Page 150 - Engineering Electromagnetics, 8th Edition

132 ENGINEERING ELECTROMAGNETICS

Note that the elements of the matrix depend on the selection of the coordinate axes in
the anisotropic material. Certain choices of axis directions lead to simpler matrices. 7
D and E (and P) are no longer parallel, and although D = 0 E + P remains
avalid equation for anisotropic materials, we may continue to use D = E only
by interpreting it as a matrix equation. We will concentrate our attention on linear
isotropic materials and reserve the general case for a more advanced text.
In summary, then, we now have a relationship between D and E that depends on
the dielectric material present,

D = E (30)
where
(31)
= 0 r
This electric flux density is still related to the free charge by either the point or integral
form of Gauss’s law:

(27)
∇ · D = ρ ν

D · dS = Q (26)
S

Use of the relative permittivity, as indicated by Eq. (31), makes consideration
of the polarization, dipole moments, and bound charge unnecessary. However, when
anisotropic or nonlinear materials must be considered, the relative permittivity, in the
simple scalar form that we have discussed, is no longer applicable.

EXAMPLE 5.4
We locate a slab of Teflon in the region 0 ≤ x ≤ a, and assume free space where
x < 0 and x > a. Outside the Teflon there is a uniform field E out = E 0 a x V/m. We
seek values for D, E, and P everywhere.
Solution. The dielectric constant of the Teflon is 2.1, and thus the electric suscepti-
bility is 1.1.
Outside the slab, we have immediately D out = 0 E 0 a x . Also, as there is no
dielectric material there, P out = 0. Now, any of the last four or five equations will
enable us to relate the several fields inside the material to each other. Thus

D in = 2.1 0 E in (0 ≤ x ≤ a)
P in = 1.1 0 E in (0 ≤ x ≤ a)

7 A more complete discussion of this matrix may be found in the Ramo, Whinnery, and Van Duzer
reference listed at the end of this chapter.