====== Electromagnetic field equations ======

There are two types of coupling between electric charge and electromagnetic field. A charge at rest creates an electrostatic field $\mathbf{E}$ from which comes the Coulomb force:

$$ \mathbf{F}_C = q\mathbf{E} $$

where charge $q$ will experience the combined electric field strength of all other charges. The second coupling is that a charge moving with velocity $\mathbf{v}$ acts as a current that gives rise to the magnetic field $\mathbf{B}$ and the associated Lorentz force:

$$ \mathbf{F}_L = q(\mathbf{v}\times \mathbf{B}) $$

Where once again the magnetic field created by all the charges must be captured. These two equations explain the effect of external field on charge. But a charge itself creates a kind of 'internal' field which can be explained by [[maxwell|Maxwell's equations]]:

$$ \nabla \times \mathbf{B} = \mu_0 \mathbf{j} + \epsilon_0 \mu_0 \frac{\partial \mathbf{E}}{\partial t} $$

$$ \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} $$

where Nabla $\nabla\times$ is the [[vector-field|curl]] operator, $\mathbf{j}$ is the electric current density, $\epsilon_0$ and $\mu_0$ are the permittivity and permeability of the vacuum, and $t$ is the time . That means $\nabla\times \mathbf{E}$ is the measure of circular motion or circulation in the three-dimensional [[vector-field|vector field]] named $\mathbf{E}$.

The first equation, known as the Ampere-Maxwell law, states that the spatial variation of the magnetic field depends on the current, and the temporal variation (oscillation) of the electric field. The second equation, known as Faraday's Law, states that the spatial variation (or circulation) of the electric field depends on the temporal variation of the magnetic field. The first also says that the electric current is the source of the magnetic field, and the source of the fast oscillation in the electric field.

The first equation can be further simplified if we remember $\epsilon_0 \mu_0=c^{-2}$ where $c$ is the speed of light. If the speed of light is very high, the product of permittivity and permeability will be so small that the second term of this equation can be ignored, unless the electric field oscillations are very fast and large. This is true in vacuum or near-vacuum. Putting the two equations together, the oscillation of one field depends on the curl of the other field, and this interdependence creates light or electromagnetic waves.

Two more equations need to be added as conditions:

$$ \nabla\cdot \mathbf{B} = 0 $$
$$ \nabla\cdot \mathbf{E} = \frac{\rho}{\epsilon_0} $$

where $\nabla\cdot$ is the [[vector-field|divergence]] operator, and for a plasma the electric space charge density is $\rho=e(n_i-n_e)$ and $n_i$ is the ion number density and $n_e$ Electron number density. That means $\nabla\cdot \mathbf{E}$ is the measure of the inward motion as the sink and the outward motion as the source in the vector field of electricity.

Here, the first condition known as Gauss's Law of Magnetism states that the magnetic field has no source or sink, meaning that the magnetic field line is always closed, ending where it starts. If we think of the magnetic field metaphorically as a fluid, then this fluid must be called incompressible, which cannot be compressed. The second condition, known simply as Gauss's Law, states that the source of the electric field is the electric charge, because the source or sink of a vector field is measured by divergence; Positive charge can be thought of as source and negative as sink. By comparing these two equations we say, there is no magnetic monopole like electric monopole (separate charges), magnetic poles are always in pairs.

Current density, like charge density in plasma, can be defined as: $\mathbf{j}=e(n_i\mathbf{v}_i-n_e\mathbf{v}_e)$ where $\mathbf{v}$ is again the velocity, and Thus $n_i\mathbf{v}_i$ is the electron flux.