Maxwell’s equations

Maxwell’s equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism , classical optics , and electrical circuits . The equations provide mathematical models for electrical, optical and radio technologies, such as power generation, electric motors, wireless communications, lenses, radar, etc. They describe how electric and magnetic fields arise from changes in charges , currents , and fields. , [note 1] Name of the equations Physicist and MathematicianJames Clerk Maxwell , who in 1861 and 1862 published an early form of the equations that included the Lorentz force law. Maxwell was the first to use the equations to propose that light is an electromagnetic phenomenon.

An important consequence of Maxwell’s equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed ( c ) in a vacuum. Known as electromagnetic radiation , these waves can occur at different wavelengths to produce a spectrum of light from radio waves to gamma rays .

There are two major forms of the equations. Micro -equations have universal applicability, but are unwieldy for common calculations. They link electric and magnetic fields to total charge and total current, which include complex charges and currents in materials at the atomic scale . Macroscopic equations define two new auxiliary fields that describe the mass behavior of matter without considering quantum phenomena like atomic scale charges and spins. However, their use requires experimentally determined parameters for an unprecedented description of the material’s electromagnetic response.

The term “Maxwell’s equation” is also often used for equivalent alternative formulations . Versions of Maxwell’s equations based on electric and magnetic scalar potentials are preferred for use in boundary value problems , analytical mechanics , or quantum mechanics to explicitly solve equations. The compatibility of Maxwell’s equation with the covariant formulation ( instead of saying space and time separately) makes special relativity manifest . commonly used in high energy and gravitational physicsMaxwell’s equations in curved spacetime are compatible with general relativity . [note 2] In fact, Albert Einstein developed special and general relativity to accommodate the irreversible speed of light, a consequence of Maxwell’s equations, with the theory that only relativistic motion has physical consequences.

The publication of the equations marked the unification of a theory for previously described phenomena : magnetism, electricity, light and related radiation. Since the mid-twentieth century, it has been understood that Maxwell’s equations do not give an accurate description of electromagnetic phenomena, but are instead a classical limitation of the more precise theory of quantum electrodynamics.

Conceptual description

Gauss’s law

Gauss’s law describes the relationship between a static electric field and the electric charges it causes : a static electric field points away from positive charges and toward negative charges, and the net of the electric field through any closed surface The outflow is proportional to the charge attached to the surface. This means that the net current passing through a closed surface generates the total charge (including the bound charge due to the polarization of the material) surrounded by that surface, divided by the dielectric (vacuum permittivity) of free space.

Gauss’s law for magnetism

Gauss’s law for magnetism states that there are no “magnetic charges” (also called magnetic monopoles) like electric charges. ^[1] Instead, the magnetic field due to the material is generated by a configuration called a dipole, and the net outflow of the magnetic field through any closed surface is zero. Magnetic dipoles are best represented as loops of electric current, but they resemble positive and negative ‘magnetic charges’, inseparably bound together, with no net ‘magnetic charge’ . The equivalent technical statement is that the sum of the total magnetic flux through any Gaussian surface is zero, or that the magnetic field is a solenoidal vector field.

Faraday’s law

Maxwell-Faraday’s version Faraday’s law of induction describes how a time-varying magnetic field produces (“brings”) an electric field. ^{[1] In} integral form, it states that the work per unit charge required to move a charge around a closed loop is equal to the rate of change of the magnetic flux through the enclosed surface.

Electromagnetic induction is the working principle behind many electric generators: for example, a rotating magnet creates a changing magnetic field, which in turn generates an electric field in a nearby coil.

Ampere’s law with Maxwell’s addition

Ampere’s law, with Maxwell’s addition, states that magnetic fields can be generated in two ways: by electric current (this was the original “Ampere’s law”) and by changing electric fields (this was “Maxwell’s addition”, which he called the displacement current). In integral form, the induced magnetic field around any closed loop is proportional to the electric current plus the displacement current (proportional to the rate of change of the electric current) through the enclosed surface.

