# MAXWELL THEORY

### M. H. Farris and C. T. Russell

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

(1)

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 However, for historical reference, equations will also be given in Gaussian units, in which the electric fields and magnetic fields have equivalent units.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

(2)

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

(3)

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

(4)

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

(5)

where µ0 is the permeability of free space and is related to 0 and the velocity of light c by the relation

(6)

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

(7)

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

(8)

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

(9)

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.

### References

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

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

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