Re: Is charge conserved between frames?

From: J.J. Simplicio (abc_at_nospam.com)
Date: 10/08/04

  • Next message: Dummy: "eleaticus" 
    Date: Fri, 08 Oct 2004 03:08:42 GMT

    "sal" <pragmatist@nospam.org> wrote in message
    news:pan.2004.10.07.20.25.56.256327@nospam.org...
    > (Warning: This is an actual physics question, so it may be considered
    > somewhat off-topic for this newsgroup... :-) )
    >
    > A circle of wire is carrying a current. The positively charged "matrix"
    > (metal lattice) of the wire is stationary in the "laboratory frame"; the
    > negative charge-carriers (electrons) are moving in a circle in the lab
    > frame.
    >
    > In the lab frame, the wire is neutral -- no net charge.
    >
    > (That's the description of the setup; everything from here down is
    > deductions.)
    >
    > In the lab frame, there's a magnetic field around the wire.
    >
    > Look at a ROTATING reference frame, in which the negative charge-carriers
    > are _stationary_ and the positively charged atoms in the metal lattice are
    > moving in a circle.
    >
    > In the rotating frame, there must be an ELECTRIC field, pointing radially
    > out from the wire. To see this, consider a positive charged particle, near
    > the wire, momentarily stationary in the rotating frame. In the lab frame
    > the particle is moving parallel to the wire, and feels a force due to the
    > B field. Therefore in the rotating frame it also feels a force; but in
    > that frame the particle is stationary so the force must be due to an E
    > field.
    >
    > Imagine a torus enclosing the wire. In the ROTATING frame, integrate the
    > outward-pointing E field, dotted into a unit vector normal to the surface,
    > over the whole torus. The E field points out everywhere, so the integral
    > will be nonzero.
    >
    > By Gauss's law, the integral (which is nonzero) is equal to the enclosed
    > charge. Therefore, in the rotating frame, there's a net POSITIVE charge
    > on the wire.
    >
    > HOW CAN THIS BE? Is charge not conserved when moving into an accelerated
    > frame? Or is Gauss's law only valid in inertial frames?
    >
    >
    > --
    > I can be contacted through http://www.physicsinsights.org
    >

    Interesting question, Sal (as usual). How would the observer in the
    rotating frame measure the total charge of the ring? One suggestion is as
    follows. The amount of charge on a point charge is frame-independent.
    Likewise, the total number of positive (or negative) point charges in the
    entire ring is frame-independent. So, if the rotating observer measures the
    total charge by adding up the charge on each individual point charge, he
    must get the same answer as the lab observer. If the total charge in the
    lab frame is zero, then the total charge in the rotating frame will also be
    zero if measured in this way.

    However, as you noted, the rotating observer will find that at each point of
    the ring there is a non-zero charge density! I am assuming that he defines
    the charge density at a point of the circular wire as the charge density
    that is measured in an inertial frame that is instantaneously co-moving with
    that point of the wire. In the comoving inertial frame, Gauss' law is valid
    and I venture to say that Gauss' law can therefore be said to hold LOCALLY
    in the rotating frame.

    So, what if the rotating observer now decides to calculate the total charge
    on the wire by integrating this nonzero charge density over the entire wire?
    I think there will be a serious difficulty with actually carrying out this
    integration in the rotating frame. In order to be a valid representation of
    the total charge of the ring, the integration must sum the charge in ALL the
    volume elements of the wire AT THE SAME TIME. But, it is well known that
    there is no way to use comoving inertial frames to synchronize clocks all
    the way around a rotating ring. So, the rotating observer will have a very
    hard time adding up the charge in each volume element 'simultaneously'!
    But, if he can't integrate over all the volume elements at the same time,
    then he cannot be sure that his integral represents the 'true' total charge.

    Of course, you might argue that it doesn't matter if the rotating observer
    adds up the charge in each volume element simultaneously. After all, the
    charge density is time independent at each point of the wire in the rotating
    frame. But, you have to keep in mind that in the rotating frame, the
    positive point charges are moving. If you add up the charge in one volume
    element at 'one time' and then add up the charge in another volume element
    'at some other time' then it could happen that some of the positive charges
    are counted twice because they happened to move from the first element to
    the second during the time interval. Or it could turn out that some of the
    positive point charges are never counted even after summing over all volume
    elements if the summation is not done at the same time. So, even though the
    charge density is time-independent, I think it is still necessary to
    integrate over all volume elements simultaneously if you want each point
    charge to contribute once and only once. But, as argued above, it appears
    to be impossible for the rotating observer to integrate over all elements
    simultaneously.

    Likewise, the rotating observer cannot apply Gauss' law globally, since that
    would require integrating over all elements of the Gaussian surface
    simultaneously.

    That's my take on it, anyway.

    JJ

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