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Let's assume that all charges flowing in both wires are either all positive or all negative. From the frame of reference of an individual charge in either wire, it would appear as if the position of all charges were static and that this could be treated as an electrostatics problem. By treating it as an electrostatics problem, though, I would expect the wires to repel one another.

If, on the other hand, we were to assume the frame of reference of either wire and were to solve for the net force due to the currents in each by treating it as a magnetostatics problem, it seems that the solution indicates that the wires would be attracted to one another.

Where is the error in this logic?

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From the frame of reference of an individual charge in either wire, it would appear as if the position of all charges were static and that this could be treated as an electrostatics problem.

This is not true. If in the rest frame of the neutral wires there is, for example, positive charge moving to the right, then if we start moving with this current we would then observe negative charge flowing to the left. Both situations produce attraction of the wires.

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  • $\begingroup$ Thanks, Aaron, that does make sense. As a follow-up question, what if the charges weren't in a neutral wire and were instead just parallel streams of moving charges in free space? $\endgroup$ – cfsnyder May 17 at 9:29
  • $\begingroup$ @cfsnyder Related video $\endgroup$ – Aaron Stevens May 17 at 12:47
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    $\begingroup$ thanks, Aaron, that video was exactly what I was looking for. $\endgroup$ – cfsnyder May 18 at 12:15
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Definitely there is one. you can simply put the Right hand law here. Using your right hand and the thumb is perpendicular to your other four fingers.(other four fingers turns in to spiral shape). The direction of your thumb should be the direction of the current. The direction of other four fingers pointing should be the induce magnetic force. The magnetic field should be spiral around the wire. the result of the two wire will stick together.

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