Lecture 11

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  • 1ECE 303 Fall 2007 Farhan Rana Cornell University

    Lecture 11

    Faradays Law and Electromagnetic Inductionand

    Electromagnetic Energy and Power Flow

    In this lecture you will learn:

    More about Faradays Law and Electromagnetic Induction

    The Non-uniqueness of Voltages in Magnetoquasistatics

    Electromagnetic Energy and Power Flow

    Electromagnetic Energy Stored in Capacitors and Inductors

    Appendix (some proofs)

    ECE 303 Fall 2007 Farhan Rana Cornell University

    Faradays Law Revisited

    adHt

    adBt

    sdE orrrrrr ...

    ==

    A closed contour

    BFaradays Law: The line integral of E-field over a closed contour is equal to ve of the time rate of change of the magnetic flux that goes through any arbitrary surface that is bounded by the closed contour

    Important Note: In electroquasistatics the line integral of E-field over a closed contour was always zero

    In magnetoquasistatics this is NOT the case

    ( ) == 0.. sdsdE rrr

  • 2ECE 303 Fall 2007 Farhan Rana Cornell University

    Electromagnetic Induction and Kirchoffs Voltage Law

    Now consider a circuit through which the magnetic flux is changing with time (Kirchoffs voltage law is violated)

    +

    +-

    -

    dtdadB

    tsdE =

    = rrrr

    ..

    dtdVRIRIsdE =+= 21. r

    r

    1R

    2R

    I

    ( ) ( ) dtd

    RRRRVI

    2121

    1++=

    V+-( )t

    +

    +-

    -

    1R

    2R

    I V+-

    Kirchoffs voltage law comes from the electroquasistaticapproximation: 0. = sdE r

    r

    0. 21 =+= VRIRIsdE rr

    ( )21 RRVI +=

    ECE 303 Fall 2007 Farhan Rana Cornell University

    Lenz Law

    +

    +-

    -

    Suppose an induced current I is flowing through the wire:

    ( ) dtd

    RRI

    21

    1+=

    1R

    2R

    IThe induced current in the wire produces its own magnetic field

    Lenz Law is just an easy way to remember in which direction the induced current flows

    The law states that the induced current will flow in a direction such that its own magnetic field opposes the time variation of the magnetic field that produced it

    Example: Suppose the magnetic flux through the wire loop shown above wasincreasing with time (so that d/dt > 0). Lenz would tell us that the induced current would flow in the clockwise direction so that its own magnetic field would oppose the increasing magnetic flux through the loop

    In the equation above this fact comes out from the negative sign on the right hand side

    ( )t

  • 3ECE 303 Fall 2007 Farhan Rana Cornell University

    Non-Uniqueness of Voltages in Magnetoquasistatics - I

    +

    +-

    -

    ( )dtdRRIsdE =+= 21. r

    r

    1R

    2R

    I

    2V1V

    Question: What is the voltages difference V2-V1 ?

    One may be tempted to write .

    112

    221

    VRIVVRIV

    ==

    00

    12 ==

    VVI

    .. which cannot be correct since we know that:

    We have:

    What went wrong? Our usual concepts of circuit theory and potentials which are based on conservative E-fields are not valid when time varying magnetic fields are present

    ( ) dtd

    RRI

    21

    1+=

    ( )t

    ECE 303 Fall 2007 Farhan Rana Cornell University

    Non-Uniqueness of Voltages in Magnetoquasistatics - II

    +

    +-

    -

    1R

    2R

    I

    2V1V

    Lets do some real experimental measurements and see what we get for V2-V1

    +-

    V

    C1

    C2

    Faradays Law for contour C1:

    voltmeter

    dtdRIRI

    tadBsdE

    =+

    =

    21

    ..rrrr

    Faradays Law for contour C2:

    112

    1 0

    ..

