Mathematics of Physics and Engineering

Equations in Material Media 175

field E so that div E = (pb + pf)/so- In the dielectric material, we have
Pb = — divP so that div(eo E + P) = pf. It is then natural to define the
vector D = eo E + P, called the electric displacement. We get the
following modification of (3.3.2):

divD = p/. (3.3.35)

Dielectric material is called linear if P is proportional to the external
field E:

P = eoXeE, D = e 0 E + e 0 XeE = eE, (3.3.36)

where Xe is called the susceptibility and e = £o(l -I- Xe), the
permittivity of the dielectric material; K = 1+Xe is called the dielectric
constant or relative permittivity. In general, e (as well as Xe and K)
is a tensor field and the value of e depends on both the location and
the direction. For time-dependent E, the value of e can also depend on
the frequency of E. In homogeneous isotropic materials, e is a constant
number so that (3.3.35) becomes div.E = Pf/£-
AMPERE'S LAW FOR MAGNETIZED MATERIALS. In material media,
there exists an internal magnetic field created by the motion of electrons
around the atoms. At distances much larger than the size of an atom,
the magnetic field produced by an electron moving around an atom is well
approximated by the field of a point magnetic dipole. These magnetic
dipoles affect the overall magnetic field in the material, just as the bound
charges in dielectrics affect the electric field.
Denote by M the density, per unit volume, of the point magnetic dipoles
in the material; this density is called magnetization. According to (3.3.33),
the overall vector potential A of the resulting magnetic field is

G

where G is the region in space occupied by the material, Q is a point in Q

and P is the point in M^3 at which the magnetic field is computed. We now

treat P as a fixed point and Q, as variable. Then

^(wod-mr

(3

3

38)