Maxwell’s addition to Ampere’s law is particularly important: it makes the set of equations mathematically consistent for non-stationary fields without changing Ampere’s and Gauss’s laws for stationary fields. ^[2] However, as a consequence, it predicts that a changing magnetic field induces an electric field and vice versa. ^[1] [3] Therefore, these equations allow self-contained “electromagnetic waves” to travel in empty space (see electromagnetic wave equations).

The speed calculated for electromagnetic waves, which can be estimated from experiments on charges and currents, ^{[note 4]} corresponds to the speed of light; In fact, light is a form of electromagnetic radiation (such as X-rays, radio waves, and others). Maxwell understood the relationship between electromagnetic waves and light in 1861, thereby integrating the principles of electromagnetism and optics.

Formulation in terms of electric and magnetic fields (in the micro or vacuum version)

Electric and magnetic field construction consists of four equations that determine the field for a given charge and current distribution. A separate law of nature, the Lorentz force law, describes how, in contrast, electric and magnetic fields act on charged particles and currents. A version of this law was included in the original equations by Maxwell, but by convention, it is no longer included. The formalism below vector calculus, the work of Oliver Heaviside, ^[4] [5] has become the standard. It is manifestly rotational invariant, and therefore mathematically more transparent in the x, y, z components than Maxwell’s original 20 equations. Relativistic formulations are even more symmetric and obviously Lorentz invariant. For similar equations expressed using tensor calculus or differential form, see Alternative formulations.

Differential and integral formulations are mathematically equivalent and both are useful. The integral formulation relates regions within a region of space to regions on the boundary and can often be used to simplify and directly calculate areas from symmetric distributions of charges and currents. On the other hand, differential equations are purely local and are a more natural starting point for computing fields in more complex (less symmetric) situations, for example using finite element analysis. ^[6]

key to notation

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 typically having a time and space dependence. sources are

• total electric charge density (total charge per unit volume), , and
• Total electric flux density (total current per unit area) , J.

Universal constants (only the first two ones explicitly in the SI units construction) are to appear in the equation:

• the permittivity of free space, ₀ , and
• Free space permeability, ₀ , and
• speed of light ,

{\displaystyle c={\frac {1}{\sqrt {\varepsilon _{0}\mu _{0}}}}}

Differential equations

In differential equations,

• The nabla symbol , , denotes three dimensional gradient , operator del ,
• The symbol (pronounced “del dot”) denotes the divergence operator,
• The × symbol (pronounced “del par”) denotes the curl operator.

Integral equation

In integral equations,

• is any fixed quantity with closed boundary surface , and
• is any fixed surface along the closed boundary curve ,

Here a fixed volume or surface means that it does not change with time. The equations are a bit easier to interpret with true, complete, and time-independent surfaces. For example, since the surface is time-independent, we can bring the difference in Faraday’s law under the integral sign:

{\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} \,,

Maxwell’s equations can probably be formulated with time-dependent surfaces and volumes using differential vars and using Gauss and Stokes formulas appropriately.

\oiint{\displaystyle_{\scriptstyle \partial \Omega }}

 The boundary is an integral surface on the surface , closed with a loop indicating the surface

\iiint _{\Omega } 

There is a volume integral over volume ,

\oint _{\partial \Sigma }

One is the line integral around the boundary curve , with indicating the loop’s curve is closed.

\iint _{\Sigma }

A surface integral on a surface is ,

The total electric charge Q enclosed in is the volume integral plus of the charge density (see the ” Macroscopic Construction” section below):

{\displaystyle Q=\iiint _{\Omega }\rho \ \mathrm {d} V,}

where dV is the quantity element.

• The net electric flux I is the surface integral of the electric flux density J passing through a fixed surface, :

{\displaystyle I=\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} ,}

where d S denotes the difference of the vector element surface area S , for the normal surface . (The vector field is sometimes denoted by A instead of S , but this conflicts with the notation of the magnetic vector potential).

