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Changing magnetic field and a point charge, seems unresolvable

  1. Aug 23, 2012 #1
    Consider a current loop with a switch and an AC source in space, there is a point charge(electron) inside this loop, my question is,

    Will the point charge feel any force when we close the switch, i.e. will the point charge feel any force in the presence of changing magnetic field ?

    If yes, then, in which direction the force will be and of what magnitude ? Since, all sorts of electric line integral pass through this point charge, Does the force depend on the shape of the loop too ?

    If no, then, I think there is a problem with in the Feynman's Lecture on physics, vol. 2, ch. 17, section. 4, A paradox.

    And don't forget the betatrons work almost on the same principle, except the electrons are already moving in a well defined loop themselves when the changing magnetic field is applied to increase their energy.

    Thanks
     
  2. jcsd
  3. Aug 23, 2012 #2

    mfb

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    It depends on the position of the charge. If the loop is a perfect circle, the magnetic field is symmetric around the center, and changing, this gives rise to a circular electric field.
    If the point charge is not in the center, it will feel a force along these electric field lines, perpendicular to the line "point charge <-> center". You can find its magnitude by integrating along the loop to get the magnetic field as a function of time and integrating over its time-derivative in the area around the center. In other words: It can be done, but unless you need the result for something practical I would not recommend it.
    If the loop is not a perfect circle, it becomes more complicated to evaluate the forces, but the basic concept is the same.
     
  4. Aug 23, 2012 #3
    Since you say there will be a force, and for a perfect circle you even suggested the direction.

    Does that mean that a point charge standing near a straight long current carrying wire which is also connected to an AC source, will also feel the force ?

    If yes then in which direction ? Because among the possible line integral for electric field, are at both sides of the point charge, and therefore both line integral will give different direction for the force, and there are many more direction if we start considering all the line integrals !

    So, my question is, are there any known solutions or experiments(other than betatron) which shows how we should proceed about it.

    Thanks
     
  5. Aug 23, 2012 #4

    mfb

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    The AC source is a source of a variable electric field - not only in the wire, but also in the environment. This can accelerate electrons, and the direction is simply the "+" side.

    Solve the Maxwell equations?
     
  6. Aug 24, 2012 #5

    I see you are going around about the problem instead of tackling it directly.

    I think I know that there will be an electric field due to changing magnetic filed from a changing current, the problem is how am I supposed to apply the maxwell's equation for a point charge ?

    Since [itex]∇ × E = - ∂B/∂t[/itex] , the curl of E is only definable for a given metal loop or a given orbit of electrons, how can we define the same for a stationary point charge ?
     
  7. Aug 24, 2012 #6

    mfb

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    Maxwells equations give you the external field, given by the AC source and wire - and your point charge is affected by this field. Unless you have a very large point charge, you can ignore the effects of this charge on the charge distribution in the wire (and the corresponding changes in the electromagnetic field of the wire).
     
  8. Aug 24, 2012 #7
    Let's suppose I calculate the magnetic field vector as a function of time, around the wire.

    Now, the problem is how am I supposed to know about the direction and magnitude of the electric field induced, for a point charge ?

    For simplicity, let's assume that magnetic field is given by the equation [itex]B = B_0 sin(\omega t)[/itex], all over the place. Now, tell me how and why does the electron should or would move in a particular direction.

    Since, according to the maxwell's third equation that I posted earlier, [itex]∇×E=−∂B/∂t[/itex], by stokes's theorem leads to [itex]\oint E. ds =−∂\phi/∂t , where, \phi = \oint B. da[/itex], Now, since the integral is only meaningful for a closed loop, how should I to use this equation for the force on a stationary point charge ?
     
  9. Aug 24, 2012 #8

    mfb

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    The field induced by a point charge? You can approximate this point charge with a small object, or use distributions. However, this field will not create a force on the charge.

    The electromagnetic field you describe does not exist, there has to be some electric component, and B cannot fill the space like that. The force on the point charge is given by [itex] F=q(E + \vec{v} \times \vec{B})[/itex] with fields E and B from your device.
     
  10. Aug 24, 2012 #9
    No, the point charge is just the test charge, the self interaction is not the point here.
    I won't argue if such electromagnetic field is possible or not, but I know you have not yet told me how can I use Faraday's law for stationary point charges.

    But here is another real scenario, consider an infinitely long current carrying wire with current [itex]I = I_0 . t[/itex], i.e it is proportional to the time(for simplicity). Now, a point charge 'q' at a distance 'r' from the current carrying wire should feel a force.

    Please tell me in which direction and of what magnitude ?

    Thanks
     
  11. Aug 24, 2012 #10

    Dale

    Staff: Mentor

    The fields for a point charge are given by the Lienard Weichert potential:
    http://en.wikipedia.org/wiki/Liénard–Wiechert_potential
     
  12. Aug 24, 2012 #11
    Thanks for your reply,

    Seems like you didn't get my question, my question is not about the field of a point charge, but the force on a point charge due to changing magnetic field.
     
  13. Aug 24, 2012 #12
    So what you do is apply those field eqn's to the source current distribution - coil or line current etc. One approach is to first evaluate the vector potential A at the charge but owing to summing over all moving charge contributions in the coil, line current etc., which is then differentiated wrt time to give an unambiguously defined E field -dA/dt wherever one wishes to evaluate it. Or compute the E, B fields directly from summing over the actual retarded E, B fields owing to each and every moving source charge (or current density usually) in that coil etc. Latter I would prefer. The Faraday expression is useful for emf calcs in a coil but as you have discovered has limited utility outside of that.
     
