# 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), $\rho$, 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

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:

$\mathbf{\nabla }×\mathbf{H}-\frac{\partial \mathbf{D}}{\partial t}=\mathbf{J}$

### The Maxwell-flux equation

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, $\mathbf{B}$.

$\mathbf{\nabla }×\mathbf{E}+\frac{\partial \mathbf{B}}{\partial t}=\mathbf{0}$

## Potential equations

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 scalar potential

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 vector potential

The magnetic field is the curl of the magnetic potential:

$\mathbf{B}=\mathbf{\nabla }×\mathbf{A}$

## Solid media

### Permittivity equation for isotropic media

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 SI (Système International), the permittivity is a defined constant: the permittivity of free space, ${\epsilon }_{0}=\frac{625000}{22468879468420441\pi }\mathrm{F}\cdot {\mathrm{m}}^{-1}\simeq 8.854187817620\cdot {10}^{-12}\mathrm{F}\cdot {\mathrm{m}}^{-1}$ (this follows from the definition of the ampere and of the meter).

### Permeability equation for isotropic media

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 SI (Système International), the permeability is a defined constant: the permeability of free space, ${\mu }_{0}=4\pi \cdot {10}^{-7}\mathrm{N}\cdot {\mathrm{A}}^{-2}\simeq 1.256637061436\cdot {10}^{-6}\mathrm{N}\cdot {\mathrm{A}}^{-2}$ (this follows from the definition of the ampere).

### Ohm's law for conductive media

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

$\mathbf{J}=\sigma \mathbf{E}$

## Energy density and flux

### Energy density

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:

$\mathcal{U}=\frac{1}{2}\left(\mathbf{E}\cdot \mathbf{D}+\mathbf{B}\cdot \mathbf{H}\right)$

—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:

$\mathbf{S}=\mathbf{E}×\mathbf{H}$