Representation in the Convention of SI Units

name	integral equation	differential equations
Gauss’s law
Gauss’s law for magnetism
Maxwell-Faraday Equation(Faraday’s law of induction)
Ampere’s Circuit Law (with Maxwell’s Sum)

Representation in the Convention of Gaussian Units

The definitions of charge, electric field, and magnetic field can be changed to simplify theoretical calculations, by absorbing dimensioned factors ₀ and μ _{0 ,} according to custom in the units of calculation. It yields the same physics, with a corresponding change in convention for the Lorentz force law, i.e. the trajectory of charged particles, or the work done by an electric motor. These definitions are often preferred in theoretical and high-energy physics, where it is natural to treat electric and magnetic fields with the same units, to simplify the appearance of electromagnetic tensors: Lorentz covalent objects integrating electric and magnetic fields. The following components shall contain the same unit and dimensions. ^[7] : viiSuch modified definitions are traditionally used with Gaussian (CGS) units. Using these definitions and conventions, colloquially “in Gaussian units”, ^[8] the Maxwell equations become: ^[9]

name	integral equation	differential equations
Gauss’s law
Gauss’s law for magnetism
Maxwell-Faraday Equation(Faraday’s law of induction)
Ampere’s Circuit Law (with Maxwell’s Sum)

Equations are especially readable when length and time are measured in corresponding units such as seconds and lightseconds, i.e. in units such as c = 1 unit of length/time. Since 1983 (see International System of Units), the meter and second have been consistent except for historical legacy because by definition c = 299 792 458 m/s (≈ 1.0 ft/nanosecond).

Also cosmetic changes, called rationalisations, are possible absorbing factors 4 depending on whether we want Coulomb ‘s law or Gauss’s law to come out nicely, see Lorentz-Heaviside units (used mainly in particle physics).

Relationship Between Differential and Integral Formulations

The equivalence of the differential and integral formulations is a result of the Gauss Divergence Theorem and the Kelvin–Stokes Theorem.

flow and divergence

• F of a vector field containing volume and its closed boundary , (respectively enclosing ) a source (+) and sink (-) . Here, F may be the E field with the source electric charge, but not the B field, which is shown to have no magnetic charge. The outgoing unit is normal n .

According to the (purely mathematical) Gauss Deviation Theorem , the boundary surface through an electric current can be written as

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

Thus the integrated version of the Gauss equation can be written as:

{\displaystyle \iiint _{\Omega }\left(\nabla \cdot \mathbf {E} -{\frac {\rho }{\varepsilon _{0}}}\right)\,\mathrm {d} V=0}

Since is arbitrary (e.g. an arbitrary small ball with an arbitrary center), it is satisfied if and only if the integrand is zero everywhere. This is the differential equation formulation of the Gauss equation up to a trivial rearrangement.

Similarly, in Gauss’s law for magnetism, by writing the magnetic flux in an integrated form, we get

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

which is satisfied for all if and only if everywhere.

\nabla \cdot {\mathbf {B}}=0

Circulation and curl

By Kelvin Stokes’ theorem we can rewrite the line integral of the fields around the closed boundary curve ( i.e. their “circulation of the fields” of the integral of a surface curl to it’s boundaries, i.e. over)

{\displaystyle \oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\iint _{\Sigma }(\nabla \times \mathbf {B} )\cdot \mathrm {d} \mathbf {S} },

Therefore the modified Ampere law can be rewritten in the integral form

{\displaystyle \iint _{\Sigma }\left(\nabla \times \mathbf {B} -\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\right)\cdot \mathrm {d} \mathbf {S} =0}.

Since can be chosen arbitrarily as an arbitrary small, e.g., oriented arbitrarily, and disc-centered arbitrarily, we conclude that the integrand is zero iff the differential equation as Ampere’s modified law is satisfied. . The equivalence of Faraday’s law in the differential and integral forms is the same.

The line integral and the curl correspond to quantities in classical fluid dynamics: the movement of a fluid is the integral of the flow velocity field of the fluid around a closed loop, and the vorticity of the fluid is the curl of the velocity field.

