Video on Magnetism and Relativity

In this video : http://www.youtube.com/watch?v=1TKSfAkWWN0 the narrator explains that the magnetic field created by moving electric particles is caused by special relativity. In the moving cat's reference frame, the electrons are spread out and the positive ions closely packed causing a net repulsive force between the wire and the positive moving cat. However in the narrator's reference frame it should be the opposite : electrons closely packed because they move relative to him and positive ions spread out. Thus in his reference frame the wire should be negatively charged and the cat should be attracted rather than being repelled. Is my understanding correct? Is the video explanation correct? Thanks
- Anonymous

That video is exactly right about the origin of magnetic fields.  

The only mistake I noticed was earlier on, in the part about how long it took to shave. If the shaver thought it took 6 minutes, the guy who says the shaver is moving would think that the shaver's clock is slow, so he'd say it took just over 6 minutes, not just under. 

Mike W.

(published on 06/17/2014)

Yeah but in the narrator's reference frame, it is the electrons that are moving so they should have their distance between them contracted making the wire negatively charged. According to this, in the narrator's reference frame, the cat should be attracted to the wire while in the cat's reference frame, it should be the opposite as shown in the video. This is a paradox. How it is solved?
- Anonymous

You wonder whether that part of the wire stays neutral in the narrator's frame (as he claims) or becomes slightly negative as the electrons start to flow. There's no general exact answer, since it depends on the rest of the circuit, but the assumption in the video- neutrality- makes the most sense. The narrator and the wire are at rest with respect to each other. The wire is net neutral in that frame. So unless there's some special reason for part of the wire to be positive and part to be negative, each part should be neutral in that frame, just as the video assumes.

Mike W.

(published on 07/05/2014)

Okay, but you didn't answer the important part of the question. At the time when the current is flowing but the cat is not moving then from the narrator's point of view we see that the electrons are moving. This would mean the space between them contracts and thus the electrons are more densely packed. Hence, the wire should be negative. But the narrator says it is neutral. Why is it neutral or was it a mistake?
- Istvan (age 25)
Aachen, Germany

Hi Istvan- I've got another answer waiting in the queue for posting. It will partially answer your question. Here's another part of the answer. To get the current flowing, there must be forces on the electrons. So that's a reminder that they do respond to forces.  If, in some part of the wire, in the wire frame, the electrons were more dense than the positive charges, there would be an electric field driving the electrons away from that region. Their density will go down. So, at least to a first approximation, the stable state will be one in which the net charge is neutral in the wire frame. If the loop is circular, that result is exact. If there are kinks in the wire, etc. then in fact there are some accelerations other than a uniform acceleration toward the center, so there are very slight charge densities along the wire, adding up to zero. For the circular loop, the inward acceleration also implies that the inner and outer part of the wire have a slight charge imbalance, but each slice is still neutral.

Mike W.

(published on 07/06/2014)

But then again, if the wire is neutral in the narrator frame, how come does he see the cat being repulsed by the electric field? There's still a paradox no?
- Anonymous

No, in his frame the cat is moving. So although there's no electrical force on the cat, there is a magnetic force on the cat. According to the cat there's no magnetic force but there is an electric force.

Mike W.

(published on 07/07/2014)

Thanks. Still not clear, though. Look at the video at 1:31 and at 1:59. They show analogous situations. In the first, the electrons move and the cat is static. In the second, the protons move and the cat is static. Somehow the gaps are distorted in the second case creating a repulsive force but no attractive force is created in the first case. You say something to the effect that in the first case the electric forces create a balance that results in neutrality. Basically electrons are pushed and drawn by the electric force until they are in balance. Why doesn't the same happen in the second case? The cat at 1:59 should expect that the electrons will arrange themselves (because of the electric force) so that the wire becomes neutral. In other words, why are the two cases different if the only change is that the role of protons was replaced by electrons and vice versa?
- Istvan (age 25)
Aachen, Germany

Before answering your specifics, let me introduce a slightly different example in which things are clearer because there's more symmetry. Instead of using a metal wire, use a tube of salt solution in which the positive and negative ions carry equal currents, flowing opposite directions around a circle.  Then obviously in the lab frame they have equal densities everywhere, by symmetry. Now as you look from the point of view of the moving cat, in one part of the circuit the positives are more spread out and in another part the negatives are more spread out due to the Lorentz contraction. If the cat is nearer the part where the positives are more spread out, he sees the force from that nearby negative charge.

Now you ask how the cat can understand, from his point of view, why the charges are not uniformly distributed around the loop. From the cat's point of view, there is no simple rotational symmetry. The different charges are moving at different speeds in different parts of the loop. So no simple symmetry argument requires that he sees local neutrality. 

Still, it would be nice to go beyond just saying that symmetry doesn't require local neutrality from the cat's viewpoint, unlike from the lab viewpoint. Say the loop is in the horizontal plane, heading north according to the cat. Then the cat sees an east-west charge imbalance, making an electric field in the east or west direction. That would affect the charge distributions, pushing positives toward one side and negatives toward the other, unless it were canceled by another force. Now the whole thing is moving north, so if there were some velocity dependent force pushing charged particles at right angles to their motion, we could end up with the cat's view being consistent. But whoever heard of such a force? Oh, wait...

Mike W.

(published on 07/07/2014)