Another Relativistic Train

Landsburg returns to the subject of the relativistic train. It no doubt reflects badly upon my character that it pains me to note that this time he seems to get it right.

UPDATE: As Arun points out in the comments, not really. There is a big magnetic field in Fred's inertial frame, and he would see it if he moved his charge or deployed a compass.]

... imagine a wire, made of protons that stay still and electrons that drift rightward; that drift is what we call a current. And imagine a nearby charged particle—call it Fred—also traveling rightward.

Now relativity tells us that Fred is allowed to think of himself as stationary, and the protons (along with you and me) as drifting off to the left. Relativity also tells us that if passengers on a moving train say the cars are 100 feet apart, then an observer at the station will say they’re closer than that. In this case (according to Fred) you and I are the passengers moving with the train of protons, and if we say they’re an angstrom apart, then Fred says they’re closer. That means Fred sees more positive charge per inch of wire than we do. If Fred himself happens to be negatively charged, he’ll be drawn toward the wire.

As far as Fred is concerned, that’s a purely electrical force, but it’s a force that you and I can’t account for on electrical grounds. So you and I call it magnetism.

UPDATE continued: The thing is, Fred sees only an electrostatic force because he doesn't bother to measure the magnetic field. A more honest way to express it is to say that part of the field we see as purely magnetic from the stationary frame appears to be an electrical field in Fred's frame.

The details are explained at the freshman physics level in Chapter 5 of Ed Purcell's "Electricity and Magnetism."

Let me use SL's example as a jumping off point for a slight puzzle. Imagine that the wire we are seeing is arranged in a circle like the train we considered once before. The numbers of electrons and protons in the wire are equal, making it electrically neutral. When the electrons are set in motion, Fred decides to amble along with them. He doesn't need to go very fast, since the average drift velocity of electrons in is much less than the speed of light - less, in fact, than the speed of a snail - a few centimeters per hour. Now George, our stationary observer measures the average distance between electrons in his frame, and its still the same as it was at rest, call it l. It ought to be, since we have the same number of electrons and they are still in the same wire. (electrons, unlike opposite ends of your typical rail car, don't mind being pushed a bit further apart).

Fred has carried his own meter stick with him, and he decides to check some distances in his frame (co-moving with the local electrons). He finds that the nearby electrons have been pushed apart by the relativistic factor gamma = 1/sqrt(1-(v/c)^2). I will call it g. The protons, as seen from his frame, are closer together by the factor 1/g. The circle of wire, firmly attached to the protons, has similarly shrunk. So here is our first puzzle: The electrons are farther apart on a track that is smaller - so how do they all fit?

If Fred is carrying a charge, he will see a force due to the fact that the local protons are more densely concentrated than their negative counterparts by a factor that amounts to (v/c)^2 x charge concentration of carriers. The resulting force scales like charge x current x v/c as one would expect from the Lorentz Force law and Biot-Savart law of magnetism.

About that puzzle...