Electromagnetic Fields and Waves

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Transcript of Electromagnetic Fields and Waves

  • More on MaxwellWave EquationsBoundary ConditionsPoynting VectorTransmission Line

    A. Nassiri - ANL

    Lecture 2Review 2

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 2

    Maxwells equations in differential form

    HBED

    == Varying E and H fields are coupled

    continuityofEquationt

    lawsFaradaydt

    lawsAmperedt

    ticsmagnetostaforlawGaussticselectrostaforlawGauss

    =

    =

    +=

    ==

    J

    BE

    DJH

    0BD

    .

    '

    '

    '.'.

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 3

    Electromagnetic waves in lossless media - Maxwells equations

    SI Units J Amp/ metre2 D Coulomb/metre2 H Amps/metre B Tesla

    Weber/metre2Volt-Second/metre2

    E Volt/metre Farad/metre Henry/metre Siemen/metre

    dtBE =dtDJH +=

    = D.0. = B

    EJHHB

    EED

    ===

    ==

    or

    or

    t

    =J.

    Maxwell

    Equation of continuity

    Constitutive relations

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 4

    Wave equations in free space

    In free space =0 J=0 Hence:

    dtBE =

    dtdtDDJH =+=

    2

    2

    t

    tt

    ttt

    o

    o

    o

    =

    =

    =

    =

    =

    EE

    D

    HBBE

    Taking curl of both sides of latter equation:

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 5

    Wave equations in free space cont.

    It has been shown (last week) that for any vector A

    where is the Laplacian operatorThus:

    2

    2

    to

    =EE

    AAA 2. =

    2

    2

    2

    2

    2

    22

    zyx

    +

    +

    =

    2

    22.

    to

    =EEE

    2

    22

    to

    =EE

    There are no free charges in free space so .E==0 and we get

    A three dimensional wave equation

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 6

    Both E and H obey second order partial differential wave equations:

    2

    22

    to

    =EE

    Wave equations in free space cont.

    2

    22

    to

    =HH

    22 secondseVolts/metr

    metreeVolts/metr

    = o

    What does this mean dimensional analysis ?

    has units of velocity-2

    Why is this a wave with velocity 1/ ?

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 7

    Uniform plane waves - transverse relation of E and H

    Consider a uniform plane wave, propagating in the z direction. E is independent of x and y

    0. =

    +

    +

    =zyx

    zyx EEEE

    00 =

    =

    yxEE

    )waveanotisconst(000,0 ===

    =

    =

    zzzyx EE

    zE

    yE

    xE

    In a source free region, .D= =0 (Gauss law) :

    E is independent of x and y, so

    So for a plane wave, E has no component in the direction of propagation. Similarly for H. Plane waves have only transverse E and H components.

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 8

    Orthogonal relationship between E and H:

    For a plane z-directed wave there are no variations along x and y:

    +

    +

    =

    yA

    xA

    xA

    zA

    zA

    yAA

    xyz

    zxy

    yzx

    a

    a

    a

    zH

    zH x

    yy

    x

    +

    = aaH

    tE

    zH

    tE

    zH

    yx

    xy

    =

    =

    tH

    zE

    tH

    zE

    yo

    x

    xo

    y

    =

    =

    to = HE Equating terms:dtDJH +=

    and likewise for :

    +

    +

    =

    =

    tE

    tE

    tE

    t

    zy

    yy

    xx aaa

    D

    Spatial rate of change of H is proportionate to the temporal rate of change of the orthogonal component of E & v.v. at the same point in space

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 9

    Orthogonal and phase relationship between E and H:

    Consider a linearly polarised wave that has a transverse component in (say) the y direction only:

    tE

    zH

    tE

    zH

    yx

    xy

    =

    =

    ( )

