Page 585 - Engineering Electromagnetics, 8th Edition

E

APPENDIX

Origins of the Complex

Permittivity

As we learned in Chapter 5, a dielectric can be modeled as an arrangement of atoms
and molecules in free space, which can be polarized by an electric field. The field
forces positive and negative bound charges to separate against their Coulomb attrac-
tive forces, thus producing an array of microscopic dipoles. The molecules can be
arranged in an ordered and predictable manner (such as in a crystal) or may exhibit
random positioning and orientation, as would occur in an amorphous material or a
liquid. The molecules may or may not exhibit permanent dipole moments (existing
before the field is applied), and if they do, they will usually have random orienta-
tions throughout the material volume. As discussed in Section 5.7, the displacement of
charges in a regular manner, as induced by an electric field, gives rise to a macroscopic
polarization, P, defined as the dipole moment per unit volume:

1 N ν
P = lim p i (E.1)

ν→0 ν
i=1
where N is the number of dipoles per unit volume and p i is the dipole moment of the
ith atom or molecule, found through
(E.2)
p i = Q i d i
Q i is the positive one of the two bound charges composing dipole i, and d i is the
distance between charges, expressed as a vector from the negative to the positive
charge. Again, borrowing from Section 5.7, the electric field and the polarization are
related through
P = 0 χ e E (E.3)

where the electric susceptibility, χ e , forms the more interesting part of the dielectric
constant:

r = 1 + χ e (E.4)

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