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C:\User\Teaching\505-506\S11\HW04'sol.doc
PHY 506
Classical Electrodynamics II
Spring 2011
Homework 4 with solutions
Problem 4.1
(to be graded of 15 points). Calculate the skin-effect contribution to the attenuation
coefficient
defined by equation (7.196) of the lecture notes, for the basic (
H
10
) mode propagating in a
waveguide with the rectangular cross-section (Fig. 7.21). Use the results to evaluate
for a 10 GHz
wave propagating in the standard X-band waveguide WR-90 (with copper walls,
a
= 23 mm,
b
= 10 mm,
and no dielectric filling) at room temperature. Compare the estimate with that for the standard coaxial
cable, at the same frequency – see Sec. 7.9.
Solution
: As discussed in class, in the
H
10
mode the electric field has just one Cartesian component, with
complex amplitude
a
x
ZH
ka
i
x
E
l
y
sin
)
(
,
while the magnetic field has two components:
a
x
H
x
H
a
x
H
a
k
i
x
H
l
z
l
z
x
cos
)
(
,
sin
)
(
.
Of those two components, only
H
x
contributes to the longitudinal (
z
) component of the time-averaged
Poynting vector
a
x
H
Z
a
kk
H
E
H
E
S
l
z
x
y
y
x
z
2
2
2
2
sin
2
2
*
*
,
which gives the total power flow along the waveguide:
.
4
2
2
3
0
0
l
z
a
b
z
H
Z
b
a
kk
S
dy
dx
P
(*)
In order to find losses per unit length, we have to integrate losses per unit area, given by Eq.
(7.206) of the lecture notes, over the cross-section’s perimeter:
.
2
1
4
2
cos
sin
2
4
)
(
)
0
(
)
(
)
(
2
4
)
(
4
2
2
0
0
2
2
2
2
0
0
2
0
2
0
2
2
0
2
0
0
loss
b
a
a
k
H
b
dx
a
x
a
x
a
k
H
dy
b
H
dy
H
dx
x
H
x
H
dl
x
H
dz
d
z
l
a
z
l
b
z
b
z
a
z
x
C
P
According to the waveguide’s dispersion relation (see Eqs. (7.126) and (7.128) of the lecture notes),
a
k
k
k
k
H
t
t
z
10
2
2
2
2
,
,
the expression in the last square brackets is just (
ka/
)
2
, so that, using Eq. (*), for the attenuation
constant we finally get

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