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PHY 506
Classical Electrodynamics II
Spring 2011
Optional Problems Set 1 with solutions
Problem O.1
. Modify the results of Homework Problem 2.1 (ii) for a superconductor microstrip line,
taking into account the London depth of magnetic field penetration into both the strip and the ground
plane.
Solution
: In this case, the expression for
L
has to be replaced with the result of PHY 505 Homework
Problem 11.2:
L
0
2
d
w
L
,
while the capacitance
C
0
per unit length remains the same as in Problem 2.1, because the electric field
penetrates into conductors (including superconductors) by a much smaller length – see Sec. 2.1 of the
lecture notes. As a result, the TEM wave propagation speed in such a line,
2
/
1
L
2
/
1
L
2
/
1
2
/
1
0
0
/
2
1
1
2
1
)
(
1
d
v
d
d
C
L
v'
,
is lower that that (
v
) of plane waves.
1
Note that in this case the “general” relation (7.109) is not satisfied,
because it hinges on the similarity of boundary conditions for the magnetic and electric fields.
Problem O.2
. For a metallic coaxial cable with the circular crosssection (Fig. 7.19 of the lecture notes),
find the lowest nonTEM mode and calculate its cutoff frequency.
Solution
:
The analysis repeats that of the hollow circular waveguide (Fig. 7.22a) up to the Bessel
equation (7.136). However, in the coaxial cable, point
= 0 is inaccessible for the field, and hence,
instead of Eq. (7.137), we have to look for the radial part
R
(
) in the form of a linear superposition of
the Bessel functions of the fist and second kind:
,...
2
,
1
,
0
,
cos
)
(
)
(
0
2
1
n
n
k
Y
c
k
J
c
f
nm
n
nm
n
nm
,
where eigenvalues
k
nm
of the transversal wave number
k
t
have to be chosen to satisfy boundary
conditions at
=
a
and
=
b
, and give nonuniform longitudinal field,
f
nm
const. For the
E
modes, the
boundary condition Eq. (7.120) yields the following system of two equations:
0
)
(
)
(
,
0
)
(
)
(
2
1
2
1
b
k
Y
c
b
k
J
c
a
k
Y
c
a
k
J
c
nm
n
nm
n
nm
n
nm
n
,
These two linear, homogeneous equations for constants
c
1,2
are compatible if
)
(
)
(
)
(
)
(
a
k
Y
b
k
J
b
k
Y
a
k
J
nm
n
nm
n
nm
n
nm
n
,
where integer index
m
= 1, 2, … numbers the roots of this characteristic equation (for each fixed angular
quantum number
n
). Introducing the dimensionless variable
nm
k
nm
a
, we can rewrite
the equation as
1
This difference is especially large
(
v’
/
v
~ 0.1) in the socalled
long
(or “
distributed
”)
Josephson junctions
with
d
~ 1 nm and
L
~ 100 nm, enabling very interesting effects of interactions between the slow TEM waves and
magneticfieldinduced “waves” of tunneling supercurrent – see, e.g., Section 6.4 of the monograph by M.
Tinkham, recommended in Sec. 6.3.
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C:\User\Teaching\505506\S11\Opt1'sol.doc
nm
n
nm
n
nm
n
nm
n
Y
a
b
J
a
b
Y
J
.
(*)
Table on the right shows results for
combination (
b
/
a
– 1)
nm
=
k
nm
(
b
–
a
), obtained by
numerical solution
2
of this equation for a few values
of ratio
b
/
a
, and for a few lowest numbers
n
and
m
.
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