Contents 1 Formulation in terms of electric and magnetic fields (microscopic or in vacuum version) 1.1 Formulation in
1.3.1 Differential equations
1.3.2
2 Relationship between differential and integral formulations 2.1 Flux and divergence 2.2 Circulation and curl 3 Conceptual descriptions 3.1 Gauss's law
3.2
4 Charge conservation 5 Vacuum equations, electromagnetic waves and speed of light 6 Macroscopic formulation 6.1 Bound charge and current 6.2 Auxiliary fields, polarization and magnetization 6.3 Constitutive relations 7 Alternative formulations 8 Relativistic formulations 9 Solutions 10 Overdetermination of Maxwell's equations 11 Limitations of the Maxwell equations as a theory of electromagnetism 12 Variations 12.1 Magnetic monopoles 13 See also 14 Notes 15 References 16 Historical publications 17 External links 17.1 Modern treatments 17.2 Other Formulation in terms of electric and magnetic fields (microscopic or
in vacuum version)[edit]
In the electric and magnetic field formulation there are four
equations. The two inhomogeneous equations describe how the fields
vary in space due to sources.
Name
Meaning
Gauss's law The electric flux leaving a volume is proportional to the charge inside. ∂ Ω displaystyle scriptstyle partial Omega E ⋅ d S = 1 ε 0 ∭ Ω ρ d V displaystyle mathbf E cdot mathrm d mathbf S = frac 1 varepsilon _ 0 iiint _ Omega rho ,mathrm d V ∇ ⋅ E = ρ ε 0 displaystyle nabla cdot mathbf E = frac rho varepsilon _ 0
∂ Ω displaystyle scriptstyle partial Omega B ⋅ d S = 0 displaystyle mathbf B cdot mathrm d mathbf S =0 ∇ ⋅ B = 0 displaystyle nabla cdot mathbf B =0 Maxwell–Faraday equation (Faraday's law of induction) The voltage induced in a closed loop is proportional to the rate of change of the magnetic flux that the loop encloses. ∮ ∂ Σ E ⋅ d ℓ = − d d t ∬ Σ B ⋅ d S 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 ∇ × E = − ∂ B ∂ t displaystyle nabla times mathbf E =- frac partial mathbf B partial t
∮ ∂ Σ B ⋅ d ℓ = μ 0 ∬ Σ J ⋅ d S + μ 0 ε 0 d d t ∬ Σ E ⋅ d S 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 ∇ × B = μ 0 ( J + ε 0 ∂ E ∂ t ) displaystyle nabla times mathbf B =mu _ 0 left(mathbf J +varepsilon _ 0 frac partial mathbf E partial t right) Formulation in
E
S I =
c
S I
B
S I displaystyle mathbf E _ mathrm SI =c_ mathrm SI mathbf B _ mathrm SI ). The Gaussian 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
Name
Gauss's law ∂ Ω displaystyle scriptstyle partial Omega E ⋅ d S = 4 π ∭ Ω ρ d V displaystyle mathbf E cdot mathrm d mathbf S =4pi iiint _ Omega rho ,mathrm d V ∇ ⋅ E = 4 π ρ displaystyle nabla cdot mathbf E =4pi rho The electric flux leaving a volume is proportional to the charge inside.
∂ Ω displaystyle scriptstyle partial Omega B ⋅ d S = 0 displaystyle mathbf B cdot mathrm d mathbf S =0 ∇ ⋅ B = 0 displaystyle nabla cdot mathbf B =0 There are no magnetic monopoles; the total magnetic flux through a closed surface is zero. Maxwell–Faraday equation (Faraday's law of induction) ∮ ∂ Σ E ⋅ d ℓ = − 1 c d d t ∬ Σ B ⋅ d S 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 ∇ × E = − 1 c ∂ B ∂ t displaystyle nabla times mathbf E =- frac 1 c frac partial mathbf B partial t The voltage induced in a closed loop is proportional to the rate of change of the magnetic flux that the loop encloses.
