Page 138 - Engineering Electromagnetics, 8th Edition

120 ENGINEERING ELECTROMAGNETICS

n

Figure 5.4 An appropriate closed path and gaussian surface are used to
determine boundary conditions at a boundary between a conductor and free
space; E t = 0 and D N = ρ S.

and normal components of D and E on the free-space side of the boundary. Both
fields are zero in the conductor. The tangential field may be determined by applying
Section 4.5, Eq. (21),

E · dL = 0
around the small closed path abcda. The integral must be broken up into four parts
b c d a
= 0
+ + +
a b c d
Remembering that E = 0 within the conductor, we let the length from a to b or c to
d be w and from b to c or d to a be h, and obtain

1
1
E t w − E N,at b h + E N,at a h = 0
2 2
As we allow h to approach zero, keeping w small but finite, it makes no
difference whether or not the normal fields are equal at a and b, for h causes these
products to become negligibly small. Hence, E t w = 0 and, therefore, E t = 0.
The condition on the normal field is found most readily by considering D N rather
than E N and choosing a small cylinder as the gaussian surface. Let the height be h
and the area of the top and bottom faces be S.Again, we let h approach zero.
Using Gauss’s law,

D · dS = Q
S
we integrate over the three distinct surfaces

+ + = Q
top bottom sides
and find that the last two are zero (for different reasons). Then

D N S = Q = ρ S S