[Kryptid (OP)]

I've begun to gain an understanding in how magnetism arises by looking at electric fields in different reference frames. However, the example I most commonly find to explain the effect has me confused. Here is how it typically goes:

Imagine a stationary negative charge sitting beside an electrically neutral wire which has a current flowing through it. From the reference frame of the negative charge, the nuclei inside of the wire are at rest and the electrons are moving to the side at some drift velocity. Due to Lorentz contraction, the electrons appear closer together than the nuclei do. Since the electrons appear closer together, there appears to be a net negative charge on the wire from the viewpoint of the stationary negative charge. Thus, the charge is repelled from a wire, despite the fact that there is no net charge on the wire itself.

This supposedly explains where the magnetic force on the negative charge comes from. However, I thought that magnetism was not supposed to exert any force on stationary electric charges (whereas this example suggests that it does)? I don't think its even supposed to attract or repel moving charges either. It only changes the trajectory of a charge moving through the field, right?

Also, how would relativity explain the origin of a magnetic field on a magnetic monopole? If a monopole is at rest in my reference frame, would I be able to measure an magnetic field around it?

[Mr. Scientist]

Yeh. The monopole has two poles in which curve into an arguably pointlike particle. If they exist, that is.

[graham.d]

I have not looked at what was written about the wire and charge in detail (it has been covered before on this site btw) through lack of time, however I would venture to say that the bit about the Lorentz contraction and the electrons being closer together is nonsense. The electron drift velocity is nothing like fast enough to have any noticeable relativistic effect.

I think, in any frame, a stationary charge will be unaffected by a neutral wire carrying current (or not). Charges moving with respect to the wire would be a different matter. A stationary charge is not affected by a magnetic field.

[JP]

If a charge moves in an electric and in a magnetic field, it experiences a force according to the cross product,
F=q (E+ vxB),
where F is the force vector, q is the charge, v is the velocity of the charged particle, E is the electric field and B is the magnetic field.  There's one other tricky bit here.  Current is actually pointing in the opposite direction of electron flow because it was originally defined before people knew what was flowing in wires.

The problem is usually stated as: Imagine you have a current carrying wire which creates a magnetic field that loops around it (in the same direction as your fingers curl if you point the thumb of your right hand along the current).  In this case, the wire is neutral, so it has no electric field.  The force equation simplifies to only the magnetic field term:

F=q vxB

An electron moving next to the wire in the direction of the current experiences a force in the direction given by the cross product of its velocity vector and the magnetic field vector of the wire at its location.  (If you don't know how to do a cross product, look up the right-hand-rule, since its easier to see with pictures than me explaining it in text.)  The net result is a repulsive force.

If you decide to hitch a ride on the electron, the electron's velocity goes to zero, so the magnetic term in the force equation is zero.  Now, the electron is moving in the same direction as the current, so it's moving opposite the electron flow.  This means the negative bits of the wire are moving more quickly with respect to you than the positive bits, so you see more of a contraction of the negative bits than the positive bits, so the wire appears to be negatively charged, which sets up an electric field pointing towards the wire, which in turn repels the electron (it's negative so it tends to move in the opposite direction the electric field points.)

I know the electron drift velocity is slow, but I think the quantity of interest here is actually current flow, so relativity applies.  If you want to talk about electrons, you'd probably need to deal with a more complicated quantum theory, but the same results should eventually pop out.

[graham.d]

jp, it was stated as a stationary charge next to the wire. The charge would feel no force from the magnetic field unless it were moving wrt the magnetic field. Relativity comes into it if you wish to change frames of reference and using the fact that Maxwell's equations are invariant with a Lorentz transformation. But I guess that was the point of the question. D'oh!

I was wrong about the effects of the Lorentz contraction not being responsible for there being effectively more electrons per unit length of wire than positive fixed charges. It was the issue of drift velocity that wrankled. It is actually very complicated because the electrons are actually zipping about quite fast in all sorts of random directions. It is not at all clear how the summation of all this movement (whose average sum is the drift) stacks up to give the right amount of net charge density so as to result in the precise force in the moving frame. If you find a correct mathematical treatment of this I would be interested to take a look.

It is much simpler with two electrons :-)

[JP]

Oops.  It's harder to see with a stationary electron.  In the initial frame, it experiences no forces, since there's no electric field and v=0. 

In the moving frame, there is still a magnetic field since there's still a current (although its strength is changed by your motion).  The electron also appears to be moving now, and so it experiences a force due to the magnetic field.  This should be balanced out by the fact that the currents Lorentz-contract as well and therefore the charge feels a balancing electric force.

[JP]

Although here's a confusing point to me:

If you go with the stationary electron and decide that it should still feel no force when you move, what happens when you move fast enough to pass the current flow?  The sign on the magnetic force should flip, since the current now seems to be flowing in the other direction...?  But then the electron would move which would violate relativity.  I think that you resolve it by saying that current flows at the speed of light--either that or I'm missing something obvious about the Lorentz contraction going on here.

[graham.d]

I'm confused too jp. As I said it is a much easier problem with two point charges. If you find a correct mathematical treatment of this, I would like to see it. I will have a think about it when time allows.

