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 describes electric charge (density), $\rho $, as the source of the electric displacement field, $\mathbf{D}$.

Here is the Maxwell-Gauß equation:

$$\mathbf{\nabla}\cdot \mathbf{D}=\rho $$The Maxwell-Ampère equation describes electric current (density), $\mathbf{J}$, and time variations of the electric displacement field, $\mathbf{D}$, as the sources of the magnetic induction field, $\mathbf{H}$.

Here is the Maxwell-Ampère equation:

$$\mathbf{\nabla}\times \mathbf{H}-\frac{\partial \mathbf{D}}{\partial t}=\mathbf{J}$$The Maxwell-flux equation asserts the conservative nature (the nondivergence) of the magnetic field, $\mathbf{B}$.

Here is the Maxwell-flux equation:

$$\mathbf{\nabla}\cdot \mathbf{B}=0$$The Maxwell-Faraday equation asserts the induction law — a varying magnetic field creates an induction potential — by relating the curl of the electric field, $\mathbf{E}$, to the time variation of the magnetic field, $\mathbf{B}$.

Here is the Maxwell-Faraday equation:

$$\mathbf{\nabla}\times \mathbf{E}+\frac{\partial \mathbf{B}}{\partial t}=\mathbf{0}$$The conservation equations (the Maxwell-flux and Maxwell-Faraday equations) make it possible to write the electric and magnetic fields, $\mathbf{E}$ and $\mathbf{B}$, from a scalar potential—the electrical potential, $\phi $—and a vector potential—the magnetic potential, $\mathbf{A}$. These are not uniquely defined but only up to a gauge choice condition.

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

$$\mathbf{E}=-\mathbf{\nabla}\phi -\frac{\partial \mathbf{A}}{\partial t}$$The magnetic field is the curl of the magnetic potential:

$$\mathbf{B}=\mathbf{\nabla}\times \mathbf{A}$$The permittivity $\epsilon $ of an isotropic medium relates the electric displacement field, $\mathbf{D}$, with the electric field, $\mathbf{E}$:

$$\mathbf{D}=\epsilon \mathbf{E}$$In vacuum, in the

The permeability $\mu $ of an isotropic medium relates the magnetic induction field, $\mathbf{H}$, with the magnetic field, $\mathbf{B}$:

$$\mathbf{H}=\frac{1}{\mu}\mathbf{B}$$In vacuum, in the

The conductivity $\sigma $ of a conductive medium relates the electric current density, $\mathbf{J}$, with the electric field, $\mathbf{E}$, through Ohm's law:

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

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

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

$$\mathbf{S}=\mathbf{E}\times \mathbf{H}$$