26 April 2004 Physics 218, Spring 2004 1
Today in Physics 218: review I
You learned a lot this semester, in principle. Here’s a laundry-list-like reminder of the first half of it:
Generally useful thingsElectrodynamicsElectromagnetic plane wave propagation in a variety of media (linear, conducting, dispersive, guides)
“The Scream,” by Edvard Munch (1893).
26 April 2004 Physics 218, Spring 2004 2
Generally useful math facts
Vector and vector-calculus product relation from the inside covers of the bookProperties of the delta functionOrthonormality of sines and cosines
2 2
0 02
02
2 2
02
2 2
0
cos cos sin sin
cos sin 0 , so
cos cos2
1 1cos sin2 2
mnmx nxdx mx nxdx
mx nxdx
t tdt
xdx t
π π
π
π ω
π
πδ
ωω ωπ
ωπ
= =
=
=
= = =
∫ ∫
∫
∫
∫,
iau ibu icuAe Be CeA B C a b c
+ =⇒ + = = =
26 April 2004 Physics 218, Spring 2004 3
Electrodynamics (as opposed to statics or quasistatics)
Beyond magneto-quasistaticsDisplacement current, and Maxwell’s repair of Ampère’s LawThe Maxwell equationsSymmetry of the equations: magnetic monopoles?
0
0 0 0
cgs units:4 0
1 4 1
MKS units:
0
c t c c t
t t
πρπ
ρε
µ µ ε
⋅ = ⋅ =∂ ∂
× = − × = +∂ ∂
⋅ = ⋅ =
∂ ∂× = − × = +
∂ ∂
Ε BB EE B J
Ε B
B EE B J
— —
— —
— —
— —
26 April 2004 Physics 218, Spring 2004 4
Electrodynamics (continued)
The Maxwell equations in matter
Boundary conditions for electrodynamics
4 0
1 4 1f
fc t c c t
πρ
π
⋅ = ⋅ =
∂ ∂× = − × = +
∂ ∂
D BB DE H J
— —
— —
( )( )
above ,above below ,below
,above ,below
,above ,below
,aboveabove
and are continuous; is discontinuous by 4 ;
ˆ is discontinuous by 4 .
In linear media , :4
00
1
f
f
f
B ED
H c
E E
B BE E
B
πσ
π
ε µε ε πσ
µ
⊥
⊥
⊥ ⊥
⊥ ⊥
×
= =
− =
− =
− =
−
K n
D E B H
,belowbelow
1 4 ˆfBcπ
µ= ×K n
26 April 2004 Physics 218, Spring 2004 5
Electrodynamics (continued)
Potentials and fieldsGauge transformations, especially the LorentzgaugeEnergy conservation in electrodynamics: Poynting’s theorem
1
1 0
Vc t
Vc t
∂= − −
∂= ×
∂⋅ + =
∂
AE
B A
A
—
—
—
( )
( )
( )
( )
mech.
2 2
mech.
2 2
4
18
0
1,4 8
EB
EB
dW c ddt
d B E ddt
u ut
c u E B
π
τπ
π π
= − × ⋅
− +
∂+ + ⋅ =
∂
= × = +
∫
∫
E B a
S
S E B
S
V
—
26 April 2004 Physics 218, Spring 2004 6
Electrodynamics (continued)
Momentum conservation in electrodynamics and the Maxwell stress tensor
( )
( )
( )
2 2
mech.
mech.
