Originally published in:

*Encyclopedia of Planetary Sciences*, edited by J. H. Shirley and
R. W. Fainbridge,

468-469, __Chapman and Hall__, New York, 1997.

The solid, liquid and gaseous states of matter with which we are most familiar are accompanied by a fourth state in the space environment. The upper atmosphere of the planets, and the entire solar system, is filled with a rarefied plasma, in which the energies of the charged particles which make up this medium, mostly protons and electrons, are so great that the electrical forces that bind these particles together as atoms are overcome. Although these plasmas are generally so rarified that they do not interact through normal collisions, and hence are termed collisionless, they do interact with each other through the electromagnetic force. The laws which describe the behavior of the electromagnetic force were first compiled into a self- consistent theory in 1864 by the Scottish physicist James Clerk Maxwell (1831-1879), and since then the theory of electromagnetism has been called Maxwell theory, and the four key equations of electromagnetism called Maxwell's laws or Maxwell's equations.

The theory of electromagnetism divides its effects on electrically charged particles into two categories, electric and magnetic. The total force acting on a charged particle is given by the vector sum of the electric and magnetic forces. This is defined as the Lorentz force, given in System Internationale (ST) and Gaussian units, respectively, by

where *F* is the electromagnetic force (in newtons), *q* is the magnitude of
electric charge of a given particle (in coulombs), *E* is the electric field
vector (in V m^{-1}), *v* is the velocity of the particle
(in m s^{-1}), *c* is the speed of light (in m s^{-1}),
and B is the magnetic field vector (in tesla). The equation on
the left side of expression (**1**) and quantities in parenthesis are in SI
units. However, for historical reference, equations will also be given in
Gaussian units, in which the electric fields and magnetic fields have
equivalent units. In Gaussian units, electric fields are measured in
statvolts, magnetic fields in gauss, and the electromagnetic force in dyne
cm^{-2}. This system of units is no longer in widespread use, but has been
utilized extensively in previous work and can be frequently found in the
older literature.

In a plasma, especially, the electric and magnetic fields which act on
charged particles are not separate entities, but are self-consistently related
to one another. There are four basic laws which govern the behavior of the
electric and magnetic fields, which are known collectively as Maxwell's
equations (e.g. Jackson, 1962; Dendy, 1990). The first of these laws is
Gauss' law, named after Carl Friedrich Gauss (1777-1855, Germany),
which relates the electric field *E* to the electric charge density _{q} of a
volume

where _{0} is the permittivity of free space.
This expression shows that the
amount of total electric flux through a given closed surface is proportional
to the amount of electric charge in the volume contained by that surface.
This also implies that a particle containing a given electric charge has an
electric field associated with it.

The second of Maxwell's equations tells of the continuity of magnetic
flux through a surface. Sometimes referred to as Gauss' law for magnetic
fields, this expression states that the magnetic field *B* is divergenceless,
and is given by

This law is similar to Gauss' law for electric fields, since it tells us about the amount of total magnetic flux through a given closed surface, which is zero. 'This states that all magnetic field lines which enter a particular closed surface must eventually leave the surface; thus there are no magnetic monopoles or sources of 'magnetic charge'.

Michael Faraday (1791-1867, England) discovered the law of electromagnetic induction, which describes how a magnetic field that changes in time can also act as a source for the electric field. Faraday's law is given by

Andre-Marie Ampere (1775-1836, France) discovered that current was a
source of the magnetic field, thus the magnetic field is related to the
current density *j* (in A m^{-2}) by

where µ_{0} is the permeability of free space and is related to
_{0} and the
velocity of light *c* by the relation

Ampere's law implies that electric charge is conserved since, if we take
the divergence of both sides of (**5**), we get

However, Maxwell noticed that · *j* = 0
is only valid for steady state
situations and that the complete relation for the continuity of electric
charge also includes the variation of the electric charge density
_{q} with
time, which is given by

With this knowledge, Maxwell modified Ampere's law to relate the magnetic field to time-varying electric fields, as well as to the current density, obtaining

The second term of Ampere's law is called the Maxwell displacement
current. Maxwell showed that it was needed in order to combine self-consistently
the laws of electromagnetism. It was for this insight
that equations (**2**), (**3**), (**4**) and (**9**) which explain the
theory of electric and magnetic fields became known as Maxwell's
equations.

These equations enable us to study electromagnetic phenomena ranging from waves, such as light waves in a vacuum, to the effect of the solar wind on a planetary magnetosphere. We note that particles having electric charge are subject to the forces which act upon them by the electric and magnetic fields. However, the electric charges contained by the particles are the source of the total electric field, and the motion of the charged particles, which creates electric current, is the source of the total magnetic field. Thus the charged particles in a plasma are constantly interacting with each other through the electric and magnetic fields which they help to create.

Dendy, R. O. (1990) *Plasma Dynamics*. Oxford: Clarendon Press.

Jackson, J. D. (1962) *Classical Electrodynamics*. New York: John Wiley.

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