Charge protection

The change of charge can be obtained as a consequence of Maxwell’s equations. The left-hand side of the modified Ampere’s law is the zero deviation by the div-curl identity. Expanding the divergence to the right, interchanging the derivatives, and applying Gauss’s law gives:

{\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)}

meaning

{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0}.

By the Gauss Divergence Theorem, it means that the rate of change of charge in a given quantity is equal to the net current flowing through the boundary:

{\displaystyle {\frac {d}{dt}}Q_{\Omega }={\frac {d}{dt}}\iiint _{\Omega }\rho \mathrm {d} V=-}\oiint_{\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.

Vacuum Equation, Electromagnetic Waves and Speed ​​of Light

In a region with no charges ( ρ = 0 ) and no currents ( J = 0 ), such as in a vacuum, reducing to Maxwell’s equation:

{\displaystyle {\begin{aligned}\nabla \cdot \mathbf {E} &=0\quad &\nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}},\\\nabla \cdot \mathbf {B} &=0\quad &\nabla \times \mathbf {B} &=\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}.\end{aligned}}}

Taking the curl (∇×) of the curl equations , and using the curl of the curl identity, we get

{\displaystyle {\begin{aligned}\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}-\nabla ^{2}\mathbf {E} =0\\\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}-\nabla ^{2}\mathbf {B} =0\end{aligned}}}

Quantity (time/length) has a dimension of ^{2 .} Defined , the standard wave equations in the above equations have the form

{\displaystyle \mu _{0}\varepsilon _{0}}{\displaystyle c=(\mu _{0}\varepsilon _{0})^{-1/2}}

{\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} =0\end{aligned}}}

Already during Maxwell’s lifetime, it was found that . Known value for and give , which is already known as the speed of light in free space. This led him to propose that light and radio waves are propagating electromagnetic waves, as this has been sufficiently confirmed. In the old SI system of units, k has a value and a defined constant, (which means by definition ) that defines the ampere and the meter. In the new SI system, only c keeps its set value, and the electron charge becomes a set value.

\varepsilon _{0}\ mu _ {0}{\displaystyle c\approx 2.998\times 10^{8}\,{\text{m/s}}}{\displaystyle \mu _{0}=4\pi \times 10^{-7}}{\displaystyle c=299792458\,{\text{m/s}}}{\displaystyle \varepsilon _{0}=8.854...\times 10^{-12}\,{\text{F/m}}}

In materials with relative permittivity, _r , and _relative permittivity , μr , the phase velocity of light becomes

v_{\text{p}}={\frac {1}{\sqrt {\mu _{0}\mu _{\text{r}}\varepsilon _{0}\varepsilon _{\text{r}}}}}

which is usually less than

Furthermore, E and B are perpendicular to each other and in the direction of wave propagation, and are in phase with each other. A sinusoidal plane wave is a special solution of these equations. Maxwell’s equations describe how these waves can physically propagate through space. A changing magnetic field produces a changing electric field through Faraday’s law. In turn, that electric field produces a changing magnetic field by combining Maxwell’s Ampere’s law. This continuous cycle allows these waves, now known as electromagnetic radiation, to travel through space at a velocity c .

Macroscopic formulation

The above equations are a subtle version of Maxwell’s equations, which express electric and magnetic fields in terms of (possibly atomic-level) charges and currents present. This is sometimes called the “normal” form, but the macroscopic version below is equally common, one of difference bookkeeping.

The microscopic version is sometimes called “a vacuum in Maxwell’s equations”: it refers to the fact that the physical medium is not created in the structure of the equations, but appears only in charge and current terms. The microscopic version was introduced by Lorentz, who attempted to use it to derive macroscopic properties of bulk matter from its microscopic components.

