Equations in Vacuum 1633.3 Maxwell's Equations and Electromagnetic Theory
Electromagnetic theory, or electromagnetism, is the branch of physics that
studies electric and magnetic fields. Numerous experiments have shown
that these two fields are physically related. Maxwell's equations provide a
mathematical model that establishes the precise relation between the elec-
tric and magnetic fields. The original publication in 1864 by the Scottish
mathematician and physicist JAMES CLERK MAXWELL (1831-1879) con-
tained 20 equations in 20 unknowns. Later, in the 1880s, J. W. Gibbs and
O. Heaviside put the equations in a more compact vector form, which we
will present below. On the basic physical level, Maxwell's equations gener-
alize a number of experimental facts; there are some physicists these days
who believe that one cannot derive Maxwell's equations, although Maxwell
himself would probably disagree. On the mathematical level, these equa-
tions admit numerous derivations, for different models and on various levels
of abstraction, and we discuss some of these derivations below.
If an equation describes a physical law, the particular form of the equa-
tion often depends on the units used to measure the corresponding phys-
ical quantities. In our discussion of electromagnetism, we will use the
International System of Units (SI). Some of the units in SI are Am-
pere (A) for electric current, Coulomb (C) for electric charge, meter (m)
for length, Newton (N) for force, second (s) for time.
3.3.1 Maxwell's Equations in Vacuum
In vacuum, the electric and magnetic fields are described mathematically
by the vectors E and B, respectively. A possible interpretation of these
vectors is through the force F exerted by the fields on a point charge q
moving with velocity v.
F = q{E + vxB). (3.3.1)The charges generating the fields can be stationary or moving; moving
charges constitute an electric current. Denote by p the density per unit
volume of free stationary charges, and by J, the density, per unit area, of
electric currents. In general, E, B, p, and J are time-dependent. Maxwell's
equations establish the connection between E, B, p, and J. We begin by
stating the equation, and then show how to derive them by combining
the basic empirical laws of electricity and magnetism with the theorems of