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There are various mathematical descriptions of the electromagnetic field that are used in the study of
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, one of the four
fundamental interaction In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions. There are four fundamental interactions known to exist: the gravitational and electro ...
s of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.


Vector field approach

The most common description of the electromagnetic field uses two three-dimensional vector fields called the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
and the
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
. These vector fields each have a value defined at every point of space and time and are thus often regarded as functions of the space and time coordinates. As such, they are often written as (electric field) and (magnetic field). If only the electric field (E) is non-zero, and is constant in time, the field is said to be an
electrostatic field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
. Similarly, if only the magnetic field (B) is non-zero and is constant in time, the field is said to be a
magnetostatic field Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equati ...
. However, if either the electric or magnetic field has a time-dependence, then both fields must be considered together as a coupled electromagnetic field using
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside 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 electric circuits ...
.


Maxwell's equations in the vector field approach

The behaviour of electric and magnetic fields, whether in cases of electrostatics, magnetostatics, or
electrodynamics In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
(electromagnetic fields), is governed by
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside 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 electric circuits ...
: : where ''ρ'' is the charge density, which can (and often does) depend on time and position, ''ε''0 is the
electric constant Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
, ''μ''0 is the
magnetic constant The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constan ...
, and J is the current per unit area, also a function of time and position. The equations take this form with the
International System of Quantities The International System of Quantities (ISQ) consists of the quantities used in physics and in modern science in general, starting with basic quantities such as length and mass, and the relationships between those quantities. This system underli ...
. When dealing with only nondispersive isotropic linear materials, Maxwell's equations are often modified to ignore bound charges by replacing the permeability and permittivity of
free space A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
with the permeability and permittivity of the linear material in question. For some materials that have more complex responses to electromagnetic fields, these properties can be represented by tensors, with time-dependence related to the material's ability to respond to rapid field changes (
dispersion (optics) In optics, and by analogy other branches of physics dealing with wave propagation, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency; sometimes the term chromatic dispersion is used for specificity to ...
,
Green–Kubo relations The Green–Kubo relations (Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for transport coefficients \gamma in terms of integrals of time correlation functions: :\gamma = \int_0^\infty \left\langle \dot(t) \dot(0 ...
), and possibly also field dependencies representing nonlinear and/or nonlocal material responses to large amplitude fields (
nonlinear optics Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in ''nonlinear media'', that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typic ...
).


Potential field approach

Many times in the use and calculation of electric and magnetic fields, the approach used first computes an associated potential: the
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
, \varphi, for the electric field, and the
magnetic vector potential In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic ...
, A, for the magnetic field. The electric potential is a scalar field, while the magnetic potential is a vector field. This is why sometimes the electric potential is called the scalar potential and the magnetic potential is called the vector potential. These potentials can be used to find their associated fields as follows: \mathbf E = - \mathbf \nabla \varphi - \frac \mathbf B = \mathbf \nabla \times \mathbf A


Maxwell's equations in potential formulation

These relations can be substituted into Maxwell's equations to express the latter in terms of the potentials. Faraday's law and
Gauss's law for magnetism In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics. It states that the magnetic field has divergence equal to zero, in other words, that it is a solenoidal vector field. It is ...
(the homogeneous equations) turn out to be identically true for any potentials. This is because of the way the fields are expressed as gradients and curls of the scalar and vector potentials. The homogeneous equations in terms of these potentials involve the divergence of the curl \nabla \cdot \nabla \times \mathbf A and the curl of the gradient \nabla \times \nabla \varphi, which are always zero. The other two of Maxwell's equations (the inhomogeneous equations) are the ones that describe the dynamics in the potential formulation. These equations taken together are as powerful and complete as Maxwell's equations. Moreover, the problem has been reduced somewhat, as the electric and magnetic fields together had six components to solve for. In the potential formulation, there are only four components: the electric potential and the three components of the vector potential. However, the equations are messier than Maxwell's equations using the electric and magnetic fields.


Gauge freedom

These equations can be simplified by taking advantage of the fact that the electric and magnetic fields are physically meaningful quantities that can be measured; the potentials are not. There is a freedom to constrain the form of the potentials provided that this does not affect the resultant electric and magnetic fields, called
gauge freedom In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct c ...
. Specifically for these equations, for any choice of a twice-differentiable scalar function of position and time ''λ'', if is a solution for a given system, then so is another potential given by: \varphi' = \varphi - \frac \mathbf A' = \mathbf A + \mathbf \nabla \lambda This freedom can be used to simplify the potential formulation. Either of two such scalar functions is typically chosen: the Coulomb gauge and the Lorenz gauge.