“Maxwell’s macroscopic equations”, also known as Maxwell’s equations, are similar to those that Maxwell introduced himself.

name	Integral Equations (SI Convention)	Differential Equations (SI Convention)	Differential Equations (Gaussian Convention)
Gauss’s law
Gauss’s law for magnetism
Maxwell-Faraday Equation (Faraday’s Law of Induction)
Ampere’s Circuit Law (with Maxwell’s Sum)

In the macroscopic equations, the effect of the bound charge Q _b and the bound current I _b is included in the displacement field D and the magnetic field H , while the equations depend only on the free charge Q _f and the free current I _{f .} It shows the division of the total electric charge Q and current I (and their densities and J ) into free and bound parts:

{\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}}}

The cost of this division is that additional fields D and H are needed to determine these fields by means of phenomenological component equations relating to the electric field E and the magnetic field B with bound charge and current.

See below for a detailed description of the difference between the microscopic equations, including total charge and current content contributions, useful in air/vacuum; ^{[Note 6]} and deal with the macroscopic equations, free charge and current, practical to use within materials.

Bound charge and current

When an electric field is applied to a dielectric material, its molecules react by creating microscopic electric dipoles – their atomic nuclei travel a short distance in the direction of the field, while their electrons travel a short 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 each molecule reacts the same way, as shown in the figure, these small motions of charge combine to form a layer of positively bound charge on one side of the material and a layer of negative charge on the other. There are. Bound charge is most easily described in terms of the material’s polarization P , its dipole moment per unit volume. if Pis uniform, then a macroscopic separation of charge only occurs at the surfaces where P enters and leaves the material. For non-uniform P, a charge is also generated in bulk. ^{[11 1]}

Somewhat similarly, constituent atoms in all materials exhibit magnetic moments that are intrinsically linked to the angular momentum of the components of the atoms, particularly their electrons. The connection for angular momentum shows a picture of an assembly of microscopic current loops. Outside the material, an assembly of such a microscopic current loop is indistinguishable from the macroscopic current circulating around the surface of the material, despite the fact that no individual charge is traveling large distances. These bound currents can be described using the magnetization M.

Very complex and granular bound charges and bound currents can, therefore, be represented on a macroscopic scale in terms of P and M, averaging these charges and currents on a sufficiently large scale so that the granularity of individual atoms is not observed. but also small enough that they change with space in the material. Thus, Maxwell’s macroscopic equations ignore many details on a fine scale that may be unimportant for understanding matters on a gross scale by computing some suitable amount of mean fields.

Auxiliary field, polarization and magnetization

The definitions of auxiliary fields are:

{\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 as microscopic bound charges and bound currents, respectively. In the case of macroscopic bound charge density _b and bound current density j _b , polarization P and magnetization M are then defined as

{\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 densities as

{\begin{aligned}\rho &=\rho _{\text{b}}+\rho _{\text{f}},\\\mathbf {J} &=\mathbf {J} _{\text{b}}+\mathbf {J} _{\text{f}},\end{aligned}}

and use the relations defined above to eliminate D and H , the “macroscopic” Maxwell’s equations reproduce the “microscopic” equations.

Constitutional relations

In order to apply ‘Maxwell’s macroscopic equation’, it is necessary to specify the relationship between the displacement field D and the electric field E as well as the magnetic field H and the magnetic field B. Equally, we have to specify the dependence of the polarization P (hence the bound charge) and the magnetization M (hence the bound current) on the applied electric and magnetic field. The equations specifying this reaction are called constitutional relations. For real-world material, constitutive relations are rarely simple, except in approx., and are usually determined by experiment. See the main article on Constitutional Relations for a detailed description. ^{[13] : 44-45}

For materials without polarization and magnetization, the constitutional relations are (by definition)

\mathbf {D} =\varepsilon _{0}\mathbf {E} ,\quad \mathbf {H} ={\frac {1}{\mu _{0}}}\mathbf {B}

where ₀ is the permittivity of free space and μ 0 is the permittivity of _free space . Since there is no bound charge, the total and free charge and current are equal.

