A Dynamical Theory of The Electromagnetic Field - Maxwell's Original Equations

Maxwell's Original Equations

In part III of "A Dynamical Theory of the Electromagnetic Field", which is entitled "General Equations of the Electromagnetic Field", Maxwell formulated twenty equations which were to become known as Maxwell's equations, until this term became applied instead to a set of four vectorized equations selected in 1884 by Oliver Heaviside, which had all appeared in "On physical lines of force".

Heaviside's versions of Maxwell's equations are distinct by virtue of the fact that they are written in modern vector notation. They actually only contain one of the original eight—equation "G" (Gauss's Law). Another of Heaviside's four equations is an amalgamation of Maxwell's law of total currents (equation "A") with Ampère's circuital law (equation "C"). This amalgamation, which Maxwell himself had actually originally made at equation (112) in "On Physical Lines of Force", is the one that modifies Ampère's Circuital Law to include Maxwell's displacement current.

Eighteen of the twenty original Maxwell's equations can be vectorized into 6 equations. Each vectorized equation represents 3 original equations in component form. Including the other two equations, in modern vector notation, they can form a set of eight equations. They are listed below:

(A) The law of total currents

(B) Definition of the magnetic potential

(C) Ampère's circuital law

(D) The Lorentz force

This force represents the effect of electric fields created by convection, induction, and by charges.

(E) The electric elasticity equation

(F) Ohm's law

(G) Gauss's law

(H) Equation of continuity of charge

Notation

is the magnetic field, which Maxwell called the "magnetic intensity".

is the electric current density (with being the total current including displacement current).

is the displacement field (called the "electric displacement" by Maxwell).

is the free charge density (called the "quantity of free electricity" by Maxwell).

is the magnetic potential (called the "angular impulse" by Maxwell).

is the electric field (called the "electromotive force" by Maxwell, not to be confused with the scalar quantity that is now called electromotive force).

is the electric potential (which Maxwell also called "electric potential").

is the electrical conductivity (Maxwell called the inverse of conductivity the "specific resistance", what is now called the resistivity).

Maxwell did not consider completely general materials; his initial formulation used linear, isotropic, nondispersive permittivity ε and permeability μ, although he also discussed the possibility of anisotropic materials.

It is of particular interest to note that Maxwell includes a term in his expression for the "electromotive force" at equation "D", which corresponds to the magnetic force per unit charge on a moving conductor with velocity . This means that equation "D" is effectively the Lorentz force. This equation first appeared at equation (77) in "On Physical Lines of Force" quite some time before Lorentz thought of it. Nowadays, the Lorentz force sits alongside Maxwell's equations as an additional electromagnetic equation that is not included as part of the set.

When Maxwell derives the electromagnetic wave equation in his 1864 paper, he uses equation "D" as opposed to using Faraday's law of electromagnetic induction as in modern textbooks. Maxwell however drops the term from equation "D" when he is deriving the electromagnetic wave equation, and he considers the situation only from the rest frame.