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Dear Radu!

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.

Bye,

Georg.

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