Coulomb gauge

The
Coulomb gauge In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...
is chosen in such a way that \mathbf \nabla \cdot \mathbf A' = 0, which corresponds to the case of magnetostatics. In terms of ''λ'', this means that it must satisfy the equation \nabla^2 \lambda = - \mathbf \nabla \cdot \mathbf A. This choice of function results in the following formulation of Maxwell's equations: \nabla^2 \varphi' = -\frac \nabla^2 \mathbf A' - \mu_0 \varepsilon_0 \frac = - \mu_0 \mathbf J + \mu_0 \varepsilon_0 \nabla\!\! \left (\! \frac \!\right ) Several features about Maxwell's equations in the Coulomb gauge are as follows. Firstly, solving for the electric potential is very easy, as the equation is a version of
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with ...
. Secondly, solving for the magnetic vector potential is particularly difficult. This is the big disadvantage of this gauge. The third thing to note, and something which is not immediately obvious, is that the electric potential changes instantly everywhere in response to a change in conditions in one locality. For instance, if a charge is moved in New York at 1 pm local time, then a hypothetical observer in Australia who could measure the electric potential directly would measure a change in the potential at 1 pm New York time. This seemingly violates causality in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
, i.e. the impossibility of information, signals, or anything travelling faster than the speed of light. The resolution to this apparent problem lies in the fact that, as previously stated, no observers can measure the potentials; they measure the electric and magnetic fields. So, the combination of ∇''φ'' and ∂A/∂''t'' used in determining the electric field restores the speed limit imposed by special relativity for the electric field, making all observable quantities consistent with relativity.


Lorenz gauge condition

A gauge that is often used is the
Lorenz gauge condition In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ...
. In this, the scalar function ''λ'' is chosen such that \mathbf \nabla \cdot \mathbf A' = - \mu_0 \varepsilon_0 \frac , meaning that ''λ'' must satisfy the equation \nabla^2 \lambda - \mu_0 \varepsilon_0 \frac= - \mathbf \nabla \cdot \mathbf A - \mu_0 \varepsilon_0 \frac . The Lorenz gauge results in the following form of Maxwell's equations: \nabla^2 \varphi' - \mu_0 \varepsilon_0 \frac = -\Box^2 \varphi' = - \frac \nabla^2 \mathbf A' - \mu_0 \varepsilon_0 \frac = -\Box^2 \mathbf A' = - \mu_0 \mathbf J The operator \Box^2 is called the
d'Alembertian In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of Mi ...
(some authors denote this by only the square \Box). These equations are inhomogeneous versions of the
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...
, with the terms on the right side of the equation serving as the source functions for the wave. As with any wave equation, these equations lead to two types of solution: advanced potentials (which are related to the configuration of the sources at future points in time), and retarded potentials (which are related to the past configurations of the sources); the former are usually disregarded where the field is to analyzed from a causality perspective. As pointed out above, the Lorenz gauge is no more valid than any other gauge since the potentials cannot be directly measured, however the Lorenz gauge has the advantage of the equations being
Lorentz invariant In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
.


Extension to quantum electrodynamics

Canonical quantization In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the greatest extent possible. Historically, this was not quit ...
of the electromagnetic fields proceeds by elevating the scalar and vector potentials; ''φ''(x), A(x), from fields to field operators. Substituting into the previous Lorenz gauge equations gives: \nabla^2 \mathbf A - \frac 1 \frac = - \mu_0 \mathbf J \nabla^2 \varphi - \frac 1 \frac = - \frac Here, J and ''ρ'' are the current and charge density of the ''
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...
field''. If the matter field is taken so as to describe the interaction of electromagnetic fields with the Dirac electron given by the four-component
Dirac spinor In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain co ...
field ''ψ'', the current and charge densities have form: \mathbf=-e\psi^\boldsymbol\psi\,\quad \rho=-e\psi^\psi \,, where ''α'' are the first three
Dirac matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...
. Using this, we can re-write Maxwell's equations as: which is the form used in
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
.


