Maxwell's equations (vector fields)
{displaystyle nabla cdot mathbf {E} ={frac {rho }{varepsilon _{0}}}}	Gauss's law
{displaystyle nabla cdot mathbf {B} =0}	Gauss's law for magnetism
{displaystyle nabla times mathbf {E} =-{frac {partial mathbf {B} }{partial t}}}	Faraday's law
{displaystyle nabla times mathbf {B} =mu _{0}mathbf {J} +mu _{0}varepsilon _{0}{frac {partial mathbf {E} }{partial t}}}	Ampère–Maxwell law

where ρ is the charge density, which can (and often does) depend on time and position,ε₀ is the electric constant, μ₀ is the magnetic constant, and J is the current per unit area, also a function of time and position. The equations take this form with theInternational System of Quantities.

Many times in the use and calculation of electric and magnetic fields, the approach used first computes an associated potential: the electric potential, {displaystyle varphi }, for the electric field, and the magnetic potential, A, for the magnetic field. The electric potential is a scalar field, while the magnetic potential is a vector field.

These equations can be simplified by taking advantage of the fact that only 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. Specifically for these equations, for any choice of a twice-differentiable scalar function of position and time λ, if (φ, A) is a solution for a given system, then so is another potential (φ′, A′) given by:

The Coulomb gauge is chosen in such a way that {displaystyle 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

This choice of function results in the following formulation of Maxwell's equations:

A gauge that is often used is the Lorenz gauge. In this, the scalar function λ is chosen such that

meaning that λ must satisfy the equation

The Lorenz gauge results in the following form of Maxwell's equations:

The operator {displaystyle Box ^{2}} is called the d'Alembertian (some authors denote this by only the square{displaystyle Box }). These equations are inhomogeneous versions of the wave equation, with the terms on the right side of the equation serving as the source functions for the wave.

Analogous to the tensor formulation, two objects, one for the field and one for the current, are introduced. In geometric algebra (GA) these are multivectors. The field multivector, known as the Riemann–Silberstein vector, is

and the current multivector is

where, in the algebra of physical space (APS) {displaystyle Cell _{3,0}(mathbb {R} )} with the vector basis {displaystyle {sigma _{k}}}. The unit pseudoscalar is {displaystyle I=sigma _{1}sigma _{2}sigma _{3}} (assuming an orthonormal basis). Orthonormal basis vectors share the algebra of the Pauli matrices, but are usually not equated with them. After defining the derivative

Maxwell's equations are reduced to a single equation,^[3]

In free space, where ε = ε₀ and μ = μ₀ are constant everywhere, Maxwell's equations simplify considerably once the language of differential geometry and differential formsis used. In what follows, cgs-Gaussian units, not SI units are used. (To convert to SI, see here.) The electric and magnetic fields are now jointly described by a 2-form F in a 4-dimensional spacetime manifold. The Faraday tensor {displaystyle F_{mu nu }} can be written as a 2-form in Minkowski space with metric signature (- + + +) as

which, as the curvature form, is the exterior derivative of the electromagnetic four-potential,

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 and the Ampère-Maxwell equation), the Hodge dual of this 2-form is needed. The Hodge 'star' dual takes a p-form to a (n − p)-form, where n is the number of dimensions. Here, it takes the 2-form (F) and gives another 2-form (in four dimensions, n − p = 4 − 2 = 2). For the basis cotangent vectors, the Hodge dual is given as (see here)

and so on. Using these relations, the dual of the Faraday 2-form is the Maxwell tensor,

Here, the 3-form J is called the electric current form or current 3-form:

with the corresponding dual 1-form:

Maxwell's equations then reduce to the Bianchi identity and the source equation, respectively:

In the literature, the current is usually defined as a 1-form (i.e. the hodge dual of the 3-form J above):

with the corresponding dual 3-form:

In terms of these forms, the Maxwell equations are:^[4]

Curved spacetime[edit]

Traditional formulation[edit]

Matter and energy generate curvature of spacetime. This is the subject of general relativity. 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 derivatives. (Whether this is the appropriate generalization requires separate investigation.) The sourced and source-free equations become (cgs-Gaussian units):

{displaystyle {4pi over c}j^{beta }=partial _{alpha }F^{alpha beta }+{Gamma ^{alpha }}_{mu alpha }F^{mu beta }+{Gamma ^{beta }}_{mu alpha }F^{alpha mu } {stackrel {mathrm {def} }{=}} nabla _{alpha }F^{alpha beta } {stackrel {mathrm {def} }{=}} {F^{alpha beta }}_{;alpha },!}

and

{displaystyle 0=partial _{gamma }F_{alpha beta }+partial _{beta }F_{gamma alpha }+partial _{alpha }F_{beta gamma }=nabla _{gamma }F_{alpha beta }+nabla _{beta }F_{gamma alpha }+nabla _{alpha }F_{beta gamma }.,}

Here,

{displaystyle {Gamma ^{alpha }}_{mu beta }!}

is a Christoffel symbol that characterizes the curvature of spacetime and ∇_α is the covariant derivative.

Formulation in terms of differential forms[edit]

The formulation of the Maxwell equations in terms of differential forms 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 dx^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

{displaystyle {mathbf {F}}={frac {1}{2}}F_{alpha beta },mathrm {d} x^{alpha }wedge mathrm {d} x^{beta }.}

• The current-vector infinitesimal 3-form J

{displaystyle {mathbf {J}}={4pi over c}left({frac {1}{6}}j^{alpha }{sqrt {-g}},epsilon _{alpha beta gamma delta }mathrm {d} x^{beta }wedge mathrm {d} x^{gamma }wedge mathrm {d} x^{delta }.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 of the matrix representing the metric tensor, g_αβ. A small computation that uses the symmetry of the Christoffel symbols (i.e., the torsion-freeness of the Levi-Civita connection) and the covariant constantness of the Hodge star operator then shows that in this coordinate neighborhood we have:

• the Bianchi identity

{displaystyle mathrm {d} {mathbf {F}}=2(partial _{gamma }F_{alpha beta }+partial _{beta }F_{gamma alpha }+partial _{alpha }F_{beta gamma })mathrm {d} x^{alpha }wedge mathrm {d} x^{beta }wedge mathrm {d} x^{gamma }=0,}

• the source equation

{displaystyle mathrm {d} {star {mathbf {F}}}={frac {1}{6}}{F^{alpha beta }}_{;alpha }{sqrt {-g}},epsilon _{beta gamma delta eta }mathrm {d} x^{gamma }wedge mathrm {d} x^{delta }wedge mathrm {d} x^{eta }={mathbf {J}},}

• the continuity equation

{displaystyle mathrm {d} {mathbf {J}}={4pi over c}{j^{alpha }}_{;alpha }{sqrt {-g}},epsilon _{alpha beta gamma delta }mathrm {d} x^{alpha }wedge mathrm {d} x^{beta }wedge mathrm {d} x^{gamma }wedge mathrm {d} x^{delta }=0.}