Parallel line currents & Lorentz transformation

[Preno]

Okay, I'm sure I must have overlooked something very trivial, so please help me with this:

Two parallel infinite line currents of equal magnitude attract each other. The current can be thought of as consisting of positive and negative particles with equal linear densities, but each moving with a different velocity (wlog let's assume the positive ones are static in some frame of reference K). Let us now move to the frame of reference K' which moves with the current so that the current in K' is zero. The force which was purely magnetic in K is now replaced with a purely electrical one - the linear densities of the positive and negative particles undergo Lorentz transformation, so that the total density in K' is non-zero.

Here's my problem: if linear densities and the velocities of particles (and hence the currents) have the same magnitude in K, then they must also have the same magnitude in K' (the transformation respects the symmetry between the two lines). But like charges repel each other. So it would seem that in K' (hence also in K), the currents actually repel each other.

 

[tiny-tim]

Hi Preno!

The current can be thought of as consisting of positive and negative particles with equal linear densities, but each moving with a different velocity …
Let us now move to the frame of reference K' which moves with the current so that the current in K' is zero. …

Sorry, I'm not following what K' is …

as you say, there are two lots of charge, moving with a different velocity.

(and don't forget the Lorentz-Fitzgerald length-contraction)

 

[Preno]

Hi Preno!


Sorry, I'm not following what K' is …

K' is the frame in which the current vanishes.

as you say, there are two lots of charge, moving with a different velocity.

Well, the line is electrically neutral, so there need to be both positive and negative charges.

(and don't forget the Lorentz-Fitzgerald length-contraction)

No, I'm not forgetting it. That's the whole point. The linear density of those charges changes when transforming to K' but it remains the same on both lines. Hence they should repel each other.