[Eternal Student]

Hi.

Why can't we translate to the frame where the electrons stay still instead?  Will it change the expected result?

    You could but it's harder.  You lose the ability to assume the density of positive charge in the wire ≈ the density of negative charge in the wire in that frame.   (I think I would just keep changing frames of reference in my head so that the wire was stationary).
    No it shouldn't change the overall result, the test charge would still be repelled, just for slightly different reasons.

Best Wishes.

[hamdani yusuf (OP)]

You could but it's harder.  You lose the ability to assume the density of positive charge in the wire ≈ the density of negative charge in the wire in that frame.   (I think I would just keep changing frames of reference in my head so that the wire was stationary).
    No it shouldn't change the overall result, the test charge would still be repelled, just for slightly different reasons.

If I assume that translation to the frame of the electrons is symmetrical to translation to the frame of the wire atoms anyway, will the test particle still be expected to experience repulsion instead of attraction?

[Eternal Student]

Hi.

    Sometimes I hate making replies, especially when they just aren't likely to say what the original poster wanted to hear.   I don't know what I can do about that, sorry.   Let's just start by saying that Magnetism is complicated and not entirely understood.   I don't claim to understand all of it.

If I assume that translation to the frame of the electrons is symmetrical to translation to the frame of the wire atoms anyway

    Why or how could you do this?   In what way is the situation symmetric or the same?

Here's a typical electrical circuit:

   
    The conventional current moves anti-clockwise around the circuit, so the electrons move clockwise.
Now if you have a test charge at the bottom of the circuit and you use a frame of reference where the positive charges of the wire are stationary, then the situation looks a certain way.     Neither the top, bottom or side wires in the circuit are moving,  a certain total length of the wire is obtained and the electrons, being free to move, will tend to spread out so as to be uniformly dense throughout the wire*.
    *This is an approximation but a very good one when the electrons have low velocity and the wire is stationary.  The electrons will tend to move to a region of lower electrical potential just by ordinary electrostatics until all regions reach an equi-potential.   In a more thorough treatment we must note that every charge creates both an electric field and, if the charge is moving, a magnetic field.   The total pattern of movement of electrons is not quite as simple as moving so as to spread out uniformly along the wire.  More generally, we actually do think that the electrons distribute themselves to be slightly more dense at the peripheral (outer edges of the wire) and slightly less dense at the core of the wire, that's how a net electric field along the wire is maintained and what drives the electrons along the wire with the usual drift velocity.    The surface density of charge also changes very slightly as you progress along the wire from the +ve end of the cell to the -ve end  -  BUT overall, this is just complicated and not making a significant difference to the overall density of electrons along the wire anyway, it is almost uniform along the wire.     For low electron velocities and a stationary wire, it's a reasonable approximation the electrons are uniformly spread along the wire.   Specifically, by experiment we notice that if we have current flowing in a wire and we bring a test charge close to it (but the test charge has 0 velocity relative to the wire) then there is no electrostatic attraction to/from the wire (no E field exists).  Conversely if we bring a compass close to such a wire then it swings (a B field does seem to exist).

   As it happens there is an overall conservation of charge in special relativity, so the total number of electrons = the total number of positive atoms in any frame of reference you choose.   The atoms weren't moving and so they are equally spread out (indeed their separation is totally determined by the lattice in which they are held anyway) and then we have seen that the electrons will tend to be equally spread out.   Overall the net density of charge in the wire at any small volume element is 0.

    Meanwhile, if you switch to a frame of reference where the electrons were stationary, then the situation looks completely different.    Presumably you meant the electrons in the bottom wire closest to your test charge were stationary because the electrons move in a loop and there's no way to have ALL the electrons stationary everywhere.   Then the electrons in the top wire are moving at almost 2v and in the side wires the electrons have non-aligned but equal valued diagonal velocities of almost √2 v.   The most striking difference is that the wires themselves are moving in this frame.   There is very little about the situation that is the same as how things look in the other frame of reference.   You can "assume symmetry" but it doesn't make it exist.  Nothing much looks the same in this new frame of reference and the approximation about the electrons distributing themselves uniformly (marked with a * earlier) doesn't hold well.
    Overall there tends to be a lower density of electrons in the bottom wire and a correspondingly higher density of electrons in the top wire (so total charge is conserved as it should be in every frame of reference).   While the positive charges were locked in a lattice and can not be re-distributed like this - both the top and bottom wire show the same length contraction and hence the same density of +ve charges is observed.

