Page 139 - Engineering Electromagnetics, 8th Edition

CHAPTER 5 Conductors and Dielectrics 121

or

D N = ρ S
Thesearethedesiredboundaryconditionsfortheconductor-to-free-spacebound-
ary in electrostatics,

D t = E t = 0 (15)

D N = 0 E N = ρ S (16)
The electric flux leaves the conductor in a direction normal to the surface, and the
value of the electric flux density is numerically equal to the surface charge density.
Equations (15) and (16) can be more formally expressed using the vector fields

E × n = 0 (17)

s

D · n = ρ s (18)

s
where n is the unit normal vector at the surface that points away from the conductor,
as shown in Figure 5.4, and where both operations are evaluated at the conductor
surface, s.Taking the cross product or the dot product of either field quantity with n
gives the tangential or the normal component of the field, respectively.
An immediate and important consequence of a zero tangential electric field in-
tensity is the fact that a conductor surface is an equipotential surface. The evaluation
of the potential difference between any two points on the surface by the line integral
leads to a zero result, because the path may be chosen on the surface itself where
E · dL = 0.
To summarize the principles which apply to conductors in electrostatic fields, we
may state that
1. The static electric field intensity inside a conductor is zero.
2. The static electric field intensity at the surface of a conductor is everywhere
directed normal to that surface.
3. The conductor surface is an equipotential surface.
Using these three principles, there are a number of quantities that may be calcu-
lated at a conductor boundary, given a knowledge of the potential field.

EXAMPLE 5.2
Given the potential,
2
2
V = 100(x − y )
and a point P(2, −1, 3) that is stipulated to lie on a conductor-to-free-space boundary,
find V , E, D, and ρ S at P, and also the equation of the conductor surface.
Solution. The potential at point P is
2
2
V P = 100[2 − (−1) ] = 300 V