Page 147 - Engineering Electromagnetics, 8th Edition

CHAPTER 5 Conductors and Dielectrics 129

Figure 5.9 (a)An incremental surface element S is shown in the interior of a
dielectric in which an electric field E is present. (b) The nonpolar molecules form dipole
moments p and a polarization P. There is a net transfer of bound charge across S.

p = Qd in each molecule, such that p and d make an angle θ with S,as indicated
in Figure 5.9b.
The bound charges will now move across S. Each of the charges associated
1
with the creation of a dipole must have moved a distance d cos θ in the direction
2
perpendicular to S. Thus, any positive charges initially lying below the surface S
1
and within the distance d cos θ of the surface must have crossed S going upward.
2
Also, any negative charges initially lying above the surface and within that distance
1
( d cos θ) from S must have crossed S going downward. Therefore, because there
2
3
are n molecules/m , the net total charge that crosses the elemental surface in an upward
direction is equal to nQd cos θ S,or
Q b = nQd · S
where the subscript on Q b reminds us that we are dealing with a bound charge and
not a free charge. In terms of the polarization, we have
Q b = P · S
If we interpret S as an element of a closed surface inside the dielectric material,
then the direction of S is outward, and the net increase in the bound charge within
the closed surface is obtained through the integral

Q b =− P · dS (22)
S