Page 367 - Electromagnetics

which for y > h is
∞+ j
˜ I 0 (ω)
ω ˜µ 2 j sin k y h
˜ i 2 − jk y y − jk x x
E (x, y,ω) =− e e dk x .
z
2π 2k y
−∞+ j
The scattered field has the form of (5.7) but must be odd. Thus A + =−A − and the
total field for y > h is
∞+ j
˜
1 I 0 (ω) 2 j sin k y h
˜ + − jk y y − jk x x
E z (x, y,ω) = 2 jA (k x ,ω) sin k y y − ω ˜µ e e dk x .
2π 2 2k y
−∞+ j
˜
+
Setting E z = 0 at z = d and solving for A we find that the total field for this case is
∞+ j
˜ I 0 (ω)
ω ˜µ e − jk y |y−h| − e − jk y |y+h|
˜ 2
E z (x, y,ω) =− −
2π 2k y
−∞+ j
2 j sin k y h e − jk y d − jk x x
− sin k y y e dk x .
2k y sin k y d
Adding the fields for the two cases we find that
∞+ j
˜
ω ˜µI 0 (ω) e − jk y |y−h|
˜ − jk x x
E z (x, y,ω) =− e dk x +
2π 2k y
−∞+ j
∞+ j
˜
ω ˜µI 0 (ω) cos k y h cos k y y sin k y h sin k y y e − jk y d − jk x x
+ + j e dk x ,
2π cos k y d sin k y d 2k y
−∞+ j
(5.8)
which is a superposition of impressed and scattered fields.

5.2 Solenoidal–lamellar decomposition
We nowdiscuss the decomposition of a general vector field into a lamellar component
having zero curl and a solenoidal component having zero divergence. This is known as a
Helmholtz decomposition.If V is any vector field then we wish to write
V = V s + V l , (5.9)

where V s and V l are the solenoidal and lamellar components of V. Formulas expressing
these components in terms of V are obtained as follows. We first write V s in terms of a
“vector potential” A as
V s =∇ × A. (5.10)
This is possible by virtue of (B.49). Similarly, we write V l in terms of a “scalar potential”
φ as
V l =∇φ. (5.11)

© 2001 by CRC Press LLC