# Special relativity and electromagnets

This Veritasium video explains how electromagnets can be explained by special relativity, and how the magnetic field surrounding a current-carrying wire can also be viewed as an electric field, if your frame of reference is moving with respect to the wire.

The example they use is a positively-charged cat, moving along a current-carrying wire in the same direction as the electron drift:

If you view this from the rest frame of the cat, then the electron's drift velocity is zero, while the protons are moving to the left. Because the protons are moving, length contraction makes it look like (to the cat) there are more of them, giving the wire a net positive charge, repelling the cat.

This makes sense and is all kinds of elegant and intuitive. It explains electromagnets in a way that depends on only three simple concepts:

1. motion is relative
2. things contract in their direction of apparent motion
3. opposite charges attract, like charges repel

Groovy. Now back up in the video. Derek says:

Now the number of protons is equal to the number of negative electrons, so overall the wire is neutral. So if there were a positively-charged cat nearby, it would experience no force from the wire at all. And even if there were a current in the wire, the electrons would just be drifting in one direction, but the density of positive and negative charges would still be the same, and so the wire would be neutral, so no force on the kitty.

Wait...what? Why is it that in the cat's frame, the protons are moving, are contracted, and the wire is charged, but in Derek's frame, the electrons are moving, but are not contracted, and the wire is still neutral?

How can you say "well, length contraction creates charge imbalances, allowing magnetic forces to be explained as electrical ones if you choose the right reference frame", but simultaneously say "but length contraction doesn't happen sometimes"? That's not elegant at all. Is there an elegant, intuitive1 explanation?

1: meaning, I've seen the math on Wikipedia and it's over my head. There is also current in wire + special relativity = magnetism where the answer to my question seems to be "the Lorentz force". OK, but that negates the elegance of the explanation above, with only three simple axioms. Are they not sufficient? If so, why?

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But the row of electrons is contracted in the lab frame. You can see that in your screencaps from the video too -- Derek sees 10 electrons per image width, whereas in the rest frame of the electrons there are only about 8½ electrons per image width. –  Henning Makholm Jul 10 '14 at 15:17
@HenningMakholm Yes, this is true in the 2nd screencap. Here's the confusing thing: just a few seconds ago, he was next to a wire with no current, and a 1:1 proton:electron ratio. Then the current begins, and still there is a 1:1 ratio. The electrons don't contract. So...length contraction happens for moving charged cats, but not for Derek? Length contraction happens for moving protons but not for moving electrons? None of that seems right. –  Phil Frost Jul 10 '14 at 16:36
–  Qmechanic Jul 10 '14 at 17:26

But the row of moving electrons is contracted in the lab frame, compared to what the cat sees. You can see that in your screencaps from the video too -- Derek sees 10 electrons per image width, whereas in the rest frame of the electrons there are only about 8½ electrons per image width.

What is potentially confusing is that as far as the electrons themselves are aware (electrons are not "aware" of anything, but never mind), they are not that the same mutual distance when they are moving as when the wire carried no current.

In other words, the row of electrons is not a rigid object. If each pair of neighboring electrons had been separated by a little rigid rod, the electrons would have to come closer together when the current starts flowing. But there are no such rods, and the row of electrons is free to stretch when the current begins to flow, and this stretching is exactly canceled out by the length contraction, such that in the lab it looks like the distance between the moving electrons is the same as the distance between the stationary protons.

What does the cat see? When the wire didn't carry current, the electrons and protons moved backwards together at the same speed (and with the same contracted spacing as the cat sees the protons have during the entire experiment). Then, when the current starts to flow, the electrons in front of the cat begin coming to a halt (with respect to the cat) slightly before those behind it. So from the cat's point of view the row of electrons get significantly stretched.

Meanwhile, Derek will see all the electrons begin to move at the same time. The cat and Derek do not agree whether two electrons changed their velocity at the same time or not -- this is relativity of simultaneity and is mathematically necessary to make length contraction consistent.