1 14 2
0
14
14
ij i j i j ij
EB
EB
EB
EB EB
T E E B B E B
d dd ddt dt
t
c
c
δπ
τ
π
π
⎛ ⎞= + − +⎜ ⎟⎝ ⎠
= −
∂+ − =
∂
≡
= =
∫ ∫p T a g
g g T
g E B
r g r E B
S V
L
◊
◊
¥
¥ ¥ ¥
—
26 April 2004 Physics 218, Spring 2004 7
Waves
Electromagnetic wavesWaves on a string
The simple solutions to the wave equationSinusoidal waves
Polarization
2 22 2 2 2,
c t c tµε µε2 2∂ ∂
∇ = ∇ =∂ ∂
E BE B2 2
2 2f f
Tx tµ∂ ∂⎛ ⎞= ⎜ ⎟
⎝ ⎠∂ ∂
( ) ( ) .f g x vt g z= ± =
( ) ( ), ,i kx t if x t Ae A Aeω δ−= =
26 April 2004 Physics 218, Spring 2004 8
Waves (continued)
Reflection and transmission of waves on a stringImpedance
( )
( )
( )}( ) ( ) ( ) ( )
1
1
2
2 1 1 2
2 1 1 2
2 1
1 2 1 2
0 :
0 :
0 , 0 , , 0 , 0 ,
2 2 ;
i k z tI I
I Ri k z tR R
i k z tT T T
R I I
T I I
f A ez f f f
f A e
f A e z f f
f ff t f t t t
z zv v Z ZA A Av v Z Z
v ZA A A Z T v Tv v Z Z
ω
ω
ω
µ
−
− −
−
− + − +
⎫= ⎪ ≤ = +⎬⎪= ⎭
= ≥ =
∂ ∂= =
∂ ∂− −
= =+ +
= = = =+ +
26 April 2004 Physics 218, Spring 2004 9
Plane electromagnetic waves in linear media
Plane electro-magnetic wavesEnergy and momentum in plane electro-magnetic wavesRadiation pressureWaves in linear media
( )02 2 2
2
2
ˆ,
ˆ ˆ,4 4 4
ˆ ˆ4
ie
E B cEu cu
E uc cc
ω
π π π
π
⋅ −= =
= = = =
= = =
k r tE E B k E
S k k
Sg k k
¥
( ) 2 2
2
ˆ
4 41 1 1
8 8
4 4
c c
u E B
c cc
µε
π πµ
επ π µεµ εµ ε
π π
=
= × = ×
⎛ ⎞= + = +⎜ ⎟
⎝ ⎠
= = × = ×
B z E
S E H E B
E D B H
Sg E H E B
¥
◊ ◊
26 April 2004 Physics 218, Spring 2004 10
Plane electromagnetic waves in linear media (continued)
The impedance of linear mediaSpaceclothBoundary conditions for reflection and transmission of electromagnetic plane waves at interfaces
4Zcπ µ
ε=
( )( )
( )
1 ,1 2 ,2 ,1 ,2
,1 ,2 ,1 ,21 2
1 0 0 2 0 0 0 0
0 0 0 0 0 01 2
0 0 0 0 0 01 2
0 01 10 0
or
1 1
1 1
Iz Rz Tz Iz Rz Tz
Ix Rx Tx Ix Rx Tx
Iy Ry Ty Iy Ry Ty
E E B B
E E B B
E E E B B B
E E E B B B
E E E B B B
ε ε
µ µ
ε ε
µ µ
µ µ
⊥ ⊥ ⊥ ⊥− = − =
− = − =
+ = + =
+ = + =
+ = + =
26 April 2004 Physics 218, Spring 2004 11
Plane electromagnetic waves in linear media (continued)
Snell’s Law
The Fresnelequations
2 1
1 2
00 0 0
00 0 0
21
2
1 2 2 1 1 2 2 1 2 1
sinsin
2 1, , : , .1 12, , : , .