    I completely agree that this is a bit weird and little hard to believe and any sensible person would want a reference or something to back this up.   Trying to keep things simple, take a look at this video   ("How special relativity fixed electromagnetism",  Science Asylum, available on YouTube)   around time index  7:00 to 7:20 where the unequal distribution of -ve charge density in the top and bottom wires is exhibited and discussed in much the same way as it was done here.   This is an example where observers in different frames of reference disagree about something that might have seemed like it should have been an invariant:  Specifically the density of electrons in the top and bottom wires are not agreed on they aren't always the same in every frame of reference, that is not an invariant.   This could take ages to think about and resolve and I haven't tried to do it myself much.   This is where I would start:   The electrons weren't all at the same place and that's the problem -  you might count all the electrons in the bottom wire at a fixed time t₀ in one reference frame.   By conservation of charge (or just plain old conservation of events),  you'd expect all those events to be mapped to distinct events in the other frame and the total number of them to be the same.  That will happen (hooray!) but, of course, there's a simultaneity problem because the electrons were not all at the same location in space.   In the new frame you have a collection of events you can count but the time co-ordinates are all different.   So you're not counting anything that looks like a density of negative charge at one fixed time (in the new frame) in the bottom wire.  If you did apply equations of motion and determine where all of those electrons would have been at one fixed time in the new frame, then some of them have left the bottom wire and are now in the side wires etc.   Overall, you really do seem to be able to get a different density of -ve charge in the bottom wire in the new frame (compared to the density in the top wire in the new frame).

Final Notes:   
1.  The movement and behaviour of a test charge is never exactly the same as you observe in real life.   This is because in real life, the test charge is making it's own electric field and when it starts moving, then it is making it's own magnetic field.   Trying to solve Maxwell's equations for the real-life situation is then extremely complicated and usually reduced to finding reasonable numerical approximations.

2.   As mentioned before,  by far the best way to consider electric and magnetic fields is just to give up on handling them separately.   A separate E field and B field is a useful way to describe what happens for some situations,  however describing the both of them with a single electromagnetic field strength tensor is by far more consistent.    There really doesn't have to be a magnetic field, it's not fundamental, it's just what the elctromagnetic field looks like in some frames of reference.   Similar comment goes for the Electric field.

3.   You asked this question earlier:

Do we have to also take time dilation into account? why or why not?

    Yes but it makes everything more complicated.   It's best if you use a Lorentz 4-force to describe the effect of an electromagnetic field.   Here you are using derivatives w.r.t. proper time, τ.   So all the effects like time dilation are already built-in and taken as a contribution to the final conventional Newtonian 3-force you would observe.
   See Wikipedia:  https://en.wikipedia.org/wiki/Four-force    especially the short section about the example of the Lorentz 4-force determined from the electromagnetic field strength tensor  F_μν.

Best Wishes.

LATE EDITING:  To try and empahsise that it's not just the density of -ve charges that varies when you change frames, it's the difference in density between top and bottom wires that isn't an invariant.   (The density of +ve charge in the top and bottom wires would also change - but they change in the same way so the difference between +ve charge density of the top and bottom wires remains invariant).

[hamdani yusuf (OP)]

The total pattern of movement of electrons is not quite as simple as moving so as to spread out uniformly along the wire.  More generally, we actually do think that the electrons distribute themselves to be slightly more dense at the peripheral (outer edges of the wire) and slightly less dense at the core of the wire, that's how a net electric field along the wire is maintained and what drives the electrons along the wire with the usual drift velocity.    The surface density of charge also changes very slightly as you progress along the wire from the +ve end of the cell to the -ve end  -  BUT overall, this is just complicated and not making a significant difference to the overall density of electrons along the wire anyway, it is almost uniform along the wire.   

We can simplify it by replacing the wire with a hollow pipe, or a bundle of thin wires electrically  isolated from one another.

