }mathbf {E} cdot mathrm {d} {boldsymbol {ell }}=-{frac {d}{dt}}iint _{Sigma }mathbf {B} cdot mathrm {d} mathbf {S} }	{displaystyle nabla times mathbf {E} =-{frac {partial mathbf {B} }{partial t}}}	{displaystyle nabla times mathbf {E} =-{frac {1}{c}}{frac {partial mathbf {B} }{partial t}}}
Ampère's circuital law (with Maxwell's addition)	{displaystyle 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} }	{displaystyle nabla times mathbf {H} =mathbf {J} _{text{f}}+{frac {partial mathbf {D} }{partial t}}}	{displaystyle nabla times mathbf {H} ={frac {1}{c}}left(4pi mathbf {J} _{text{f}}+{frac {partial mathbf {D} }{partial t}}right)}

Unlike the "microscopic" equations, the "macroscopic" equations separate out the bound charge Q_b and bound current I_b to obtain equations that depend only on the free chargesQ_f and currents I_f. This factorization can be made by splitting the total electric charge and current as follows:

Correspondingly, the total current density J splits into free J_f and bound J_b components, and similarly the total charge density ρ splits into free ρ_f and bound ρ_b parts.

The cost of this factorization is that additional fields, the displacement field D and the magnetizing field H, are defined and need to be determined. Phenomenological constituent equations relate the additional fields to the electric field E and the magnetic B-field, often through a simple linear relation.

For a detailed description of the differences between the microscopic (total charge and current including material contributes or in air/vacuum)^{[note 4]} and macroscopic (freecharge and current; practical to use on materials) variants of Maxwell's equations, see below.

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 adielectric material its molecules respond by forming microscopic electric dipoles – their atomic nuclei move a tiny distance in the direction of the field, while theirelectrons move a tiny distance in the opposite direction. This produces amacroscopic 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 chargeon 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 P of the material, its dipole moment per unit volume. If P is uniform, a macroscopic separation of charge is produced only at the surfaces where P enters and leaves the material. For non-uniform P, a charge is also produced in the bulk.^[11]

Somewhat similarly, in all materials the constituent atoms exhibit magnetic moments that are intrinsically linked to the angular momentum of the components of the atoms, most notably their electrons. The 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 currents can be described using the magnetization M.^[12]

The very complicated and granular bound charges and bound currents, therefore, can be represented on the macroscopic scale in terms of P and 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.

The definitions (not constitutive relations) of the auxiliary fields are:

where P is the polarization field and M is the magnetization field which are defined in terms of microscopic bound charges and bound currents respectively. The macroscopic bound charge density ρ_b and bound current density J_b in terms of polarization P andmagnetization M are then defined as

If we define the total, bound, and free charge and current density by

and use the defining relations above to eliminate D, and H, the "macroscopic" Maxwell's equations reproduce the "microscopic" equations.

In order to apply 'Maxwell's macroscopic equations', it is necessary to specify the relations between displacement field D and the electric field E, as well as themagnetizing field H and the magnetic field B. Equivalently, we have to specify the dependence of the polarisation P (hence the bound charge) and the magnetisation M(hence the bound current) on the applied electric and magnetic field. The equations specifying this response are called constitutive relations. 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.^[13]:44–45

For materials without polarisation and magnetisation, the constitutive relations are (by definition)^[5]:2

where ε₀ is the permittivity of free space and μ₀ the 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 polarisation and magnetisation. More generally, for linear materials the constitutive relations are^[13]:44–45

where ε is the permittivity and μ the permeability of the material. For the displacement field 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 10¹¹ V/m are much higher than the external field. For the magnetizing field {displaystyle mathbf {H} }, however, the linear approximation can break down in common materials like iron leading to phenomena likehysteresis. Even the linear case can have various complications, however.

• For homogeneous materials, ε and μ are constant throughout the material, while for inhomogeneous materials they depend on location within the material (and perhaps time).^[14]:463

• For isotropic materials, ε and μ are scalars, while for anisotropic materials (e.g. due to crystal structure) they are tensors.^{[13]:421[14]:463}

• Materials are generally dispersive, so ε and μ depend on the frequency of any incident EM waves.^{[13]:625[14]:397}

Even more generally, in the case of non-linear materials (see for example nonlinear optics), D and P are not necessarily proportional to E, similarly H or M is not necessarily proportional to B. In general D and H depend on both E and 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 E and 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 Ohms law in the form