[PmbPhy (OP)]

I don't know if this is the right place to put this but I have a challenge for all of you.

If you have a straight current carrying wire which has zero charge density in the rest frame of the wire and then transform to a frame moving parallel to the wire then the charge density in this new frame is no longer zero.

Now think of a current carrying loop which when not rotating about its axis of symmetry and which is normal to the plane of the loop the loop is uncharged. If you now start rotating it about this axis then will it then become charged? If so then where does this charge come from? If not then why not?

[Kryptid]

If you have a straight current carrying wire which has zero charge density in the rest frame of the wire and then transform to a frame moving parallel to the wire then the charge density in this new frame is no longer zero.

I've heard something about this being related to the origin of magnetism. However, I'm uncertain exactly what you mean by charge density. Do you mean net charge?

Now think of a current carrying loop which when not rotating about its axis of symmetry and which is normal to the plane of the loop the loop is uncharged. If you now start rotating it about this axis then will it then become charged? If so then where does this charge come from? If not then why not?

I know that conservation of charge must prevent any appearing out of nowhere, so I say you don't get any net charge. This situation also isn't completely analogous to the linear one. In a rotating ring, some parts of the ring will be coming towards me, some away from me and some parallel to me with different velocities relative to me. I'm guessing that has something to do with the solution to this problem? Or does length contraction make the whole ring decrease in diameter slightly?

[chiralSPO]

I don't have an answer, but I will start by pointing out that in the first scenario, the two reference frames (where the wire is stationary and where the wire is moving at a set velocity) are equivalent (both being inertial rest frames), while for the second scenario, one of the possibilities is a rotating frame of reference, which is not equivalent to the resting case.

What I don't understand is the bit about charge density. Are you saying that the electric charge of the wire is different in the different frames of reference? Is there a current involved? Charge carrier density? Is the observer a charged particle?

[PmbPhy (OP)]

I don't have an answer, but I will start by pointing out that in the first scenario, the two reference frames (where the wire is stationary and where the wire is moving at a set velocity) are equivalent (both being inertial rest frames), while for the second scenario, one of the possibilities is a rotating frame of reference, which is not equivalent to the resting case.

While both frames are inertial and thus equivalent, what should be measured in each frame is not. This is due to the fact that the source of the magnetic field is the 4-current. This is a 4-vedctor whose time component is charge density and whose spatial component is current density. Thus what is purely current in one frame is a combination of charge density and current density in another frame.

Total charge is always conserved though. Consider a square loop of current which is non-charged in its rest frame. In a frame moving parallel to one of its sides, two of its sides will have charge on them, one negative and the other positive, the total of which is zero.

The physics for the linear case is described in the Feynman lectures. I've studied this in great detail over the years. See:
http://www.newenglandphysics.org/physics_world/em/rotating_magnet.htm

What I don't understand is the bit about charge density. Are you saying that the electric charge of the wire is different in the different frames of reference? Is there a current involved? Charge carrier density? Is the observer a charged particle?

The total charge is the same, i.e. zero. Its the distribution of the charge that's different.

In special relativity the term observer is almost always used synonymously as the inertial frame of reference. So you can always replace "observer" with "inertial frame of reference" in SR.

[dutch]

I don't have an answer, but I will start by pointing out that in the first scenario, the two reference frames (where the wire is stationary and where the wire is moving at a set velocity) are equivalent (both being inertial rest frames), while for the second scenario, one of the possibilities is a rotating frame of reference, which is not equivalent to the resting case.

What I don't understand is the bit about charge density. Are you saying that the electric charge of the wire is different in the different frames of reference? Is there a current involved? Charge carrier density? Is the observer a charged particle?

Let me put some math and a couple concepts from Special Relativity in here if anyone wants to see it. Specifically, relativistic velocity addition and Lorentz contraction (also note the contraction implies a concurrent and equal magnitude time dilation).

Let the charge densities be Q+/L  and Q-/L. Let Q+ + Q- = 0

In an uncharged metal wire with no net movement of either the positive or negative charges the wire will not have an effect on a nearby test charge. The positive and negative charge densities cancel.

However, if the negative charges in the wire move at velocity a perpendicular to the test charge down the wire then the charge density appears contracted as it moves by the test charge. If this was the only effect the below would be true:

Q+/L + γa Q-/L  ≠ 0    where γa = 1 / √(1 - (v/c)²)

This is not correct yet because while the electrons do undergo Lorentz Contraction when moving at velocity v these electrons are still evenly spaced throughout the wire (lookup Bell's spaceship paradox and think about the connect between electrons accelerating to a distance d apart relative to the rest frame of the wire). The electrons will repel (like they always do) until evenly spaced throughout the wire counteracting the Lorentz Contraction. With this idea added the expression for the rest frame is:

Q+/L + γa Q-/(γa L)     →   Q+/L + Q-/ L = 0    (I will use the left form for a reason that becomes apparent below)

Now we need to consider the situation where the test charge is moving at velocity v parallel to the wire and the negative charges are moving at velocity a. We cannot use classical velocity addition in the form of a + v = b. We must use Relativistic velocity addition (a + v) / (1 + a v/c²) = b. The expression we get for the charge density directly below the test charge is:

γv Q+/L + γav Q-/(γa L)

Now we need to figure out what γav is. Let's first set the speed of light to c =1 units so we don't have to carry c around in the math. Next let's input b = (a + v) / (1 + a v) into γ

γav = 1 / √(1 - ((a + v) / (1 + a v))²) = (1 + a v) γa  γv      (run through the Algebra. It's not hard but it takes a couple steps)

γv Q+/L + (1 + a v) γa  γv Q-/(γa L)  →   γv Q+/L + (1 + a v)  γv Q-/ L

Now  γv Q+/L +  1  γv Q-/ L = 0  but we are left with another term still dependent on a :  a v  γv Q-/ L

There is a net effect dependent on a, v, and Q-/L  which will cause a force perpendicular to the wire. This dependence remains because γav / γa ≠ γv. The dependency on velocity a no longer cancels.