Geometric algebra formulations

Analogous to the tensor formulation, two objects, one for the field and one for the current, are introduced. In
geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ...
(GA) these are
multivector In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra of a vector space . This algebra is graded, associative and alternating, and consists of linear combinations of simple -vectors ...
s. The field multivector, known as the
Riemann–Silberstein vector In mathematical physics, in particular electromagnetism, the Riemann–Silberstein vector or Weber vector named after Bernhard Riemann, Heinrich Martin Weber and Ludwik Silberstein, (or sometimes ambiguously called the "electromagnetic field") is ...
, is \mathbf = \mathbf + Ic\mathbf = E^k\sigma_k + IcB^k\sigma_k and the current multivector is c \rho - \mathbf = c \rho - J^k\sigma_k where, in the
algebra of physical space In physics, the algebra of physical space (APS) is the use of the Clifford algebra, Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a ...
(APS) C\ell_(\R) with the vector basis \. The unit
pseudoscalar In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not. Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. T ...
is I=\sigma_1\sigma_2\sigma_3 (assuming an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
). Orthonormal basis vectors share the algebra of the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
, but are usually not equated with them. After defining the derivative \boldsymbol = \sigma^k \partial_k Maxwell's equations are reduced to the single equation In three dimensions, the derivative has a special structure allowing the introduction of a cross product: \boldsymbol\mathbf = \boldsymbol \cdot \mathbf + \boldsymbol \wedge \mathbf = \boldsymbol \cdot \mathbf + I \boldsymbol \times \mathbf from which it is easily seen that Gauss's law is the scalar part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation. After expanding and rearranging, this can be written as \left( \boldsymbol \cdot \mathbf - \frac \right)- c \left( \boldsymbol \times \mathbf - \mu_0 \varepsilon_0 \frac - \mu_0 \mathbf \right)+ I \left( \boldsymbol \times \mathbf + \frac \right)+ I c \left( \boldsymbol \cdot \mathbf \right)= 0 We can identify APS as a subalgebra of the
spacetime algebra In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra . According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of spec ...
(STA) C\ell_(\mathbb), defining \sigma_k=\gamma_k\gamma_0 and I=\gamma_0\gamma_1\gamma_2\gamma_3. The \gamma_\mus have the same algebraic properties of the
gamma matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
but their matrix representation is not needed. The derivative is now \nabla = \gamma^\mu \partial_\mu. The Riemann–Silberstein becomes a bivector F = \mathbf + Ic\mathbf = E^1\gamma_1\gamma_0 + E^2\gamma_2\gamma_0 + E^3\gamma_3\gamma_0 -c(B^1\gamma_2\gamma_3 + B^2\gamma_3\gamma_1 + B^3\gamma_1\gamma_2), and the charge and current density become a vector J = J^\mu \gamma_\mu = c \rho \gamma_0 + J^k \gamma_k = \gamma_0(c \rho - J^k \sigma_k). Owing to the identity \gamma_0 \nabla = \gamma_0\gamma^0 \partial_0 + \gamma_0\gamma^k\partial_k = \partial_0 + \sigma^k\partial_k = \frac\dfrac + \boldsymbol, Maxwell's equations reduce to the single equation


Differential forms approach


Field 2-form

In
free space A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
, where and are constant everywhere, Maxwell's equations simplify considerably once the language of
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
and
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s is used. In what follows, cgs-Gaussian units, not
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
are used. (To convert to SI, see
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Television * Here TV (formerly "here!"), a ...
.) The electric and magnetic fields are now jointly described by a
2-form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
F in a 4-dimensional
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
manifold. The Faraday tensor F_ (
electromagnetic tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. ...
) can be written as a 2-form in Minkowski space with metric signature as \begin \mathbf & \equiv \fracF_ \mathrmx^ \wedge \mathrmx^ \\ & = B_x \mathrmy \wedge \mathrmz + B_y \mathrmz \wedge \mathrmx + B_z \mathrmx \wedge \mathrmy + E_x \mathrmx \wedge \mathrmt + E_y \mathrmy \wedge \mathrmt + E_z \mathrmz \wedge \mathrmt \end which, as the
curvature form In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie algeb ...
, is the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
of the
electromagnetic four-potential An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. W ...
, \mathbf = \mathrm \mathbf = ( \partial_ A_ ) \mathrmx^ \wedge \mathrmx^ The source free equations can be written by the action of the exterior derivative on this 2-form. But for the equations with source terms (
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it st ...
and the Ampère-Maxwell equation), the
Hodge dual In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the a ...
of this 2-form is needed. The Hodge star operator takes a ''p''-form to a ()-form, where ''n'' is the number of dimensions. Here, it takes the 2-form (''F'') and gives another 2-form (in four dimensions, ). For the basis cotangent vectors, the Hodge dual is given as (see ) ( \mathrmx \wedge \mathrmy ) = - \mathrmz \wedge \mathrmt ,\quad ( \mathrmx \wedge \mathrmt ) = \mathrmy \wedge \mathrmz, and so on. Using these relations, the dual of the Faraday 2-form is the Maxwell tensor, \mathbf = - B_x \mathrmx \wedge \mathrmt - B_y \mathrmy \wedge \mathrmt - B_z \mathrmz \wedge \mathrmt + E_x \mathrmy \wedge \mathrmz + E_y \mathrmz \wedge \mathrmx + E_z \mathrmx \wedge \mathrmy


