Why doesn't length contraction create an electric force in the negative charges' frame?

[Ja4Coltrane]

I am giving a little lecture on the conceptual aspect of relativity tomorrow and was wondering something.
Consider a wire carying positive charges moving to the right and negative charges moving to the left. Say that a positive charge is placed above the wire and this positive charge moves at the some velocity as the positive charges within the wire.
Now I know that what happens is from the frame of the negative charges there is a magnetic force and from the frame of the positive charges there is an electric force due to length contraction acting on the free positive charge but my question is this: why doesn't length also contract for the frame of the negative charges? It seems to me that the positive charges should contract and therefore create an electric force which opposes the magnetic force. Does it have something to due with the fact that the charge on top is moving too? Any help is GREATLY appreciated.

 

[pervect]

You should, in general, expect there to be exacty one frame in which the electric field is zero.

In your example, you are assuming that this is a frame where the positive charges are moving right and the negative charges are left.

Therefore you are correct in thinking that there will be electric fields in both a frame comoving with the positive charges and a frame comoving with the negative charges.

What you probably want to do is therefore work out the total force (electric and magnetic) and show that it transforms covariantly.

To do that, one of the things you need to know how forces should transform under special relativity. I'm assuming that you do know this or that at least it would do you some good to think about it. If you don't know or get stuck, ask some more questions.