From: jos.bergervoet@xs4all.nl   
      
   On 2/12/2017 5:27 PM, John Heath wrote:   
   > On Thursday, December 29, 2016 at 11:21:45 AM UTC-5, Jos Bergervoet wrote:   
   >> On 12/29/2016 8:59 AM, John Heath wrote:   
       ...   
     ...   
   >>> Two wires from floor to ceiling 1 inch apart with current moving up in   
   >>> both wires. The force is attractive.   
   >>   
   >> Exactly, that's magnetic force. And the electric force   
   >> between the parallel moving charges (the electrons) is   
   >> repulsive, so that's the opposite.   
   >>   
   >>> There are as many electrons as   
   >>> there are protons in the copper wire. As current starts to flow up in   
   >>> both wires the electrons see the other electron as non relativistic as   
   >>> they are both moving up . However the electrons see the protons as   
   >>> length contracted therefore there is and attractive Coulomb force   
   >>> between the wires.   
   >>   
   >> You now have stationary particles in addition to the   
   >> parallel moving charges. The electric force between the   
   >> moving charges is still repulsive and my only claim   
   >> was that that one is opposite to the magnetic force!   
   >   
   > I wish this exchange of ideas were closer to 900 MHz vs 2.4 GHz as I   
   > could use some help in this area. I suspect you feel my pain.   
      
   Actually I might be missing your exact point (why not consider   
   5.8GHz?)   
      
   But we certainly can look at the two currents for these   
   frequencies. There are basically two situations to look at:   
      1) We could consider currents without phase delay over the   
   whole length of two long parallel wires. The analog of   
   parallel or anti-parallel static currents would be   
   in-phase alternating currents or 180 degrees out-of-phase   
   currents, in both cases changing direction simultaneously   
   everywhere along the length of the wires. (In practice this   
   is not a behavior that is easy to create!)   
      2) We could have waves propagating over the wires, again   
   with the currents either in-phase or 180 deg. out of phase,   
   and with the waves moving in the same direction. (This is   
   the more natural situation in a transmission line.)   
      
      In case 1) the fields of each wire can be expressed in   
   Hankel functions of the radial distance to the wire, at   
   small distances equal to the 1/r fields of the static case.   
   So with increasing distance between the wires, the case with   
   attractive force for DC currents (equal-direction currents)   
   will turn into repulsive force and then back to attractive   
   (and in between also alternating forces with the double   
   frequency of the current!). There are no unbalanced charges   
   in the wires so we have no radial E-fields, the only force is   
   from the B-field.   
      
      In case 2) we basically have the two-wire transmission   
   line. Opposite currents require the differential-mode   
   solution for the signal, in-phase currents the common-mode   
   solution. These are both approximately TEM solutions (the   
   common-mode solution only very approximately) which means   
   that: a) the fields are shaped more or less like static   
   fields, and b) the forces from radial E-fields and from   
   the B-fields now approximately cancel each other. The latter   
   follows from the fact that d rho/dt = dI/dx, the continuity   
   equation for the charge. This gives equal charges where   
   there are equal currents (current and voltage are in phase   
   in a transmission line) and as we saw above: that means   
   opposite electric and magnetic forces! That they are also   
   (approximately) equal in magnitude is easy to verify from   
   the equations.   
      
   So, at RF frequencies, with transmission line behavior of   
   the currents, the force is substantially smaller than for   
   a pure DC current (without voltage). Of course if you add   
   a DC voltage as well (with the ratio to the current equal   
   to the characteristic impedance of the transmission line)   
   then you will have the same cancellation for DC as well.   
      
   --   
   Jos   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)