Page 146 - Engineering Electromagnetics, 8th Edition

128 ENGINEERING ELECTROMAGNETICS

This displacement against a restraining force is analogous to lifting a weight
or stretching a spring and represents potential energy. The source of the energy is
the external field, the motion of the shifting charges resulting perhaps in a transient
current through a battery that is producing the field.
The actual mechanism of the charge displacement differs in the various dielectric
materials. Some molecules, termed polar molecules, have a permanent displacement
existing between the centers of “gravity” of the positive and negative charges, and
each pair of charges acts as a dipole. Normally the dipoles are oriented in a random
way throughout the interior of the material, and the action of the external field is to
align these molecules, to some extent, in the same direction. A sufficiently strong
field may even produce an additional displacement between the positive and negative
charges.
A nonpolar molecule does not have this dipole arrangement until after a field is
applied. The negative and positive charges shift in opposite directions against their
mutual attraction and produce a dipole that is aligned with the electric field.
Either type of dipole may be described by its dipole moment p,asdeveloped in
Section 4.7, Eq. (36),
p = Qd (20)
where Q is the positive one of the two bound charges composing the dipole, and d is
the vector from the negative to the positive charge. We note again that the units of p
are coulomb-meters.
If there are n dipoles per unit volume and we deal with a volume ν, then there
are n ν dipoles, and the total dipole moment is obtained by the vector sum,
n ν

p total = p i
i=1
If the dipoles are aligned in the same general direction, p total may have a significant
value. However, a random orientation may cause p total to be essentially zero.
We now define the polarization P as the dipole moment per unit volume,
1 n ν
P = lim p i (21)

ν→0 ν
i=1
with units of coulombs per square meter. We will treat P as a typical continuous field,
even though it is obvious that it is essentially undefined at points within an atom
or molecule. Instead, we should think of its value at any point as an average value
taken over a sample volume ν—large enough to contain many molecules (n ν in
number), but yet sufficiently small to be considered incremental in concept.
Our immediate goal is to show that the bound-volume charge density acts like
the free-volume charge density in producing an external field; we will obtain a result
similar to Gauss’s law.
To be specific, assume that we have a dielectric containing nonpolar molecules.
No molecule has a dipole moment, and P = 0 throughout the material. Somewhere in
the interior of the dielectric we select an incremental surface element S,as shown
in Figure 5.9a, and apply an electric field E. The electric field produces a moment