∮ ∂ Σ B ⋅ d ℓ = 1 c ( 4 π ∬ Σ J ⋅ d S + d d t ∬ Σ E ⋅ d S ) 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) ∇ × B = 1 c ( 4 π J + ∂ E ∂ t ) 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) that the loop encloses. Key to the notation[edit] 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, each generally having a time and location dependence. The sources are the total electric charge density (total charge per unit volume), ρ, and the total electric current density (total 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. Differential equations[edit] 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.
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. For example, since the surface is time-independent, we can bring the differentiation under the integral sign in Faraday's law: d d t ∬ Σ B ⋅ d S = ∬ Σ ∂ B ∂ t ⋅ d S , displaystyle frac d dt iint _ Sigma mathbf B cdot mathrm d mathbf S =iint _ Sigma frac partial mathbf B partial t cdot mathrm d mathbf S ,,
∂ Ω displaystyle scriptstyle partial Omega is a surface integral over the boundary surface ∂Ω, with the loop indicating the surface is closed ∭ Ω displaystyle iiint _ Omega is a volume integral over the volume Ω, ∮ ∂ Σ displaystyle oint _ partial Sigma is a line integral around the boundary curve ∂Σ, with the loop indicating the curve is closed. ∬ Σ displaystyle iint _ Sigma is a surface integral over the surface Σ, The total electric charge Q enclosed in Ω is the volume integral over Ω of the charge density ρ (see the "macroscopic formulation" section below): Q = ∭ Ω ρ d V , displaystyle Q=iiint _ Omega rho mathrm d V, where dV is the volume element. The net electric current I is the surface integral of the electric current density J passing through a fixed surface, Σ: I = ∬ Σ J ⋅ d S , displaystyle I=iint _ Sigma mathbf J cdot mathrm d mathbf S , where dS denotes the vector element of surface area S, normal to
surface Σ. (
Relationship between differential and integral formulations[edit] The equivalence of the differential and integral formulations are a consequence of the Gauss divergence theorem and the Kelvin–Stokes theorem. Flux and divergence[edit] Volume Ω and its closed boundary ∂Ω, containing (respectively enclosing) a source (+) and sink (−) of a vector field F. Here, F could be the E field with source electric charges, but not the B field, which has no magnetic charges as shown. The outward unit normal is n. The "sources of the fields" (i.e. their divergence) can be determined from the surface integrals of the fields through the closed surface ∂Ω. I.e. the electric flux is ∂ Ω displaystyle scriptstyle partial Omega E ⋅ d S = ∭ Ω ∇ ⋅ E d V 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 ∭ Ω ( ∇ ⋅ E − ρ ϵ 0 ) d V = 0 displaystyle iiint _ Omega left(nabla cdot mathbf E - frac rho epsilon _ 0 right),mathrm d V=0 Since Ω can be chosen arbitrarily, e.g. as an arbitrary small ball
with arbitrary center, this implies that the integrand must be zero,
which is the differential equations formulation of Gauss equation up
to a trivial rearrangement.
∂ Ω displaystyle scriptstyle partial Omega B ⋅ d S = ∭ Ω ∇ ⋅ B d V = 0 displaystyle mathbf B cdot mathrm d mathbf S =iiint _ Omega nabla cdot mathbf B ,mathrm d V=0 . Circulation and curl[edit] Surface Σ with closed boundary ∂Σ. F could be the E or B fields. 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 ∮ ∂ Σ B ⋅ d ℓ = ∬ Σ ( ∇ × B ) ⋅ d S displaystyle oint _ partial Sigma mathbf B cdot mathrm d boldsymbol ell =iint _ Sigma (nabla times mathbf B )cdot mathrm d mathbf S , 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 ∬ Σ ( ∇ × B − μ 0 ( J + ϵ 0 ∂ E ∂ t ) ) ⋅ d S = 0 displaystyle iint _ Sigma left(nabla times mathbf B -mu _ 0 left(mathbf J +epsilon _ 0 frac partial mathbf E partial t right)right)cdot mathrm d mathbf S =0 . Since Σ can be chosen arbitrarily, e.g. as an arbitrary small,
arbitrary oriented, and arbitrary centered 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.
