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 19 203/17/2013

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

03/17/2013

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

03/17/2013

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

03/17/2013

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)

03/17/2013

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

03/17/2013

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

03/17/2013

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

03/17/2013

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

03/17/2013

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

03/17/2013

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 ,

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PHY 712 Spring 2014 -- Lecture 19 19

Solution for m=n=12 2

2 2 2mn

m nk k

a b

k

w

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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

03/17/2013

PHY 712 Spring 2014 -- Lecture 19 2103/17/2013

Resonant cavity

z x

y

dz

by

ax

0

0

0

2222

222

22

1

d

p

b

n

a

m

b

n

a

m

d

pk