phy 712 spring 2014 -- lecture 191 phy 712 electrodynamics 10-10:50 am mwf olin 107 plan for lecture...
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PHY 712 Spring 2014 -- Lecture 19 1
PHY 712 Electrodynamics10-10:50 AM MWF Olin 107
Plan for Lecture 19:
Start reading Chap. 8 in Jackson.
A. Examples of waveguides
B. TEM, TE, TM modes
C. Resonant cavities
03/17/2013
PHY 712 Spring 2014 -- Lecture 19 3
Reminder: Topic choices for “computational project” – suggestions available upon request-- due Monday March 24th?
03/17/2013
PHY 712 Spring 2014 -- Lecture 19 4
Fields near the surface on an ideal conductor
22
2
Suppose for an isotropic medium:
Maxwell's equations in terms of and :
0 0
0 ,
Plan
b
b
b
t t
t t
D E J E
H E
E H
H EE H E
F F E H
0
e wave form for :
ˆ, where i i tR It e n in
c k r
E
E r E k k
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PHY 712 Spring 2014 -- Lecture 19 5
Fields near the surface on an ideal conductor -- continued
1
2 1For
112
112
:systemour For
2/12
2/12
IR
b
bI
b
bR
nc
nc
nc
nc
ˆ ˆ/ /0,
1ˆ ˆ, , ,
i i tt e e
n it t t
c
k r k rE r E
H r k E r k E r03/17/2013
PHY 712 Spring 2014 -- Lecture 19 6
Fields near the surface on an ideal conductor -- continued
icinnc
nc
nc
IR
IR
11
limit, In this
1
2 1For
00
ˆ ˆ/ /0,
, , ,
1ˆ ˆ, , ,
1ˆ ˆ, , , ,
i i tt e e
it t t
n it t t
c
n it t t t
c
k r k rE r E
D r E r E r
H r k E r k E r
B r H r k E r k E r03/17/2013
PHY 712 Spring 2014 -- Lecture 19 7
Fields near the surface on an ideal conductor -- continued
ˆ ˆ/ /0,
, , ,
1ˆ ˆ, , ,
1ˆ ˆ, , , ,
i i tt e e
it t t
n it t t
c
n it t t t
c
k r k rE r E
D r E r E r
H r k E r k E r
B r H r k E r k E r
ˆ ˆ/ /0
Note that we can express the results in terms of the field:
,
1 ˆ, ,2
i i tt e e
it t
k r k r
H
H r H
E r k H r
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PHY 712 Spring 2014 -- Lecture 19 8
Boundary values for ideal conductor
At the boundary of an ideal conductor, the E and H fields decay in the direction normal to the interface.
tit
eet tii
,ˆ2
1,
,
:conductor theInside/ˆ
0/ˆ
rHkrE
HrH rkrk
k̂H0
n̂
Ideal conductor boundary condit
0 0
ions:
S S n E n H
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PHY 712 Spring 2014 -- Lecture 19 9
Waveguide terminology• TEM: transverse electric and magnetic (both E and H
fields are perpendicular to wave propagation direction)• TM: transverse magnetic (H field is perpendicular to
wave propagation direction)• TE: transverse electric (E field is perpendicular to wave
propagation direction)
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PHY 712 Spring 2014 -- Lecture 19 10
Wave guides
Coaxial cable TEM modes
Top view:
a
b
m ez
0
0
for :medium inside equations sMaxwell'
BEB
EBE
εiω
i
ba
(following problem 8.2 in Jackson’s text)
Inside medium, m e assumed
to be real
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PHY 712 Spring 2014 -- Lecture 19 11
Electromagnetic waves in a coaxial cable -- continued
ˆ cos ˆ sinˆ
ˆ sinˆ cosˆ
ˆ
ˆ
for solution Example
0
0
yxφ
yxρ
φB
ρE
tiikz
tiikz
eaB
eaE
baTop view:
a
b
m er
f
zBES ˆ22
1
: ),(with medium cablethin vector wiPoynting22
0*
aBavg
00
:Find
BE
k
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PHY 712 Spring 2014 -- Lecture 19 12
Electromagnetic waves in a coaxial cable -- continuedTop view:
a
b
m er
f
a
baBdd
avg
b
a
lnˆ
:material cablein power averaged Time22
02
0
zS
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PHY 712 Spring 2014 -- Lecture 19 13
Analysis of rectangular waveguide
Boundary conditions at surface of waveguide: Etangential=0, Bnormal=0
z x
y
Cross section view b a
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PHY 712 Spring 2014 -- Lecture 19 14
Analysis of rectangular waveguide
z x
y
tiikz
zyx
tiikzzyx
eyxEyxEyxE
eyxByxByxB
zyxE
zyxB
ˆ,ˆ,ˆ,
ˆ,ˆ,ˆ,
2222
b
n
a
mk
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PHY 712 Spring 2014 -- Lecture 19 15
Maxwell’s equations within the pipe:
0.
0.
yxz
yxz
BBikB
x y
EEikE
x y
.
.
.
zy x
zx y
y xz
EikE i B
y
EikE i B
xE E
i Bx y
.
.
.
zy x
zx y
y xz
BikB i E
y
BikB i E
xB B
i Ex y
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PHY 712 Spring 2014 -- Lecture 19 16
Solution of Maxwell’s equations within the pipe:
2 22 2
2 2
Combining Faraday's Law and Ampere's Law, we find that each field
component must satisfy a two-dimensional Helmholz equation:
( , ) 0.xkx y
E x y
0
2 22 2 2
For the rectangular wave guide discussed in Section 8.4 of your
text a solution for a TE mode can have
( , ) 0 and ( , ) cos cos ,
with
S
:
z z
mn
m x n yE x y B x y B
a b
m nk k
a b
ome of the other field components are:
and .x y y x
k kB E B E
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PHY 712 Spring 2014 -- Lecture 19 17
TE modes for retangular wave guide continued:
0
02 2 2 2
02 2 2 2
( , ) 0 and ( , ) cos cos ,
cos sin ,
sin
z z
zx y
zy x
m x n yE x y B x y B
a b
Bi i n m x n yE B B
k k y b a bm na b
Bi i mE B B
k k x am na b
cos .m x n y
a b
tangential
normal
0 because: ( ,0) ( , ) 0
and
Check boundary conditions:
(0, ) ( , ) 0
0
.
x x
y y
E x E x b
E y E a y
E
B
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PHY 712 Spring 2014 -- Lecture 19 18
Solution for m=n=1
b
yn
a
xmamByxiE
b
yn
a
xmbnByxiE
b
yn
a
xmByxB
bn
amy
bn
amx
z
cossin/
,
sincos/
,
coscos,
220
220
0
yxBz ,
yxiEx , yxiEy ,
03/17/2013
PHY 712 Spring 2014 -- Lecture 19 20
d
pk
kzyxBkzyxBzyxB
kzyxEkzyxEzyxE
ezyxEzyxEzyxE
ezyxBzyxBzyxB
iii
iii
tizyx
tizyx
cos,or sin,,,
cos,or sin,,, :generalIn
ˆ,,ˆ,,ˆ,,
ˆ,,ˆ,,ˆ,,
zyxE
zyxB
20
Resonant cavity
z x
y
dz
by
ax
0
0
0
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