Conceptual descriptions[edit]
Gauss'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. (Not to scale.) The Maxwell–Faraday version of
0 = ∇ ⋅ ∇ × B = μ 0 ( ∇ ⋅ J + ε 0 ∂ ∂ t ∇ ⋅ E ) = μ 0 ( ∇ ⋅ J + ∂ ρ ∂ t ) displaystyle 0=nabla cdot nabla times mathbf B =mu _ 0 left(nabla cdot mathbf J +varepsilon _ 0 frac partial partial t nabla cdot mathbf E right)=mu _ 0 left(nabla cdot mathbf J + frac partial rho partial t right) i.e. ∂ ρ ∂ t + ∇ ⋅ J = 0 displaystyle frac partial rho partial t +nabla cdot mathbf J =0 . By the Gauss
d d t Q Ω = d d t ∭ Ω ρ d V = − displaystyle frac d dt Q_ Omega = frac d dt iiint _ Omega rho mathrm d V=- ∂ Ω displaystyle scriptstyle partial Omega J ⋅ d S = − I ∂ Ω . displaystyle mathbf J cdot rm d mathbf S =-I_ partial Omega . In particular, in an isolated system the total charge is conserved.
Vacuum equations, electromagnetic waves and speed of light[edit]
Further information: Electromagnetic wave equation, Inhomogeneous
electromagnetic wave equation, and
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) and B = B0 sin(−ωt + k ⋅ r) In a region with no charges (ρ = 0) and no currents (J = 0), such as
in a vacuum,
∇ ⋅ E = 0 ∇ × E = − ∂ B ∂ t , ∇ ⋅ B = 0 ∇ × B = 1 c 2 ∂ E ∂ t . displaystyle begin aligned nabla cdot mathbf E &=0quad &nabla times mathbf E &=- frac partial mathbf B partial t ,\nabla cdot mathbf B &=0quad &nabla times mathbf B &= frac 1 c^ 2 frac partial mathbf E partial t .end aligned Taking the curl (∇×) of the curl equations, and using the curl of the curl identity ∇ × (∇ × X) = ∇(∇·X) − ∇2X we obtain the wave equations 1 c 2 ∂ 2 E ∂ t 2 − ∇ 2 E = 0 1 c 2 ∂ 2 B ∂ t 2 − ∇ 2 B = 0 displaystyle begin aligned frac 1 c^ 2 frac partial ^ 2 mathbf E partial t^ 2 -nabla ^ 2 mathbf E =0\ frac 1 c^ 2 frac partial ^ 2 mathbf B partial t^ 2 -nabla ^ 2 mathbf B =0end aligned which identify c = 1 μ 0 ε 0 = 2.99792458 × 10 8 m/s displaystyle c= frac 1 sqrt mu _ 0 varepsilon _ 0 =2.99792458times 10^ 8 , text m/s with the speed of light in free space. In materials with relative permittivity, εr, and relative permeability, μr, the phase velocity of light becomes v p = 1 μ 0 μ r ε 0 ε r displaystyle v_ text p = frac 1 sqrt mu _ 0 mu _ text r varepsilon _ 0 varepsilon _ text r which is usually[note 3] less than c.
In addition, E and B are perpendicular to each other and to the
direction of wave propagation, and are in phase with each other. A
sinusoidal plane wave is one special solution of these equations.