[hamdani yusuf (OP)]

Going back to your original frame of reference, the distance between the metal atoms would have been contracted slightly while the distance between the electrons would have been increased slightly compared to the frame I have just used.   Overall there would have been a net +ve charge density in the wire and that would have created an Electric field that repelled the test particle.

If in the next experiment the velocities are doubled to 2v, the classical Lorentz force would be quadrupled, because the electric current is doubled, so is the relative velocity between the test particle and the wire.
Do we get the same results when using length contraction method? Do we have to also take time dilation into account? why or why not?

Have you tried to calculate the repulsive force when v is 1 mm/s?
What happens to the force if the velocities are doubled?
What must be done to make the force attractive instead of repulsive?

[hamdani yusuf (OP)]

Here's a typical electrical circuit:

   

Let's make the bottom horizontal wire much longer and free to move horizontally. Electrical contacts with vertical wires use carbon brushes. Will your reasoning still hold?

[Eternal Student]

Hi.

Have you tried to calculate the repulsive force when v is 1 mm/s?
What happens to the force if the velocities are doubled?
What must be done to make the force attractive instead of repulsive?

   No.  I'm also sorry if the previous reply wasn't all that well structured.   I seem to have CoVid and can't concentrate.  I'll be handling light topics for a few days.

Best Wishes.

[Bored chemist]

I seem to have CoVid

Get well soon.

[Spring Theory]

A deterministic explanation of magnetic force needs to begin with electric charge. A charge can be thought as a rotating pulse in space from a particle's charge dipoles (bear with me here). A positive charge can be thought of a compression of space or a pulse of negative curvature. A negative charge can be thought of as a decompression of space or a pulse of positive curvature.

Like curvatures create interfering or repelling forces and opposite curvatures cause intensifying or attractive forces. This is the nature of electric charge.

Magnetic force is the result of spacial effects perpendicular to the electric effects. This is why the charge and its partner in magnetic force causes it to curve in trajectory. It is a transverse curvature.

Likewise, similar magnetic fields create repulsion and opposite magnetic fields result in attraction.

And now I have linked gravitational forces (curvature) and electromagnetic forces rather elegantly.

[Origin]

. A positive charge can be thought of a compression of space or a pulse of negative curvature. A negative charge can be thought of as a decompression of space or a pulse of positive curvature.

That can't be correct it seems.  If space was curving then all matter would be effected, but only charged particles are.

[hamdani yusuf (OP)]

A charge can be thought as a rotating pulse in space from a particle's charge dipoles

A rotation requires an axis. An electrostatic charge doesn't seem to have any.

[Spring Theory]

. A positive charge can be thought of a compression of space or a pulse of negative curvature. A negative charge can be thought of as a decompression of space or a pulse of positive curvature.

That can't be correct it seems.  If space was curving then all matter would be effected, but only charged particles are.

The total effect of curvature is still the total mass or average mass of the particle. The dipole is a point like pulse of curvature. For negative particles, the convex curvature pulse subtracts from the overall curvature mass but not enough to make the mass less than zero.

For positive particles the concave curvature pulse adds a bit of gravitational curvature to the total mass of the particle.

The effect on neutral particles is a pulse of positive or negative mass, but overall gravitational effect is just due to the net mass. 

Charged particles however create an attraction when a convex pulse meets a concave pulse from opposite directions because the result is a curvature that is intensified. A repulsion is created when like pulses interact because the result is the curvature is interference.

Of course this is my theory, hence located here in the speculative board.

[Spring Theory]

A charge can be thought as a rotating pulse in space from a particle's charge dipoles

A rotation requires an axis. An electrostatic charge doesn't seem to have any.

All charged particles have a magnetic moment due to its "intrinsic" spin.

An intrinsic spin requires an intrinsic axis and hence a source for an instrisic pulse.

[Origin]

The total effect of curvature is still the total mass or average mass of the particle. The dipole is a point like pulse of curvature. For negative particles, the convex curvature pulse subtracts from the overall curvature mass but not enough to make the mass less than zero.

That is just word salad.

For positive particles the concave curvature pulse adds a bit of gravitational curvature to the total mass of the particle.

