Page 152 - Engineering Electromagnetics, 8th Edition

134 ENGINEERING ELECTROMAGNETICS

the tangential components by using

E · dL = 0

around the small closed path on the left, obtaining
E tan 1 w − E tan 2 w = 0

The small contribution to the line integral by the normal component of E along
the sections of length h becomes negligible as h decreases and the closed path
crowds the surface. Immediately, then,

(32)
E tan 1 = E tan 2
Evidently, Kirchhoff’s voltage law is still applicable to this case. Certainly we have
shown that the potential difference between any two points on the boundary that are
separated by a distance w is the same immediately above or below the boundary.
If the tangential electric field intensity is continuous across the boundary, then
tangential D is discontinuous, for

D tan 1 D tan 2
= E tan 1 = E tan 2 =
1 2
or
D tan 1 1
= (33)
D tan 2 2
The boundary conditions on the normal components are found by applying
Gauss’s law to the small “pillbox” shown at the right in Figure 5.10. The sides are
again very short, and the flux leaving the top and bottom surfaces is the difference

D N1 S − D N2 S = Q = ρ S S
from which

(34)
D N1 − D N2 = ρ S
What is this surface charge density? It cannot be a bound surface charge density,
because we are taking the polarization of the dielectric into effect by using a dielectric
constant different from unity; that is, instead of considering bound charges in free
space, we are using an increased permittivity. Also, it is extremely unlikely that any
free charge is on the interface, for no free charge is available in the perfect dielectrics
we are considering. This charge must then have been placed there deliberately, thus
unbalancing the total charge in and on this dielectric body. Except for this special
case, then, we may assume ρ S is zero on the interface and

D N1 = D N2 (35)