Name
Gauss's law ∂ Ω displaystyle scriptstyle partial Omega D ⋅ d S = ∭ Ω ρ f d V displaystyle mathbf D cdot mathrm d mathbf S =iiint _ Omega rho _ text f ,mathrm d V ∇ ⋅ D = ρ f displaystyle nabla cdot mathbf D =rho _ text f ∇ ⋅ D = 4 π ρ f displaystyle nabla cdot mathbf D =4pi rho _ text f
∂ Ω displaystyle scriptstyle partial Omega B ⋅ d S = 0 displaystyle mathbf B cdot mathrm d mathbf S =0 ∇ ⋅ B = 0 displaystyle nabla cdot mathbf B =0 ∇ ⋅ B = 0 displaystyle nabla cdot mathbf B =0 Maxwell–Faraday equation (Faraday's law of induction) ∮ ∂ Σ E ⋅ d ℓ = − d d t ∬ Σ B ⋅ d S displaystyle oint _ partial Sigma mathbf E cdot mathrm d boldsymbol ell =- frac d dt iint _ Sigma mathbf B cdot mathrm d mathbf S ∇ × E = − ∂ B ∂ t displaystyle nabla times mathbf E =- frac partial mathbf B partial t ∇ × E = − 1 c ∂ B ∂ t displaystyle nabla times mathbf E =- frac 1 c frac partial mathbf B partial t
∮ ∂ Σ H ⋅ d ℓ = ∬ Σ J f ⋅ d S + d d t ∬ Σ D ⋅ d S 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 ∇ × H = J f + ∂ D ∂ t displaystyle nabla times mathbf H =mathbf J _ text f + frac partial mathbf D partial t ∇ × H = 1 c ( 4 π J f + ∂ D ∂ 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 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: Q = Q f + Q b = ∭ Ω ( ρ f + ρ b ) d V = ∭ Ω ρ d V I = I f + I b = ∬ Σ ( J f + J b ) ⋅ d S = ∬ Σ J ⋅ d S displaystyle begin aligned 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 aligned Correspondingly, the total current density J splits into free Jf and bound Jb components, and similarly the total charge density ρ splits into free ρf and bound ρb 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 (free charge and current; practical to use on materials) variants of Maxwell's equations, see below. Bound charge and current[edit] Main articles: Current density, Bound charge, and Bound current 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 material its molecules respond by forming microscopic electric dipoles – their atomic nuclei move a tiny distance in the direction of the field, while their electrons 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 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 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. Auxiliary fields, polarization and magnetization[edit] The definitions (not constitutive relations) of the auxiliary fields are: D ( r , t ) = ε 0 E ( r , t ) + P ( r , t ) H ( r , t ) = 1 μ 0 B ( r , t ) − M ( r , t ) displaystyle begin aligned 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 aligned 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 ρb and bound current density Jb in terms of polarization P and magnetization M are then defined as ρ b = − ∇ ⋅ P J b = ∇ × M + ∂ P ∂ t displaystyle begin aligned rho _ text b &=-nabla cdot mathbf P \mathbf J _ text b &=nabla times mathbf M + frac partial mathbf P partial t end aligned If we define the total, bound, and free charge and current density by ρ = ρ b + ρ f , J = J b + J f , displaystyle begin aligned rho &=rho _ text b +rho _ text f ,\mathbf J &=mathbf J _ text b +mathbf J _ text f ,end aligned and use the defining relations above to eliminate D, and H, the
"macroscopic"
