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Page 1: Today in Physics 218: review I - University of Rochesterdmw/phy218/Lectures/Lect_82b.pdf · Today in Physics 218: review I ... kdk n ω ω ==< 26 April 2004 ... Microsoft PowerPoint

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).

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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

+ =⇒ + = = =

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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

— —

— —

— —

— —

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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

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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

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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

¥

¥ ¥ ¥

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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−= =

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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

− −

− + − +

⎫= ⎪ ≤ = +⎬⎪= ⎭

= ≥ =

∂ ∂= =

∂ ∂− −

= =+ +

= = = =+ +

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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

¥

◊ ◊

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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

⊥ ⊥ ⊥ ⊥− = − =

− = − =

+ = + =

+ = + =

+ = + =

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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

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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 = = …

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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⎡ ⎤⎢ ⎥⎣ ⎦

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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

+ +

+ +

+ +

+ ⊥ +

+ − −=

+ + +

=+ + +

= =

= =

+ =

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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

−⎛ ⎞⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟= = + −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠⎝ ⎠

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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

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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

−= =

+ +

= ⎯⎯⎯→ + =

− +=

+ +

= =

⎧+⎪

⎪= ⎨⎪ +⎪⎩

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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

=

=

=

+ + =

= +− −

=

−≡ ≅ +

− +

≡ =− +

= +

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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= = <

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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

= ⇒

= ⇒

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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

=

=

∂ ∂−= =

∂ ∂− −

∂ ∂−= =

∂ ∂− −

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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

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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

= − −

= −

= = >−

= = − <