In the Lorentz Force equation from Relativity

F = q E + q v X B       where B is the magnetic field, E is the electric field, F is the force, and q is the charge

The magnetic field of a long wire is,  B = μo I /(2π r)   μo is a constant, I is the current (dependent on velocity a), and 2π r is constant for the test charge set at distance r from the wire. This gives the same dependency of B on velocity a (current I depends on a) and matches our result above. Magnetism is explained by Relativity and is a result of changes in simultaneity and length contraction/time dilation as viewed by different reference frames. While the E and B fields will now contain γv,  F = γv m a in relativity and this again matches the Lorentz Force Law.

For current in a loop all this math works the same. Because of a continuous change in simultaneity of a current going around in a loop (the electrons accelerate to move in a circle) loops of current can repel or attract depending on the direction of the currents, the charge of the current, and the orientation of the loops. The charges will still net cancel in the rest frame because they evenly distribute around the loop that doesn't contract. However, for moving charges the change in simultaneity for both loops (similar to the blue shift/red shift that occurs for light when simultaneity changes under acceleration) can cause a net repulsion or attraction force. The magnetic force of two loops is.

F = ∇ (m • B)   where m is the magnetic moment of a small loop. If the magnetic fields m and B are both loops both depend on the current (and velocity) of the charges in the loop.

Other forces of nature should also have "magnetic" like effects because of relativity. For example, frame dragging and gravitational waves are analogous to magnetic (or electromagnetic) effects but it's for gravity instead of EM. Frame dragging and other similar gravitational effects are often described with a set of formal analogies called gravitomagnetics. The only force strong enough and with a range effectively long enough for us to notice a magnetic effect in everyday life is Electromagnetism. Gravity definitely has similar effects noticed in very sensitive experiments (gravity is weak compared to EM) and the other two forces are so short-ranged it's hard to test the effects.

https://en.wikipedia.org/wiki/Gravitoelectromagnetism

Finally, all magnetic fields with a net zero electrical field do completely net to zero globally even if a net attractive effect occurs on say one side of the current loop because there will be a region with a net repulsive effect on the other side. Monopoles might possibly exist according to some theories but they haven't been seen. Magnetism is like a redistribution due to relativistic effects seen in other reference frames instead of charge appearing out of nowhere.

Is there a current involved? Charge carrier density? Is the observer a charged particle?

I understand the confusion. When I first read PmbPhy's post it didn't make sense because there was information missing on the setup (I believe we weren't told about currents being present).

[PmbPhy (OP)]

I understand the confusion. When I first read PmbPhy's post it didn't make sense because there was information missing on the setup (I believe we weren't told about currents being present).

Your post merely repeated the math I linked to in my previous post. It's all given in: http://www.newenglandphysics.org/physics_world/em/rotating_magnet.htm

Did you read it before responding?

And as far as any information missing on the currents, recall the opening post

Now think of a current carrying loop which when not rotating about its axis of symmetry and which is normal to the plane of the loop the loop is uncharged.

By the way, regarding your comment that while the electrons do undergo Lorentz Contraction when moving at velocity v, electrons don't undergo Lorentz contraction because point particles cannot contract. What contracts is the distance between electrons.

I didn't read most of your post after that because I saw no use in it. The magnetic fields have nothing to do with the answer. The answer to the question posed in the opening post can be found on the internet and in certain EM textbooks, none of which uses most of what you posted.

[dutch]

From my original post unaltered. Only made bold to point something out......

However, if the negative charges in the wire move at velocity a perpendicular to the test charge down the wire then the charge density appears contracted as it moves by the test charge. If this was the only effect the below would be true:

Q+/L + γa Q-/L  ≠ 0    where γa = 1 / √(1 - (v/c)²)

This is not correct yet because while the electrons do undergo Lorentz Contraction when moving at velocity v these electrons are still evenly spaced throughout the wire (lookup Bell's spaceship paradox and think about the connect between electrons accelerating to a distance d apart relative to the rest frame of the wire). The electrons will repel (like they always do) until evenly spaced throughout the wire counteracting the Lorentz Contraction. With this idea added the expression for the rest frame is:

Let me put some math and a couple concepts from Special Relativity in here if anyone wants to see it. Specifically, relativistic velocity addition and Lorentz contraction (also note the contraction implies a concurrent and equal magnitude time dilation).

The charges will still net cancel in the rest frame because they evenly distribute around the loop that doesn't contract. However, for moving charges the change in simultaneity for both loops (similar to the blue shift/red shift that occurs for light when simultaneity changes under acceleration) can cause a net repulsion or attraction force.