D = ε 0 E , H = 1 μ 0 B displaystyle mathbf D =varepsilon _ 0 mathbf E ,quad mathbf H = frac 1 mu _ 0 mathbf B 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 D = ε E , H = 1 μ B displaystyle mathbf D =varepsilon mathbf E ,,quad mathbf H = frac 1 mu mathbf B 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 H displaystyle mathbf H , 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, 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 J f = σ E . displaystyle mathbf J _ text f =sigma mathbf E ,. Alternative formulations[edit] For an overview, see Mathematical descriptions of the electromagnetic field. For the equations in quantum field theory, see quantum electrodynamics. Following is a summary of some of the numerous other ways to write the microscopic Maxwell's equations, showing they can be formulated using different mathematical formalisms. In addition, we formulate the equations using "potentials". Originally they were introduced as a convenient way to solve the homogeneous equations, but it was originally thought that all the observable physics was contained in the electric and magnetic fields (or relativistically, the Faraday tensor). The potentials play a central role in quantum mechanics, however, and act quantum mechanically with observable consequences even when the electric and magnetic fields vanish (Aharonov–Bohm effect). See the main articles for the details of each formulation. SI units are used throughout. Formalism Formulation Homogeneous equations Inhomogeneous equations Vector calculus Fields 3D Euclidean space + time ∇ ⋅ B = 0 displaystyle begin aligned nabla cdot mathbf B =0end aligned ∇ × E + ∂ B ∂ t = 0 displaystyle begin aligned nabla times mathbf E + frac partial mathbf B partial t =0end aligned ∇ ⋅ E = ρ ε 0 displaystyle begin aligned nabla cdot mathbf E &= frac rho varepsilon _ 0 end aligned ∇ × B − 1 c 2 ∂ E ∂ t = μ 0 J displaystyle begin aligned nabla times mathbf B - frac 1 c^ 2 frac partial mathbf E partial t &=mu _ 0 mathbf J end aligned Potentials (any gauge) 3D Euclidean space + time B = ∇ × A displaystyle begin aligned mathbf B &=mathbf nabla times mathbf A end aligned E = − ∇ φ − ∂ A ∂ t displaystyle begin aligned mathbf E &=-mathbf nabla varphi - frac partial mathbf A partial t end aligned − ∇ 2 φ − ∂ ∂ t ( ∇ ⋅ A ) = ρ ε 0 displaystyle begin aligned -nabla ^ 2 varphi - frac partial partial t left(mathbf nabla cdot mathbf A right)&= frac rho varepsilon _ 0 end aligned ( − ∇ 2 + 1 c 2 ∂ 2 ∂ t 2 ) A + ∇ ( ∇ ⋅ A + 1 c 2 ∂ φ ∂ t ) = μ 0 J displaystyle begin aligned left(-nabla ^ 2 + frac 1 c^ 2 frac partial ^ 2 partial t^ 2 right)mathbf A +mathbf nabla left(mathbf nabla cdot mathbf A + frac 1 c^ 2 frac partial varphi partial t right)&=mu _ 0 mathbf J end aligned Potentials (Lorenz gauge) 3D Euclidean space + time B = ∇ × A displaystyle begin aligned mathbf B &=mathbf nabla times mathbf A \end aligned E = − ∇ φ − ∂ A ∂ t displaystyle begin aligned mathbf E &=-mathbf nabla varphi - frac partial mathbf A partial t \end aligned ∇ ⋅ A = − 1 c 2 ∂ φ ∂ t displaystyle begin aligned mathbf nabla cdot mathbf A &=- frac 1 c^ 2 frac partial varphi partial t \end aligned ( − ∇ 2 + 1 c 2 ∂ 2 ∂ t 2 ) φ = ρ ε 0 displaystyle begin aligned left(-nabla ^ 2 + frac 1 c^ 2 frac partial ^ 2 partial t^ 2 right)varphi &= frac rho varepsilon _ 0 end aligned ( − ∇ 2 + 1 c 2 ∂ 2 ∂ t 2 ) A = μ 0 J displaystyle begin aligned left(-nabla ^ 2 + frac 1 c^ 2 frac partial ^ 2 partial t^ 2 right)mathbf A &=mu _ 0 mathbf J end aligned
