(^178) Maxwell's Equations
For most linear materials, \xm\ ls close to zero. In linear homogeneous
isotropic paramagnetic materials, Xm > 0 and fj, > IIQ. For example, alu-
minum has Xm «2x 10-5. In linear homogeneous isotropic diamagnetic
materials, Xm < 0 and fi < HQ. For example, both copper and water have
Xm ^ —10~^5 (we write « rather than = because the precise values are
sensitive to the temperature of the materials and the frequency of the field;
the above values correspond to room temperature and a time-homogeneous
field).
The main examples of nonlinear materials are ferromagnetics and
superconductors. In ferromagnetics, the induced magnetic field is much
stronger than in vacuum. In superconductors the induced magnetic field is
essentially zero.
Let us summarize Maxwell's equations in material media:
(3.3.47)
(3.3.48)
(3.3.49)
(3.3.50)
where pf is the density, per unit volume, of the free electric charges, and Je
is the density, per unit area, of the externally produced electric currents.
These equations describe the electromagnetic field at distances much larger
than atomic size (atomic size is usually taken as 10-10 meters.) In Section
6.3.4, we further investigate Maxwell's equations for some simple models,
both in vacuum and in material media.
At sub-atomic distances, of the order of e^2 /(4nemec^2 ) « 2.8 •
10~^17 m, Maxwell's equations are no longer applicable, and quantum
electrodynamics takes over. A quantum-theoretic analog of Maxwell's
equations was suggested in 1954 by two American physicists, CHEN-NING
YANG (b. 1922) and ROBERT L. MILLS (1927-1999). Mathematical anal-
ysis of these Yang-Mills equations is mostly an open problem and is
outside the scope of our discussion. In fact, this analysis is literally a
million-dollar question, being one of the seven Millennium Problems an-
nounced by the Clay Mathematics Institute in 2000; for details, see the
book The Millennium Problems by K. Devlin, 2002.
d\vD =
divB =
curlJ3 =
curl if :
:P/;
0;
dB
at
= Je +)
dD
dt'