Current 3-form, dual current 1-form

Here, the 3-form J is called the ''electric current form'' or '' current 3-form'': \mathbf = \rho\, \mathrmx \wedge \mathrmy \wedge \mathrmz - j_x \mathrmt \wedge \mathrmy \wedge \mathrmz - j_y \mathrmt \wedge \mathrmz \wedge \mathrmx - j_z \mathrmt \wedge \mathrmx \wedge \mathrmy with the corresponding dual 1-form: = -\rho\, \mathrmt + j_x \mathrmx + j_y \mathrmy + j_z \mathrmz Maxwell's equations then reduce to the Bianchi identity and the source equation, respectively: where d denotes the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
– a natural coordinate- and metric-independent differential operator acting on forms, and the (dual)
Hodge star In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of ...
operator is a linear transformation from the space of 2-forms to the space of (4 − 2)-forms defined by the metric in
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
(in four dimensions even by any metric conformal to this metric). The fields are in
natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
where . Since d2 = 0, the 3-form J satisfies the conservation of current (
continuity equation A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. ...
): \mathrm=\mathrm^2\mathbf=0. The current 3-form can be integrated over a 3-dimensional space-time region. The physical interpretation of this integral is the charge in that region if it is spacelike, or the amount of charge that flows through a surface in a certain amount of time if that region is a spacelike surface cross a timelike interval. As the exterior derivative is defined on any
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, the differential form version of the Bianchi identity makes sense for any 4-dimensional manifold, whereas the source equation is defined if the manifold is oriented and has a Lorentz metric. In particular the differential form version of the Maxwell equations are a convenient and intuitive formulation of the Maxwell equations in general relativity. ''Note:'' In much of the literature, the notations \mathbf and \mathbf are switched, so that \mathbf is a 1-form called the current and \mathbf is a 3-form called the dual current.


Linear macroscopic influence of matter

In a linear, macroscopic theory, the influence of matter on the electromagnetic field is described through more general linear transformation in the space of 2-forms. We call C:\Lambda^2\ni\mathbf\mapsto \mathbf\in\Lambda^ the constitutive transformation. The role of this transformation is comparable to the Hodge duality transformation. The Maxwell equations in the presence of matter then become: \mathrm\mathbf = 0 \mathrm\mathbf = \mathbf where the current 3-form J still satisfies the continuity equation . When the fields are expressed as linear combinations (of
exterior product In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of th ...
s) of basis forms ''θ''''p'', \mathbf = \fracF_\mathbf^p\wedge\mathbf^q. the constitutive relation takes the form G_ = C_^F_ where the field coefficient functions are antisymmetric in the indices and the constitutive coefficients are antisymmetric in the corresponding pairs. In particular, the Hodge duality transformation leading to the vacuum equations discussed above are obtained by taking C_^ = \fracg^g^ \varepsilon_ \sqrt which up to scaling is the only invariant tensor of this type that can be defined with the metric. In this formulation, electromagnetism generalises immediately to any 4-dimensional oriented manifold or with small adaptations any manifold.