∂ [ i B j k ] = ∇ [ i B j k ] = 0 ∂ [ i E j ] + ∂ B i j ∂ t = ∇ [ i E j ] + ∂ B i j ∂ t = 0 displaystyle begin aligned partial _ [i B_ jk] &=\nabla _ [i B_ jk] &=0\partial _ [i E_ j] + frac partial B_ ij partial t &=\nabla _ [i E_ j] + frac partial B_ ij partial t &=0end aligned 1 h ∂ i h E i = ∇ i E i = ρ ϵ 0 − 1 h ∂ i h B i j − 1 c 2 ∂ ∂ t E j = − ∇ i B i j − 1 c 2 ∂ E j ∂ t = μ 0 J j displaystyle begin aligned frac 1 sqrt h partial _ i sqrt h E^ i &=\nabla _ i E^ i &= frac rho epsilon _ 0 \- frac 1 sqrt h partial _ i sqrt h B^ ij - frac 1 c^ 2 frac partial partial t E^ j &=&\-nabla _ i B^ ij - frac 1 c^ 2 frac partial E^ j partial t &=mu _ 0 J^ j \end aligned Potentials space (with topological restrictions) + time spatial metric independent of time B i j = ∂ [ i A j ] = ∇ [ i A j ] displaystyle begin aligned B_ ij &=partial _ [i A_ j] \&=nabla _ [i A_ j] end aligned E i = − ∂ A i ∂ t − ∂ i ϕ = − ∂ A i ∂ t − ∇ i ϕ displaystyle begin aligned E_ i &=- frac partial A_ i partial t -partial _ i phi \&=- frac partial A_ i partial t -nabla _ i phi \end aligned − 1 h ∂ i h ( ∂ i ϕ + ∂ A i ∂ t ) = − ∇ i ∇ i ϕ − ∂ ∂ t ∇ i A i = ρ ϵ 0 − 1 h ∂ i ( h h i m h j n ∂ [ m A n ] ) + 1 c 2 ∂ ∂ t ( ∂ A j ∂ t + ∂ j ϕ ) = − ∇ i ∇ i A j + 1 c 2 ∂ 2 A j ∂ t 2 + R i j A i + ∇ j ( ∇ i A i + 1 c 2 ∂ ϕ ∂ t ) = μ 0 J j displaystyle begin aligned - frac 1 sqrt h partial _ i sqrt h left(partial ^ i phi + frac partial A^ i partial t right)&=\-nabla _ i nabla ^ i phi - frac partial partial t nabla _ i A^ i &= frac rho epsilon _ 0 \- frac 1 sqrt h partial _ i left( sqrt h h^ im h^ jn partial _ [m A_ n] right)+ frac 1 c^ 2 frac partial partial t left( frac partial A^ j partial t +partial ^ j phi right)&=\-nabla _ i nabla ^ i A^ j + frac 1 c^ 2 frac partial ^ 2 A^ j partial t^ 2 +R_ i ^ j A^ i +nabla ^ j left(nabla _ i A^ i + frac 1 c^ 2 frac partial phi partial t right)&=mu _ 0 J^ j \end aligned Potentials (Lorenz gauge) space (with topological restrictions) + time spatial metric independent of time B i j = ∂ [ i A j ] = ∇ [ i A j ] E i = − ∂ A i ∂ t − ∂ i ϕ = − ∂ A i ∂ t − ∇ i ϕ ∇ i A i = − 1 c 2 ∂ ϕ ∂ t displaystyle begin aligned B_ ij &=partial _ [i A_ j] \&=nabla _ [i A_ j] \E_ i &=- frac partial A_ i partial t -partial _ i phi \&=- frac partial A_ i partial t -nabla _ i phi \nabla _ i A^ i &=- frac 1 c^ 2 frac partial phi partial t \end aligned − ∇ i ∇ i ϕ + 1 c 2 ∂ 2 ϕ ∂ t 2 = ρ ϵ 0 − ∇ i ∇ i A j + 1 c 2 ∂ 2 A j ∂ t 2 + R i j A i = μ 0 J j displaystyle begin aligned -nabla _ i nabla ^ i phi + frac 1 c^ 2 frac partial ^ 2 phi partial t^ 2 &= frac rho epsilon _ 0 \-nabla _ i nabla ^ i A^ j + frac 1 c^ 2 frac partial ^ 2 A^ j partial t^ 2 +R_ i ^ j A^ i &=mu _ 0 J^ j \end aligned Differential forms Fields Any space + time d B = 0 d E + ∂ B ∂ t = 0 displaystyle begin aligned dB&=0\dE+ frac partial B partial t &=0\end aligned d ∗ E = ρ ϵ 0 d ∗ B − 1 c 2 ∂ ∗ E ∂ t = μ 0 J displaystyle begin aligned d*E= frac rho epsilon _ 0 \d*B- frac 1 c^ 2 frac partial *E partial t = mu _ 0 J\end aligned Potentials (any gauge) Any space (with topological restrictions) + time B = d A E = − d ϕ − ∂ A ∂ t displaystyle begin aligned B&=dA\E&=-dphi - frac partial A partial t \end aligned − d ∗ ( d ϕ + ∂ A ∂ t ) = ρ ϵ 0 d ∗ d A + 1 c 2 ∂ ∂ t ∗ ( d