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Transcript of Today in Physics 218: review I - University of dmw/phy218/Lectures/Lect_82b.pdfToday in Physics 218:...

  • 26 April 2004 Physics 218, Spring 2004 1

    Today in Physics 218: review I

    You learned a lot this semester, in principle. Heres 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

    = =

    =

    =

    = = =

    ,

    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 Maxwells repair of Ampres 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

    = =

    = = +

    = =

    = = +

    BB EE B J

    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

    = =

    = = +

    D BB DE H J

    ( )( )

    above ,above below ,below

    ,above ,below

    ,above ,below

    ,aboveabove

    and are continuous; is discontinuous by 4 ;

    is discontinuous by 4 .

    In linear media , :4

    00

    1

    f

    f

    f

    B ED

    H c

    E E

    B BE E

    B

    = =

    =

    =

    =

    K n

    D E B H

    ,belowbelow

    1 4 fB c

    = K n

  • 26 April 2004 Physics 218, Spring 2004 5

    Electrodynamics (continued)

    Potentials and fieldsGauge transformations, especially the LorentzgaugeEnergy conservation in electrodynamics: Poyntings 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

    =

    +

    + + =

    = = +

    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

    = + +

    =

    + =

    = =

    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 t 2 2

    = =

    E BE B2 2

    2 2f f

    Tx t =

    ( ) ( ) .f g x vt g z= =

    ( ) ( ), ,i kx t if x t Ae A Ae = =

  • 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

    + +

    = = +=

    = =

    = =

    = =+ +

    = = = =+ +

  • 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

    ,

    ,4 4 4

    4

    ie

    E B cEu cu

    E uc cc

    = =

    = = = =

    = = =

    k r tE E B k E

    S k k

    Sg k k

    ( ) 2 2

    2

    4 41 1 1

    8 8

    4 4

    c c

    u E B

    c cc

    =

    = =

    = + = +

    = = =

    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

    4Zc

    =

    ( )( )

    ( )

    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

    = =

    = =

    + = + =

    + = + =

    + = + =

  • 26 April 2004 Physics 218, Spring 2004 11

    Plane electromagnetic waves in linear media (continued)

    Snells 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

    =

    = = =

    = =

    + +

    = =

    + +

    =

    = = =

    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

    = =

    2

    1arcsin 1IC

    nn

    >

    ( )2 cos 0,1,2,tmdn m

    m

    = =

  • 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

    +

    +

    = =

    =

    =

    =

    = = 1221 22

    .m

    m 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

    + +

    + +

    + +

    + +

    + =

    + + +

    =+ + +

    = =

    = =

    + =

  • 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 2

    = =

    2

    2 2 2 24 .

    tz c t c 2

    = +

    E E E

    1 21 221 2 41 1cd

    = = +

  • 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

    8

    i

    z

    k i kc c

    k k i k Zc c

    c k ek iB cE E

    c k E e

    = + = =

    = + = =

    += =

    =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

    = =

    + +

    = + =

    +=

    + +

    = =

    +

    = +

  • 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

    =

    =

    =