    ( )vtzfvEt

    EvtzfEE

    oy

    oy

    =

    =

    tH

    zE

    tH

    zE

    yo

    x

    xo

    y

    =

    =

    xo

    y EH

    =

    Similarly

    ( ) ( )

    yo

    x

    y

    oox

    EH

    vE

    vtzfvEconstzvtzfvEH

    =

    =

    =+= dz

    H x

    =

    H and E are in phase and orthogonal

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 10

    The ratio of the magnetic to electric fields strengths is:

    which has units of impedance

    yo

    x EH

    = xo

    y EH

    =

    ===

    +

    +o

    yx

    yx

    HE

    HH

    EE22

    22

    =metreampsmetreVolts

    //

    ==

    =

    37712010

    361

    1049

    7

    o

    o

    cH

    EBE

    ooo===

    1

    Note:

    and the impedance of free space is:

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 11

    Orientation of E and H

    For any medium the intrinsic impedance is denoted by

    and taking the scalar product

    so E and H are mutually orthogonal

    y

    x

    x

    y

    HE

    HE

    ==

    0

    .

    ==

    +=

    yxxy

    yyxx

    HHHH

    HEHE

    HE

    ( )( )

    2

    22

    H

    HH

    HEHE

    z

    xyz

    xyyxz

    aHE

    a

    aHE

    =

    =

    = ( )( )( )xyyxz

    zxxzy

    yzzyx

    BABA

    BABA

    BABA

    +

    +=

    a

    a

    aBA

    Taking the cross product of E and H we get the direction of wave propagation

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 12

    A horizontally polarised wave

    Sinusoidal variation of E and H E and H in phase and orthogonal

    Hy Ex

    xo

    y EH

    =

    HE

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 13

    A block of space containing an EM plane wave

    Every point in 3D space is characterised by Ex, Ey, Ez

    Which determine

    Hx, Hy, Hz and vice versa

    3 degrees of freedom

    HEHy

    Ex

    yo

    x

    xo

    y

    EH

    EH

    =

    =

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 14

    Power flow of EM radiation

    Energy stored in the EM field in the thin box is:

    HEHy

    Exdx

    Area A

    ( ) xAuuUUU HEHE dddd +=+=

    2

    22

    2

    Hu

    Eu

    oH

    E

    =

    =

    xAE

    xAHEU o

    d

    d22

    d

    2

    22

    =

    +=

    xo

    y EH

    = Power transmitted through the box is dU/dt=dU/(dx/c)....

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 15

    Power flow of EM radiation cont.

    This is the instantaneous power flow Half is contained in the electric component Half is contained in the magnetic component

    E varies sinusoidal, so the average value of S is obtained as:

    ( )2

    22W/md

    ddd

    ExA

    cxAE

    tAUS

    o====

    xAEU dd 2=

    ( )

    ( )

    ( )( )

    2sinRMS

    sin

    2sinE

    222

    2

    22

    oo

    o

    o

    o

    EvtzEES

    vtzES

    vtzE

    ==

    =

    =

    S is the Poynting vector and indicates the direction and magnitude of power flow in the EM field.

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 16

    Example

    The door of a microwave oven is left open estimate the peak E and H strengths in the aperture of the door. Which plane contains both E and H vectors ? What parameters and

    equations are required?

    222

    W/mHES

    ==

    Power-750 W Area of aperture - 0.3 x 0.2 m impedance of free space - 377 Poynting vector:

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 17

    Tesla2.775.5104B

    A/m75.53772170H

    V/m171,22.0.3.0

    750377

    WattsA

    7

    22

    ===

    ===

    ===

    ===

    H

    E

    APowerE

    HAESAPower

    o

  • Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 18

    Constitutive relations

    permittivity of free space 0=8.85 x 10-12 F/m permeability of free space o=4x10-7 H/m Normally r (dielectric constant) and r

    vary with material are frequency dependant

    For non-magnetic materials mr ~1 and for Fe is ~200,000 er is normally a few ~2.25 for glass at optical frequencies

    a