An alternative view on the microscopic equations is that they are macroscopic equations along with the statement that the vacuum behaves like a perfect linear “material” without additional polarization and magnetism. More generally, the constitutional relations for linear materials are:

\mathbf {D} =\varepsilon \mathbf {E} \,,\quad \mathbf {H} ={\frac {1}{\mu }}\mathbf {B}

where is the permittivity and μ is the permittivity of the material . The linear approximation to the displacement field D is usually excellent because the interatomic electric fields of materials are very high, of the order of 10 ¹¹ V/m for the most extreme electric fields or temperatures (high power pulsed lasers) achievable in the laboratory. compared to the area. For magnetic fields however, the linear approximation can break down in common materials such as iron leading to phenomena such as hysteresis. However, there can be various complications in the linear case as well.

• For homogeneous materials, and μ are constant throughout the material, whereas for nonhomogeneous materials they depend on location within the material (and perhaps time). ^[14] : 463
• For isotropic materials, and μ are scalar, whereas for anisotropic materials (such as due to crystal structure) they are tensors. ^{[13] : 421 [14] : 463}
• Materials are typically dispersive, so 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 e.g. nonlinear optics), d and p are not necessarily proportional to e , similarly h or m are not necessarily proportional to b . In general d and h depend on both e and b on space and time, and possibly on other physical quantities .

In applications one also has to describe how free currents and charge densities in terms of E and B possibly behave with other physical quantities such as pressure, and the mass, number density, and velocity of charge-carrier particles. For example, the original equation given by Maxwell (see History of Maxwell’s equation) contains Ohm’s law as

\mathbf {J} _{\text{f}}=\sigma \mathbf {E} \,.

Alternative formulation

The following is a summary of several other mathematical formalisms for writing microscopic Maxwell’s equations, with columns separating the two homogeneous Maxwell equations from two inhomogeneous equations involving charge and current. Each formulation has direct versions in terms of electric and magnetic fields, and indirect ones in terms of electric potential and vector potential A. Potentials were introduced as a convenient way to solve homogeneous equations, but it was thought that all observable physics was rooted in electric and magnetic fields (or relativistically, the Faraday tensor). However, potentials play a central role in quantum mechanics, and act with quantum mechanically observable consequences even when electric and magnetic fields are missing (Aharonov–Bohm effect).

Each table describes a formality. See the main article for details of each formulation. SI units are used throughout.

formulation	homogeneous equation	inhomogeneous equation
farm3D Euclidean Space + Time
Capacity (any gauge)3D Euclidean Space + Time
Capacity (Lorenz gauge)3D Euclidean Space + Time

formulation	homogeneous equation	inhomogeneous equation
farmspace + timetime independent spatial metrics
abilityspace (with occasional restrictions) + timetime independent spatial metrics
Capacity (Lorenz gauge)space (with occasional restrictions) + timetime independent spatial metrics

formulation	homogeneous equation	inhomogeneous equation
farmany place + time
Capacity (any gauge)Any location (with occasional restrictions) + time
Potential (Lorenz gauge)Any location (with occasional restrictions) + timetime independent spatial metrics

Relative formulation

Maxwell equations can also be formulated on spacetime such as Minkowski space where space and time are considered on the same level. Direct spacetime formulations reveal that Maxwell’s equations are relativistically invariant. Because of this symmetry, the electric and magnetic fields are considered on the same level and recognized as components of the Faraday tensor. This reduces the four Maxwell equations to two, which simplifies the equations, although we can no longer use the familiar vector formulation. In fact the Space + Time formulation has Maxwell equations not Galileo invariant and Lorentz invariance as a hidden symmetry. It was a major source of inspiration for the development of the theory of relativity. In fact, even the formulation which treats space and time separately, is not a non-relativistic approximation and only describes the same physics by renaming the variables. For this reason, relativistic invariant equations are also commonly called Maxwell’s equations.

