Page 322 - Electromagnetics

Figure 4.25: TM plane-wave field incident on a material cylinder.

secondaryor scattered fields, which are standing waves within the cylinder and outward
traveling waves external to the cylinder. Although we have not yet learned how to write
the secondaryfields in terms of the impressed sources, we can solve for the fields as a
boundaryvalue problem. The total field must obeythe boundaryconditions on tangen-
tial components at the interface between the cylinder and surrounding free space. We
need not worryabout the effect of the secondarysources on the source of the primary
field, since bydefinition impressed sources cannot be influenced bysecondaryfields.
The scattered field can be found using superposition. When excited bya TM impressed
field, the secondaryfield is also TM. The situation for TE excitation is similar. By
decomposing the impressed field into TE and TM components, we maysolve for the
scattered field in each case and then superpose the results to determine the complete
solution.
We first consider the TM case. The impressed electric field maybe written as

˜
˜ i
˜
E (r,ω) = ˆ zE 0 (ω)e − jk 0 x = ˆ zE 0 (ω)e − jk 0 ρ cos φ (4.356)
while the magnetic field is, by(4.223),

˜
˜
E 0 (ω) − jk 0 x E 0 (ω) − jk 0 ρ cos φ
ˆ
˜ i
H (r,ω) =−ˆ y e =−(ˆρ sin φ + φ cos φ) e .
η 0 η 0
Here k 0 = ω(µ 0 0 ) 1/2 and η 0 = (µ 0 / 0 ) 1/2 . The scattered electric field takes the form
of a nonuniform cylindrical wave (4.353). Periodicityin φ implies that k φ is an integer,
say k φ = n. Within the cylinder we cannot use any of the functions N n (kρ), H (2) (kρ),
n
or H (1) (kρ) to represent the radial dependence of the field, since each is singular at the
n
(1)
origin. So we choose B (kρ) = J n (kρ) and B ρ (ω) = 0 in (4.355). Physically, J n (kρ) rep-
n
resents the standing wave created bythe interaction of outward and inward propagating
waves. External to the cylinder we use H (2) (kρ) to represent the radial dependence of the
n
secondaryfield components: we avoid N n (kρ) and J n (kρ) since these represent standing
waves, and avoid H (1) (kρ) since there are no external secondarysources to create an
n
inward traveling wave.
© 2001 by CRC Press LLC