PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 18:

PHY 712 Spring 2014 -- Lecture 18 102/28/2014

PHY 712 Electrodynamics10-10:50 AM MWF Olin 107

Plan for Lecture 18:

Finish reading Chap. 7; start Chap. 8

A. Summary of results for plane waves

B. Electromagnetic waves in an ideal conductor

C. TEM electromagnetic modes



Review: Electromagnetic plane waves in isotropic medium with real permeability and permittivity: m e.

,tμε,t

c

n,t

cnet ctni c

rEkrEkrB

ErE rk

ˆˆ

με , 22ˆ

0

kEkE

S ˆ2

1ˆ2

: wavesneticelectromag planefor vector Poynting

2

0

2

0

c

navg

2

02

1

: wavesneticelectromag planefor density Energy

Eavg

u


Reflection and refraction between two isotropic media

m’ e’

me

k’ki kRi R

q

z

x

1TR that Note

cos

cos

'

''ˆ

ˆ'T

ˆ

ˆR

:ance transmitte,Reflectanc2

0

0

2

0

0

in

n

E

E

E

E

iii

R

i

R

zS

zS

zS

zS


Reflection and refraction between two isotropic media -- continued

m’ e’

me

k’ki kRi R

q

z

x

1sin

sin 'sincos'

' ' mediumin propagateslonger no field refracted

,'

sinfor ,' If

sin'cos'

:interfaceat condition Matching

02

2222

10

222

i

inininin

n

niinn

innn

,tμε,t

c

n,t

cnet ctni c

rEkrEkrB

ErE rk

ˆˆ

με ,

:each waveFor 22ˆ

0

ctniz

i

i

c

cn

eet

rkErE ||02

2

ˆ

0

1sin

sin

','

Total internal reflection:


sin'cos' : thatNote

cos''

cos

cos2'

cos''

cos

cos''

cos

222

0

0

0

0

innn

nin

in

E

E

nin

nin

E

E

ii

R

For s-polarization

sin'cos' : thatNote

coscos''

cos2'

coscos''

coscos''

222

0

0

0

0

innn

nin

in

E

E

nin

nin

E

E

ii

R

For p-polarization

PHY 712 Spring 2014 -- Lecture 18 702/28/2014 PHY 712 Spring 2013 -- Lecture 19 7

nn

n

E

E

nn

nn

E

E

ii

R

''

2'

''

''

0

0

0

0

'

'

''

2

'

''T

''

''

R

:ance transmitte,Reflectanc

2

2

0

0

2

2

0

0

n

n

nn

n

n

n

E

E

nn

nn

E

E

i

i

R

Special case: normal incidence (i=0, q=0)


Extension to complex refractive index n= nR + i nI

1R

:''for that Note

''

''

''

''

R

:incidence normalat eReflectanc

''' real, , ' Suppose

22

22

2

2

0

0

nnn

nnn

nnn

nn

nn

E

E

innnn

RI

IR

IR

i

R

IR


Fields near the surface on an ideal conductor

ticin

IRtii

b

b

b

Reet

cinnet

tt

tt

rkrk

rk

ErE

kkErE

E

HEFF

EEH

HE

HE

EH

EJED

ˆ/0

/ˆ

0

2

22

,

ˆ re whe ,

:for form wavePlane

, 0

0 0

: and of in terms equations sMaxwell'

:medium isotropican for Suppose


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 r



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 r z

r||

0



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 z

r||

0

ˆ ˆ/ /0

Note that the field is larger than field so we can write:

,

1 ˆ, ,2

i i tt e e

it t

k r k r

H E

H r H

E r k H r


Boundary values for ideal conductor

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 r kE0

At the boundary of an ideal conductor, the E and H fields decay in the direction normal to the interface, the field directions are in the plane of the interface.

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)


TEM wavesTransverse electric and magnetic (both E and H fields are perpendicular to wave propagation direction)

BkEk

rEkrEkrB

ErE rk

ˆ0ˆ

ˆˆ

με ,

:areTEM modes neticelectromag normal"" the

medium; conducting-non aor within space free In the

22ˆ

0

,tμε,tc

n,t

cnet ctni c


Wave guides

Coaxial cable TEM modes

Simple optical pipe TE or TM modes


Comment on HW #11

1. Consider an infinitely long wire with radius a, oriented along the z axis. There is a steady uniform current inside the wire. Specifically the current is along the z-axis with the magnitude of J0 for ρ ≤ a and zero for ρ > a, where ρ denotes the radial parameter of the natural cylindrical coordinates of the system. a. Find the vector potential (A) for all ρ. b. Find the magnetic flux field (B) for all ρ.

Solution to problem using PHY 114 ideas In this case, it is convenient to solve part b first.

Top view for r < a

Top view for r > a

BB

J0


Top view for r < a

Top view for r > a

BB

0

20 0

0 0

0 0

2 20 0

2

2

ˆ2

ˆ4

B

B

a

J

J

J

J

B d J dA

B A

A

φ

z

0

20 0

20 0

20 0

20 0

2

2

ˆ2

ln /ˆ

2

B J a

J a

J a

B

aJ a

B d

φ

J dA

z

B A

A

Comment on HW #11 -- continued



0 02

0 0

20 0

1

2 3

ˆ for

0 for

for

0 for

for ( ) 4

( ) for

Choosing constants from

Alternative treatment using differential equ

continuity

ations:

1

ln

req

z

z

J a

a

J a

a

JC a

A

C C

A

a

zA

2 20 0 0 0

20 0

uirements:

for 4 4( )

( / ) for

( )

ln2

z

z

J J aa

AJ a

a a

A

B φ

J0


Comment on HW #12

w

r

A sphere of radius a carries a uniform surface charge distribution . s The sphere is rotated about a diameter with constant angular velocity w. Find the vector potential A and magnetic field B both inside and outside the sphere.

30

*1

*1

( )( ) .

4 | |

( ) for ( )

0 otherwise

1 4ˆNote that: ( ) ( )

| | 2 1

ˆand: ( ) ( ) .

l

lm lmllm

lm lm lm

d r

r a r a

rY Y

l r

rd Y Y

r

J rA r

r r

ω rJ r

r rr r

r r r r



3 30 0

0

0 4

3

0 4

2

3

( ) 4( ) ' ' ( )

4 | | 4 3

for

( )3 for

2 for

( )3 3 for

a rd r r dr r a

r r

a r a

ar a

r

a r a

ar a

r

J r ω r

A rr r

A r ω r

ω

B rr ω r ω

Documents

PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 18: