## classical physics vs relativity: parallel electron beams

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### Re: classical physics vs relativity: parallel electron beams

pardon me for interupting and I'm not sure whether to post this here or elsehwere or cross post but to 'Bob' and his relative motion and whomever else is involved as the observers
it's a school of fish
we, planets, atoms, 'scale' it
we move through these fractal Birkeland currents

I can't even bring half of it back...
miss you Dean.
flyingcloud

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### Re: classical physics vs relativity: parallel electron beams

Physicist:
It is strange that somehow the physicist cannot comprehend that when mag. field is considered the gamma factor from denominator disappears and the energy becomes the same independent of the capacitor rotation as it should. Also, somewhere I have read that the energy is not covariant but depends on velocity. An equation E2/cc - pp=const comes to mind. Do not know where the negative kinetic energy statement comes from. Another strange comment. Of course; F=qE always in freshmen or advanced courses and q is an absolute invariant. I think that further discussion is pointless. Good luck with such a strange physics. engineer

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### Re: classical physics vs relativity: parallel electron beams

engineer wrote:Physicist:
It is strange that somehow the physicist cannot comprehend that when mag. field is considered the gamma factor from denominator disappears and the energy becomes the same independent of the capacitor rotation as it should.

But engineer, you were telling me that there's a factor of gamma in the denominator for both orientations, no?

Also, somewhere I have read that the energy is not covariant but depends on velocity. An equation E2/cc - pp=const comes to mind.

Don't confuse covariant with invariant. The energy is covariant AND depends on velocity, as your equation demonstrates.

Do not know where the negative kinetic energy statement comes from.

It comes from expanding the gamma in the denominator as a power series in v. The term with the v^2, corresponding to the kinetic energy, comes in with a minus sign.

Another strange comment. Of course; F=qE always in freshmen or advanced courses and q is an absolute invariant.

The correct statement is F = q(E + v x B). The division of this Lorentz force into electric and magnetic forces is frame-dependent. The charge q is certainly invariant, but the electric field E isn't.

I think that further discussion is pointless. Good luck with such a strange physics.

Thanks engineer for an interesting discussion. But I maintain that the strange physics is what you get when you put the gamma in the denominator Physicist
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### parallel electron beams vs real Birkeland filaments

Going back to my original question about parallel electron beams, below is an explanation by Physicist which I haven't yet evaluated.

But it occurred to me that in a real-life Birkeland filament it is not just negative particles going in one direction, following the magnetic field lines, but positive particles going in the other direction, right?

Hence we have an analog with the classical experiment involving current in parallel wires. Without particles of the opposite charge going in the opposite direction, there is no "wire" for the negative particles to be attracted to.

So real Birkeland filaments in space are attracted to each other. But pure beams of electrons travelling at the same speed will not be, unless Physicist is correct here:

Physicist wrote:The solution to this conundrum lies in how the electromagnetic field tensor transforms under a Lorentz transformation.

The simple frame of reference is when both lines of charge (electrons, if you like) are at rest. There is no magnetic field because there is no current, and the the lines of charge repel each other: simple electrostatic repulsion.

The less simple frame of reference is the frame in which both lines of charge are moving, thereby constituting currents. The current does indeed generate a magnetic field, and the lines of charge, being like currents, are attracted to each other. But before we come to the conclusion that two different things happen in two different frames of reference, let's have a look at that electrostatic repulsion in the new frame of reference. Is it still the same? Well, no it's not. The reason why is that the two lines of current appear Lorentz-contracted, and therefore their line charge densities are greater than they are in the simple frame. So the electrostatic repulsion is greater than it was in the simple frame - and if you do the math you find out that this increase in electrostatic repulsion exactly cancels out the magnetic attraction. So the physics is exactly the same in both frames.

The lesson to be learned is that the electric and magnetic fields are not relativistically invariant objects. They are like components of a vector (in fact they are components of an antisymmetric tensor of rank 2) - by rotating the axes, you can mix them up. But however you mix them up, the physics always comes out the same. The names have been changed, but the plot is the same.
trevbus

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### parallel electron beams & Weber's force law

Parallel electron beam attraction is the only thing I've seen so far that seems incompatible with Weber's force law.

Assis writes, p. 86 of Weber's Electrodynamics (Weber force law → Ampère force law derivation):
We first note that this expression was obtained utilizing neutral current elements, which means that (4.24) does not need to be valid, for instance, for two electron beams.

Accelerators show that two beams of particles of like charge and velocity "focus" (analogously to how parallel current elements attract). See eq. (12) of "Space Charge" by K. Schindl. That eq. (12) doesn't seem derivable from Weber's law (∵ there is no relative velocity between the two beams' electrons); Schindl derives it from Lorentz's force law.

Beginning of the abstract:
The Coulomb forces between the charged particles of a high-intensity beam in an accelerator create a self-field which acts on the particles inside the beam like a distributed lens, defocusing in both transverse planes. A beam moving with speed v is accompanied by a magnetic field which partially cancels the electrostatic defocusing effect, with complete cancellation at c, the speed of light.
Geremia

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