# Purcell's electromagnetism vs. Bell's spaceship paradox and getting lost in Special Relativity again

In Bell's spaceship paradox a thread between equally accelerating spaceships breaks, because length contraction shortens the spaceships and the thread but not the distance between them. Wikipedia does not explain what happens when the observer accelerates instead of the spaceships, but I assume the thread does not break then.

Compare this with Purcell's description of electromagnetism replicated here and linked from here. The magnetic force on a (positive) test charge moving along a live wire is nothing but relativistic electrostatic force — the latter created due to a relativistically reduced distance between (negative) charges.

Moving the test charge past the wire is similar to the spaceship situation except the observer accelerates not the spaceships and the thread does not break. But why then should the distance between the negative charges in the wire shrink, but not the distances between the positive ones? Relativistically, the whole wire gets shorter in the frame of the moving test charge, but this would then decrease the average distance between all charges, not only the (in this case) negative charges.

Puzzled! Can someone explain this?

Similar, but I don't understand neither answer :-(

Your first example involves acceleration, and your second one doesn't. Special relativity distinguishes between inertial and noninertial frames of reference.

Moving the test charge past the wire is similar to the spaceship situation except the observer accelerates not the spaceships and the thread does not break.

The observer in the magnetic example has a velocity, not necessarily an acceleration.

Let's say observer observes initially positive charges not moving and negative charges moving. The observer is standing next to a electric wire where current is flowing. And then the observer starts to accelerate.

The accelerating observer observes the positive charges gaining more speed, and from that it follows that said observer observes distances between positive charges decreasing.

The same accelerating observer may observe the negative charges gaining speed or losing speed, depending on the direction of the acceleration. Therefore said observer may observe distances between negative charges decreasing or increasing.

Oh yes, speed of electrons relative to protons inside an electric wire is very small. So you are right, distances between both protons and electrons shrink, according to almost any accelerating observer.

But not exactly the same amount of shrinking occurs, as the speeds of protons and electrons are not exactly the same.

• Your explanation is the same as Purcell's as replicated in the link I posted. You are stating that the density of negative and positive charges changes independently of each other, exactly what I don't understand. The wire undergoes length contraction as a whole, so density should change for both charges equally. Getting more charges of one kind due to their distance length contracting contradicts the solution to the spaceship paradox where empty space does in fact not contract. So if the distance gets smaller, it seems there must be more charges of this one kind. Where do the come from? Commented Dec 26, 2017 at 17:24
• Where do they come from? From other parts of the wire. If the wire forms a loop, then from the other side of the loop. If the wire is straight ... then the wire is an antenna, where charges slosh around. "Andromeda paradox" is relevant in this case. Wikipedia has something about that paradox. Commented Dec 26, 2017 at 23:26