Maxwell's equationsFrom Wikipedia, the free encyclopediaFor thermodynamic relations, see Maxwell relations. For the history of the equations, see History of Maxwell's equations.

Electromagnetism

Electricity

Magnetism

Electrostatics [show]

Magnetostatics [show]

Electrodynamics [hide]

Lorentz force law

Electromagnetic induction

Faraday's law

Lenz's law

Displacement current

Maxwell's equations

Electromagnetic field

Electromagnetic radiation

Maxwell tensor

Poynting vector

Liénard–Wiechert potential

Jefimenko's equations

Eddy current

London equations

Electrical network [show]

Covariant formulation [show]

Scientists[show]

V

T

E

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies. Maxwell's equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents. They are named after the physicist and mathematician James Clerk Maxwell, who published an early form of those equations between 1861 and 1862.

The equations have two major variants. The "microscopic" set of Maxwell's equations uses total charge and total current, including the complicated charges and currents in materials at the atomic scale; it has universal applicability but may be infeasible to calculate. The "macroscopic" set of Maxwell's equations defines two new auxiliary fields that describe large-scale behaviour without having to consider these atomic scale details, but it requires the use of parameters characterizing the electromagnetic properties of the relevant materials.

The term "Maxwell's equations" is often used for other forms of Maxwell's equations. For example, space-time formulations are commonly used in high energy and gravitational physics. These formulations, defined on space-time rather than space and time separately, are manifestly [note 1]compatible with special and general relativity. In quantum mechanics and analytical mechanics, versions of Maxwell's equations based on theelectric and magnetic potentials are preferred.

Since the mid-20th century, it has been understood that Maxwell's equations are not exact but are a classical 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 or virtual photons.

Contents

  [hide] 

1 Formulation in terms of electric and magnetic fields 2 Conventional formulation in SI units 3 Relationship between differential and integral formulations

o 3.1 Flux and divergenceo 3.2 Circulation and curlo 3.3 Time evolution

4 Conceptual descriptionso 4.1 Gauss's lawo 4.2 Gauss's law for magnetismo 4.3 Faraday's lawo 4.4 Ampère's law with Maxwell's addition

5 Vacuum equations, electromagnetic waves and speed of light 6 "Microscopic" versus "macroscopic"

o 6.1 Bound charge and currento 6.2 Auxiliary fields, polarization and magnetizationo 6.3 Constitutive relations

7 Equations in Gaussian units 8 Alternative formulations 9 Solutions 10 Limitations for a theory of electromagnetism 11 Variations

o 11.1 Magnetic monopoles 12 See also 13 Notes 14 References 15 Historical publications 16 External links

o 16.1 Modern treatmentso 16.2 Other

Formulation in terms of electric and magnetic fields[edit]

The powerful and most widely familiar form of Maxwell's equations, whose formulation is due to Oliver Heaviside, in the vector calculus formalism, is used throughout unless otherwise explicitly stated.

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

The equations introduce the electric field E, a vector field, and the magnetic field B, a pseudovector field, where each generally have time-dependence. The sources of these fields are electric charges and electric currents, which can be expressed as local densities namely charge density ρ and current density J. 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.

In the electric-magnetic field formulation there are four equations. Two of them describe how the fields vary in space due to sources, if any; electric fields emanating from electric charges in Gauss's law, and magnetic fields as closed field lines not due to magnetic monopoles in Gauss's law for magnetism. The other two describe how the fields "circulate" around their respective sources; the magnetic field "circulates" around electric currents and time varying electric fields in Ampère's law with Maxwell's addition, while the electric field "circulates" around time varying magnetic fields in Faraday's law.

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.

Conventional formulation in SI units[edit]

The equations in this section are given in the convention used with SI units. Other units commonly used are Gaussian units based on the cgs system,[1] Lorentz–Heaviside units (used mainly inparticle physics), and Planck units (used in theoretical physics). See below for the formulation with Gaussian units.

Name Integral equations Differential equationsMeanin

g

Gauss's law

The electric

field

leaving a

volume is

proportional to

the charge inside.

Gauss's law for magnet

ism

There are

no magnetic

monopoles; the total

magnetic flux

piercing a closed surface is zero.

Maxwell–

Faraday

equation (

Faraday's law

of inducti

on)

The voltage accumul

ated around a closed

circuit is proportional to

the time rate of change of the

magnetic flux it encloses

.

Ampère's

circuital

law (wi

Electric currents

and changes

in

th Maxwe

ll's additio

n)

electric fields are

proportional to

the magnetic field

circulating about the area

they pierce.

where 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, a local description of the fields,

the nabla symbol ∇ denotes the three-dimensional gradient operator, and from it

the divergence operator is ∇· the curl operator is ∇×.

