Magnetism and radiation as a consequence of SR

[jainabhs]

Please refer the doc attached.
The document is available at http://physics.weber.edu/schroeder/mrr/MRRhandout.pdf

It explains magnetism and radiation as a consequence of SR.
In the section for magnetism, it is depicted that, with respect to test charge Q, the distance between the -q charges gets contracted. That is fine. But why do they say that distance between +q charges has increased. Why is that?
According to me, only the distance between -q charges should decrease and the distance between +q charges should remain constant as they are not in motion w.r.t test charge Q. And length contraction for -q charges is sufficient to explain charge imbalance in the conductor,in turn, applying attraction to test charge Q

Please explain...thanks in anticipation

 

[pervect]

Here's how you work out the charge distribution. Assume an ideal wire, so that its potential is zero. A real wire will have some small resistance which complicates the analysis, but it can be safely ignored by setting the resistance to zero and eliminating this complicating factor.

Solve for the electric field given the boundary conditions that the potential of the wire is zero. This is very easy - the electric field is the gradient of the potential, and the potential can be taken to be zero everywhere (on and off the wire). Note that this makes an additional assumption - that the wire started out with no net charge in order for us to say it had a zero potential.

So now that we've established that the electric field around a perfectly conducting uncharged wire is zero regardles of whether it's carrying currnet, you can find the charge densities and how they transform by Gauss law. The correct solution for charge densities will have no electric field in the laboratory frame. Thus in the laboratory frame, the two charge densities must be equal. This leads to the results described in the paper when viewed in the rest frame of the wire.