# Do parallel currents attract/repel in vacuum?

My poor understanding of attraction/repulsion between parallel currents:

-if you have 2 symmetrical current densities moving in parallel (e.g. Current carrying wires) they will attract/repel each other depending on the sign

-assuming the direction of current is the same, the [charges in the] currents will appear static relative to one another

-if you boost into a reference frame moving with the currents, the surrounding static charge density (air whatever) will experience length contraction; however, the parallel current will be unaffected (because its moving at the same speed as your reference frame)

-since the static charge is length contracted in the direction of current, it will exert a stronger [repulsive] force in a direction perpendicular to the current flow (the electrons are more tightly packed)

-the added forces from contracting static charges on either side of the moving current would normally cancel, but because the parallel current was not length contracted, the net change in force is in a direction towards the other wire, so they attract

-if the currents flow in opposite directions, length contraction will be even stronger on the other current than the static charge (v - (-v) = 2v; v - 0 = v) so the wires repel

QUESTION: -assuming the above is true, what would cause parallel currents to attract in a vacuum? (I.e. With all other charge sufficiently far removed, so that the force due to length contraction is negligible). Is there something in the vacuum other than charge that can contract to create the asymmetry?

BONUS QUESTION: -if two currents are moving in parallel and there are no other charges present (between them or around them) is this just the same as one current? Or will the cuolomb force between the currents take over so the densities slowly repel? What if they move in opposite directions?

If you have two infinite parallel wires with equal size and positive x-direction electron currents $I=Anev$, where $A$ is the wire cross section area, $n$ is the electron concentration, e is the electronic charge and $v$ is the drift velocity in negative x-direction of the electrons, there will be a magnetic attractive force per unit length $$F=\frac{\mu_0 I^2}{2\pi d^2}$$ When you change into a Galilean frame of reference with speed $-u$ relative to the rest frame of the wire, the electron drift velocity in the wire will be reduced by $u$ (first term) but an additional current arises due to the positive metal ions in the wires (second term) $$I^{'}=Ane(v-u)+Aneu$$ where the second term of the current is the positive current of the positive metal ions with concentration $n$ which is the same as the electron concentration. Thus you can see that, in the moving frame of reference, the total current $I^{'}$ in the wires will stay the same, even if the velocity becomes equal to the electron drift velocity in the wires ($u=v$ ). Therefore the magnetic force between the wires given by the first equation stays the same in the moving reference frame.