The sources are taken to be

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

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

In the integral equations, a description of the fields within a region of space,

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

Σ is any fixed open surface with boundary curve ∂Σ,

 is a surface integral over the surface ∂Ω (the oval indicates the surface is closed and not open),

 is a volume integral over the volume Ω,

 is a surface integral over the surface Σ,

 is a line integral around the curve ∂Σ (the circle indicates the curve is closed).

Here "fixed" means the volume or surface do not change in time. Although it is possible to formulate Maxwell's equations with time-dependent surfaces and volumes, this is not actually

necessary: the equations are correct and complete with time-independent surfaces. The sources are correspondingly the total amounts of charge and current within these volumes and surfaces, found by integration.

The volume integral of the total charge density ρ over any fixed volume Ω is the total electric charge contained in Ω:

where dV is the differential volume element, and

the net electric current is the surface integral of the electric current density J, passing through any open fixed surface Σ:

where dS denotes the differential vector element of surface area S normal to surface Σ. (Vector area is also denoted by A rather than S, but this conflicts with the magnetic potential, a separate vector field).

The "total charge or current" refers to including free and bound charges, or free and bound currents. These are used in the macroscopic formulation below.

Relationship between differential and integral formulations[edit]

The differential and integral formulations of the equations are mathematically equivalent, by the divergence theorem in the case of Gauss's law and Gauss's law for magnetism, and by the Kelvin–Stokes theorem in the case of Faraday's law and Ampère's law. Both the differential and integral formulations are useful. The integral formulation can often be used to simplify and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential formulation is a more natural starting point for calculating the fields in more complicated (less symmetric) situations, for example using finite element analysis.[2]

Flux and divergence[edit]

Closed volume Ω and its boundary ∂Ω, enclosing a source (+) and sink (−) of a

vector field F. Here, F could be the E field with source electric charges,

but not the Bfield which has no magnetic charges as shown. The outward unit

normal is n.

The "fields emanating from the sources" can be inferred from the surface integrals of the fields through the closed surface ∂Ω, defined as the electric flux

 and magnetic flux  , as well as their

respective divergences ∇ · E and ∇ · B. These surface integrals and divergences are connected by the divergence theorem.

Circulation and curl[edit]

Open surface Σ and boundary ∂Σ. Fcould be the E or B fields. Again, n is theunit

normal. (The curl of a vector field doesn't literally look like the "circulations", this is a

heuristic depiction).

The "circulation of the fields" can be interpreted from the line integrals of the fields around the closed curve ∂Σ:

where dℓ is the differential vector element of path length tangential to the path/curve, as well as their curls:

These line integrals and curls are connected by Stokes' theorem, and 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.

Time evolution[edit]

The "dynamics" or "time evolution of the fields" is due to the partial derivatives of the fields with respect to time:

These derivatives are crucial for the prediction of field propagation in the form of electromagnetic waves. Since the surface is taken to be time-independent, we can make the following transition in Faraday's law:

see differentiation under the integral sign for more on this result.

Conceptual descriptions[edit]

Gauss's law[edit]

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[edit]

Gauss's law for magnetism states that there are no "magnetic charges" (also called magnetic monopoles), analogous to electric charges.[3] 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 flux through any Gaussian surface is zero, or that the magnetic field is a solenoidal vector field.

Faraday's law[edit]

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.[3] 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 magnetcreates a changing magnetic field, which in turn generates an electric field in a nearby wire.

Ampère's law with Maxwell's addition[edit]

Magnetic core memory (1954) is an application of Ampère's law.

Each corestores one bit of data.

Ampère's law with Maxwell's addition states that magnetic fields can be generated in two ways: by electric 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 shows that not only does a changing magnetic field induce an electric field, but also a changing electric field induces a magnetic field.[3][4] 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.

Vacuum equations, electromagnetic waves and speed of light[edit]

Further information: Electromagnetic wave equation, Inhomogeneous electromagnetic wave equation and Sinusoidal plane-wave solutions of the electromagnetic wave equation

This 3D diagram shows a plane linearly polarized wave

propagating from left to right with the same wave equations

whereE = 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 εr and relative permeability μr, the phase velocity of light becomes

which is usually 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.

"Microscopic" versus "macroscopic"[edit]

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.

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

Name Integral equations Differential equations

Gauss's law

Gauss's law for

magnetism

Maxwell–Faraday equation

(Faraday's

law of induction)

Ampère's circuital law (with Maxwell's addition)

Unlike the "microscopic" equations, the "macroscopic" equations separate out the bound charge Qb and bound current Ib to obtain equations that depend only on the free charges Qf and currents If. This factorization can be made by splitting the total electric charge and current as follows:

Correspondingly, the total current density J splits into free Jf and