  14. Aug 24, 2012 #13
    Thankyou very much for the clear cut explanation.

    And if one uses the vector potential A to calculate electric field as [itex]\vec{E}=-∂\vec{A}/∂t[/itex] (since scalar potential is zero for currents in a wire), does it mean that the effect is due to retardation of electric field of the moving electrons in the current carrying wire ?

    Since, I thought that Lienard and wiechert potentials are retarded, i.e. A in above equation works only if we consider the retardation effects while formulating the A itself.
     
  15. Aug 24, 2012 #14
    Well by examination of those potential and field eqn's it's seen that A itself, evaluated at the field point (where the test charge is) depends on just charge velocity and direction, but evaluated at the retarded time computed via radial distance between source charge/current element and field point. If the source is 'slowly time-varying' (wavelength >> radial distance involved) one can mostly safely ignore differences in retarded time and simply sum for the total A first assuming no retardation, then time differentiate in one go. Handy for nice symmetrical structures like a circular coil or straight wire. When high frequencies are involved (say for electrically large antenna field calcs) this approximation fails and it's best to work by summing over the source currents using the field equations directly, which requires knowledge of retarded acceleration of source charge/currents in addition to velocity, at each source location. There are plenty of internet ham radio sites and the like, and of course textbooks giving worked examples.
    [I left out discussing the scalar potential phi. It certainly contributes whenever there are non-uniform phase variations involved - say in dipole antennas, but except for near-field problems, it can be factored out in many cases. You will get examples here: http://www.ece.rutgers.edu/~orfanidi/ewa/] [Broken]
     
    Last edited by a moderator: May 6, 2017
  16. Aug 24, 2012 #15
    Well you put it completely and nicely, and yes I know just like relativistic effects when β is appreciably less we can ignore retardation of motion ! (i.e we can work in coulomb gauge)

    And are you aware of any experiments that shows this very effect, I mean moving a stationary free electron/proton(if something like this exist) by changing the magnetic fields.

    Thanks in advance
     
    Last edited by a moderator: May 6, 2017
  17. Aug 24, 2012 #16

    Dale

    Staff: Mentor

    That is just given by the Lorentz force.
    http://en.wikipedia.org/wiki/Lorentz_force

    There is no additional force due to the time derivative of the field.
     
  18. Aug 25, 2012 #17
    I know Lorentz Force does not contain any time derivative of E or B field, but time derivative of one can produce other according to electromagnetism of Maxwell's equations, which of course then validly comes under the formula of Lorentz Force.

    My concern is, if we have an electric field due to changing magnetic field as [itex]\vec{E}=-∂\vec{A}/∂t[/itex], then how is it any different than the Faraday's law [itex]∇×\vec{E}=-∂\vec{B}/∂t[/itex]. Just insert [itex] \vec{B}=∇×\vec{A}[/itex]

    The above problem is there (i.e ambiguity of E field at a point due to changing magnetic field), until we introduce the Green's function for elementary current densities and integrate for the whole current density setup. Then again, why do we need E and B of Maxwell's equations if the potentials(phi and A) are more fundamental and less confusing, that is, the model in which we integrate the spherical effect of currents and charges seems more practical than the abstract concept of E and B fields.
     
    Last edited: Aug 25, 2012
  19. Aug 25, 2012 #18
    By defining B=Curl(A) and E=-(Del(V)+dA/dt) we automatically satisfy Faradays law and gauss' law of magnetism.
    V (or Phi, the scalar potential) and A are not more fundamental, they are not directly observable and do not have a one to one relation with their electromagnetic fields.
    I do not see what is so unresolvable here.

    Edit:After re-reading the original post I am confused what you are asking, first you say you are confused about the potential formulation of EM, and then you go on to say something about the "ambiguity of E field..", without finishing that sentence you continue on to say that the potential formulation is more fundamental then the E and B field one. Could you restate your question please?
     
    Last edited: Aug 25, 2012
  20. Aug 25, 2012 #19

    Dale

    Staff: Mentor

    It isn't any different. When you work with potentials instead of fields then Faraday's law is an identity (as is Gauss' law for magnetism). I.e. you can write down fields E and B such that Faraday's law is violated, but you cannot write down potentials V and A such that Faraday's law is violated.
     
  21. Aug 25, 2012 #20
    Sorry universal_101 about delayed reply, but thanks for the compliments. :smile:
    None off hand, but especially with toroidal transformers no other explanation really fits the behavior. If I have read between the lines right here, your chief interest is in resolving the Feynman disk paradox referred to in #1. There, a mechanical imbalance of angular momentum arises when the central superconducting coil 'thaws' and a decaying dI/dt -> -dA/dt solenoidal E field acting on the peripheral charges, yet with no commensurate back-reaction on the coil. The usual explanation (Feynman doesn't provide it there) posits a compensating physically real angular momentum stored in the initial distribution of crossed static E and B fields mostly exterior to the physical structure. A variant with analysis is at http://au.search.yahoo.com/r/_ylt=A...n.edu/~mcdonald/examples/feynman_cylinder.pdf
    Am I right in thinking your main concern is with the mechanical imbalance, as the notion that the superposition of purely static and independently generated fields creates real field momentum seems fishy? Then you are far from alone but the book-keeping works out in such scenarios. I have looked at certain situations involving precessional motions where the book-keeping balance sure seems hard to find, but won't elaborate here.
     
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