Each table describes a formality.

formulation	homogeneous equation	inhomogeneous equation
farmminkowski space
Capacity (any gauge)minkowski space
Capacity (Lorenz gauge)minkowski space
farmAny spacetime
Capacity (any gauge)Any spacetime (with topological restrictions)
Capacity (Lorenz gauge)Any spacetime (with topological restrictions)

formulation	homogeneous equation	inhomogeneous equation
farmAny spacetime
Capacity (any gauge)Any spacetime (with topological restrictions)
Capacity (Lorenz gauge)Any spacetime (with topological restrictions)

• In the tensor calculus formulation, the electromagnetic tensor F _αβ is an antisymmetric covariant order 2 tensor; Four possible , one _α , is a covariant vector; The current, J ^α , is a vector; The square brackets, [ ] , denote the antisymmetry of the indices; is the derivative of the alpha coordinate, with respect ^to x _alpha . In Minkowski space coordinates are chosen with respect to an inertial frame; ( x ^α ) = ( ct , x , y , z )_{, so that the metric tensor used} to raise and lower the indices is αβ = diag(1,−1,−1,−1) . minkowskiThe D’Alembert operator on space is of the form vector formulation = _α ^. In general spacetimes, the coordinate system X ^alpha is arbitrary, the covariant derivative alpha _, Ricci tensor, R _αβ and the raising and decreasing indices are defined on the basis of the Lorentzian metric, G _αβ and the operator ‘Alembert D’ as is defined ⁱⁿ = _alpha alpha, The topological restriction is that the second real cohomology group of the space vanishes (see differential form formulation for an explanation). This is violated for Minkowski space, which has one line removed, which can model a (flat) spacetime with a point-like monopole on the line’s complement.
• In the differential form construction on arbitrary space-time, f =1/2F _αβ d x ^α d x ^β is the electromagnetic tensor that is assumed to be 2-form, A = A _α d x ^α is potential 1-form,current is 3-form, d is external derivative, andis Hodge Star defined. On forms (by its orientation, i.e. its sign) based on the Lorentzian metric of spacetime. In the special case of 2-forms such as F , the Hodge stardepends only on the metric tensor for the local scale. This means that, as formulated, the differential form field equations are conformally invariant, but the Lorenz gauge condition breaks the conformal irreversibility. the operator

{\displaystyle J=-J_{\alpha }{\star }\mathrm {d} x^{\alpha }}{\displaystyle {\star }}{\displaystyle {\star }}{\displaystyle \Box =(-{\star }\mathrm {d} {\star }\mathrm {d} -\mathrm {d} {\star }\mathrm {d} {\star })}

• is the d’Alembert–Laplace–Beltrami operator on an arbitrary 1-forms on Lorentzian space. The topological condition is again that the second real cohomology group is ‘trivial’ (meaning that its form follows from a definition). This condition from isomorphism with the second De Rum cohomology means that every closed 2-form is exact.

Other formalities include geometric algebra formulations and a matrix representation of Maxwell’s equations. Historically, a quaternary formulation was used.

Solution

Maxwell’s equations are partial differential equations that relate electric and magnetic fields to each other and to electric charges and currents. Often, charges and currents themselves are dependent on electric and magnetic fields through the Lorentz force equation and constitutive relations. All of these form a set of coupled partial differential equations that are often very difficult to solve: the solutions encompass all the diverse phenomena of classical electromagnetism. Some general comments follow.

For any differential equation, boundary conditions ^{[17] [18] [19]} and initial conditions ^[20] are necessary for a unique solution. For example, despite there being no charge and no current anywhere in spacetime, there are obvious solutions for which e and b are zero or constant, but there are also non-trivial solutions corresponding to electromagnetic waves. In some cases, Maxwell’s equations are solved throughout space, and the boundary conditions are given as asymptotic limits at infinity. ^[21] In other cases, Maxwell’s equations are solved in a finite region of space, with suitable conditions at the boundary of that region, for example an artificial absorbing boundary that represents the rest of the universe,^[22] [23] or periodic boundary conditions, or walls that separate a small region from the outside world (such as with waveguides or cavity resonators). ^[24]

Jefimenko’s equations (or the closely related Leonard–Weichert potential) are clear solutions to Maxwell’s equations for the electric and magnetic fields created by any distribution of charges and currents. This assumes specific initial conditions to obtain the so-called “retarded solution”, where the only fields present are those created by the charges. However, Jefimenko’s equations are unusable in situations when charges and currents themselves are affected by the fields created by them.

