This is patently false and if one thinks about it, it is something anyone who has taken an introductory physics class in which magnetism and electric currents has observed, although granted without realizing it.
The canonical place to find this treatment is in a book by Purcell. But this book is expensive and I'm not so happy with it, except for its lucid treatment of this subject.
There is a resource available on the web, a PDF file (see attached) written by Daniel Schroeder of Weber State University. I have hijacked the file since I would have essentially typed in the same thing and it wouldn't have been any better. However, I will sketch the relevant information for the simple case below. This file includes a much broader set of examples. He also has a simplified talk here, but there is no guarantee that this URL will last forever.
In essence, the point is that magnetism is actually an electric force in a different reference frame.
Specifically, consider a single charged particle of charge q moving parallel to a wire with current I. You can calculate the magnetic field caused by the wire
Because the Lorentz force is
But what happens if you jump to the frame where the particle is not moving? With a velocity of zero, the magnetic force is zero. Yet the object does feel a force, specifically an attractive one. So in this frame, magnetism is simple not there. So from where comes the force?
Well, recall that the charge in the wire is now moving with respect to the frame containing the charge. Moving frames means Lorentz contractions. Expand the image (stolen from Schroeder) for the simplified analysis.
(see attachment for larger figure)
So we see in the frame in which the charge is not moving, we see an electric force that is numerically identical to the magnetic force seen in the other frame.
There is no magnetic force in a frame where the charge is not moving. Without relativity, there is no charge difference on the wire, so there is no electrostatic force. It is only because of relativity that this new frame observes an electric charge on the wire..
Case closed.
Note that there are more possible cases...for instance, you could do two parallel wires. That's more complicated because of the additional charges in the the test-charge's wire, but that's just fine.
The other thing to remember is that the drift velocity of electrons in wires is on the order of millimeters per second. That's quite a big difference from the speed of light's 300,000,000 m/s. Yet I've done this experiment, as has anyone who has used a current balance in a Freshman physics lab. The forces are small...typically the same as the force of gravity on 100 mg of mass for a 1 ampere current and about 1 cm separation. But remember that the electric force is about 1039 times more powerful than gravity, so an itsy-bitsy, tiny Lorentz contraction can be observed.
This effect is absolutely pure intellectual gold....
