Is Relativistic Electromagnetism Affecting a Test Charge Above a Moving Wire?

[Jimmy87]

I have recently come across relativistic electromagnetism so I am very new to it. The situation described to me was one where a positive test charge moves above a wire with a velocity V which matches the current in the wire in both magnitude and direction. Apparently, from the test charge's reference frame, the attractive force on it is strictly Coulmobic owing to the fact that from its own reference frame there is no relative motion between itself and the flowing positive charges in the wire, is that right? That seems so strange! Can you really say from the test charge's frame that there is no magnetic force even though its moving? The electrostatic attraction is supposedly caused by length contraction of the negative charges which are stationary in the other reference frame but have relative motion from the test charge's frame. How can you get length contraction to be this significant at such low velocities (i.e. drift velocity)? Also, does this relativistic attractive Coulomb force exactly match the magnetic force in terms of magnitude? Is all this phenomena really true, i.e. can it be proven that a charge moving outside a wire really feels relativistic effects? Any help answering these questions is greatly appreciated!

 

[lugita15]

Here is an excerpt from Purcell discussing how the magnetic force arises from the electrostatic force and relativity.

 

[WannabeNewton]

Can you really say from the test charge's frame that there is no magnetic force even though its moving?

But the statement "it's moving" has no absolute meaning. In the rest frame of the charge it is of course at rest; in the rest frame of some other observer the charge may be in motion but the latter is not any more special than the former.