Induced Electric Field

  1. Hello all,

    In Faraday's law we have that the variation of the magnetic flux in time
    will generate an induced electric field that loops on itself.

    My question concerns this induced electric field.

    If we imagine a bar magnet moving through a circular surface and consider
    the electric field induced on the circle at the boundary of this surface.
    (Note: the circle may be imaginary or an actual circle made of wire)

    1. What is the electric potential of this induced field (measured on the
    rest frame of the surface) for one complete circular loop?

    2. Since it is a loop, if we placed an imaginary test charge on it, what
    would be the energy that this charge would acquire (or work done to move
    this charge around the loop)? You can also imagine a superconducting wire
    with no resistance going around the induced electric field.

    Basically I don't understand the ideal of an electric field looping on
    itself. In my mind when we move along an electric field line we must have a
    delta in the potential. So is this delta in potential uniquely defined? What
    if I do one lap or two or infinite laps? what would the delta potential be?
     
  2. jcsd

  3. >the variation of the magnetic flux in time
    >will generate an induced electric field (...)
    >My question concerns this induced electric field.
    >What is the electric potential of this induced field (...)


    That's the point: There is NO electric potential for this induced
    field. If we had one, i.e., if we had E = - grad Phi, that would
    entail curl E = 0, instead of Faraday's curl E = - d_t B, where the
    r.h.s. is nonzero in the situation you describe.
     
  4. So when this electric field appears due to the d_t B if I place a test
    charge there and circulate the field I will do no work?

    "Bossavit" <bossavit@lgep.supelec.removethis.fr> wrote in message
    news:ge176p$ku0$1@fb07-hees.theo.physik.uni-giessen.de...
    >
    > >the variation of the magnetic flux in time
    > >will generate an induced electric field (...)
    > >My question concerns this induced electric field.
    > >What is the electric potential of this induced field (...)

    >
    > That's the point: There is NO electric potential for this induced
    > field. If we had one, i.e., if we had E = - grad Phi, that would
    > entail curl E = 0, instead of Faraday's curl E = - d_t B, where the
    > r.h.s. is nonzero in the situation you describe.
     
  5. ElectromagStudent schrieb:
    > Hello all,
    >
    > In Faraday's law we have that the variation of the magnetic flux in time
    > will generate an induced electric field that loops on itself.
    >
    > My question concerns this induced electric field.
    >
    > If we imagine a bar magnet moving through a circular surface and consider
    > the electric field induced on the circle at the boundary of this surface.
    > (Note: the circle may be imaginary or an actual circle made of wire)
    >
    > 1. What is the electric potential of this induced field (measured on the
    > rest frame of the surface) for one complete circular loop?
    >
    > 2. Since it is a loop, if we placed an imaginary test charge on it, what
    > would be the energy that this charge would acquire (or work done to move
    > this charge around the loop)? You can also imagine a superconducting wire
    > with no resistance going around the induced electric field.
    >
    > Basically I don't understand the ideal of an electric field looping on
    > itself. In my mind when we move along an electric field line we must have a
    > delta in the potential. So is this delta in potential uniquely defined? What
    > if I do one lap or two or infinite laps? what would the delta potential be?


    The Maxwell equations are a linear set, so for the fundamentals one is
    able to understand everything in a linear setting.

    Assume a constant magnetic field rising with linear in time t with
    constant rate b

    B = t (0,0,b).

    This fixes the induced part of the electric field by the law of
    induction (t = c*time)

    dB/dt + curl E = 0

    Near to the origin of space and time you may choose eg a circular
    electric field around the origin

    E = 1/2 (-y, x, 0)

    Of course, electric fields around other origins or superpositions with
    constant fields yield the same curl

    E = 1/2 (-(y-y0) s , (x-x0)(1-s)), s,x0,y0 some real parameter.

    So the actual electric field in such a situation will be specified by
    boundary or symmetry conditions of the real situation. This may be a non
    constant magnetic field to fix a midpoint. Or you have boundary
    conditons of conducting surfaces - eg the superconducting ring with a
    slit to measure the electric line integral along the wire aka the
    induced voltage over the slit.

    Of course, there exists a 4-potential for this situation.

    A = (A0,A1,A2,A3) = 1/2 t (0 , -(y-y0) s, (x-x0)(1-s), 0)

    with B and E given by

    B = curl spacepart A = curl(A1,A2,A3)

    E = spacepart d/dt A

    Scalar potentials for B and E do not exist, since line integrals of
    (-y,x,0) depend on the contour of integration chosen.

    In a closed normal conducting wire the time changing magnetic field is
    transferring energy to the electron gas by induction heating up the wire.

    In a superconducting ring the rising magnetic field - assumed starting
    at zero strength - is inducing a surface current distribution.

    The resulting magnetic field is a topologically determined solution of
    the time-linear Maxwell boundary problem with the effect, that the
    magnetic flux through the superconducting torus up to its outermost
    contuour is exactly zero. The inner part of the torus is free of any
    fields.

    In the superconducting surface Maxwells equation are satisfied by an
    exponential decaying distribution of the surface current and the
    shielded outer magnetic field (Meissner effect).

    These topological implications of Maxwells equations cannot be
    understood locally in terms of fields and potentials. They are deeply
    connected to the quantum mechanical description of electric charges and
    currents and the aspect of electromagnetic fields as a locally gauge
    field for linear and angular momentum and of charged quantum particles.

