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.
The behaviour of electric and magnetic fields, whether in cases of electrostatics, magnetostatics, or electrodynamics (electromagnetic fields), is governed by Maxwell-Heaviside's equations: 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.
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 with the permeability and permittivity of the linear material in question.
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.
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:
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, i.e. the impossibility of information, signals, or anything travelling faster than the speed of light.
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.
Analogous to the tensor formulation, two objects, one for the electromagnetic field and one for the current density, are introduced.
These basis vectors share the algebra of the Pauli matrices, but are usually not equated with them, as they are different objects with different interpretations.
In three dimensions, the derivative has a special structure allowing the introduction of a cross product:
By Einstein notation, we implicitly take the sum over all values of the indices that can vary within the dimension.
That F is a closed form, and the exterior derivative of its Hodge dual is the current 3-form, express Maxwell's equations:[4]
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.
When the fields are expressed as linear combinations (of exterior products) of basis forms θi,
The formulation of the Maxwell equations in terms of differential forms can be used without change in general relativity.
Using this basis and cgs-Gaussian units we define The epsilon tensor contracted with the differential 3-form produces 6 times the number of terms required.
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: An elegant and intuitive way to formulate Maxwell's equations is to use complex line bundles or a principal U(1)-bundle, on the fibers of which U(1) acts regularly.
The principal U(1)-connection ∇ on the line bundle has a curvature F = ∇2, which is a two-form that automatically satisfies dF = 0 and can be interpreted as a field strength.
Since there is no electric field either, the Maxwell tensor F = 0 throughout the space-time region outside the tube, during the experiment.
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).
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.
For every scalar function of position and time λ(x, t), the potentials can be changed by a gauge transformation as
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 Lorenz gauge condition has the advantage of being Lorentz invariant and leading to Lorentz-invariant equations for the potentials.
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.
obviously consistent with special relativity, even with just a glance at the equations—using covariant and contravariant four-vectors and tensors.
For the field formulation of Maxwell's equations in terms of a principle of extremal action, see electromagnetic tensor.
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.