Page 153 - Engineering Electromagnetics, 8th Edition

CHAPTER 5 Conductors and Dielectrics 135

or the normal component of D is continuous. It follows that

1 E N1 = 2 E N2 (36)
and normal E is discontinuous.
Equations (32) and (34) can be written in terms of field vectors in any direction,
along with the unit normal to the surface as shown in Figure 5.10. Formally stated,
the boundary conditions for the electric flux density and the electric field strength at
the surface of a perfect dielectric are

(D 1 − D 2 ) · n = ρ s (37)
which is the general statement of Eq. (32), and
(E 1 − E 2 ) × n = 0 (38)

generally states Eq. (34). This construction was used previously in Eqs. (17) and (18)
for a conducting surface, in which the normal or tangential components of the fields
are obtained through the dot product or cross product with the normal, respectively.
These conditions may be used to show the change in the vectors D and
E at the surface. Let D 1 (and E 1 ) make an angle θ 1 with a normal to the surface
(Figure 5.11). Because the normal components of D are continuous,

(39)
D N1 = D 1 cos θ 1 = D 2 cos θ 2 = D N2
The ratio of the tangential components is given by (33) as

D tan 1 D 1 sin θ 1 1
= =
D tan 2 D 2 sin θ 2 2
or

2 D 1 sin θ 1 = 1 D 2 sin θ 2 (40)

n

Figure 5.11 The refraction of D at a
dielectric interface. For the case shown,
1 > 2 ; E 1 and E 2 are directed along D 1
and D 2 , with D 1 > D 2 and E 1 < E 2 .