Alternate metric signature

In the particle physicist's sign convention for the
metric signature In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...
, the potential 1-form is \mathbf = \phi\, \mathrmt - A_x \mathrmx - A_y \mathrmy - A_z \mathrmz . The Faraday curvature 2-form becomes \begin \mathbf \equiv & \fracF_ \mathrmx^ \wedge \mathrmx^ \\ = & E_x \mathrmt \wedge \mathrmx + E_y \mathrmt \wedge \mathrmy + E_z \mathrmt \wedge \mathrmz - B_x \mathrmy \wedge \mathrmz - B_y \mathrmz \wedge \mathrmx - B_z \mathrmx \wedge \mathrmy \end and the Maxwell tensor becomes = - E_x \mathrmy \wedge \mathrmz - E_y \mathrmz \wedge \mathrmx - E_z \mathrmx \wedge \mathrmy - B_x \mathrmt \wedge \mathrmx - B_y \mathrmt \wedge \mathrmy - B_z \mathrmt \wedge \mathrmz. The current 3-form J is \mathbf = - \rho\, \mathrmx \wedge \mathrmy \wedge \mathrmz + j_x \mathrmt \wedge \mathrmy \wedge \mathrmz + j_y \mathrmt \wedge \mathrmz \wedge \mathrmx + j_z \mathrmt \wedge \mathrmx \wedge \mathrmy and the corresponding dual 1-form is = -\rho\, \mathrmt + j_x \mathrmx + j_y \mathrmy + j_z \mathrmz . The current norm is now positive and equals = (\rho^2 + j_x^2 + j_y^2 + j_z^2)\,(1) with the canonical
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...
(1) = \mathrmt \wedge \mathrmx \wedge \mathrmy \wedge \mathrmz.


Curved spacetime


Traditional formulation

Matter and energy generate curvature of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
. This is the subject of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. Curvature of spacetime affects electrodynamics. An electromagnetic field having energy and momentum also generates curvature in spacetime. Maxwell's equations in curved spacetime can be obtained by replacing the derivatives in the equations in flat spacetime with
covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differe ...
s. (Whether this is the appropriate generalization requires separate investigation.) The sourced and source-free equations become ( cgs-Gaussian units): j^ = \partial_ F^ + _ F^ + _ F^ \ \stackrel\ \nabla_ F^ \ \stackrel\ _ \, \! and 0 = \partial_ F_ + \partial_ F_ + \partial_ F_ = \nabla_ F_ + \nabla_ F_ + \nabla_ F_.\, Here, _ is a
Christoffel symbol In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing dist ...
that characterizes the curvature of spacetime and ∇''α'' is the covariant derivative.


Formulation in terms of differential forms

The formulation of the Maxwell equations in terms of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s can be used without change in general relativity. The equivalence of the more traditional general relativistic formulation using the covariant derivative with the differential form formulation can be seen as follows. Choose local coordinates ''x''''α'' which gives a basis of 1-forms d''x''''α'' in every point of the open set where the coordinates are defined. Using this basis and cgs-Gaussian units we define *The antisymmetric field tensor ''F''''αβ'', corresponding to the field 2-form F \mathbf = \fracF_ \,\mathrmx^ \wedge \mathrmx^. *The current-vector infinitesimal 3-form J \mathbf = \left ( \frac j^ \sqrt \, \varepsilon_ \mathrmx^ \wedge \mathrmx^ \wedge \mathrmx^. \right) The epsilon tensor contracted with the differential 3-form produces 6 times the number of terms required. Here ''g'' is as usual the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the matrix representing the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
, ''g''''αβ''. A small computation that uses the symmetry of the
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
(i.e., the torsion-freeness of the
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...
) and the covariant constantness of the
Hodge star operator In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
then shows that in this coordinate neighborhood we have: *the Bianchi identity \mathrm\mathbf = 2(\partial_ F_ + \partial_ F_ + \partial_ F_)\mathrmx^\wedge \mathrmx^ \wedge \mathrmx^ = 0, *the source equation \mathrm = \frac_\sqrt \, \varepsilon_\mathrmx^ \wedge \mathrmx^ \wedge \mathrmx^ = \mathbf, *the continuity equation \mathrm\mathbf = _ \sqrt \, \varepsilon_\mathrmx^\wedge \mathrmx^ \wedge \mathrmx^ \wedge \mathrmx^ = 0.


