# Non-zero charge density due to Lorentz contraction in current carrying wires

In trying to answer this question I came across the following problem. The original question relates to the idea that what looks like a magnetic field in one reference frame, ends up as an electrostatic field in another frame through special relativity.

I'm going to imagine that the wire is just two blobs, one of positive charge, the other of negative charge (same magnitude of charge density), and I can choose to move them relative to one another along the length of the wire (Basically I'm ignoring all of the real physics of currents in wires to get a pristine situation). We have a test charge; we're going to describe the world in it's rest frame.

Case 1

Both the positive and negative charge densities are at rest, in the test particle's frame. Obviously there is no electric force on the test charge, since the two charge distributions cancel each other out. There is no magnetic force either

Case 2

Positive charge density is moving with velocity +v, negative charge density is moving with velocity −v. By symmetry, any relativistic effects on the the charge density must affect both densities in the same way; therefore no electric force. But we do have moving charges, so we'd see a magnetic force.

Case 3

Positive charge is stationary relative to our test charge; negative charges are moving relative the positive charge and our test charge; length contraction yields higher negative charge density, and an electrostatic force on the test particle.

Case (2) would seem to imply that we only see pure magnetic fields in a special symmetric inertial frame -- that frame where the effects of length contraction serve to make the two charge densities exactly the same.

However, Case (3) is the normal lab measurement situation. I'm not sure if anyone has tried to put an upper bound on the magnitudes of any electrostatic forces related to running a real current through a real wire. If I run a computations based on a numerical example for drift velocity I get a result that the "extra" linear charge density for a 1mm diameter copper wire is <<1 one electron per meter; a deviation that would be very hard to measure experimentally.

If this logic holds we'd have:

1. Theoretically, lab measurements of real wires do have some electrostatic forces involved, although these forces are way too small to have significant effects (and are probably smaller than thermal effects),
2. And we have to go to a reference frame where both charge densities are moving in order to see pure magnetic effects;

I guess that this could be true, but I've never heard anyone refer to it in the application of special relativity to classical E&M; is there a flaw in my logic?

• – Dave
Oct 30 '13 at 19:28