Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

[$J satisfies the two-dimensional Lapl-ace equa-tion, is bounded
and \rani-shes at infinity because of the finite extent of the
sources. Hence, from Liouvi.lleLs theorem of function theory
[Q,] = 0, i.e. @ C continuous. Differentiating (2.15a) with respect
to x and (2.15b) with respect to y and adding we infer along the
same lines thxt a$ /az j.s coiltinuous. Conversely differentiating
C
C2i15a) with respect to y and (2.15b) with respect to x and sub-
tracting one obtains oQM continuous. Fina1l.y differentiate (2.16a)
with rbspec-t to x and (2.1.6b) with respect to y and add. It re-
sults that ('licr) a(oQ )/az is con.tinuous 01- 34 /2z is
r! M
continous
if cf tends to a constant value at both sides.
Summarizing: - 1
+EJ a~ M
-- 1
a az
continuous
I Conti.nuity conditions I
This result shows that the TE- and TM-field satisfy disioinl ingab;
k.bLrJ
boundary conditions. Hence, they are comp1etel.y independent and
not coupled.
Frc~n (2.17~) follows
@ (z = 4-0) = 0 (2.18;
M
This boundary coilditioll has a serious drawback for the TM-field:
As a result within the conductor no TM-field can be excited by
external sources.
From (2.18) follows via (2.13) fM(o) = 0. Now
K.iultip:lyi.ng (2.9a) by ~ c f and ~ integrating ) ~ over z from 0 Co z,
integra-tion by parts yj-elds in virtue of fM(o) = 0
From (2.20) and (.2.19). fo1.3.oi..ls for yet?]. Er,equencies