    The kind of solutions of Maxwell equations with superconducting
    boundaries depend on the topological genus of surface. There is a
    correspondence of the vector space of current distributions on surfaces
    without poles and zeros on the one hand and the number of holes in the
    surface of the other hand.

    Any spherelike object has no holes and so there is no nonsingular
    current distribution to shield an external or internal magnetic field:
    Spheres, cylinders, cubes dont show super currents.

    Any topologically toruslike object has two linear independent current
    distibutions as solutions of Maxwells equations.

    The first solution - a flow along the rim around the axis without any
    carge winding through the hole - is the boundary (and the physical
    source) of external magnetic fields with internal zero field.

    The second - a flow through the hole without any charge rotating arund
    the axis - generates an internal magnetic field with zero field in the
    outer space like a closed ore infinite selenoid.

    Funny things and a branch of pure mathematics. It shows the way by which
    the many degrees of freedom, the Maxwell equations don't fix, can be
    used to model the charge and current field distibutions in ideal boundaries.

    --

    Roland Franzius
     
  6. ElectromagStudent <test@hotmail.com> wrote:

    > Hello all,
    >
    > In Faraday's law we have that the variation of the magnetic flux in time
    > will generate an induced electric field that loops on itself.
    >
    > My question concerns this induced electric field.
    >
    > If we imagine a bar magnet moving through a circular surface and consider
    > the electric field induced on the circle at the boundary of this surface.
    > (Note: the circle may be imaginary or an actual circle made of wire)
    >
    > 1. What is the electric potential of this induced field (measured on the
    > rest frame of the surface) for one complete circular loop?
    >
    > 2. Since it is a loop, if we placed an imaginary test charge on it, what
    > would be the energy that this charge would acquire (or work done to move
    > this charge around the loop)? You can also imagine a superconducting wire
    > with no resistance going around the induced electric field.
    >
    > Basically I don't understand the ideal of an electric field looping on
    > itself. In my mind when we move along an electric field line we must have a
    > delta in the potential. So is this delta in potential uniquely defined? What
    > if I do one lap or two or infinite laps? what would the delta potential be?


    What you need to understand is that there is
    in general no potential.
    There is a simple undergraduate experiment to demonstrate this.
    Take a long thin coil, and feed it altenating current.
    Loop a wire (or a string of resistors) around it.
    Alternating current will be induced.
    What is the potential over a given resistor?
    When you try to measure it with an AC voltmeter
    you will find that the result depends
    on the way the leads are arranged.
    Hence no unique well defined potential difference.

    Jan
     
  7. ElectromagStudent <test@hotmail.com> wrote:

    > So when this electric field appears due to the d_t B if I place a test
    > charge there and circulate the field I will do no work?


    The E-field will do work.
    To get down to practicalities you might try to understand
    how a tokamak device works.

    Jan

    --
    A: Because it messes up the order in which people normally read text.
    Q: Why is it such a bad thing?
    A: Top-posting.
    Q: What is the most annoying thing on usenet?



    > "Bossavit" <bossavit@lgep.supelec.removethis.fr> wrote in message
    > news:ge176p$ku0$1@fb07-hees.theo.physik.uni-giessen.de...
    > >
    > > >the variation of the magnetic flux in time
    > > >will generate an induced electric field (...)
    > > >My question concerns this induced electric field.
    > > >What is the electric potential of this induced field (...)

    > >
    > > That's the point: There is NO electric potential for this induced
    > > field. If we had one, i.e., if we had E = - grad Phi, that would
    > > entail curl E = 0, instead of Faraday's curl E = - d_t B, where the
    > > r.h.s. is nonzero in the situation you describe.
     
  8. >So when this electric field appears due to the d_t B if I place a test
    >charge there and circulate the field I will do no work?


    No, there is some work involved indeed. But it cannot be expressed in
    terms of some electric potential, which in the situation you describe
    does not exist.

    Consider this highly idealized setup: A ring, made of some magic
    material which contains only ONE electron and will not let it escape
    outside, but will let it go freely around the loop (no drag). At t =
    0, B = 0 and E = 0 all over, and the electron is idle, somewhere.
    For t > 0, let B change, by the effect of some outside agency. (Your
    hand moving a magnet, for instance.) This creates an electric field E,
    which obeys the equation rot E = - dB/dt, among others. The charge is
    pushed by E, and accelerates. After the first lap, which takes some
    time T > 0 to complete, it has acquired kinetic energy, equal [give or
    take a minus sign that I'm too lazy to think about] to the flux of B,
    at time T, through the loop.

    This energy has been pumped out from the agency, whatever it is, that
    caused the built-up of B. [You may be able to devise an experiment in
    which you'd *feel* the difference it makes, when you move the magnet
    around, whether some conductor is close to you or not.]

    The force on the electric charge has two parts, Coulomb's and Lorentz's,
    but only the former does work.

    You can't determine the force from Faraday's equation alone, extra
    information is needed about the *divergence* of E, whereas Faraday only
    tells you about its *curl*. But when the charge performs a closed loop,
    you still can compute the *work* performed by this force, which is that
    thing called "circulation" of E around the loop, and depends on the
    curl of E only, by the Stokes theorem. After Faraday's law, this
    circulation is the flux of dB/dt across the loop, which you know if
    you know how B evolves in time.
     
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