Classical electrodynamics as the curvature of a line bundle

An elegant and intuitive way to formulate Maxwell's equations is to use complex
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the '' tangent bundle'' is a way of organisi ...
s or a principal U(1)-bundle, on the fibers of which
U(1) In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...
acts regularly. The principal U(1)- connection ∇ on the line bundle has a
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
F = ∇2 which is a two-form that automatically satisfies and can be interpreted as a field-strength. If the line bundle is trivial with flat reference connection ''d'' we can write and with A the
1-form In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ...
composed of the
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
and the
magnetic vector potential In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic ...
. In quantum mechanics, the connection itself is used to define the dynamics of the system. This formulation allows a natural description of the
Aharonov–Bohm effect The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential (φ, A), despite being confine ...
. In this experiment, a static magnetic field runs through a long magnetic wire (e.g., an iron wire magnetized longitudinally). Outside of this wire the magnetic induction is zero, in contrast to the vector potential, which essentially depends on the magnetic flux through the cross-section of the wire and does not vanish outside. Since there is no electric field either, the Maxwell tensor throughout the space-time region outside the tube, during the experiment. This means by definition that the connection ∇ is flat there. However, as mentioned, the connection depends on the magnetic field through the tube since the
holonomy In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geomet ...
along a non-contractible curve encircling the tube is the magnetic flux through the tube in the proper units. This can be detected quantum-mechanically with a double-slit electron diffraction experiment on an electron wave traveling around the tube. The holonomy corresponds to an extra phase shift, which leads to a shift in the diffraction pattern.


Discussion

Following are the reasons for using each of such formulations.


Potential formulation

In advanced classical mechanics it is often useful, and in quantum mechanics frequently essential, to express Maxwell's equations in a ''potential formulation'' involving the
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
(also called
scalar potential In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
) ''φ'', and the magnetic potential (a
vector potential In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector field v, a ''vecto ...
) A. For example, the analysis of radio antennas makes full use of Maxwell's vector and scalar potentials to separate the variables, a common technique used in formulating the solutions of differential equations. The potentials can be introduced by using the Poincaré lemma on the homogeneous equations to solve them in a universal way (this assumes that we consider a topologically simple, e.g.
contractible space In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that ...
). The potentials are defined as in the table above. Alternatively, these equations define E and B in terms of the electric and magnetic potentials which then satisfy the homogeneous equations for E and B as identities. Substitution gives the non-homogeneous Maxwell equations in potential form. Many different choices of A and ''φ'' are consistent with given observable electric and magnetic fields E and B, so the potentials seem to contain more, ( classically) unobservable information. The non uniqueness of the potentials is well understood, however. For every scalar function of position and time , the potentials can be changed by a
gauge transformation In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
as \varphi' = \varphi - \frac, \quad \mathbf A' = \mathbf A + \mathbf \nabla \lambda without changing the electric and magnetic field. Two pairs of gauge transformed potentials and are called ''gauge equivalent'', and the freedom to select any pair of potentials in its gauge equivalence class is called
gauge freedom In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct c ...
. Again by the Poincaré lemma (and under its assumptions), gauge freedom is the only source of indeterminacy, so the field formulation is equivalent to the potential formulation if we consider the potential equations as equations for gauge equivalence classes. The potential equations can be simplified using a procedure called
gauge fixing In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...
. Since the potentials are only defined up to gauge equivalence, we are free to impose additional equations on the potentials, as long as for every pair of potentials there is a gauge equivalent pair that satisfies the additional equations (i.e. if the gauge fixing equations define a slice to the gauge action). The gauge-fixed potentials still have a gauge freedom under all gauge transformations that leave the gauge fixing equations invariant. Inspection of the potential equations suggests two natural choices. In the
Coulomb gauge In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...
, we impose which is mostly used in the case of magneto statics when we can neglect the term. In the
Lorenz gauge In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ha ...
(named after the Dane
Ludvig Lorenz Ludvig Valentin Lorenz (; 18 January 1829 – 9 June 1891) was a Danish physicist and mathematician. He developed mathematical formulae to describe phenomena such as the relation between the refraction of light and the density of a pure transp ...
), we impose \mathbf \nabla \cdot \mathbf A + \frac \frac = 0\,. The Lorenz gauge condition has the advantage of being Lorentz invariant and leading to Lorentz-invariant equations for the potentials.


Manifestly covariant (tensor) approach

Maxwell's equations are exactly consistent with
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
—i.e., if they are valid in one inertial reference frame, then they are automatically valid in every other inertial reference frame. In fact, Maxwell's equations were crucial in the historical development of special relativity. However, in the usual formulation of Maxwell's equations, their consistency with special relativity is not obvious; it can only be proven by a laborious calculation. For example, consider a conductor moving in the field of a magnet.Albert Einstein (1905) ''On the electrodynamics of moving bodies'' In the frame of the magnet, that conductor experiences a ''magnetic'' force. But in the frame of a conductor moving relative to the magnet, the conductor experiences a force due to an ''electric'' field. The motion is exactly consistent in these two different reference frames, but it mathematically arises in quite different ways. For this reason and others, it is often useful to rewrite Maxwell's equations in a way that is "manifestly covariant"—i.e. ''obviously'' consistent with special relativity, even with just a glance at the equations—using covariant and contravariant four-vectors and tensors. This can be done using the EM tensor F, or the 4-potential A, with the
4-current In special and general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the electric current density. Also known as vector current, it is used in the geometric context of ''four-dimensional spa ...
J – see
covariant formulation of classical electromagnetism The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformatio ...
.