Numerical methods for differential equations can be used to calculate approximate solutions to Maxwell’s equations when exact solutions are impossible. These include the finite element method and the finite-difference time-domain method. ^{[17] [19] [25] [26] [27]} For more details, see Computational Electromagnetics.

Overdetermining Maxwell’s Equations

Maxwell’s equations appear to be overdetermined , in that they contain six unknowns ( three components of E and B ), but eight equations (one for each of two Gauss’s laws, three vectors each for Faraday’s and Ampere’s laws). Constituent). (Currents and charges are not unknown, subject to charge conservation being independently specified.) This is related to a certain finite type of redundancy in Maxwell’s equations: it can be proved that Faraday’s law and Ampere’s law Any system that satisfies both automatically satisfies both. Gauss’s law, as long as the system has an initial state, and assuming conservation of charge and the non-existence of magnetic monopoles. ^[28] [29]This explanation was first introduced by Julius Adams Stratton in 1941. ^[30]

Although it is possible to simply ignore two of Gauss’s laws in a numerical algorithm (apart from the initial conditions), the imperfect accuracy of the calculations can lead to ever-increasing violations of those laws. By introducing dummy variables showing these violations, the four equations ultimately do not become overdetermined. The resulting formulation can lead to more precise algorithms that take all four laws into account. ^[31]

Both identities , which reduce eight equations to six independent equations, are the true cause of overdeterminism. ^[32] [33] or definitions of linear dependence for PDEs may be referred to.

{\displaystyle \nabla \cdot \nabla \times \mathbf {B} \equiv 0,\nabla \cdot \nabla \times \mathbf {E} \equiv 0}

Equally, overdispersion can be seen as the conservation of electric and magnetic charge, as they are required in the derivation described above but implied by two of Gauss’s laws.

For linear algebraic equations, one can make ‘good’ rules for rewriting the equations and the unknowns. Equations can be linearly dependent. But in differential equations and in particular PDEs, one needs suitable boundary conditions, which do not depend on the equations in such an obvious way. Even more so, if one rewrites them in terms of vector and scalar potentials, the equations tend to be overestimated due to gauge fixing.

Maxwell’s equations as the classical limit of QED

Maxwell’s equations and the Lorentz force law (along with the rest of classical electromagnetism) are exceptionally successful in explaining and predicting a wide variety of phenomena. However they do not account for quantum effects and hence their area of ​​applicability is limited. Maxwell’s equations are considered the classical limit of quantum electrodynamics (QED).

Some observed electromagnetic phenomena are inconsistent with Maxwell’s equations. These include photon–photon scattering and many other phenomena related to photons or virtual photons, “non-classical light” and quantum entanglement of electromagnetic fields (see quantum optics). For example, quantum cryptography cannot be described by Maxwell’s principle, almost not even. The approximate nature of Maxwell’s equations becomes more and more clear over the extremely strong field regime (see Euler–Heisenberg Lagrangian) or at extremely small distances.

Finally, Maxwell’s equations cannot explain any phenomenon that involves individual photons interacting with quantum matter, such as the photoelectric effect, Planck’s law, Duane–Hunt law, and single-photon light detectors. However, many such phenomena can be predicted using the half-way principle of quantum matter associated with a classical electromagnetic field, either as an external field or as expected of charge current and density on the right hand side of Maxwell’s equations. with price.

Shift

Popular variations on the Maxwell equations as the classical theory of the electromagnetic field are relatively rare because the standard equations have withstood the test of time remarkably.

Magnetic monopole

Maxwell’s equations assume that the universe has an electric charge, but no magnetic charge (also called a magnetic monopole). In fact, despite extensive discoveries, magnetic charge has never been observed, ^{[note 7]} and may not exist. If they did exist, then both Gauss’s law and Faraday’s law would need to be modified for magnetism, and the resulting four equations would be completely symmetric under the exchange of electric and magnetic fields.