    IRVVVRIV

    tadBsdE

    ===

    =

    rrrr

    +== dt

    dRR

    RVVV 21

    112

    This value for V2-V1 would actually be measured experimentally if the measurement is done as shown above

    ( )t

  • 4ECE 303 Fall 2007 Farhan Rana Cornell University

    Non-Uniqueness of Voltages in Magnetoquasistatics - III

    +

    +-

    -

    1R

    2R

    I

    2V1V

    Lets do another real experimental measurement and see what we get for V2-V1

    +-

    V

    C1

    C3

    Faradays Law for contour C1:

    voltmeter

    dtdRIRI

    tadBsdE

    =+

    =

    21

    ..rrrr

    Faradays Law for contour C3:

    212

    2 0

    ..

    IRVVVRIV

    tadBsdE

    ===

    =rrrr

    +== dt

    dRR

    RVVV 21

    212

    This value for V2-V1 would actually be measured experimentally if the measurement is done as shown above

    Lesson: The result of a voltage measurement depends on how exactly you do the measurement when time varying magnetic fields are present

    ( )t

    ECE 303 Fall 2007 Farhan Rana Cornell University

    Inductors and Faradays Law - I

    Consider an inductor with a time varying current:

    +_

    ( )tIInductance = L

    C

    The magnetic flux is given by:

    ( ) ( )tILt =

    Consider Faradays law over the closed contour C:

    adHt

    adBt

    sdE orrrrrr

    ... =

    =

    ( )tV

    ( ) ( ) ( )dt

    tIdLdt

    tdtV ==

  • 5ECE 303 Fall 2007 Farhan Rana Cornell University

    Inductors and Faradays Law - II

    +_

    ( )tIInductance = L

    CConsider Faradays law over the new closed contour C:

    ( ) ( )( )2

    .

    21.

    ...

    vertical

    vertical

    tVsdE

    dttdsdEtV

    adHt

    adBt

    sdE o

    =

    =+

    =

    =

    rr

    rr

    rrrrrr

    ( )tV

    Question: is there electric field anywhere else besides at the terminals??

    There must be E-field inside the loop !!

    This E-field is not very easy to determine analytically.

    ECE 303 Fall 2007 Farhan Rana Cornell University

    The Current-Charge Continuity Equation in Electromagnetism - I

    Start from:tEJH

    +=rrr

    Take divergence on both sides: ( ) ( )t

    EJH +=

    rrr ...0

    ( )t

    EJ +=

    rr ..0

    Use Gauss law: = Er.

    To get:t

    J = r. Current-charge continuity equation

    What does this equation mean??

  • 6ECE 303 Fall 2007 Farhan Rana Cornell University

    Suppose the current density is spatially varying as shown below

    x

    x xx +( )xJx ( )xxJx +The difference in current density at x and x+x must result in piling up of charges in the infinitesimal strip .

    ( ) ( ) ( )t

    xtxtxxJtxJ xx =+ ,,,

    ( ) ( ) ( )t

    xtxx

    txJtxxJ xx

    =+ ,,,

    ( ) ( )t

    txx

    txJx

    = ,,

    Now generalize to 3D and get the desired result:

    ( ) ( )t

    trtrJ = ,,.

    rrr

    The Current-Charge Continuity Equation in Electromagnetism - II

    ECE 303 Fall 2007 Farhan Rana Cornell University

    A continuity equation of the form:

    ( ) ( )t

    trtrJ = ,,.

    rrr

    establishes a relation between the flow rate of a quantity (in the present case the flow rate of charge density) and the time rate of change of the quantity

    In the next few slides we will try to find a power-energy continuity equation for the electromagnetic field

    A continuity equation is in fact a conservation law

    For example, the current-charge continuity equation expresses charge conservation. It says that charge cannot just disappear. In the integral form it becomes:

    ( ) ( ) ( )dt

    tdQdVtrt

    adtrJ == ,., rrrr

    If there is a net inflow of charge into a closed volume then that must result in an increase in the total charge Q inside that closed volume

    J

    closed volume

    The Current-Charge Continuity Equation in Electromagnetism - III

  • 7ECE 303 Fall 2007 Farhan Rana Cornell University

    The Electromagnetic Power-Energy Continuity Equation - I

    We know that electric and magnetic fields have energy. Let W be the energy density (i.e. energy per unit volume) of the electromagnetic field

