Electromagnetic relations

Maxwell's equations

Maxwell's four equations, respectively called the Maxwell-Gauß, Maxwell-Ampère, Maxwell-flux and Maxwell-Faraday equations for ease of reference, can be grouped in various ways in two sets of two equations: the Maxwell-Gauß and Maxwell-Ampère equations are source equations, in that they relate the fields to their sources, while the Maxwell-flux and Maxwell-Faraday equations are conservation equations, in that they assert certain conservation properties on the fields, thereby guaranteeing the existence of certain potentials. Insofar as the separation makes sense, the Maxwell-Gauß and Maxwell-Faraday equations deal primarily with electric phenomena, and the Maxwell-Ampère and Maxwell-flux equations with magnetic phenomena. Furthermore, the Maxwell-Gauß and Maxwell-flux equations are divergence equations (and impose one scalar condition on the fields) whereas Maxwell-Ampère and Maxwell-Faraday are curl equations (and impose three independent scalar conditions on the fields).

The Maxwell-Gauß equation

The Maxwell-Gauß equation describes electric charge (density), $ρ$ , as the source of the electric displacement field, $D$ .

Here is the Maxwell-Gauß equation:

\nabla \cdot D = ρ

The Maxwell-Ampère equation

The Maxwell-Ampère equation describes electric current (density), $J$ , and time variations of the electric displacement field, $D$ , as the sources of the magnetic induction field, $H$ .

Here is the Maxwell-Ampère equation:

\nabla \times H - \frac{\partial D}{\partial t} = J

The Maxwell-flux equation

The Maxwell-flux equation asserts the conservative nature (the nondivergence) of the magnetic field, $B$ .

Here is the Maxwell-flux equation:

\nabla \cdot B = 0

The Maxwell-Faraday equation

The Maxwell-Faraday equation asserts the induction law — a varying magnetic field creates an induction potential — by relating the curl of the electric field, $E$ , to the time variation of the magnetic field, $B$ .

Here is the Maxwell-Faraday equation:

\nabla \times E + \frac{\partial B}{\partial t} = 0

Potential equations

The conservation equations (the Maxwell-flux and Maxwell-Faraday equations) make it possible to write the electric and magnetic fields, $E$ and $B$ , from a scalar potential—the electrical potential, $φ$ —and a vector potential—the magnetic potential, $A$ . These are not uniquely defined but only up to a gauge choice condition.

The electric scalar potential

The electric field is the negative gradient of the electric potential minus the time variation of the magnetic potential:

E = - \nabla φ - \frac{\partial A}{\partial t}

The magnetic vector potential

The magnetic field is the curl of the magnetic potential:

B = \nabla \times A

Solid media

Permittivity equation for isotropic media

The permittivity $ε$ of an isotropic medium relates the electric displacement field, $D$ , with the electric field, $E$ :

D = ε E

In vacuum, in the SI (Système International), the permittivity is a defined constant: the permittivity of free space, $ε_{0} = \frac{625000}{22468879468420441 π} F \cdot m^{-1} ≃ 8.854187817620 \cdot 10^{-12} F \cdot m^{-1}$ (this follows from the definition of the ampere and of the meter).

Permeability equation for isotropic media

The permeability $μ$ of an isotropic medium relates the magnetic induction field, $H$ , with the magnetic field, $B$ :

H = \frac{1}{μ} B

In vacuum, in the SI (Système International), the permeability is a defined constant: the permeability of free space, $μ_{0} = 4 π \cdot 10^{-7} N \cdot A^{-2} ≃ 1.256637061436 \cdot 10^{-6} N \cdot A^{-2}$ (this follows from the definition of the ampere).

Ohm's law for conductive media

The conductivity $σ$ of a conductive medium relates the electric current density, $J$ , with the electric field, $E$ , through Ohm's law:

J = σ E

Energy density and flux

Energy density

The density of electromagnetic energy in a linear medium (that is, one where the electric displacement field, $D$ , is proportional to the electric field $E$ , and similarly the magnetic induction field, $H$ , is proportional to the magnetic field $B$ ; for example, vacuum), is given by:

U = \frac{1}{2} (E \cdot D + B \cdot H)

—where the first term represents electric energy and the second one magnetic energy.

Energy flux: the Poynting vector

The Poynting vector, representing the density of energy flux outside those brought by electric currents, is given by:

S = E \times H