ϕ + ∂ A ∂ t ) = μ 0 J displaystyle begin aligned -d*left(dphi + frac partial A partial t right)&= frac rho epsilon _ 0 \d*dA+ frac 1 c^ 2 frac partial partial t *left(dphi + frac partial A partial t right)&=mu _ 0 J\end aligned Potential (Lorenz Gauge) Any space (with topological restrictions) + time spatial metric independent of time B = d A E = − d ϕ − ∂ A ∂ t d ∗ A = − ∗ 1 c 2 ∂ ϕ ∂ t displaystyle begin aligned B&=dA\E&=-dphi - frac partial A partial t \d*A&=-* frac 1 c^ 2 frac partial phi partial t \end aligned ∗ ( − Δ ϕ + 1 c 2 ∂ 2 ∂ t 2 ϕ ) = ρ ϵ 0 ∗ ( − Δ A + 1 c 2 ∂ 2 A ∂ 2 t ) = μ 0 J displaystyle begin aligned *left(-Delta phi + frac 1 c^ 2 frac partial ^ 2 partial t^ 2 phi right)&= frac rho epsilon _ 0 \*left(-Delta A+ frac 1 c^ 2 frac partial ^ 2 A partial ^ 2 t right)&=mu _ 0 J\end aligned where In the vector formulation on Euclidean space + time, φ is the electrical potential, and A is the vector potential. Relativistic formulations[edit]
For the equations in special relativity, see classical
electromagnetism and special relativity and covariant formulation of
classical electromagnetism.
For the equations in general relativity, see
Formalism Formulation Homogeneous equations Inhomogeneous equations
∂ [ α F β γ ] = 0 displaystyle partial _ [alpha F_ beta gamma ] =0 ∂ α F α β = μ 0 J β displaystyle partial _ alpha F^ alpha beta =mu _ 0 J^ beta Potentials (any gauge) Minkowski space F α β = 2 ∂ [ α A β ] displaystyle F_ alpha beta =2partial _ [alpha A_ beta ] 2 ∂ α ∂ [ α A β ] = μ 0 J β displaystyle 2partial _ alpha partial ^ [alpha A^ beta ] =mu _ 0 J^ beta Potentials (Lorenz gauge) Minkowski space F α β = 2 ∂ [ α A β ] ∂ α A α = 0 displaystyle begin aligned F_ alpha beta &=2partial _ [alpha A_ beta ] \partial _ alpha A^ alpha &=0end aligned ∂ α ∂ α A β = μ 0 J β displaystyle partial _ alpha partial ^ alpha A^ beta =mu _ 0 J^ beta Fields Any spacetime ∂ [ α F β γ ] = ∇ [ α F β γ ] = 0 displaystyle begin aligned partial _ [alpha F_ beta gamma ] &=\nabla _ [alpha F_ beta gamma ] &=0end aligned 1 − g ∂ α ( − g F α β ) = ∇ α F α β = μ 0 J β displaystyle begin aligned frac 1 sqrt -g partial _ alpha ( sqrt -g F^ alpha beta )&=\nabla _ alpha F^ alpha beta &=mu _ 0 J^ beta end aligned Potentials (any gauge) Any spacetime (with topological restrictions) F α β = 2 ∂ [ α A β ] = 2 ∇ [ α A β ] displaystyle begin aligned F_ alpha beta &=2partial _ [alpha A_ beta ] \&=2nabla _ [alpha A_ beta ] end aligned 2 − g ∂ α ( − g g α μ g β ν ∂ [ μ A ν ] ) = 2 ∇ α ( ∇ [ α A β ] ) = μ 0 J β displaystyle begin aligned frac 2 sqrt -g partial _ alpha ( sqrt -g g^ alpha mu g^ beta nu partial _ [mu A_ nu ] )&=\2nabla _ alpha (nabla ^ [alpha A^ beta ] )&=mu _ 0 J^ beta end aligned Potentials (Lorenz gauge) Any spacetime (with topological restrictions) F α β = 2 ∂ [ α A β ] = 2 ∇ [ α A β ] ∇ α A α = 0 displaystyle begin aligned F_ alpha beta &=2partial _ [alpha A_ beta ] \&=2nabla _ [alpha A_ beta ] \nabla _ alpha A^ alpha &=0end aligned ∇ α ∇ α A β − R β α A α = μ 0 J β displaystyle nabla _ alpha nabla ^ alpha A^ beta -R^ beta _ alpha A^ alpha =mu _ 0 J^ beta Differential forms Fields Any spacetime d F = 0 displaystyle mathrm d F=0 d ⋆ F = μ 0 J displaystyle mathrm d star F=mu _ 0 J Potentials (any gauge) Any spacetime (with topological restrictions) F = d A displaystyle F=mathrm d A d ⋆ d A = μ 0 J displaystyle mathrm d star mathrm d A=mu _ 0 J Potentials (Lorenz gauge) Any spacetime (with topological restrictions) F = d A d ⋆ A = 0 displaystyle begin aligned F&=mathrm d A\mathrm d star A&=0end aligned ⋆ ◻ A = μ 0 J displaystyle star Box A=mu _ 0 J In the tensor calculus formulation, the electromagnetic tensor Fαβ