    ( ) =trS ,rr

    ( ) =trW ,r Suppose we had a vector that expressed the energy flow rate (or energy flux) in Joules/(m2-sec) for the electromagnetic field

    Energy density (units: Joules/m3)

    Energy flow rate (units: Joules/(m2-sec))

    Then the conservation of electromagnetic energy must be expressed in a continuity equation of the form:

    ( ) ( )t

    trWtrS = ,,.

    rrr

    Compare with the current-charge continuity equation:

    ( ) ( )t

    trtrJ = ,,.

    rrr

    ECE 303 Fall 2007 Farhan Rana Cornell University

    The Electromagnetic Power-Energy Continuity Equation - II

    Start from the two Maxwells equations:

    tEJH

    tHE o

    +=

    =rrr

    rr

    Take the dot-product of the first equation with and of the second equation with to get:

    Er

    Hr

    ( )( )

    tEEEJEH

    tHHHE o

    +=

    =

    rrrrrr

    rrrr

    .21..

    .21.

    Subtract the above two to get:

    ( ) ( )t

    EEEJt

    HHEHHE o

    =rrrrrrrrrr .

    21..

    21..

  • 8ECE 303 Fall 2007 Farhan Rana Cornell University

    The Electromagnetic Power-Energy Continuity Equation - II

    Use the vector identity on the left hand side: ( ) ( ) ( ) A.BB.ABA. rrrrrr = ( ) EJEEHH

    tHE o

    rrrrrrvr.

    2.

    2..- +

    +

    =

    ( ) ( )t

    EEEJt

    HHEHHE o

    =rrrrrrrrrr .

    21..

    21..

    Define: ( ) ( ) ( )trHtrEtrS ,,, rrrrrr =

    ( ) ( ) ( ) ( ) ( )

    +=

    2,.,

    2,.,, trEtrEtrHtrHtrW o

    rrrrrrrr

    To get:

    And get:

    ( ) ( ) ( ) ( )trEtrJt

    trWtrS ,.,,,.rrrrrrr +

    =This is what we wanted (other than the last term on the right hand side)

    ECE 303 Fall 2007 Farhan Rana Cornell University

    The Electromagnetic Power-Energy Continuity Equation - III

    is called the Poynting vector and it is the energy flow per second per unit area

    points in the direction of power flow

    has the units of Joules/(m2-sec) or Watt/m2

    ( )trS ,rr

    is the energy density of electromagnetic field

    has the units of Joules/m3

    has two parts:

    ( )trW ,r

    Poynting Vector:

    Energy Density: ( ) ( ) ( ) ( ) ( )

    +=

    2,.,

    2,.,, trEtrEtrHtrHtrW o

    rrrrrrrr

    ( )trW ,r

    ( ) ( )2

    ,., trHtrHorrrr

    ( ) ( ) ( )trHtrEtrS ,,, rrrrrr =

    ( )trS ,rr( )trS ,rr

    ( )trW ,r

    ( ) ( )2

    ,., trEtrErrrr

    Energy density of the magnetic field

    Energy density of the electric field

  • 9ECE 303 Fall 2007 Farhan Rana Cornell University

    ( ) ( ) ( ) ( )trEtrJt

    trWtrS ,.,,,.rrrrrrr +

    =

    The Electromagnetic Power-Energy Continuity Equation - IVPower dissipation:

    The last term in the continuity equation J.E indicates power dissipation

    Its units are Joules/(m3-sec)

    This term represents the energy that is lost from the electromagnetic field

    Essentially what is happening is that if the medium is conductive then in the presence of electric fields currents will be produced and the last term represents the usual I 2R losses in conductors or resistors

    Integral Form: ( ) ( ) ( ) ( )+= dVtrEtrJdVtrWtadtrS ,.,,.,

    rrrrrrrr

    S

    closed volume

    Net power flow into a closed volume

    =Rate of increase of the total electromagnetic energy inside the closed volume

    +Rate of the total energy loss through dissipation

    ECE 303 Fall 2007 Farhan Rana Cornell University

    Example: Parallel Plate Capacitor

    V+ -

    = V = 0area = A

    ( )

    ( ) ( )dV

    xxxE

    dxVx

    x ==

    =

    21

    ++++++++

    --------

    0 x

    d

    Electric field energy density = ( ) ( ) 222

    .