cos 1 1 sincos cos
I R
T
I
II R T T R I
II R T T R I
TI
I I
k v nk v n
EE E E
EE E E
nn
Z Z n n
θ θθθ
αβαβ αβ
α βα β α β
θα θ
θ θ
β µ ε µ ε ε ε
=
= = =
−⊥ = =
+ +
−= =
+ +
⎛ ⎞= ≅ − ⎜ ⎟
⎝ ⎠
= = ≅ =
I
T
E k k k
E k k k
26 April 2004 Physics 218, Spring 2004 12
Plane electromagnetic waves in linear media (continued)
Total internal reflectionPolarization on reflectionInterference in layers of linear mediaTransmission and reflection in stratified linear media, viewed as a boundary-value problem
2
1tan .IB
nn
θ β= =
2
1arcsin 1IC
nn
θ⎛ ⎞
> ⎜ ⎟⎝ ⎠
( )2 cos 0,1,2,tm
dn mm
θλ = = …
26 April 2004 Physics 218, Spring 2004 13
Plane electromagnetic waves in linear media (continued)
Matrix formulation of the fields at the interfaces in stratified linear media
11, 1 1
1 1
11,
1 1
,1 ,2 ,21 1 11
1 1 1,1 ,2 ,2
, 1,11 2
,1 , 1
111 2
4 1cos cos
1cos
cos sin /sin cos
TE T T
TMT
pp
p
p
Yc Z
Y
E E Ei YM
iYH H H
EEM M M
H H
mM M M M
ε πθ θµ
εµ θ
δ δδ δ
+
+
= =
=
⎡ ⎤ ⎡ ⎤ ⎡ ⎤−⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= ≡⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ =⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦
= = 12
21 22.
mm m⎡ ⎤⎢ ⎥⎣ ⎦
26 April 2004 Physics 218, Spring 2004 14
Plane electromagnetic waves in linear media (continued)
Characteristic matrix formulation of reflected and transmitted fields and intensityExamples:• Single interface• Plane-parallel
dielectric in vacuum
• Multiple quarter-wave stacks
11 0 12 0 1 21 22 1
11 0 12 0 1 21 22 1
0
11 0 12 0 1 21 22 1
1, 2
,
, 1, 1 2
0,
2
1
p p
p p
p p
R
I
T p p
I
m Y m Y Y m m Yr
m Y m Y Y m m Y
Ytm Y m Y Y m m Y
Sr
S
S Yt
YS
ρ
τ
τ ρ
+ +
+ +
+ +
⊥
⊥
+ ⊥ +
⊥
+ − −=
+ + +
=+ + +
= =
= =
+ =
26 April 2004 Physics 218, Spring 2004 15
Plane electromagnetic waves in conductors
Electromagnetic waves in conductorsAttenuation of the waves, and an electronic analogyPenetration of waves into conductors: skin depth
.2 2ε ρετπσ π
= =
2
2 2 2 24 .
tz c t cεµ πσµ2∂ ∂ ∂
= +∂∂ ∂
E E E
1 21 221 2 41 1cd πσκ µε ω εω
−⎛ ⎞⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟= = + −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠⎝ ⎠
26 April 2004 Physics 218, Spring 2004 16
Plane electromagnetic waves in conductors (continued)
Good and bad conductorsRelative phase of E and B of waves in conductors
( )
( )0 0 0
22 20
good, bad.4 42 2
1 , good,
2, , bad.2
ˆ8
i
z
k i kc c
k k i k Zc c
c k ek iB cE E
c k E e
φ
κ
εω εωσ σπ ππωµσ πωµσ
κ
ω πσ µ σκ µε κ κε
κω ω
πµ ω−
= + = =
= + ≅ ≅= =
+= =
=S z
26 April 2004 Physics 218, Spring 2004 17
Plane electromagnetic waves in conductors (continued)
Reflection from conducting surfaces
The characteristic matrix of a conducting layer
( )
( )( )
( )
0 0 0 0
1 2 1good1 2 1 22
2
11 1 1
1
1 1
11 1
1 11
2 1,1 1
21 ,
1 2 21 2 2
1 and ,
21 good
2 bad
T I R I
R
I
E E E E
ck i
II
ckY k d
ick
ic c
ββ β
µ µ πσβ γ γε µ ω ε µ ω
γ γ
γ γ
δµ ω
πωµ σ
πσ µωµ εε
−= =
+ +
= ⎯⎯⎯→ + =
− +=
+ +
= =
⎧+⎪
⎪= ⎨⎪ +⎪⎩
26 April 2004 Physics 218, Spring 2004 18
Plane electromagnetic waves in dispersive media
Motion of bound electrons in matter, and the frequency dependence of the dielectric constantDispersion relationsOrdinary and anomalous dispersion
( )( )
( )( )
( )
220 02
2
2 21
02 22
22 2 2 21
2
22 2 2 21
41
Dilute gas: ,
21 ,
42 ,
.2
i t
eM j
e j j j
z
M j j
e j j j
M j j
e j j j
qd x dx x E edt mdt
fNqm i
I z I e
fNqcn km
fNqm c
cn n i
ω
α
γ ω
πε
ω ω γ ω
ω ωπω ω ω γ ω
γ ωπ ωα κ
ω ω γ ω
αω
−
=
−
=
=
+ + =
= +− −
=
−≡ ≅ +
− +
≡ =− +
= +
∑
∑
∑
26 April 2004 Physics 218, Spring 2004 19
Plane electromagnetic waves in dispersive media (continued)
Semiclassical theory of conductivityConductivity and dispersion in metals and in very dilute conductorsLight propagation in very dilute conductors: group velocity, plasma frequency
( )
20
02 2
0 0
02 22
2 02 2
2
2 2
1
metals, gases.
