Maxwell's Equations - Alternative Formulations of Maxwell's Equations

Alternative Formulations of Maxwell's Equations

Following is a summary of the numerous other ways to write the equations, in vacuum, showing they can be collected together in simpler and more unified formulae, though in terms of more complicated mathematics. See the main articles for the details of each formulation. SI units are used, not Gaussian.

Formulation	Homogeneous equations		Nonhomogeneous equations
Vector calculus (fields)
Vector calculus (potentials, any gauge)	identities
QED, vector calculus (potentials, Lorenz gauge)	identities
Tensor calculus (potentials, Lorenz gauge)	identities
Tensor calculus (fields)
Differential forms (fields)			(current 1-form) (current 3-form)
Spacetime algebra (fields)
Algebra of physical space (fields)

where

is the D'Alembert operator. Following are the reasons for using such formulations:

• Potential formulation approach: 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 (also called scalar potential) φ, and the magnetic potential A, (also called vector potential). These are defined such that:

Many different choices of A and φ are consistent with a given E and B, making these choices physically equivalent – a flexibility known as gauge freedom. Suitable choice of A and φ can simplify these equations, or can adapt them to suit a particular situation.

• Manifestly covariant (tensor) approach: Maxwell's equations are exactly consistent with special relativity—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 Maxwell's equations, their consistency with special relativity is not obvious; it can only be proven by a laborious calculation that involves a seemingly miraculous cancellation of different terms.

For example, consider a conductor moving in the field of a magnet. 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 J – see covariant formulation of classical electromagnetism.

• 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 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 of the fields, obtained via the Lagrangian principle of least action, from the "interaction term" A J (introduced through gauge covariant derivatives), coupling the field to matter. For the field formulation of Maxwell's equations in terms of a principle of extremal action, see electromagnetic tensor.

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 by choosing a preferred time direction. To have physical degrees of freedom propagated by these field equations, one must include a kinetic term F *F for A; and take into account the non-physical degrees of freedom which can be removed by gauge transformation A → A' = A − dα. See also gauge fixing and Faddeev–Popov ghosts.

• Geometric algebra summarizes the entire content of Maxwell's equations into a single equation, using the Riemann–Silberstein multivector F and the four-current J.