is an antisymmetric covariant rank 2 tensor; the four-potential, Aα,
is a covariant vector; the current, Jα, is a vector; the square
brackets, [ ], denote antisymmetrization of indices; ∂α is the
derivative with respect to the coordinate, xα. In Minkowski space
coordinates are chosen with respect to an inertial frame; (xα) =
(ct,x,y,z), so that the metric tensor used to raise and lower indices
is ηαβ = diag(1,−1,−1,−1). The d'Alembert operator on
⋆ displaystyle star is the
⋆ displaystyle star depends on the metric tensor only for its local scale. This means
that, as formulated, the differential form field equations are
conformally invariant, but the
◻ = ( − ⋆ d ⋆ d − d ⋆ d ⋆ ) displaystyle Box =(- star mathrm d star mathrm d -mathrm d star mathrm d star ) is the d'Alembert–
Other formalisms include the geometric algebra formulation and a
matrix representation of Maxwell's equations. Historically, a
quaternionic formulation[15][16] was used.
Solutions[edit]
∇ ⋅ ∇ × B ≡ 0 , ∇ ⋅ ∇ × E ≡ 0 displaystyle nabla cdot nabla times mathbf B equiv 0,nabla cdot nabla times mathbf E equiv 0 , which reduce eight equations to six independent ones, are the true
reason of overdetermination.[32]:161-162[33]
Limitations of the Maxwell equations as a theory of
electromagnetism[edit]
While
Electronics portal
Book: Maxwell's equations Algebra of physical space
Fresnel equations
Gravitoelectromagnetism
Interface conditions for electromagnetic fields
Moving magnet and conductor problem
Riemann–Silberstein vector
Notes[edit] ^
References[edit] ^ David J Griffiths (1999). Introduction to electrodynamics (Third
ed.). Prentice Hall. pp. 559–562.
ISBN 0-13-805326-X.
^ Bruce J. Hunt (1991) The Maxwellians, chapter 5 and appendix,
Cornell University Press
^ "IEEEGHN: Maxwell's Equations". Ieeeghn.org. Retrieved
2008-10-19.
^ Šolín, Pavel (2006).
Further reading can be found in list of textbooks in electromagnetism Historical publications[edit] On Faraday's Lines of Force – 1855/56 Maxwell's first paper (Part 1 & 2) – Compiled by Blaze Labs Research (PDF) On Physical Lines of Force – 1861 Maxwell's 1861 paper describing magnetic lines of Force – Predecessor to 1873 Treatise James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field", Philosophical Transactions of the Royal Society of London 155, 459–512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.) A Dynamical Theory Of The Electromagnetic Field – 1865 Maxwell's 1865 paper describing his 20 Equations, link from Google Books. J. Clerk Maxwell (1873) A Treatise on
Maxwell, J.C., A Treatise on
The developments before relativity:
External links[edit] Hazewinkel, Michiel, ed. (2001) [1994], "Maxwell equations", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 maxwells-equations.com — An intuitive tutorial of Maxwell's equations. Mathematical aspects of Maxwell's equation are discussed on the Dispersive PDE Wiki. Modern treatments[edit]
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