[yor_on]

Can't help this, sorry if I 'butt in' here.
To be honest I have to admit that I 'hate' magnetism

The definition of how a permanent magnet can attract a iron nail involves a lot of slight of hand to me.
First of all there is the definition of 'work done'

Every time we assume that nail to become attracted we interpose an assumed 'work done' by some deity forcing either the magnet or the nail closer to explain why it attracts or repels. Assume a magnet in 'free fall' coagulating in deep space from those precocious substances then? Meeting that gorgeous nail also 'coagulated' free-falling in a intersecting orbit?

(You know, the hitchhikers guide

And then the explanation of the permanent magnets molecules 'electron-orbits moving'?
As to create that needed electromagnetic field I presume? 
And I who thought they weren't moving at all??
Silly me

It seems to me that to explain permanent magnetism we need something without those two definitions? Or do you believe that all magnetic attraction/repelling crave a deity 'placing' them together first, attracting or repelling them? And that electrons suddenly can 'move' and 'orbit' to create this 'field'?

The field created by a permanent magnet exists without those definitions and will be noticed at all times. To compare it to some chair stopping your atoms to fall through due to 'electromagnetic bounding' of them does not explain 'matter' and neither does those definitions explain permanent magnets.

Both gravity and magnetism are still undefined by mainstream physics to me.

--

'Work done' is a principle I can see, but the idea of a permanent magnet interacting with that nail? And no 'work done' while both change course in their free fall?

[graham.d]

If you Google for Relativistic Electromagnetism it points to the work of Edward Purcell who demonstrated a simplified way to look at the effects of relativity in electromagnetism, without performing Lorentz Transformations on Maxwells equations and showing they are invariant. Unless you've done a lot of work with Vector fields and Tensors the latter is quite hard and not particularly insightful. The Purcell work is much more insightful but unfortunately lacks rigour; when you know the answer you can then see how to make the right assumptions to justify it. Covering the statistical issues to do with carrier drift in a conductor is far too complex but it does try to cover movement of a line of charges next to a line of fixed cahrges of opposite polarity. The arguments have a lot of holes in them. Did you spot the electronics pun there?

[Pmb]

Imagine a stationary negative charge sitting beside an electrically neutral wire which has a current flowing through it. From the reference frame of the negative charge, the nuclei inside of the wire are at rest and the electrons are moving to the side at some drift velocity. Due to Lorentz contraction, the electrons appear closer together than the nuclei do. Since the electrons appear closer together, there appears to be a net negative charge on the wire from the viewpoint of the stationary negative charge. Thus, the charge is repelled from a wire, despite the fact that there is no net charge on the wire itself.

This is confused a bit on Lorentz contractions. To be precise you have confused what length is contracted. The distance between the electrons does behave as a rigid body so using the contraction formula is a bit tricky. Suppose you are at rest in the rest frame, S, of the ion lattice (i.e. nuclei of wire) and there is zero current flow. You can think of this as two sets of charge distributions; one for the lattice ions (i.e. the nuclei) and those of the electrons which we can think of as being at rest. In this frame there is no electric of magnetic field. A charged particle will therefore have no force acting on it. Now let us speed up the electrons so that the charge density does not change. We can do this by accelerating all the charges with the same exact acceleration from an initial speed of zero to a final speed, v.. When the acceleration period is over the distance between the charges will remain unchanged. However in a frame, S’, which is moving with the final rest frame of the electrons the charge density will have changed. As it turns out this leads to a total charge density change in this frame. For details of this see the section “Charged Density on a Moving Wire” at  see http://www.geocities.com/physics_world/em/rotating_magnet.htm

The main reason this happens is that, although all electrons have the same acceleration as measured in S’, they don’t start/stop accelerating at the same time. This means a change in charge density.

I have not looked at what was written about the wire and charge in detail (it has been covered before on this site btw) through lack of time, however I would venture to say that the bit about the Lorentz contraction and the electrons being closer together is nonsense. The electron drift velocity is nothing like fast enough to have any noticeable relativistic effect.

That is incorrect. It is a mistake to assume that since the charges are moving slow that relativistic effects can be ignored. This is a good counter example in fact. See link above. Some good text books explain this as well. In fact the Feynman lectures explain this fact. See “The Feynman Lectures on Physics - Volume II,” Feynman, Leighton and Sands, Addison Wesley, 1977, pages 13-7 to 13-12.

As far as SR and EM - Let a neutral current carrying wire be at rest in S. Let there be a test charged particle moving parallel to the wire. It will therefore be moving in a magnetic field and there will therefore be a force on the wire. In this frame the force is magnetic. Now let us look at this from the rest frame, S', of our moving at this from test charges rest frame. There will still be a force on this charge as measured in S'. However since the charge is at rest and the magnetic force must be zero. The force on it now must be purely electric. What is the source of this electric field? The source is the wire. In frame S' there is now a non-zero charge density and that creates an electric field. The charge density can be found by using the current 4-vector J. The temperal component of J is the charge density. The spatual component is the current density 3-vector. If you were to transform this vector frome S coordinates to S' coordinates you'd see that the charge density changes from zero to some non-zero value.

The relativistic way of saying this is that current in one frame is charge density in another frame!

[graham.d]

[...] That is incorrect [...]

Yes, I agree pmb. I said so in a subsequent post to correct my error there. However, I have not seen this treatment of charge moving in a wire done with rigor. I will look at your reference when time permits.