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Yessss. I'm grasping it now. So I wonder, as you transition from "no current" to "some current", you must either (from a frame where the electrons are stationary) decrease the electron density, or (from a frame where the protons are stationary) manage to get the electrons moving while seemingly countering the effects of length contraction. When I think about this, I wonder if it also explains Faraday's law of induction, in that although the current-carrying wire in the lab frame is neutral, time varying currents can induce voltages. Is that the right track? –  Phil Frost Jul 10 '14 at 17:03
@PhilFrost: From the cat's frame, some electrons are removed from the wire during the transition -- because the cat sees electrons begin leaving the front end of the wire before new electrons begin moving into its back end. (Of course it is an impossibly idealized assumption that the current starts instantaneously from the wire's rest frame, but the net outcome is the same for less sharp transitions). –  Henning Makholm Jul 10 '14 at 17:08
@PhilFrost: I think a more complex analysis is necessary to explain Faraday's law -- a naive application seem to lead to an effect of the wrong sign. However, one also has to take into account that changes in the electic field propagate with finite speed, and that the electrostatic repulsion from a moving charge in the direction of its movement is less than that of a stationary charge. (In the transverse direction it is the same). This is not part of Coulomb's law but has to be derived using relativity -- it's the only way for things to fit together in a consistent way mathematically. –  Henning Makholm Jul 10 '14 at 17:16
I think it takes more than 1 comment's worth of words to phrase my question, so: Special relativity and inductance –  Phil Frost Jul 10 '14 at 17:34
@PhilFrost, it's an interesting path you're on but keep in mind that, for this problem, we're assuming steady currents. Once you allow for acceleration (time changing current), things get much more complicated. In the steady current case, the mobile electrons are all assumed to be in the same inertial frame of reference. However, this isn't the case when the mobile electrons are accelerating. We must then consider momentarily co-moving reference frames and, perhaps, assign a different one to each electron. –  Alfred Centauri Jul 10 '14 at 22:46

Yet, the video suggests that a non-moving charged cat in proximity to a current-carrying wire will experience no force one way or the other.

First, consider the current carrying wire without concern for the cat.

We stipulate that in the frame of reference in which the wire is at rest, the wire is electrically neutral.

The above is crucial. If there is a positively charged cat at rest with respect to the wire, there is no Lorentz force acting on the cat since

(1) the wire is electrically neutral in this frame

(2) the cat is at rest in this frame

Now, stipulate that the cat is moving along with the drift electrons. Then, in the frame in which the wire is at rest, the cat is acted on by a magnetic force only since, in this frame, the wire is, by stipulation, electrically neutral.

However, in the frame in which the cat is at rest, then, compared to the rest frame of the wire, we have

(1) the drift electron density is smaller

(2) the fixed positive charge density is larger

Thus, the wire is no longer electrically neutral in this frame and there is an electric force only acting on the cat.

In other relatively moving frames, there are both electric and magnetic forces acting on the cat.

Wait...what? Why is it that in the cat's frame, the protons are moving, are contracted, and the wire is charged, but in Derek's frame, the electrons are moving, but are not contracted, and the wire is still neutral?

You're not thinking clearly here. The following three statements can all be true without logical contradiction:

(1) Electrons in the wire are moving in Derek's frame

(2) The moving electron density is greater (their spacing is contracted) in Derek's frame compared to some (but not all$^1$) relatively moving frames.

(3) The wire is electrically neutral in Derek's frame.

1: in relatively moving frames in which the mobile electrons have less speed, their density is less than in Derek's frame - in relatively moving frames in which the mobile electrons have more speed, their density is greater than in Derek's frame

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Makes sense, but I don't see how if you say "well, length contraction alters charge densities" you can then stipulate that in the frame of reference in which the wire is at rest, it's electrically neutral. How do we define "at rest" for the wire, anyway? The frame where the protons aren't moving? That seems rather arbitrary. Why not define it as the frame where the electrons aren't moving? That's what the video did, and in this situation the cat experiences a force. What gives? –  Phil Frost Jul 10 '14 at 12:42
@PhilFrost, it's not clear to me why you would think it isn't valid to stipulate that the wire is electrically neutral in its rest frame. It's not a logical contradiction, it doesn't materially change the result, and it's convenient. Also, it seems to me that "the frame of reference in which the wire is at rest" should need no explaining. For example, imagine that there is a dot or a band painted on the surface of the wire... –  Alfred Centauri Jul 10 '14 at 13:00
Because of length contraction. If I'm the cat, moving with the electrons, then the protons are moving and they contract, becoming more dense. But if I'm the guy in the lab, the electrons are moving...but they somehow don't contract and become more dense. Huh? –  Phil Frost Jul 10 '14 at 13:12
Also, please see question edits. –  Phil Frost Jul 10 '14 at 13:20
The moving electrinns are cordially invited to be denser in the lab frame, but that would create forces driving them apart, thus "fleeing" from the wire along the current until neutrality is attained. Of course, with an infinite wire we may run into different theoretical trouble (we couldn't get the current running at the full length at once, we would have waves propagating, ...) –  Hagen von Eitzen Jul 10 '14 at 18:30