    =dVxExE xx

    Total electric field energy = ( ) ( ) ( ) ( )2

    2

    21

    22.

    CV

    dAdVdAxExE xx

    =

    =

    dAC =

    For all (linear) capacitors the total stored electric field energy equals CV 2/2

    Recall that:

  • 10

    ECE 303 Fall 2007 Farhan Rana Cornell University

    Example: Parallel Plate Inductory

    x

    zK

    d

    W

    zK

    xH

    WIK =

    WIKHx ==

    Wd

    IdHL oxo ll ==

    Magnetic field energy density =2

    22.

    =WIHH oxxo

    Total magnetic field energy = ( ) ( )2

    2

    21

    22.

    LI

    WdWIdWHH oxxo

    =

    = ll

    The structure has length in the z-directionRecall that:

    For all (linear) inductors the total stored magnetic field energy equals LI 2/2

    (L is the total inductance)

    ECE 303 Fall 2007 Farhan Rana Cornell University

    Power Flow in Electroquasistatics and Circuit Theory - I

    Consider a complex electrical circuit shown below:

    +_1V

    1I

    +_2V 2I

    + _3V

    3I

    A complex electrical circuit

    If you wanted to calculate the total electrical power P going into the circuit you would do the following sum:

    44332211 IVIVIVIVIVPn

    nn ++ ==

    +_4V

    4I

    Where does this formula comes from?

  • 11

    ECE 303 Fall 2007 Farhan Rana Cornell University

    Power Flow in Electroquasistatics and Circuit Theory - II

    +_1V

    1I

    +_2V 2I

    + _3V

    3I

    A complex electrical circuit+_

    4V4I

    Start from the Poynting Vector: ( ) ( ) ( )trHtrEtrS ,,, rrrrrr =Total power flow into the closed volume

    closed volume

    ( ) === adHEadSP rrrrr ..In electroquasistatics: =Er

    Therefore: ( ) = adHP rr .Use the vector identity: ( ) ( ) ( )FFF rrr +=

    ( ) ( ) = adHadHP rrrr .. To get:

    In magnetoquasistatics: JHrr =

    ECE 303 Fall 2007 Farhan Rana Cornell University

    Power Flow in Electroquasistatics and Circuit Theory - III

    +_1V

    1I

    +_2V 2I

    + _3V

    3I

    A complex electrical circuit+_

    4V4I

    closed volume

    ( ) ( ) = adHadHP rrrr .. 0

    ( )==

    =

    nnn IVadJP

    adHP

    rr

    rr

    .

    .

    =n

    nn IVP

    curl of a vector integrated over a closed surface is always zero

    Just what we wanted to prove

  • 12

    ECE 303 Fall 2007 Farhan Rana Cornell University

    ( ) ( )

    QV

    VQVQ

    adEadE

    adEadEadE

    dVadE

    dVEEdVE

    dVEEU

    SS

    SSS

    SSSSSS

    SSSSSS

    SSS

    21

    21

    21

    .21.

    21

    .21.

    21.

    21

    21.

    21

    ..21.

    21

    2.

    2211

    21

    21

    2121

    2121

    21

    =

    +=

    =

    =

    +=

    = =

    =

    ++++

    ++++

    ++

    rrrr

    rrrrrr

    rr

    rrr

    rr

    V

    S

    S1

    S2

    0

    0

    V1=V

    V2=0

    Appendix: Energy Stored in Capacitors is QV/2

    ECE 303 Fall 2007 Farhan Rana Cornell University

    I

    S

    ( )( )

    I

    sdAI

    dVJA

    dVHA

    dVHAadAH

    dVHAAHdVAH

    dVHHU

    C

    S

    S

    SS

    SS

    S

    o

    21

    .21

    .21

    .21

    .21.

    21

    ..21.

    21

    2.

    =

    =

    =

    =

    + =

    += =

    =

    rr

    rr

    rr

    rrrrr

    rrrrrr

    rr

    C

    0

    Appendix: Energy Stored in Inductors is I/2

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