Differential forms approach

Gauss's law for magnetism and the Faraday–Maxwell law can be grouped together since the equations are homogeneous, and be seen as
geometric Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
''identities'' expressing the ''field'' F (a 2-form), which can be derived from the ''4-potential'' A. Gauss's law for electricity and the Ampere–Maxwell law could be seen as the ''dynamical
equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (V ...
'' of the fields, obtained via the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
principle of
least action The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the ''action'' of a mechanical system, yields the equations of motion for that system. The principle states that ...
, from the "interaction term" AJ (introduced through
gauge Gauge ( or ) may refer to: Measurement * Gauge (instrument), any of a variety of measuring instruments * Gauge (firearms) * Wire gauge, a measure of the size of a wire ** American wire gauge, a common measure of nonferrous wire diameter, es ...
covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differe ...
s), coupling the field to matter. For the field formulation of Maxwell's equations in terms of a principle of extremal
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
, see
electromagnetic tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. ...
. Often, the time derivative in the Faraday–Maxwell equation motivates calling this equation "dynamical", which is somewhat misleading in the sense of the preceding analysis. This is rather an artifact of breaking relativistic
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
by choosing a preferred time direction. To have physical degrees of freedom propagated by these field equations, one must include a
kinetic term In physics, a kinetic term is the part of the Lagrangian that is bilinear in the fields (and for nonlinear sigma models, they are not even bilinear), and usually contains two derivatives with respect to time (or space); in the case of fermions, ...
for A, and take into account the non-physical degrees of freedom that can be removed by
gauge transformation In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
. See also
gauge fixing In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...
and
Faddeev–Popov ghost In physics, Faddeev–Popov ghosts (also called Faddeev–Popov gauge ghosts or Faddeev–Popov ghost fields) are extraneous fields which are introduced into gauge quantum field theories to maintain the consistency of the path integral formu ...
s.


Geometric calculus approach

This formulation uses the algebra that
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
generates through the introduction of a distributive, associative (but not commutative) product called the
geometric product In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the g ...
. Elements and operations of the algebra can generally be associated with geometric meaning. The members of the algebra may be decomposed by grade (as in the formalism of differential forms) and the (geometric) product of a vector with a ''k''-vector decomposes into a -vector and a -vector. The -vector component can be identified with the inner product and the -vector component with the outer product. It is of algebraic convenience that the geometric product is invertible, while the inner and outer products are not. The derivatives that appear in Maxwell's equations are vectors and electromagnetic fields are represented by the Faraday bivector F. This formulation is as general as that of differential forms for manifolds with a metric tensor, as then these are naturally identified with ''r''-forms and there are corresponding operations. Maxwell's equations reduce to one equation in this formalism. This equation can be separated into parts as is done above for comparative reasons.


See also

*
Ricci calculus In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be ...
*
Electromagnetic wave equation The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous fo ...
*
Speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
*
Electric constant Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
*
Magnetic constant The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constan ...
*
Free space A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
*
Near and far field The near field and far field are regions of the electromagnetic (EM) field around an object, such as a transmitting antenna, or the result of radiation scattering off an object. Non-radiative ''near-field'' behaviors dominate close to the ante ...
*
Electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
*
Electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
*
Quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
*
List of electromagnetism equations This article summarizes equations in the theory of electromagnetism. Definitions Here subscripts ''e'' and ''m'' are used to differ between electric and magnetic charges. The definitions for monopoles are of theoretical interest, although re ...


Notes


References

* * (with worked problems in Warnick, Russer 2006 ) * * {{cite book, last1=Doran, first1=Chris, last2=Lasenby, first2=Anthony, title=Geometric Algebra for Physicists, date=2007, publisher=Cambridge Univ. Press, isbn=978-0-521-71595-9 Electromagnetism Mathematical physics