Nope, charge has nothing to do with a gravitational field, charge has to do with the electric field.

Charged particles however create an attraction when a convex pulse meets a concave pulse from opposite directions because the result is a curvature that is intensified. A repulsion is created when like pulses interact because the result is the curvature is interference.

More word salad it seems.

Of course this is my theory, hence located here in the speculative board.

This is clearly not a theory, this what is referred to as a WAG.

[puppypower]

Here is an interesting twist nobody may have seen. A magnet has two poles. However, magnetic monopoles has never been seen in the lab or in space. How can two nothings; two monopoles, add to something?

The analogy is like taking two fairies, which do not exist, but if we combine these two nonexistent entities, we can make a real unicorn appear. There is a conceptual problem with this tradition of two nonexistent monopoles per magnet.

In magnetic iron, the magnetism comes from how the outer electrons of iron are arranged in the orbitals. The magnetic phase of iron is not the lowest energy state of iron. The magnetic phase has potential, due to be in a state with more unpaired electrons.

Normally lowest energy would occur from two opposite spin electrons per orbital. With magnet iron, extra electrons remain unpaired. It is not about two imaginary things; monopoles, but one electron replacing two electrons. We get residual potential that appears to extend the range of the orbitals, out toward infinity. 

In another topic, I once did a thought experiment of a wave tank with two wave generators, one on each side of the tank, each 180 degree out of phase. Even with energy being pumped into the tank by each generator, the center of the tank will appear still as waves cancel and hide the energy.

We can get this hidden energy back by placing a partition in the center of the tank, so the two sets of waves cannot cancel.  Or we can also shut off one of the two wave generators, so we have only one wave generator, to help amplify the energy via standing waves. This allows for magnetic iron and iron pieces, to all stick together like polymers in macro-space.

The electron is an elementary particle, meaning it is one thing that cannot broken down any further. According to the traditions, it has two properties; mass and negative charge. But since we cannot break down the electron to isolate these two separate properties, implied by the traditions, these two things do not exist, in the classic way, within the electron. If you could break the electron down to mass and charge, the electron would not be an elementary particle.

To solve this paradox, two properties of the electron need to be part of a single thing, like part of a unified force, that can blend mass and negative charge to where they are interchangeable. This way you always get one thing, as implied by an elementary particle.

[Origin]

The electron is an elementary particle, meaning it is one thing that cannot broken down any further.

OK.

According to the traditions

Not tradition, experimental evidence.

it has two properties; mass and negative charge.

That is not correct.  There is at least also spin.

But since we cannot break down the electron to isolate these two separate properties, implied by the traditions,

Again, this is not tradition.  We are not talking about religion we are talking about science.

implied by the traditions, these two things do not exist, in the classic way, within the electron.

Of course they do!

If you could break the electron down to mass and charge, the electron would not be an elementary particle.

Obviously.

To solve this paradox,

What paradox?  Your confusion is not a paradox for us.

[Spring Theory]

That is just word salad.

Read it a few times and it will help you grasp the concept. It will be way over your head the first time.

[hamdani yusuf (OP)]

All charged particles have a magnetic moment due to its "intrinsic" spin.

An intrinsic spin requires an intrinsic axis and hence a source for an instrisic pulse.

But the charge is not because of the rotation.
What's the evidence for the pulse? What's the frequency and duty cycle?

[Bored chemist]

That is just word salad.

Read it a few times and it will help you grasp the concept. It will be way over your head the first time.

Reading it repeatedly will not help.
It can not help because your word salad has phrases in it that do not have a meaning. (e.g. " dipole is a point like pulse of curvature"
You have been asked to explain them.
You failed to do so.


Do not try to tell us that it is our fault that you refuse to make sense.

[Spring Theory]

That is just word salad.

Read it a few times and it will help you grasp the concept. It will be way over your head the first time.

Reading it repeatedly will not help.
It can not help because your word salad has phrases in it that do not have a meaning. (e.g. " dipole is a point like pulse of curvature"
You have been asked to explain them.
You failed to do so.


Do not try to tell us that it is our fault that you refuse to make sense.

Try asking some meaningful questions instead of pontificating.