|MadSci Network: Physics|
Good question, and you are right: Special Relativity deals with this. I will not go into the details of the calculation, but will present you with some physical arguments, using only the well-known effects of Lorentz contraction and time dilation.
Consider first a simpler problem. Say there are two equal point charges, moving along some axis and having some finite initial distance to each other. As seen from an observer with respect to whom the charges are at rest (e.g. in the centre of mass system), they repel each other due to their like charge, so their distance increases with time. Another observer who is moving with respect to the centre of mass will also see the charges repel, but at a slightly lower rate. Taking SR into account, this is plausible: time dilation lets the movement appear more slowly! But how can we understand this in terms of electric and magnetic fields? Well, the second observer sees two moving charges, so there are two parallel currents flowing along the axis of movement. Parallel currents attract each other due to the Lorentz force, and this is how the ``slowdown'' of the repulsion comes about. The purely electric field in one frame of reference has been transformed into a mixture of electric and magnetic fields in the other frame!
Now maybe you can already guess what happens in the experiment that you have described. Let us first examine a single wire with some current I flowing through it. In the rest frame of the wire (which is defined by the rest frame of the positive ions in the crystal lattice of the metal) there is nothing but a magnetic field which is zero when the current is zero. If I=0, the negative charge of the electron gas in the metal is exactly compensated by the positive charge of the lattice ions. This is also the case when I is finite, after taking Lorentz contraction into account: The moving electron gas is of course Lorentz contracted, so the negative charge density would be higher than in the I=0 case, leading to an electric field. But as there is no electric field, the charge density in the wire must have been ``adapted'' in the moment the current was turned on so that the Lorentz contracted density matches again the positive charge of the lattice ions. Tricky, hm? Now imagine an observer moving along the wire with exactly the drift velocity of the electrons. They would certainly see a magnetic field due to the flow of positive ions in the other direction, but they would also see an electric field due to the Lorentz transformation of positive and negative charge densities in the wire: Now they do not compensate each other any more! Fortunately, this is again exactly what SR tells us if we apply the correct transformation rules to the initial magnetic field -- an electric field will appear.
Ok, now we are ready for the full problem. In the rest frame of the moving electrons within the parallel wires, there is still a magnetic field present, this time generated by the flow of positive ions. Thus the two wires will still attract, because the positive ions of one wire move in the magnetic field of the other (Lorentz force!). But those fields do not have any impact on the electrons: they don't move! However, there is also an electric field which comes about in the way I have described in the previous paragraph. The two wires carry an overall positive charge density and consequently there will be an additional repulsive force between them, weakening the Lorentz attraction.
Unfortunately such questions are seldomly discussed in textbooks on electrodynamics, and that's why many people are confused by problems like the one you have stated. I would like you to keep one thing in mind: Special Relativity lies at the heart of electromagnetism. One can't live without the other.
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