Page 149 - Engineering Electromagnetics, 8th Edition

CHAPTER 5 Conductors and Dielectrics 131

In order to make any real use of these new concepts, it is necessary to know the
relationship between the electric field intensity E and the polarization P that results.
This relationship will, of course, be a function of the type of material, and we will
essentially limit our discussion to those isotropic materials for which E and P are
linearly related. In an isotropic material, the vectors E and P are always parallel,
regardless of the orientation of the field. Although most engineering dielectrics are
linear for moderate-to-large field strengths and are also isotropic, single crystals may
be anisotropic. The periodic nature of crystalline materials causes dipole moments to
be formed most easily along the crystal axes, and not necessarily in the direction of
the applied field.
In ferroelectric materials, the relationship between P and E not only is nonlin-
ear, but also shows hysteresis effects; that is, the polarization produced by a given
electric field intensity depends on the past history of the sample. Important examples
of this type of dielectric are barium titanate, often used in ceramic capacitors, and
Rochelle salt.
The linear relationship between P and E is

P = χ e 0 E (28)
where χ e (chi) is a dimensionless quantity called the electric susceptibility of the
material.
Using this relationship in Eq. (25), we have
D = 0 E + χ e 0 E = (χ e + 1) 0 E
The expression within the parentheses is now defined as
r = χ e + 1 (29)
This is another dimensionless quantity, and it is known as the relative permittivity,or
dielectric constant of the material. Thus,
D = 0 r E = E (30)
where

= 0 r (31)
and is the permittivity. The dielectric constants are given for some representative
materials in Appendix C.
Anisotropic dielectric materials cannot be described in terms of a simple suscep-
tibility or permittivity parameter. Instead, we find that each component of D may be
a function of every component of E, and D = E becomes a matrix equation where
D and E are each 3 × 1 column matrices and is a 3 × 3 square matrix. Expanding
the matrix equation gives

D x = xx E x + xy E y + xz E z
D y = yx E x + yy E y + yz E z
D z = zx E x + zy E y + zz E z