41 ,
1
1
e
e e
pp
e
p
g p
Nf qm i
Nf q Nf qi
m m
Nf qk
mc
c cv ck n
dv c cdk
σγ ω
σ σγ ω
ω πω ωω
ω
ω ω
ω ω ω
=−
≅ ≅
⎛ ⎞⎜ ⎟= − ≡⎜ ⎟⎝ ⎠
= = = >−
= = − <
, always, in nondispersive media.d c ck dk nω ω= = <
26 April 2004 Physics 218, Spring 2004 20
Guided waves
Metallic waveguidesLight propagation in hollow conductive waveguides
0 00 2
22
0 00 2
22
0 00 2
22
0 00 2
22
.
z zx
z zy
z zx
z zy
E BiE kx c y
kc
E BiE ky c x
kc
B EiB kx c y
kc
B EiB ky c x
kc
ωω
ωω
ωω
ωω
⎛ ⎞∂ ∂= +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠−
⎛ ⎞∂ ∂= −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠−
⎛ ⎞∂ ∂= −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠−
⎛ ⎞∂ ∂= +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠−
0
0
0 TE waves
0 TM wavesz
z
E
B
= ⇒
= ⇒
26 April 2004 Physics 218, Spring 2004 21
Guided waves (continued)
The TE modes of rectangular metal waveguides
0 0
0 00 02 2
2 22 2
0 00 02 2
2 22 2
cos cos ,
, 0,1,2,... (but not both 0)
, ,
, .
z
z zx y
z zx y
n ym xB Ba b
m n
B Bi iE Ec y c x
k kc c
B Bik ikB Bx y
k kc c
ππ
ω ωω ω
ω ω
=
=
∂ ∂−= =
∂ ∂− −
∂ ∂−= =
∂ ∂− −
26 April 2004 Physics 218, Spring 2004 22
Guided waves (continued)
Waveguide modes, e.g. TE:
( )
220
2 2 2
2
22 2
22 2 2
22 2
ˆsin cos cos8
ˆcos sin cos
cos sin
ˆsin cos .
n yB i m m x m xa a a bc k
n y n yn m xb a b b
n yk n m xb a bc k
n ym m xb a b
πω π π ππ ω
π ππ π
πω π π
ω
ππ π
⎡ ⎛= ⎢ ⎜⎝−⎢⎣
⎞+ ⎟⎠
⎛ ⎡ ⎤⎜+ ⎢ ⎥⎜ ⎣ ⎦⎝−
⎤⎞⎡ ⎤ ⎥⎟+ ⎢ ⎥ ⎟⎣ ⎦ ⎥⎠ ⎦
S x
y
z
26 April 2004 Physics 218, Spring 2004 23
Guided waves (continued)
Dispersion and cut-off in waveguidesMassive photons?The real reason there are no TEM modes in hollow conducting waveguidesTEM modes in coaxial waveguides
2 2 2 2 2
2 2 2
2
2
2 2
2 2
1
1
1
mn
mn
g mn
m nkc a b
ccv c
k
dv c cdk
ω π π
ωωω
ω
ω ωω ω ω
= − −
= −
= = >−
= = − <